Introduction to Reidemeister torsion and twisted
Alexander polynomials
Teruaki Kitano
Soka University
Nov. 03-07, 2014 in SNU
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
1 / 226
This slide was prepared for continuous lectures (5 hours and half
) in Seoul National University.
There are lots of works related with Reidemeister torsion and
Alexander polynomials. Then it is hard that we refer everything.
This is a crash course of them.
There are lots of results I can not mention here.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
2 / 226
Torsion invariants can be defined for a pair of (X , ρ) satisfying some
condition where
X is a finite CW-complex,
ρ : π1 X → GL(V ) is a representation.
There are several tyes of torsion invariants.
Reidemeister torsion:combinatorial and finite dimensional,
Ray-Singer torsion(analytic torsion):anaytic and finite
dimensional,
combinatorial L2 -torsion: combinatorial and infinite dimensional,
analytic L2 -torsion: anaytic and infinite dimensional.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
3 / 226
Reidemeister torsion can be classified into two types:
abelian Reidemeitser torsion for an abelian representation
(1-dimensional representation),
twisted Redemeister torsion for a higher dimensional
representation.
Remark
In this lecture we simply call both of them Reidemeister torsion.
Another direction to define a torsion type invariant is Whitehead
torsion, and Whitehead group. It is related with algebraic
K-theory. We do not mention here.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
4 / 226
History before Milnor
In 1935-1940
K. Reidemeister, Homotopieringe und Linsenräume, Hamburger
Abhandlungen, 11 (1935).
W. Franz, Ueber die Torsion einer Ueberdeckung, Juurnal für die
reine und angew. Math. 173 (1935).
G. de Rham, Sur les complexes avec automorphismes,
Commentarii Math. Helv., 12 (1939).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
5 / 226
Fundamental idea of Reidemeister torsion
F: a field (for example, C, R, Q(t), Z/pZ).
”Redemeister torsion τ is the determinant.”
More precisely,
C is a finite dimensional vector space over F with a basis c.
∂ : C → C a linear map.
det ∂ is make sense but not strong. Because it often takes zero.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
6 / 226
Assume ∂ 2 = 0 (ker∂ ⊃ im∂).
We can consider its homology.
We put more strong assumption: ker∂ = im∂ (acyclic condition).
Then we can get another basis b of C associated to ∂ (not
unique).
τ (C ) is determinant of the transformation matrix from c to b.
This is only depending on (C , c).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
7 / 226
More little precisely
C = C∗ = Ceven ⊕ Codd
∂ = ∂1 ⊕ ∂2 : Ceven ⊕ Codd → Codd ⊕ Ceven ∼
= Ceven ⊕ Codd
We fix bases ceven and codd
Under some constrcution we can get another basis (beven , beven )
of C by using any beven of im∂2 and bodd of im∂1 .
det((beven , bodd ) → codd )
τ (C ) =
det((beven , beven ) → ceven )
τ (C ) is well-defined for (C , c).
Remark
We need to twist the homology by a representation for acyclicity.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
8 / 226
We follow the definition of Reidemeister torsion by Milnor.
J. Milnor, Two complexes which are homeomorphic but
combinatorially distinct, Ann. of Math. (2) 74 (1961), 575–590.
J. Milnor, A duality theorem for Reidemeister torsion, Ann. of
Math. (2) 76 (1962), 137–147.
R. H. Fox and J. Milnor, Singularities of 2-spheres in 4-space
and cobordism of knots, Osaka J. Math. 3 (1966), 257–267.
J. Milnor, Infinite cyclic coverings, 1968 Conference on the
Topology of Manifolds (Michigan State Univ., E. Lansing, Mich.,
1967), 115–133 Prindle, Weber & Schmidt, Boston, Mass.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
9 / 226
Survey and collected works:
J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72
(1966), 358–426.
J. Milnor, John Milnor Collected Papers: Volume II: The
Fundamental Group, Amer. Math. Soc. 2008.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
10 / 226
torsion of a chain complex
a chain complex C∗
∂
∂m−1
∂
∂
m
2
1
0 −→ Cm −→
Cm−1 −→ Cm−2 −→ . . . −→
C1 −→
C0 −→ 0.
Definition
Zq = ker ∂q ⊂ Cq
Bq = Im∂q+1 ⊂ Zq ⊂ Cq
∂q
Because 0 −→ Zq −→ Cq −→ Bq−1 −→ 0 is exact, then we have
Cq ∼
= Zq ⊕ Bq−1
(not canonical).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
11 / 226
Definition
C∗ is acyclic if
Bq = Zq (that is, Hq (C∗ ) = 0)
for any q = 0, 1, . . . , m.
Remark
a chain complex C∗ is acyclic if and only if C∗ is an exact sequence.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
12 / 226
We assume a basis cq of Cq is given for any q.
We also take a basis bq on the q-th boundary Bq for any q.
On this exact sequence
∂q
0 −→ Zq −→ Cq −→ Bq−1 −→ 0
As this is an exact sequence, by taking a lift b̃q−1 of bq−1 ，(bq , b̃q−1 )
is a basis on Cq . We have
Cq ∼
= Bq ⊕ Bq−1
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
13 / 226
Let b = {b1 , · · · , bn }, c = {c1 , · · · , cn } be two bases of a vector
space V over F.
∑
There exists a non-singular matrix P = (pij ) s.t. bj =
Pji ci .
Definition
P is called the transformation matrix from c to b denoted by (b/c).
Its determinant det P is denoted by [b/c].
Under( definitions )
bq , b̃q−1 /cq : the transformation matrix from cq to (bq , b̃q−1 ).
[
]
(
)
bq , b̃q−1 /cq : its determinant det bq , b̃q−1 /cq .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
14 / 226
Lemma
The determinant [bq , b̃q−1 /cq ] is independent on the choice of a lift
b̃q−1 . Hence we can simply write [bq , b̃q−1 /cq ] to it.
Proof.
Assume b̂q−1 is another lift of bq−1 on Ci . Here
0 −→ Zq −→ Cq −→ Bq−1 −→ 0
is an exact sequence, then a difference between any vector of b̂q−1
and its corresponding vector of b̃q−1 belongs to Zq = Bq . Then by
the definition of det,
[
] [
]
bq , b̃q−1 /cq = bq , b̂q−1 /cq
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
15 / 226
Definition
The torsion τ (C∗ ) of a chain complex C∗ is defined by
∏
q:odd [bq , bq−1 /cq ]
τ (C∗ ) = ∏
∈ F \ {0}.
q:even [bq , bq−1 /cq ]
Lemma
The torsion τ (C∗ ) is independent of the choice of bq .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
16 / 226
Proof.
Assume b′q is another basis of Bq .
In the definition of τ (C∗ ), the difference between bq and b′q is related
to the followings only two parts:
[ ′
]
[
]
bq , bq−1 /cq = [bq , bq−1 /cq ] b′q /bq
[
]
[bq+1 , b′ q /cq+1 ] = [bq+1 , bq /cq+1 ] b′q /bq
[
]
Since b′q /bq appears in the both of the denominator and the
numerator of the definition, they are cancelled.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
17 / 226
How it is cancelled?
We show two exmaple.
C∗ : 0 → C4 → C3 → C2 → C1 → C0 → 0.
In this case
[b4 , b3 /c4 ][b2 , b1 /c2 ][b0 , b−1 /c0 ]
[b3 , b2 /c3 ][b1 , b0 /c1 ]
[b3 /c4 ][b2 , b1 /c2 ][b0 /c0 ]
=
[b3 , b2 /c3 ][b1 , b0 /c1 ]
τ (C∗ ) =
Only b4 appears once, but this is zero. In this case the number of the
denominator and the number of numerator are not the same.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
18 / 226
C∗ : 0 → C3 → C2 → C1 → C0 → 0.
In this case
[b2 , b1 /c2 ][b0 , b−1 /c0 ]
[b3 , b2 /c3 ][b1 , b0 /c1 ]
[b2 , b1 /c2 ][b0 /c0 ]
=
[b2 /c3 ][b1 , b0 /c1 ]
τ (C∗ ) =
Only b3 appears once, but this is also zero. In this case the number
of the denominator and the number of numerator are same.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
19 / 226
Lemma
Assume
0 → C∗′ → C∗ → C∗′′ → 0
is an exact sequence of chain complexes and a basis (c′i , c′′i ) of C∗ as
a union of bases on others. If two of C∗′ , C∗ , C∗′′ are acyclic, then the
third one is also acyclic and
τ (C∗ ) = ±τ (C∗′ )τ (C∗′′ ).
Remark
This lemma is applied to the Mayer-Vietoris argument.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
20 / 226
Why does ± appear in the right hand side?
To define the torsions we use
C∗′ ∼
= Z∗′ ⊕ B∗′ ,
C∗ ∼
= Z∗ ⊕ B∗ ,
′′ ∼ ′′
C∗ = Z∗ ⊕ B∗′′ .
On the other hand, to get this formula, we use
C∗ ∼
= C∗′ ⊕ C∗′′ ∼
= Z∗′ ⊕ B∗′ ⊕ Z∗′′ ⊕ B∗′′ .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
21 / 226
Geometric situation
X :a finite CW-complex
X̃ → X :a universal covering
We take a lift of CW-structure on X to X̃ .
We may assume π1 (X ) acts cellularly on X̃ as covering
transformations from the right-hand side by cellular
approximation theorem.
A chain complex
C∗ (X̃ ; Z) = ⊕Ci (X̃ ; Z)
where each chain module Ci (X̃ ; Z) is a Z-module generated by i-cells
of X̃ .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
22 / 226
By the action of π1 (X ), C∗ (X̃ ; Z) is a chain complex of right
Z[π1 (X )]-modules．
To get acyclic complex, we consider a chain complex with local
coefficient.
Take a representation
ρ : π1 (X ) → GL(V )
where V is a l-dimensional vector space over a field F.
By this representation, Vρ = V can be a left Z[π1 (X )]-module．
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
23 / 226
Then we take
C∗ (X ; Vρ ) = C∗ (X̃ ; Z) ⊗Zπ1 (X ) Vρ
where
σ̃x ⊗ v = σ̃ ⊗ ρ(x)v
for
σ̃ is a lifted cell of X̃ ,
x ∈ π1 (X ),
v ∈ Vρ .
Definition
We call ρ an acyclic representation if C∗ (M; Vρ ) is acyclic, namely all
homology groups vanish; H∗ (M; Vρ ) = 0.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
24 / 226
To define τ (C∗ (X ; Vρ )), we need bases of Cq (X ; Vρ ) for any q.
Take all q-cells of X :
σ1 , . . . , σk ,
Lift them onto X̃ :
σ̃1 , . . . , σ̃k ,
It gives a basis of C∗ (X̃ ; Z) as a Z[π1 (X )]-module．
Take a basis of V :
e1 , . . . , el .
From the definition of Ci (X ; Vρ ),
σ̃1 ⊗ e1 , . . . , σ̃1 ⊗ el , . . . , σ̃k ⊗ e1 , . . . , σ̃k ⊗ el
gives a basis of Cq (X ; Vρ ). By using these bases, τ (C∗ (X ; Vρ )) can
be defined.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
25 / 226
Definition
Reidemeister torsion of X for ρ : π1 (X ) → GL(V ) is defined by
τρ (X ) = τ (C∗ (X ; Vρ )) ∈ F \ {0}.
Under fixing a CW-complex structure of X , we prove the
well-definedness of τρ (X )．
To do it,
how to take lifts of q-cells of X on X̃ ．
how to order q-cells.
how to take a basis of V .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
26 / 226
Lemma
If we take another lift of a cell, τρ (X ) is changed only by the
multiplication of det(ρ(π1 (X ))) ⊂ F \ {0}.
Proof.
We take the set of the q-cells {σ1 , . . . , σk } and one lift for each cell
as {σ̃1 , . . . , σ̃k }.
Another lift σ1 can be represented as σ̃1 x for x ∈ π1 (X ). Then the
basis of Ci (X ; Vρ ) is changed as
{σ̃1 x ⊗ e1 , . . . , σ̃1 x ⊗ el , . . . , σ̃k ⊗ e1 , . . . , σ̃k ⊗ el }
={σ̃1 ⊗ ρ(x)e1 , . . . , σ̃1 ⊗ ρ(x)el , . . . , σ̃k ⊗ e1 , . . . , σ̃k ⊗ el }
Then τρ (X ) is changed by only multiples of det ρ(x).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
27 / 226
Lemma
If the order of cells σ1 , . . . , σk is changed, τρ (X ) is invariant up to
sign.
Proof.
If we change the order to {σi1 , σi2 , · · · , σik }, then the basis of
Cq (X ; Vρ ) is changed by the corresponding permutation. It is clear
that τρ (X ) is changed only by the signature of the permutation.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
28 / 226
Clearly the value of τ is varied by changing a basis of V ．Then we
make some classes of bases as follows.
Definition
Two bases {e1 , · · · , el } and {e1′ , · · · , el′ } belong to the same class if
∧
e1 ∧ · · · ∧ el = e1′ ∧ · · · ∧ el′ ∈ l V ∼
= F.
Lemma
If another basis e1′ , . . . , el′ of V which belongs to the same class of
{e1 , · · · , el }, then τρ (X ) is invariant.
Proof.
It is easy to see τρ (X ) is invariant under base change of V in the
same class by [{e1′ , . . . , el′ }/{e1 , . . . , el }] = 1.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
29 / 226
Under fixing a class of bases of V , more strongly we have the
following.
Theorem (Reidemeister, Milnor, ...)
τρ (X ) is invariant under subdivision of a cell structure up to the
multiplications of ± det(ρ(π1 (X ))). Further τρ (X ) is a simple
homotopy invariant up to ± det(ρ(π1 (X ))).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
30 / 226
How to prove Reidemeister torsion is invariant under subdivisons ?
Let X be a finite CW-complex and X ′ its subdivision.
There exists a chain map
C∗ = C∗ (X ; Vρ ) → C∗′ = C∗ (X ′ ; Vρ )
We have a exact sequence of chain complexes:
0 → C∗ → C∗′ → C̄∗ = C∗′ /C∗ → 0.
We get τ (C∗′ ) = ±τ (C∗ )τ (C̄∗ ).
We can prove τ (C̄∗ ) = ±1. Then τ (C∗ ) = ±τ (C∗′ ).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
31 / 226
Remark
If dim V is even, to replace cells is even times. Then sign of
τρ (X ) is invariant.
If any matrix in ρ(π1 (X )) has determinant 1, namely,
det(ρ(π1 (X ))) = {1}, then the sign of τρ (X ) is well-defined.
Then we consider usually Reidemeister torsion for
ρ : π1 (X ) → SL(2; F), or SL(2l; F).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
32 / 226
Example:X = S 1
S 1 = 0-cell b ∪ 1-cell x
R → S 1 :its universal cover
ρ : π1 (S 1 ) → SL(2; C)
V = C2 with a canonical basis
( )
( )
1
0
e1 =
, e2 =
.
0
1
We identify this 1-cell x with a generator x of π1 (S 1 , b).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
33 / 226
Then we can take bases of C1 (S 1 ; Vρ ), C0 (S 1 ; Vρ ) as follows:
C1 = ⟨ x̃ ⊗ e1 , x̃ ⊗ e2 ⟩, C0 = ⟨ b̃ ⊗ e1 , b̃ ⊗ e2 ⟩
where x̃, b̃ are lifts of x, b of S 1 on the universal cover R1
respectively．
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
34 / 226
Under these bases,
∂1 (x̃ ⊗ ei ) = (b̃x − b̃) ⊗ ei
= b̃x ⊗ ei − b̃ ⊗ ei
= b̃ ⊗ ρ(x)ei − b̃ ⊗ ei
= (1 ⊗ ρ(x) − 1 ⊗ 1)(b̃ ⊗ ei ).
Assume
det(ρ(x) − E ) ̸= 0,
then C∗ is an acyclic chain complex．
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
35 / 226
Further by taking
b1 = {0}, b0 = {∂1 (x̃ ⊗ e1 ), ∂1 (x̃ ⊗ e2 )},
we obtain
[b1 , b̃0 /c1 ]
[b0 /c0 ]
1
=
det(ρ(x) − E )
1
.
=
2 − tr(ρ(x))
τρ (S 1 ) =
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
36 / 226
Example:X = T 2
π1 (T 2 ) = ⟨ x, y | xyx −1 y −1 = 1 ⟩
ρ : π1 (T 2 ) → SL(2; C)
A cell-structure of T 2 :
0-cell b,
1-cells x, y ,
2-cell σ
We can identify two 1-cells x, y with the generators of π1 (T 2 ).
Remark
Since π1 (T 2 ) is abelian，then ρ(x) and ρ(y ) are simulatiously upper
triangulable and conjugate．
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
37 / 226
∂2 (σ̃ ⊗ ei ) = (x̃ + ỹ x − x̃y − ỹ ) ⊗ ei
= x̃ ⊗ (E − ρ(y ))ei + ỹ ⊗ (ρ(x) − E )ei ,
∂1 (x̃ ⊗ ei ) = (b̃x − b̃) ⊗ ei
= b̃ ⊗ (ρ(x) − E )ei ,
∂1 (ỹ ⊗ ei ) = (b̃y − b̃) ⊗ ei
= b̃ ⊗ (ρ(y ) − E )ei .
Assume
det(ρ(x) − E ) ̸= 0 (and then det(ρ(y ) − E ) ̸= 0),
then C∗ is an acyclic chain complex．
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
38 / 226
We put
b2 = {0}, b1 = {∂2 (σ̃⊗e1 ), ∂2 (σ̃⊗e2 )}, b0 = {∂1 (x̃ ⊗e1 ), ∂1 (x̃ ⊗e2 )}.
[b1 , b̃0 /c1 ]
[b̃1 /c2 ][b0 /c0 ]
det(ρ(x) − E )
=
1 · det(ρ(x) − E )
=1
τρ (T 2 ) =
Remark
Reidemeister torsion of even-dimensional closed manifold is always
trivial.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
39 / 226
Classification of L(p, q)
Historically the motivation to introduce Reidemeister torsion is the
classification of 3-dimensional lens spaces.
p, q is a pair of coprime nutural numbers．
2π
ω = e p ∈ C: a p-th primitive root of unity.
S 3 ⊂ C2 = R4 : the unit sphere．
Tp = ⟨ t | t p ⟩ ∼
= Z/p ．
Tp acts on S 3 ;
t(z1 , z2 ) = (ωz1 , ω q z2 ).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
40 / 226
Definition
L(p, q) = S 3 /Tp
Remark
π1 (L(p, q)) ∼
= Tp
CW-complex structure of L(p, q)
0-cell:σ0 = (1, 0),
1-cell:σ1 = {(e iθ , 0) | 0 < θ < 2π
},
p
√
2-cell:σ2 = {(z1 , 1 − |z1 |2 ) | |z1 | < 1},
√
3-cell:σ3 = {(z1 , e iθ 1 − |z1 |2 ) | 0 < θ <
Teruaki Kitano (Soka University)
Reidemeister torsion
2π
, |z1 |
p
< 1}.
Nov. 03-07, 2014 in SNU
41 / 226
Fact
L(p, q) ∼
=PL L(p, q ′ ) if and only if q ′ ≡ ±q, or ±qq ′ ≡ 1 mod p.
L(p, q) ∼
=top L(p, q ′ ) if and only if L(p, q) ∼
=PL L(p, q ′ ).
L(p, q) ≃homotopy L(p, q ′ ) if and only if there exists c ∈ Z s. t.
qq ′ ≡ c 2 mod p, or −qq ′ ≡ c 2 mod p.
Remark
To see q ′ ≡ ±q, or ±qq ′ ≡ 1 mod p implies L(p, q) ∼
=PL L(p, q ′ ), we
construct a PL-homeomorphism between them. Conversely we can
prove that L(p, q) ∼
=PL L(p, q ′ ) implies q ′ ≡ ±q, or ± qq ′ ≡ 1 mod p
by using Reidemeister torsion．
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
42 / 226
Example;p = 7
L(7, 1) is homotopy equivalent to L(7, 2), because
1 · 2 ≡ 32 mod 7．
L(7, 1) is not PL-homeomorphic to L(7, 2), because 1 ̸≡ ±2 mod
7 and 1 · 2 ̸≡ 1 mod 7.
To show that L(7, 1) is not PL-homeomorphic to L(7, 2), we compute
Reidemeister torsions of them.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
43 / 226
We can compute Reidemeister torsion of L(p, q) as follows.
^
Boudary operators of C∗ (L(p,
q); Z) are
∂σ3 = (t r − 1)σ2 ,
∂σ2 = (t p−1 + t p−2 + · · · + t + 1)σ1
∂σ1 = (t − 1)σ0
^
where σ0 , σ1 , σ2 , σ3 are cells in L(p,
q) = S 3 .
Remark
Because Z/p ∼
= Tp is an abelian group, then we can consider its
action from the left.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
44 / 226
Here we take a tensor product this chain complex with C = ⟨1⟩C by
π1 (L(p, q) ∼
= Tp → ⟨ ω | ω p = 1 ⟩ ⊂ U(1). Now
ρa : π1 (L(p, q)) ∋ t 7→ ξ = ω a ∈ U(1)
Then it is easy to see
t p−1 + t p−2 + · · · + t + 1 7→ξ p−1 + ξ p−2 + · · · + ξ + 1
=0.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
45 / 226
Hence，the boundary operators of C∗ (L(p, q); Cρa ) are
∂σ3 = (ξ r − 1)σ2 ,
∂σ2 = 0,
∂σ1 = (ξ − 1)σ0 .
Along the definition,
c3 = σ1 , c2 = σ2 , c1 = σ1 , c0 = σ0 .
b3 = 0, b2 = (ξ r − 1)σ2 , b1 = 0, c0 = (ξ − 1)σ0 .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
46 / 226
Then
τρa (L(p, q)) =
det[b3 , b2 /c3 ] det[b1 , b0 /c1 ]
det[b2 , b1 /c2 ] det[b0 /c0 ]
det[b̃2 /c3 ] det[b̃0 /c1 ]
det[b2 /c2 ] det[b0 /c0 ]
1·1
= r
.
(ξ − 1) · (ξ − 1)
=
Theorem
τρ (L(p, q)) =
Teruaki Kitano (Soka University)
1
(ξ − 1)(ξ r − 1)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
47 / 226
Since τρ (L(p, q)) is well-defined up to ±ω s , then we consider its
absolute value |τρ (L(p, q))|.
Consider all choice of a ρ : π1 (L(p, q)) ∼
= Tp → ⟨ ω ⟩，the set of
the values of |τρ (L(p, q))|.
|τρ (L(7, 1))| = 1.33, 0.41, 0.26.
|τρ (L(7, 2))| = 0.74, 0.59, 0.33.
This implies L(7, 1) is not PL-isomorphic(in particular, not
homeomorphic) to L(7, 2).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
48 / 226
Reidemeiste torsion is an order.
Let M be a finitely generated module over a principal ideal domain
P. Then
M∼
= P/(p1 ) ⊕ P/(p2 ) ⊕ · · · ⊕ P/(pk )
where pi ∈ P.
Definition
The product ideal (p1 · · · pk ) ⊂ P is called the order of M and it is
denoted by ord(M) = (p1 · · · pk ).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
49 / 226
Here we consider a representation
ρ : π1 (L(p, q)) ∋ t 7→ ξ ∈ ⟨ω⟩ ⊂ U(1) as
ρ : π1 (L(p, q)) → GL(1; Z[ω])
and a chain complex C∗ (L(p, q); Z[ω]ρ ). Now the boundary operators
of this chain complex C∗ (L(p, q); Z[ω]ρ ) are
∂σ3 = (ξ r − 1)σ2 ,
∂σ2 = 0,
∂σ1 = (ξ − 1)σ0 .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
50 / 226
Then，
H3 (L(p, q); Z[ω]ρ ) = 0 = Z[ω]/(1),
H2 (L(p, q); Z[ω]ρ ) = Z[ω]/(ξ r − 1),
H1 (L(p, q); Z[ω]ρ ) = 0 = Z[ω]/(1),
H0 (L(p, q); Z[ω]ρ ) = Z[ω]/(ξ − 1).
Hence
1
(ξ − 1)(ξ r − 1)
ord(H1 (L(p, q); Z[ω]ρ ))ord(H3 (L(p, q); Z[ω]ρ ))
=
.
ord(H0 (L(p, q); Z[ω]ρ ))ord(H2 (L(p, q); Z[ω]ρ ))
τρ (L(p, q)) =
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
51 / 226
Alexander polynomial of a knot
There are some definitions of Alexander polynomial．We introduce
two definitions;
Apply Fox’s free differential to a presentation of G (K ).
the order of the Alexander module
Reidemeister torsion
Remark
Along these definition a twisted Alexander polynomial can be defined.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
52 / 226
Definition
An integral group ring of a group G is a ring
∑
ng g | ng ∈ Z}.
ZG = { a finite formal sum
g ∈G
Here for any element
∑
ng g the number of ng ̸= 0 is finite．
g ∈G
∑
ng g +
g ∈G
∑
ng g ·
g ∈G
Teruaki Kitano (Soka University)
∑
mg g =
g ∈G
∑
g ∈G
mg g =
∑
(ng + mg )g
g ∈G
∑∑
(nh · mh−1 g )g
g ∈G h∈G
Reidemeister torsion
Nov. 03-07, 2014 in SNU
53 / 226
Remark
The unit is 1 = 1(∈ Z) × 1(∈ G )．
We can also define QG over Q，and RG over R．
A group ring ZG is a commutative ring if and only if G is a
commutative group．
Example
G = Z = ⟨ t ⟩．Then for any element of ZZ = Z⟨ t ⟩, it is a form of
∑
nk t k . This is a Laurent polynomial of t. From here we can
identify ZZ = Z⟨ t ⟩ with Z[t, t −1 ]．
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
54 / 226
Let Fn = ⟨x1 , · · · , xn ⟩ be the free group generated by {x1 , · · · , xn }.
Recall Fox’s free differentials.
∂
∂
,...,
: ZFn → ZFn .
∂x1
∂xn
Characterization of Free differentials
∂
∂
,...,
have following properties．
∂x1
∂xn
1
They are linear over Z
{
1 (i = j)
∂
2
For any i, j ，
(xi ) = δij =
.
∂xj
0 (i ̸= j)
3
For any g , g ′ ∈ Fn ，
∂
∂
∂
(gg ′ ) =
(g ) + g
(g ′ ). (Leibniz
∂xj
∂xj
∂xj
rule)
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
55 / 226
Lemma
The followings hold．
∂
(1) = 0.
∂xj
∂
∂
(g −1 ) = −g −1
(g ) for any g ∈ Fn .
∂xj
∂xj
∂ k
(xj ) = 1 + xj + · · · + xjk−1 (k > 0).
∂xj
∂ k
(xj ) = −(xj−1 + · · · + xjk ) (k < 0).
∂xj
∂
g k − 1 ∂g
(g k ) =
for any g ∈ Fn , k > 0.
∂xj
g − 1 ∂xj
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
56 / 226
Proposition (Fundamental formula of free differentials)
For any w ∈ ZFn ，it holds that
w −1=
n
∑
∂w
j=1
Teruaki Kitano (Soka University)
∂xj
(xj − 1).
Reidemeister torsion
Nov. 03-07, 2014 in SNU
57 / 226
K is a knot in S 3
E (K ) is its exterior which is the complement of open tubular
neighborhood of K .
G (K ) = π1 E (K ) ∼
= π1 (S 3 − K ) the knot group of K .
We fix a presentation of its knot group G (K )．
⟨x1 , . . . , xn | r1 , . . . , rn−1 ⟩.
Here we do not assume it is a Wirtinger presentation. We only
assume that
deficiency = (the number of generators) − (the number of relators)
= 1.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
58 / 226
The abelianization of G (K )
α : G (K ) → Z = ⟨ t ⟩.
By using the above fixed presentation, an epimorphism
Fn = ⟨x1 , . . . , xn ⟩ ↠ G (K )
is defined. Further we consider a ring homomorphism
ZFn → ZG (K )
induced from Fn → G (K ).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
59 / 226
Definition
(n − 1) × n-matrix A is defined by
))
( (
(
)
∂ri
A = α∗
∈ M (n − 1) × n; Z[t, t −1 ]
∂xj
Here α∗ : ZG (K ) → Z⟨t⟩ = Z[t, t −1 ]. This matrix A is called the
Alexander matrix of G (K ).
Remove the k-th column from A and this (n − 1) × (n − 1)-matrix is
called Ak ．
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
60 / 226
Lemma
There exist k s.t. α∗ (xk ) − 1 ̸= 0 ∈ Z[t, t −1 ]．
Proof.
If α(xk ) = 1 for any k, α : G (K ) → Z is the trivial homomorphism,
not an epimorphism.
Lemma
For any k, l ，
(α∗ (xl ) − 1) detAk = ± (α∗ (xk ) − 1) detAl
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
61 / 226
Proof.
We can assume k = 1, l = 2．
For ri = 1 ∈ ZG (K ), we apply the fundamental formula,
n
∑
∂ri
(xj − 1)
0 = ri − 1 =
∂xj
j=1
Apply α∗ to both sides，
n
∑
j=1
Teruaki Kitano (Soka University)
(
α∗
∂ri
∂xj
)
(α∗ (xj ) − 1) = 0
Reidemeister torsion
Nov. 03-07, 2014 in SNU
62 / 226
Then
(
(α∗ (x1 ) − 1)α∗
∂ri
∂x1
)
=−
n
∑
(
α∗
j=2
∂ri
∂xj
)
(α∗ (xj ) − 1)
In the
replace the first column
( A)2 (removed the second
( column),
)
∂ri
∂ri
α∗ ∂x
to (α∗ (x1 ) − 1)α∗ ∂x
.
1
1
We write Ã2 to this obtained matrix ．
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
63 / 226
Take its determinant;
( )
( )
( )
∂r1
∂r1
(α∗ (x1 ) − 1) α∗ ∂r1
α
.
.
.
α
∗
∗
∂x1
∂x3
∂xn
.
.
..
..
detÃ2 = .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(
)
)
(
)
(
∂r
∂r
∂r
n−1
n−1
n−1
(α∗ (x1 ) − 1) α∗
α∗ ∂x3
. . . α∗ ∂xn ∂x1
= (α∗ (x1 ) − 1) detA2
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
64 / 226
On the other hand,
(
)
n
( ) ∑
∂r1
∂r1
−
α∗
(α∗ (xj ) − 1) . . . α∗ ∂x
n
∂xj
j=2
..
..
detÃ2 = .
.
.
.
.
n
)
(
)
∑ ( ∂r
n−1
∂rn−1 −
α∗
(α∗ (xj ) − 1) . . . α∗ ∂xn ∂xj
j=2
( )
( )
( )
∂r1
∂r1
∂r1
α∗ ∂x
α
.
.
.
α
∗ ∂x3
∗ ∂xn
j
n
∑
..
..
=−
(α∗ (xj ) − 1) (.
) . . .(. . . . . .). . . . . .
)
(.
j=2
∂rn−1
α∗ ∂r∂xn−1
. . . α∗ ∂r∂xn−1
α∗ ∂x
n
3
j
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
65 / 226
( )
( )
( )
∂r1
∂r1
α∗ ∂r1
α∗ ∂x3
. . . α∗ ∂x
∂x2
n
.
.
..
..
= −(α∗ (x2 ) − 1) .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(
)
(
)
(
)
∂rn−1
∂rn−1 α∗ ∂r∂xn−1
α∗ ∂x3
. . . α∗ ∂xn
2
= −(α∗ (x2 ) − 1)detA1
Therefore
(α∗ (x1 ) − 1) detA2 = −(α∗ (x2 ) − 1)detA1 .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
66 / 226
Proposition
Up to ±t s (s ∈ Z), the rational expression
detAk
α∗ (xk ) − 1
is independent of the choice of a presentation of G (K ). Namely it is
an invariant of a group.
Proof.
It can be directly checked for Tietze transformations.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
67 / 226
Tietze transformations
Let G be a group and ⟨x1 , · · · , xk |r1 , · · · , rl ⟩ a presentation of G .
Theorem (Tietze)
A presentation ⟨x1 , · · · , xk |r1 , · · · , rl ⟩ of a group G can be
transformed to any other presentation of G by an application of a
finite sequence of of the following two type operations and their
inverse:
To add a consequence r of the ralators r1 , · · · , rl to the set of
relators. The resulting presenation is ⟨x1 , · · · , xk |r1 , · · · , rl , r ⟩.
To add a new generator x and a new relator xw −1 where w is
any word in x1 , · · · , xk . The resulting presentation is
⟨x1 , · · · , xk , x|r1 , · · · , rl , xw −1 ⟩.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
68 / 226
For a knot K , we can take some special presentation, which is called a
Wirtinger presentation of G (K ), from a regular diagram on the plane.
Definition
If we take a Wirtinger presentation of G (K ) (in this case α(xj ) = t)，
The denominator is always t − 1. Then the numerator is an invariant
of G (K ) up to ±t s ．This is the Alexander polynomial ∆K (t) of K .
detAk
∆K (t)
=
.
t −1
t −1
Remark
Alexander polynomial is well-defined up to ±t s .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
69 / 226
Trefoile knot 31
31
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
70 / 226
G (31 ) = ⟨x, y | r = xyx(yxy )−1 ⟩
The relator r goes to
r = xyx(yxy )−1
= xyxy −1 x −1 y −1
7→ xy −1 ∈ G (31 )/[G (31 ), G (31 )] ∼
= Z,
then we get
xy −1 = 1
in G (31 )/[G (31 ), G (31 )].
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
71 / 226
Hence the abelianization is given by
α : G (31 ) ∋ x, y 7→ t ∈ ⟨t⟩.
Here
∂
∂
(r ) =
(xyx(yxy )−1 )
∂x
∂x
∂
∂
=
(xyx) − xyx(yxy )−1 (yxy )
∂x
∂x
∂
∂
=
(xyx) − r (yxy ).
∂x
∂x
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
72 / 226
In ZG (31 )( and then Z[t, t −1 ], r = 1. Then
)
∂ (
∂
(xyx(yxy )−1 =
((xyx − (yxy ))
∂x
∂x
in Z[t, t −1 ].
Therefore we can compute free differentials for r = xyx − yxy instead
of r = xyx(yxy )−1 .
Then
∂
∂
∂
(xyx − yxy ) =
(xyx) −
(yxy )
∂x
∂x
∂x
= 1 + xy − y
7→ α∗ (1 + xy − y )
= t 2 − t + 1 ∈ Z[t, t −1 ]
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
73 / 226
Similarly
∂
∂
∂
(xyx − yxy ) =
(xyx) −
yxy
∂y
∂y
∂y
= x − 1 − yx
7→ α∗ (x − 1 − yx)
= −(t 2 − t + 1) ∈ Z[t, t −1 ]
Hence
(
)
A = (t 2 − t + 1) −(t 2 − t + 1) ,
detA2
detA1
t2 − t + 1
=−
=
.
t −1
t −1
t −1
We change this presentation to ⟨x, y , z | xyx(yxy )−1 , xyz −1 ⟩
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
74 / 226
Then its Alexander matrix is
( 2
)
(t − t + 1) −(t 2 − t + 1) 0
A=
1
t
−1
then
t2 − t + 1
detA1
=
,
t −1
t −1
detA2
t2 − t + 1
=−
,
t −1
t −1
detA3
t(t 2 − t + 1) + (t 2 − t + 1)
=
t2 − 1
t2 − 1
2
t −t +1
=
.
t −1
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
75 / 226
Figure-eight knot 41
41
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
76 / 226
From the diagram of G (41 ) we can get its Wirtinger presentation.
⟨ x1 , x2 , x3 , x4 | x4 = x1 x3 x1−1 , x2 = x3 x1 x3−1 , x2 x1 x2−1 x4−1 ⟩
=⟨ x1 , x3 | (x3 x1 x3−1 )x1 (x3 x1 x3−1 )−1 (x1 x3 x1−1 )−1 ⟩
=⟨ x1 , x3 | x3 x1 x3−1 x1 x3 x1−1 x3−1 x1 x3−1 x1−1 ⟩
=⟨ x1 , x3 | x1−1 x3 x1 x3−1 x1 x3 x1−1 x3−1 x1 x3−1 ⟩ (conjugated by x1−1 )
=⟨ x, y | wxw −1 = y ⟩.
where x = x1 , y = x3 , w = x −1 yxy −1 .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
77 / 226
Further the abelianization
α : G (41 ) → ⟨ t ⟩
is given by α(x) = α(y ) = t ．
Then we have
∂
∂w
∂x
∂w
(wxw −1 y −1 ) =
+w
− wxw −1
∂x
∂x
∂x
∂x
∂w
= (1 − y )
+w
( ∂x
)
∂w
7→ α∗ (1 − y )
+ α∗ (w )
∂x
)
(
∂w
= (1 − t)α∗
+ 1.
∂x
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
78 / 226
Here
(
α∗
∂w
∂x
)
(
)
∂ −1 −1
= α∗
(x yxy )
∂x
= α∗ (−x −1 + x −1 y )
= −t −1 + 1.
Then
(
α∗
)
∂
−1 −1
(wxw y ) = (1 − t)(−t −1 + 1) + 1
∂x
= −t −1 + 1 + 1 − t(−t −1 + 1)
= −t −1 + 1 + 1 + 1 − t
= −t −1 + 3 − t.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
79 / 226
Similarly
(
α∗
∂
(wxw −1 y −1 )
∂y
)
(
)
∂w
= α∗ (1 − y )
−1
∂x
= (1 − t)(t −1 − 1) − 1
= t −1 − 3 + t.
Hence
(
)
A = −t −1 + 3 − t t −1 − 3 + t .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
80 / 226
detA1
t −1 − 3 + t
=
α∗ (x1 ) − 1
t −1
1 (−t 2 + 3t − 1)
.
=−
t
t −1
detA2
t −1 − 3 + t
=−
α∗ (x2 ) − 1
t −1
1 (−t 2 + 3t − 1)
.
=
t
t −1
Up to ±1,
Teruaki Kitano (Soka University)
∆K (t) = −t 2 + 3t − 1.
Reidemeister torsion
Nov. 03-07, 2014 in SNU
81 / 226
Remark
In the computation of free differentials, after applying α∗ ，
(
)
(
)
∂
∂
−1 −1
−1
α∗
(wxw y ) = α∗
(wxw − y )
∂x
∂x
)
(
∂
= α∗
(wx − yw ) ,
∂x
then we can apply free differentials for wxw −1 − y , or wx − yw
instead of wxw −1 y −1 ．
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
82 / 226
If the deficiency is less than or equal to 0,
G (K ) = ⟨ x1 , . . . , xn | r1 , . . . , rm ⟩ (m > n − 1)
A is an m × n-matrix，and Ak is an m × (n − 1)-matrix. Then in the
case, Ak is not a square matrix. Then in this case, we consider all
(n − 1) × (n − 1)-minors and take its gcd, which is written by Qk .
Now we have the following.
Proposition
For any k s.t. α∗ (xk ) − 1 ̸= 0,
τα (K ) =
Qk
α∗ (xk ) − 1
is independent of the choice of presentation up to ±t s (s ∈ Z)．
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
83 / 226
Milnor’s theorem
Let
E (K ) = S 3 − N(K )
be an exterior of K ．Here N(K ) is an open tubular neighborhood of
K in S 3 . Let
α : G (K ) → T = ⟨t⟩
be the abelianization. Here we consider α as
α : G (K ) → T ⊂ GL(1; Q(t)).
an 1-dimensional representation over Q(t)．
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
84 / 226
We take its universal cover
Ẽ (K ) → E (K ).
We assume G (K ) acts on Ẽ (K ) from the right. Then
C∗ (E (K ); Q(t)) = C∗ (Ẽ (K ); Z) ⊗Zπ1 (X ) Q(t).
By using this representation α, Reidemeister torsion of E (K )
τα (E (K )) = τ (C∗ (E (K ); Q(t))) ∈ Q(t) \ {0}
can be defined up to ±t s ．
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
85 / 226
Theorem (Milnor)
τα (E (K )) =
Teruaki Kitano (Soka University)
∆K (t)
.
t −1
Reidemeister torsion
Nov. 03-07, 2014 in SNU
86 / 226
Proof.
We take a 2-dimensional CW-complex WK associated to a Wirtinger
presentation of G (K )．Namely, from the presentation,
G (K ) = ⟨x1 , . . . , xn | r1 , . . . , rn−1 ⟩
we define
0-cell:b0 .
1-cell:x1 , . . . , xn .
2-cell:d1 , . . . , dn−1 where di is attached to x1 , . . . , xn by relator
ri ．
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
87 / 226
Remark
It is well known that E (K ) can be collapsed to WK ．
Since Reidemeister torsion is a simple homotopy invariant，it is
enough to prove
∆K (t)
τα (WK ) =
t −1
for WK . It can be done by direct computation.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
88 / 226
Duality of Alexander polynomial
Proposition (Seifert)
Up to ±t s , ∆K (t −1 ) = ∆K (t).
Proof.
We take a Seifert surface of K with genus h and its Seifert matrix V .
∆K (t) is defined by ∆K (t) = det(T V − tV ).
Then we obtain
∆K (t −1 ) = det(T V − t −1 V )
= (−t −1 )2h det(−t T V + V )
(
)
= t −2h det T (−tV + T V )
= t −2h det(T V − tV )
= t −2h ∆K (t).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
89 / 226
We prove the following proposition by using Reidemeister torsion.
First we prove the duality of the torsion.
Over a field F we take an acyclic chain complex C∗ :
∂
∂m−1
∂
m
0 → Cm →
Cm−1 → · · · → C1 →1 C0 → 0.
Definition
Its dual chain complex C∗♯ of C∗
∂♯
♯
∂m−1
∂♯
m
0 → Cm →
Cm−1 → · · · → C1 →1 C0 → 0
where
♯
∗
.
: Cq♯ → Cq−1
Cq♯ = Hom(Cm−q , F), ∂q♯ = (−1)q+1 ∂m−q+1
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
90 / 226
For any basis cq of Cq ，we take its dual basis c♯q = c∗m−q of
Cq♯ = Hom(Cm−q , F)．Now we have the following.
Lemma
m+1
τ (C∗♯ ) = ±τ (C∗ )(−1)
Teruaki Kitano (Soka University)
Reidemeister torsion
.
Nov. 03-07, 2014 in SNU
91 / 226
On Z[t, t −1 ], we can consider an involution
¯ : Z[t, t] ∋ f (t) 7→ f (t) = f (t −1 ) ∈ Z[t, t −1 ].
Then for a right Z[t, t −1 ]-module A，we can take its dual module
A∗ = Hom(A, Z[t, t −1 ])．
On A∗ a structure of Z[t, t −1 ]-module can be defined as follows．For
any homomorphism ϕ : A → Z[t, t −1 ] and any f (t) ∈ Z[t, t −1 ]，
(f (t) · ϕ)(a) = ϕ(a · f (t −1 )).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
92 / 226
Here we assume that
E (K ) is a 3-dimensional simplicial complex．
E (K )′ is the dual decomposition of E (K )．
From this decomposition a cell-decomposition of ∂E (K ) can be
induced，denoted by ∂E (K )′ ．On Ẽ (K ), Z⟨ t ⟩ = Z[t, t −1 ] acts
from the right．Similarly on Ẽ (K )′ , Z⟨ t ⟩ = Z[t, t −1 ] acts from the
right．We take the abelianization
α : G (K ) → ⟨ t ⟩.
We consider it as
α : G (K ) → GL(1; Z[t, t −1 ]).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
93 / 226
Lemma
As a Z[t, t −1 ]-module，C3−q (E (K ); Z[t, t −1 ]α ) is isomorphic to
the dual of Cq (E (K )′ , ∂E (K )′ ; Z[t, t −1 ]α )．
Further boundary operator
∂3−q : C3−q (E (K ); Z[t, t −1 ]α ) → C3−q−1 (E (K ); Z[t, t −1 ]α )
is the dual of
∂q−1 : Cq+1 (E (K )′ , ∂E (K )′ ; Z[t, t −1 ]α ) → Cq (E (K )′ , ∂E (K )′ ; Z[t, t −1 ]α ).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
94 / 226
Proof
Now Ẽ (K ) is the universal cover of E (K ), and
C∗ (E (K ); Z[t, t −1 ]α ) = C∗ (Ẽ (K ); Z) ⊗ZG (K ) Z[t, t −1 ]α .
Here over these local coefficients, an intersection pairing
Cq (E (K ); Z[t, t −1 ]α ) × C3−q (E (K )′ , ∂E (K )′ ; Z[t, t −1 ]α ) → Z[t, t −1 ]
can be considered．
For any c ⊗ f (t) ∈ Cq (Ẽ (K ); Z) ⊗Z[t,t −1 ] Z[t, t] and
c ′ ⊗ f ′ (t) ∈ C3−q (Ẽ (K )′ , ∂E (K )′ ; Z) ⊗Z[t,t −1 ] Z[t, t], by extending the
intersection on Cq (Ẽ (K ); Z) ⊗ Cq (Ẽ (K )′ , ∂E (K )′ ; Z) to the
Z[t, t −1 ]-coefficients (c, c ′ )f (t)f ′ (t) is defined．
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
95 / 226
Remark
It can be extended over Q(t) naturally．
Here for C3−q (E (K )′ , ∂E (K )′ ; Z[t, t −1 ]α ) ∋ c ′ ，we take a map
∑
c 7→
(c, c ′ x)α(x).
x∈G (K )
The number of x ∈ G (K ) with (c, c ′ x) ̸= 0 is finite, then this map is
well-defined. This gives a map
[·, c ′ ] : C3−q (E (K ); Z[t, t −1 ]α ) →
Hom(Cq (E (K )′ ,∂E (K )′ ; Z[t, t −1 ]α ); Z[t, t −1 ]).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
96 / 226
Here
[cy , c ′ ] =
∑
(cy , c ′ x)α(x)
x∈G (K )
=
∑
(c, c ′ xy −1 )α(xy −1 )α(y )
x∈G (K )
=
∑
(c, c ′ x)α(x)α(y )
x∈G (K )
= [c, c ′ ]α(y ).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
97 / 226
On the other hand,
[c, c ′ y ] =
∑
(c, c ′ yx)α(x)
x∈G (K )
=
∑
(c, c ′ yx)α(y )−1 α(yx)
x∈G (K )
=
∑
(c, c ′ x)α(x)α(y )−1
x∈G (K )
= [c, c ′ ]α(y )−1
= [c, c ′ ]α(y ).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
98 / 226
Further it holds that
[∂c, c ′ ] =
∑
(∂c, c ′ x)α(x)
x∈G (K )
=
∑
(c, ∂c ′ x)α(x)
x∈G (K )
= [c, ∂c ′ x].
Therefore we have the following.
Proposition
Up to ±t s ,
τα (E (K )) = τα (E (K ), ∂E (K )).
Remark
The coefficients can be extended from Z [t, t −1 ] to Q(t)．
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
99 / 226
Proof
Because C∗ (E (K ); Q(t)α ) is acyclic，τα (E (K )) can be defined．Then
τα (E (K )) = τ (C∗ (E (K ); Q(t)α ))
= ±τ (C∗♯ (E (K )′ , ∂E (K )′ ; Q(t)α ))
= ±τα (E (K ), ∂E (K )).
Proposition
τα (E (K )) = ±τα (E (K ), ∂E (K )).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
100 / 226
Proof
The restriction of α : G (K ) → ⟨ t ⟩ ⊂ GL(1; Q(t)) on π1 (∂E (K )) is
written by the same α. We consider a short exact sequence
0 → C∗ (∂E (K ); Q(t)α ) → C∗ (E (K ); Q(t)α )
→ C∗ (E (K ), ∂E (K ); Q(t)α ) → 0.
Now C∗ (E (K ); Q(t)α ) and C∗ (E (K ), ∂E (K ); Q(t)α ) are acyclic, then
C∗ (∂E (K ); Q(t)α ) is also acyclic.
Then
τα (E (K )) = ±τα (E (K ), ∂E (K ))τα (∂E (K )).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
101 / 226
Because ∂E (K ) = T 2 ，then
τα (∂E (K )) = ±1
We have
τα (E (K )) = ±τα (E (K ), ∂E (K )).
Theorem
τα (E (K )) = ±τα (E (K )).
Proof.
τα (E (K )) = ±τα (E (K ), ∂E (K )) (by duality),
= ±τα (E (K )) (by short exact sequence).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
102 / 226
We apply this theorem for a knot.
Corollary
Up to ±t s , we have
Teruaki Kitano (Soka University)
∆K (t −1 ) = ∆K (t).
Reidemeister torsion
Nov. 03-07, 2014 in SNU
103 / 226
Proof of Duality
By Milnor’s theorem,
τα (E (K )) =
∆K (t)
.
t −1
Further α : G (K ) ∋ xi 7→ t −1 ∈ ⟨t⟩ is also a representation, then
∆K (t −1 )
t −1 − 1
∆K (t −1 )
= −t
.
t −1
τᾱ (E (K )) =
Hence up to ±t s , we get
∆K (t) = ∆K (t −1 ).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
104 / 226
Reidemeister torsion and Alexander polynomial
Alexander polynomial of K is Reidemeister torsion of E (K ).
τα (E (K )) =
∆k (t)
.
t −1
Reidemeister torsion is described by orders of twisted homology.
Alexander polynomial ∆K (t) can be the order of some twisted
homology:
∆K (t) = ord(H1 (E (K ); Q[t, t −1 ]α ))
where α : G (K ) → GL(1; Q[t, t −1 ]).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
105 / 226
Twisted Alexander polynomial is defined by
Fox’s free differentials
Reidemeister torsion
orders
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
106 / 226
History of Twisted Alexander polynomial
In the debut of twisted Alexander polynomial
X. S. Lin, Representations of knot groups and twisted Alexander
polynomials, Acta Mathematica Sinica, English Series, 17
(2001), No.3, pp. 361–380
for a knot by using a Seifert surface.
M. Wada, Twisted Alexander polynomial for finitely presentable
groups, Topology 33 (1994), 241–256.
for a finitely presentable group with an epimorphism onto Z.
B. Jiang and S. Wang, Twisted topological invariants associated
with representations, in Topics in knot theory (Erzurum, 1992),
NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 399, Kluwer
Acad. Publ., Dordrecht (1993).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
107 / 226
Remark
We follow the definition due to Wada, because it is most
computable.
Twisted Alexander polynomial can be defined for a general linear
representation over Euclidean domain.
For simplicity we consider this invariant only for a knot with a
2-dimensional unimodular representation over a field F.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
108 / 226
Let K be a knot K and ⟨x1 , . . . , xn | r1 , . . . , rn−1 ⟩ a presentation
with deficieny one.
let α : G (K ) → Z = ⟨t⟩ be the abelianization homomorphism
Let ρ : G (K ) → SL(2; F) a representation.
ρ and α naturally induce ring homomorphisms
ρ∗ : ZG (K ) → ZSL(2; F) ∼
= M(2; F),
α∗ : ZG (K ) → ZZ ∼
= Z[t, t −1 ]
where M(2; F) is the matrix algebra of 2 × 2 matrices over F.
ρ∗ ⊗ α∗ : ZG (K ) → M(2; F) ⊗ Z[t, t −1 ] ∼
= M (2; F[t, t −1 ]) is an
induced ring homomorphism.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
109 / 226
Let Fn = ⟨x1 , . . . , xn ⟩ denote the free group.
Let
(
)
Φ : ZFn → M 2; F[t, t −1 ]
be the composite of
the surjection ZFn → ZG (K ) induced by the presentation
(
)
the ring homomorphism ρ∗ ⊗ α∗ : ZG (K ) → M 2; F[t, t −1 ] .
Let
∂
∂
,...,
: ZFn → ZFn
∂x1
∂xn
be the Fox’s free differentials.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
110 / 226
Definition
The (n − 1) × n matrix Aρ whose (i, j) component is the 2 × 2 matrix
(
)
(
)
∂ri
Φ
∈ M 2; F[t, t −1 ] ,
∂xj
This is called the twisted Alexander matrix associated to ρ (and the
fixed presentation).
Remark
(
(
))
(
)
Aρ ∈ M (n − 1) × n; M 2; F[t, t −1 ] = M 2(n − 1) × 2n; F[t, t −1 ] .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
111 / 226
To obtain the square matrix, we remove one column from Aρ .
Let us denote by Aρ,k the (n − 1) × (n − 1) matrix obtained
from Aρ by removing the k-th column.
We may regard Aρ,k as a 2(n − 1) × 2(n − 1) matrix with
coefficients in F[t, t −1 ].
The following two lemmas are the foundations of the definition of the
twisted Alexander polynomial.
Lemma
det Φ(xk − 1) ̸= 0 for some k.
Lemma
For any j, k,
det Aρ,k det Φ(xj − 1) = det Aρ,j det Φ(xk − 1).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
112 / 226
From the above two lemmas, we can define the twisted Alexander
polynomial of G (K ) associated ρ : G (K ) → SL(2; F) to be a rational
expression as follows.
Definition
∆K ,ρ (t) =
detAρ,k
detΦ(xk − 1)
for k s.t det Φ(xk − 1) ̸= 0.
Remark
Up to a factor of t s (s ∈ Z), this is an invariant of a knot group
G (K ) with ρ. Namely, it does not depend on the choices of a
presentation. Hence we can consider it as a knot invariant.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
113 / 226
In general, the twisted Alexander polynomial ∆K ,ρ (t) depends on ρ.
However the following proposition is known.
Definition
Two representations ρ and ρ′ are conjugate if there exists
S ∈ SL(2; F) s.t. ρ(x) = Sρ′ (x)S −1 .
Proposition
If ρ and ρ′ are conjugate, then ∆K ,ρ (t) = ∆K ,ρ′ (t).
Proof.
The twisted Alexander matricies are conjugate.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
114 / 226
Example:trivial represenatation
Let ρ : G (K ) → SL(2; C) be a trivial representation. That is, for any
element x ∈ G (K ), ρ(x) = E ∈ SL(2; C). Then ρ ⊗ α is just direct
sum of two copies of α;
(
)
α(x)
0
α ⊕ α : G (K ) ∋ x 7→
∈ GL(2; C(t))
0
α(x)
Hence we get
∆K ,α⊕α (t) =
∆K (t) ∆K (t)
·
.
t −1 t −1
Remark
If a representation ρ = ρ1 ⊕ ρ2 , then
∆K ,ρ (t) = ∆K ,ρ2 (t) · ∆K ,ρ2 (t).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
115 / 226
Growth preriod in 1990’s
T. Kitano, Twisted Alexander polynomial and Reidemeister
torsion, Pacific J. Math. 174 (1996), no. 2, 431–442.
Twisetd Alexander polynomial can be defined to be
Reidemeister torsion.
P. Kirk and C. Livingston, Twisted Alexander invariants,
Reidemeister torsion, and Casson-Gordon invariants, Topology
38, (1999), no. 3, 635–661.
Twisted Alexander polynomial can be defined to be the order of
twisted Alexander module.
After 2000, .....lots of many...
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
116 / 226
Two survey papers:
S. Friedl and S. Vidussi, A survey of twisted Alexander
polynomials, The mathematics of knots, p45–94, Springer,
Heidelberg, 2011.
T. Morifuji, Representation of knot groups into SL(2; C) and
twisted Alexander polynomials, to be published in the book
“Handbook of Group Actions (Vol I)”, Higher Educational Press
and International Press, Beijing-Boston.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
117 / 226
Twisted Alexander polynomial is a polynomial ?
By definition, it is not clear that a twisted Alexander polynomial is a
polynomial. However, under a generic assumption on ρ, the twisted
Alexander polynomial becomes a Laurent polynomial.
Proposition (K.-Morifuji)
If ρ : G (K ) → SL(2; F) is not an abelian representation, then ∆K ,ρ (t)
is a Laurent polynomial with coefficients in F.
Remark
For any abelian representation ρ : G (K ) → SL(2; F), ∆K ,ρ (t) is not a
Laurent polynomial. In this case it can be described by the Alexander
polynomial as the one for the trivial representation.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
118 / 226
Example:figure-eight knot 41
Here we consider an irreducible representation of
G (41 ) = ⟨ x, y | wx = yw ⟩
where w = x −1 yxy −1 .
Remark
Here the generators x and y are conjugate by w . This is the point to
study SL(2; C)-representation for 2-bridge knot.
For simplicity, we write X to ρ(x) for x ∈ G (K ).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
119 / 226
Lemma
Let X , Y ∈ SL(2, C). If X and Y are conjugate and XY ̸= YX , then
there exists P ∈ SL(2; C) s.t.
(
)
(
)
s 1
s 0
−1
−1
PXP =
, PYP =
.
0 1/s
u 1/s
Proof.
Linear algebra.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
120 / 226
For any representation, by taking conjugate, we may assume that its
representative of this conjugacy class is given by
ρs,u : G (41 ) → SL(2; C) (s, u ∈ C \ {0})
(
where
X =
)
(
)
s 1
s 0
,Y =
0 1/s
u 1/s
Remark
Because
1
tr X = s + , tr X −1 Y = 2 − u,
s
then the sapce of the conjugacy classes of the irreducible
represenations can be parametrized by the traces of X , X −1 Y ．
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
121 / 226
We compute the matrix
R = WX − YW = ρ(w )ρ(x) − ρ(y )ρ(w )
to get the defining equations of the space of the conjugacy classes.
We compute each entry of R = (Rij ):
R11 = 0,
R12 = 3 − s12 − s 2 − 3u + su2 + s 2 u + u 2 ,
2
R21 = −3u + su2 + s 2 u + 3u 2 − us 2 − s 2 u 2 − u 3 = −uR12 ,
R22 = 0.
Hence R12 = 0 is the equation and it gives the space R̂ of the
conjugacy classes of the irreducible representations.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
122 / 226
Proposition
R̂ = {(s, u) ∈ (C \ {0})2 | R12 = 0}
This equation
3−
1
u
2
−
s
−
3u
+
+ s 2u + u2 = 0
s2
s2
can be solved in u:
u=
−1 + 3s 2 − s 4 ±
Teruaki Kitano (Soka University)
√
1 − 2s 2 − s 4 − 2s 6 + s 8
.
2s 2
Reidemeister torsion
Nov. 03-07, 2014 in SNU
123 / 226
Here we compute Fox’s free differential for wx − yw in stead of
wxw −1 y −1 .
∂(wx − yw )
= (1 − y )(x −1 − wx) − 1.
∂y
Therefore we obtain
Aρ,1
(
)
∂(wx − yw )
=Φ
∂y
= (E − tY )(t −1 X −1 − tWX ) − E .
We substitute
u=
−1 + 3s 2 − s 4 ±
√
1 − 2s 2 − s 4 − 2s 6 + s 8
2s 2
to each entry and compute its determinant.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
124 / 226
Here we get the following (not depend on the choice of u):
det A1 =
3
3t
1
3s
2
2
−
−
+
6
+
+
2s
−
− 3st + t 2
t 2 st
t
s2
s
On the other hand, we obtain
det(tX − E ) = t 2 − (s + 1/s)t + 1.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
125 / 226
Finnaly we obtain
1
2 (1 + s 2 )
−
+1
t2
st
(
)
1 2
1
= 2 (t − 2 s +
t + 1)
t
s
1
= 2 (t 2 − 2(tr X )t + 1).
t
∆41 ,ρs,u (t) =
Remark
Because ρs,u is not abelian, then ∆41 ,ρs,u (t) is a Laurent
polynomial.
Because 41 is fibered，then ∆41 ,ρs,u (t) is monic．
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
126 / 226
Example:a torus knot
We consider that ∆K ,ρ (t) is a Laurant polynomial valued function on
the space of conjugacy classes of SL(2; C)-irreducible representations.
Let K = T (p, q) be a torus (p, q)-knot.
Theorem (K.-Morifuji)
∆T (p,q),ρ (t) is a locally constant, that is, a constant function on each
connected component of the space of SL(2; C)-representations.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
127 / 226
Let K be a torus (p, q)-knot.
Let G (p, q) = ⟨x, y |x p = y q ⟩.
Let m be the meridian given by x −r y s where ps − qr = 1.
We write z to x p = y q . z is a center element of the infinite
order.
Let ρ : G (p, q) → SL(2; C) be an irreducible representation.
The center of SL(2; C) is {±E }.
Lemma
Z = ρ(z) = ±E .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
128 / 226
Because Z = ±E , then
X p = ±E , Y q = ±E .
Here we may assume the eigenvalues of X and Y are given by
λ±1 = e ±
√
−1πa/p
, µ±1 = e ±
√
−1πb/q
,
where 0 < a < p, 0 < b < q. Hence we have
X p = (−E )a , Y q = (−E )b
In any case we have
X 2p = Y 2q = E .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
129 / 226
Now we get
tr X = 2 cos
πa
πb
, tr Y = 2 cos .
p
q
Proposition (D. Johnson)
The conjugacy class of the irreducible representation ρ is
uniquely determined for fixed pair (tr X , tr Y , tr M).
Any representative ρa,b can be parametrized by (a, b, tr M) where
0 < a < p, 0 < b < q.
a ≡ b mod 2.
sb
tr M ̸= 2 cos π( ra
p ± q ).
πb
a
tr X = 2 cos πa
p , tr Y = 2 cos q , Z = (−E ) .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
130 / 226
By using this proposition, we see each connected component of the
conjugacy classes can be parametrized by tr M under fixing (a, b).
Recall G (p, q) = ⟨x, y | r = x p y −q ⟩. By applying Fox’s differentials,
∂r
= 1 + x + · · · + x p−1 .
∂x
Then we get
∆T (p,q),ρ (t)
=
∂r
Φ( ∂x
)
Φ(y − 1)
=
(1 + λt q + · · · + λp−1 t (p−1)q )(1 + λ−1 t q + · · · + λ−(p−1) t −(p−1)q )
1 − (µ + µ−1 )t p + t 2p
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
131 / 226
We consider the case of torus (2, q)-knot more simply. Here the
connected components consists of q−1
components parametrized by
2
odd integer b with 0 < b < q. Then the twisted Alexander
polynomial is given by
∏
( 2
)(
)
(
)
∆K ,ρb (t) = t 2 + 1
t − ξk t 2 − ξ¯k ,
0<k<q, k:odd, k̸=b
√
where ξk = exp( −1πk/q).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
132 / 226
In partiquar, for 31 = T (2, 3), there is just one connected component
and we see that
∆K ,ρ (t) =
t6 + 1
= t2 + 1
t4 − t2 + 1
holds for any irreducible representation ρ.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
133 / 226
Theorem (Cha, Goda-Morifuji-K.)
If K is fibered, then ∆K ,ρ is monic.
J. C. Cha, Fibred knots and twisted Alexander invariants, Trans.
Amer. Math. Soc. 355 (2003), no. 10, 4187–4200
H. Goda, T. Kitano and T. Morifuji, Reidemeister torsion,
twisted Alexander polynomial and fibered knots, Comment.
Math. Helv. 80 (2005), no. 1, 51–61.
Using properties of Reidemeister torion
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
134 / 226
S. Friedl and S. Vidussi, Twisted Alexander polynomials detect
fibered 3-manifolds, Ann. of Math. (2) 173 (2011), no. 3,
1587–1643.
Theorem (Friedl-Vidussi)
The converse is true.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
135 / 226
N. Dunfield, S. Friedl and N. Jackson, Twisted Alexander
polynomials of hyperbolic knots, Exp. Math. 21 (2012), no. 4,
329–352.
Let K be a hyperbolic knot.
Let ρ0 : G (K ) → SL(2; C) be a lift of holonomy representation
with tr (m) = 2.
If K is a fiber knot of genus g , then twisted Alexander
polynomial ∆K ,ρ0 (t) is monic polynomial of degree 4g − 2,
DFJ-conjecture
The converse is true.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
136 / 226
Epimorphism between knot groups
We study epimorphisms, namely surjective homomorphisms, between
knot groups.
Definition
For two knots K1 , K2 , we write K1 ≥ K2 if there exists an
epimorphism φ : G (K1 ) → G (K2 ) which preserves a meridian.
We start from a simple example 85 ≥ 31 .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
137 / 226
31
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
138 / 226
85
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
139 / 226
85 ≥ 31
They have the following presentations:
G (85 ) = ⟨y1 , y2 , y3 , y4 , y5 , y6 , y7 , y8 |y7 y2 y7−1 y1−1 , y8 y3 y8−1 y2−1 ,
y6 y4 y6−1 y3−1 , y1 y5 y1−1 y4−1 ,
y3 y6 y3−1 y5−1 , y4 y7 y4−1 y6−1 ,
y2 y8 y2−1 y7−1 ⟩.
G (31 ) = ⟨x1 , x2 , x3 | x3 x1 x3−1 x2−1 , x1 x2 x1−1 x3−1 ⟩.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
140 / 226
If generators are mapped to the following words:
y1 7→ x3 , y2 7→ x2 , y3 7→ x1 , y4 7→ x3 ,
y5 7→ x3 , y6 7→ x2 , y7 7→ x1 , y8 7→ x3 .
then any relator in G (85 ) maps to trivial element in G (31 ).
y7 y2 y7−1 y1−1 7→ x1 x2 x1−1 x3−1 = 1, . . .
Then this gives an epimorphism from G (85 ) onto G (31 ). Clearly this
preseves a meridian. Therefore, we can write
85 ≥ 31 .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
141 / 226
The geometric reason why there exists an epimorphism from G (85 )
to G (31 ) is
85 has a period 2, namely, it is invariant under some π-rotation
of S 3 ,
31 is its quotient knot.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
142 / 226
When and how there exists an epimorphism between knot groups ?
There are some geometric situations as follows.
To the trivial knot ⃝:
For any knot K , then there exists an epimorphism
α : G (K ) → G (⃝) = Z.
This is just the abelianization
G (K ) → G (K )/[G (K ), G (K )] ∼
= Z.
This can be realized a collapse map with degree 1.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
143 / 226
From any composite knot to each of factor knots:
There exist two epimorphisms
G (K1 ♯K2 ) → G (K1 ), G (K1 ♯K2 ) → G (K2 ).
They are also just induced by collapse maps with degree 1.
Degree one maps: Explain precisely later.
Periodic knots: Let K be a knot with period n. Its quotient map
(S 3 , K ) → (S 3 , K ′ ) induces an epimorphism
G (K ) → G (K ′ )
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
144 / 226
For any knot K , we take the composite knot K ♯K̄ . Then there
exist epimorphisms
G (K ♯K̄ ) → G (K ).
This epimorphism is induced from a quotient map
(S 3 , K ♯K̄ ) → (S 3 , K )
of a reflection (S 3 , K ♯K̄ ), whose degree is zero.
Ohtsuki-Riley-Sakuma construction between 2-bridge links : We
do not mention precisely here.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
145 / 226
A proper map
φ : (E (K1 ), ∂E (K1 )) → (E (K2 ), ∂E (K2 ))
induces an homorphism
φ∗ : H3 (E (K1 ), ∂E (K1 ); Z) → H3 (E (K2 ), ∂E (K2 ); Z).
Definition
A degree of φ is defined to be the integer d satisfying
φ∗ [E (K1 ), ∂E (K1 )] = d[E (K2 ), ∂E (K2 )]
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
146 / 226
The following is seen by taking a finite covering.
Lemma
If φ∗ : G (K1 ) → G (K2 ) is induced from a degree d map, then this
degree d can be divisible by the index n = [G (K2 ) : φ∗ (G (K1 ))].
Namely d/n is an integer.
In particular if d = 1, then n should be 1. Therefore we get
Proposition
If there exists a degree one map
φ : (E (K1 ), ∂E (K1 )) → (E (K2 ), ∂E (K2 )),
then φ induces an epimorphism
φ∗ : G (K1 ) → G (K2 ).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
147 / 226
Remark
There exists an epimorphism induced from
a non zero degree map, but not degree one map,
a degree zero map
For example, the epimorphism
G (K ♯K̄ ) → G (K )
is induced from a degree zero map.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
148 / 226
Proposition
The relation K ≥ K ′ gives a partial order on the set of the knots.
Namely,
1
K ≥K
2
K ≥ K ′, K ′ ≥ K ⇒ K = K ′
3
K ≥ K ′, K ′ ≥ K ” ⇒ K ≥ K ”
The only one non trivial claim is,
K ≥ K ′, K ′ ≥ K ⇒ K = K ′.
Here are two key facts to prove it.
A knot group G (K ) is Hopfian, namely any epimorphism
G (K ) → G (K ′ ) is an isomorphism.
Any meridian preserving isomorphism G (K ) → G (K ”) can be
induced by a homeomorphism E (K ) → E (K ′ ).
E (K ) determines its knot type of K .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
149 / 226
Computation for the knot table
T. Kitano, M. Suzuki and M. Wada, Twisted Alexander
polynomials and surjectivity of a group homomorphism, Algebr.
Geom. Topol. 5 (2005), 1315–1324.
Erratum: Algebr. Geom. Topol. 11 (2011), 2937–2939
A criterion for the non-existence.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
150 / 226
T. Kitano and M. Suzuki, A partial order in the knot table,
Experiment. Math. 14 (2005), no. 4, 385–390.
Corrigendum to: ”A partial order in the knot table”. Exp.
Math. 20 (2011), no. 3, 371.
the knots with up to 10-crossings (=Reidemeister-Rolfsen table)
K. Horie, T. Kitano, M. Matsumoto and M. Suzuki, A partial
order on the set of prime knots with up to 11 crossings, J. Knot
Theory Ramifications 20 (2011), no. 2, 275–303.
Errata: A partial order on the set of prime knots with up to 11
crossings. J. Knot Theory Ramifications 21 (2012), no. 4,
1292001, 2 pp.
the knots with up to 11-crossings.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
151 / 226
Fundamental tools to do are
Alexander polynomial
Twisted Alexander polynomial
Computer
By using two invariant, we can prove the non-exisitence of
epimorphisms. By using computer, we find it for the rest. The list
with up to 11-crossings of the partial order are the following.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
152 / 226
KS+HKMS
85 , 810 , 815 , 818 , 819 , 820 , 821 ,
91 , 96 , 916 , 923 , 924 , 928 , 940 ,
105 , 109 , 1032 , 1040 , 1061 , 1062 ,
1063 , 1064 , 1065 , 1066 , 1076 , 1077 ,
1078 , 1082 , 1084 , 1085 , 1087 , 1098 ,
1099 , 10103 , 10106 , 10112 , 10114 ,
10139 , 10140 , 10141 , 10142 , 10143 ,
10144 , 10159 , 10164
Teruaki Kitano (Soka University)
Reidemeister torsion























≥ 31
Nov. 03-07, 2014 in SNU
153 / 226
11a43 , 11a44 , 11a46 , 11a47 , 11a57 ,
11a58 , 11a71 , 11a72 , 11a73 , 11a100 , 11a106 ,
11a107 , 11a108 , 11a109 , 11a117 , 11a134 , 11a139 ,
11a157 , 11a165 , 11a171 , 11a175 , 11a176 ,
11a194 , 11a196 , 11a203 , 11a212 , 11a216 , 11a223 ,
11a231 , 11a232 , 11a236 , 11a244 , 11a245 , 11a261 ,
11a263 , 11a264 , 11a286 , 11a305 , 11a306 , 11a318 ,
11a332 , 11a338 , 11a340 , 11a351 , 11a352 , 11a355 ,
11n71 , 11n72 , 11n73 , 11n74 , 11n75 , 11n76 , 11n77 ,
11n78 , 11n81 , 11n85 , 11n86 , 11n87 , 11n94 ,
11n104 , 11n105 , 11n106 , 11n107 , 11n136 ,
11n164 , 11n183 , 11n184 , 11n185 ,
918 , 937 , 940 , 958 , 959 , 960 ,
10122 , 10136 , 10137 , 10138 ,
11a5 , 11a6 , 11a51 , 11a132 , 11a239 , 11a297 , 11a348 ,
11a349 , 11n100 , 11n148 , 11n157 , 11n165
Teruaki Kitano (Soka University)
Reidemeister torsion














































≥ 31
≥ 41
Nov. 03-07, 2014 in SNU
154 / 226
11n78 , 11n148 ≥ 51
1074 , 10120 , 10122 , 11n71 , 11n185 ≥ 52
11a352 ≥ 61
11a351 ≥ 62
11a47 , 11a239 ≥ 63
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
155 / 226
To decide the partial order relations, how many cases do we have to
consider ?
up to 10 crossings
number of knots : 249
number of cases : 249 P2 = 61, 752
up to 11 crossings
number of knots : 249 + 552 = 801
number of cases : 801 P2 =640,800
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
156 / 226
Criterions on the existence of epimorphims
The following fact is well known for Alexander polynomial.
Proposition
If K1 ≥ K2 , then ∆K1 (t) can be divisible by ∆K2 (t).
This can be generalized to the twisted Alxander polynomial as follows.
Theorem (K.-Suzuki-Wada)
If K1 ≥ K2 realized by φ : G (K1 ) → G (K2 ), then ∆K1 ,ρ2 φ (t) can be
divisible by ∆K2 ,ρ2 (t) for any representation ρ2 : G (K2 ) → SL(2; F).
By using these criterion over a finite prime field, we have checked the
non-existence. For the rest, we can find epimorphisms between knot
groups by using a computer.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
157 / 226
Minimal elements
By our results, the following problem appears naturally.
Problem
If K1 ≥ K2 , then the crossing number of K1 is greater than the one of
K2 ?
Theorem (Agol-Liu)
Any knot group G (K ) surjects onto only finitely many knot groups.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
158 / 226
By using the Kawauchi’s imitation theory. The next theorem is
proved.
Theorem (Kawauchi)
For any knot K , there exists a knot K̃ such that
K̃ is a hyperbolic knot
there exists an epimorphism from G (K̃ ) onto G (K ) induced by a
degree one map.
On the other hand, the following fact is known.
Fact
For any torus knot K , if there exists an epimorphism
φ : G (K ) → G (K ′ ), then K ′ is also a torus knot.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
159 / 226
Now we can consider a Hasse diagram, which is an oriented graph,
for this partial ordering as follows.
a vertex : each prime knot
an oriented ege : if K1 ≥ K2 , then we draw it from the vertex of
K1 to the one of K2 .
More generally the following problem arises.
Problem
How can we describe the Hasse diagram of the prime knots under
this partial order ?
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
160 / 226
This Hasse diagram is not so simple as follows.
Fact
For any two knots K1 and K2 , there exists a knot K such that
K ≥ K1 and K ≥ K2 .
It can be done by the imitation theory, too. We take K to be an
imitation of K1 ♯K2 . Then there exists epimorphisms
G (K ) → G (K1 ♯K2 )
By composing epimorphisms
G (K1 ♯K2 ) → G (K1 ), G (K2 )
we get two epimorphisms
G (K ) → G (K1 ), G (K ) → G (K2 )
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
161 / 226
In our list, we can see that the knots
31 , 41 , 51 , 52 , 61 , 62 , 63
are minimal elements in the set of prime knots with up to
11-crossings.
Here in fact, we can prove that they are minimal in the set of all
prime knots.
Now we can see
Theorem (K.-Suzuki)
They are minimal elements in the set of all prime knots.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
162 / 226
Outline of Proof.
First we recall the following facts on a fibered knot.
Fact
If K1 ≥ K2 and K1 is fibered, then K2 is also fibered and
g (K1 ) ≥ g (K2 ).
Fact
Let K , K ′ are fibered knots of the same genus. If K ≥ K ′ then
K = K ′.
Fact
31 , 41 are the only two fibered knots of genus one.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
163 / 226
31 and 41 are minimal.
We assume that 31 ≥ K and 31 ̸= K . By the above fact, K is a
fibered knot of the genus one. It means that K = 41 . However we
can see 31 ≱ 41 easily by direct computation.
For the case of 41 , it is similar argument.
51 and 62 are minimal.
Here it is seen that 51 , 62 are fibered knots of genus two. If 51 ≥ K ,
then K is a fibered knot of the genus two. Then 51 ≥ K implies that
g (K ) ≤ 2. If the genus of K is one, then K = 31 , or 41 . But it is
impossible. Then the genus is two and 51 = K by the above fact.
For 62 , it is easily seen by the same argument.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
164 / 226
52 and 61 are minimal.
Here they are 2-bridge knots;
52 = S(7, 3), 61 = S(9, 2).
We can prove it by the next theorems.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
165 / 226
Theorem (Boileau-Boyer)
If S(p, q) ≥ S(p ′ , q ′ ) and p ̸= p ′ , then p = kp ′ where k > 1.
Theorem (Boileau-Boyer-Reid-Wang)
If it holds that S(p, q) ≥ K , then K is also a 2-bridge knot.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
166 / 226
Boileau-Boyer-Reid-Wang proved the following.
Proposition (BBRW)
Any epimorphism between 2-bridge hyperbolic knots is always
induced from a non zero degree map.
On the other hand, there are some interesting example as follows.
Example
1059 , 10137 are 3-bridge hyperbolic knots.
1059 , 10137 ≥ 41 .
There is no non-zero degree map between them. Namely any
epimorphism between them is induced from a degree zero map.
To see that there are no non-zero degree maps, we have to study the
structure of Alexander modules.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
167 / 226
1059
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
168 / 226
10137
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
169 / 226
The following facts are well known in the theory of surgeries on
compact manifolds. For examples, see in the book by Wall.
Fact
If there exits an epimorphism
φ∗ : G (K ) → G (K ′ )
induced from a non zero degree map(a degree one map), then its
induced epimorphism
H1 (Ẽ (K ); Q) → H1 (Ẽ (K ′ ); Q)
between their Alexander modules is split over Q(Z).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
170 / 226
Example
We can see the followings by similarly observing Alexander modules.
924 ≥ 31 :any epimorphism between them is induced from an only
degree zero map.
11a5 ≥ 41 :any epimorphism between them is induced from an
only degree zero map.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
171 / 226
How do these knots are ?
Remark
Here 1059 , 10137 , 924 are Montesinos knots.
1059 = M(−1; (5, 2), (5, −2), (2, 1))
10137 = M(0; (5, 2), (5, −2), (2, 1))
924 = M(−1; (3, 1), (3, 2), (2, 1))
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
172 / 226
How there exists an epimorphism between them ?
Recall the geometric observation by Ohtsuki-Riley-Sakuma.
Here we assume that
φ : G (K ) → G (K ′ )
is an epimorphism.
We take a simple closed curve γ ⊂ S 3 ∪ K which belongs to
Kerφ ⊂ G (K ). Then if γ is an unknot in S 3 , by taking the surgery
along γ, we get a new knot K̃ in S 3 such that there exists an
epimorphism G (K̃ ) → G (K ′ ).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
173 / 226
We can apply this construction to 41 ♯4¯1 = 41 ♯41 . First we recall that
there exists an epimorphism
G (41 ♯4¯1 ) → G (41 )
which is a quotient map of a reflection. Then it is induced from a
degree zero map. By surgery along some simple closed curve, we get
both of
G (1059 ) → G (41 ),
and
G (10137 ) → G (41 ),
More generally we can see the following. It was not written
explicitely, but essentially in the paper by Ohtsuki-Riley-Sakuma.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
174 / 226
Proposition
For any 2-bridge knot K , there exists a Montesinos knot K̃ such that
there exists an epimorphism
G (K̃ ) → G (K )
induced from a degree zero map E (K̃ ) → E (K ).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
175 / 226
Return to the list of knots with up to 10-crossings. We can find
epimorphisms explicitely, but not find all epimorphisms if there exist.
For the epimorpshism we could find, the following partial order
relations can be realized by epimorphisms induced from degree zero
maps.
}
810 , 820 , 924 , 1062 , 1065 , 1077 ,
≥ 31
1082 , 1087 , 1099 , 10140 , 10143
1059 , 10137 ≥ 41
In this list, Montesinos knots appear as follows.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
176 / 226
Return to the list of knots with up to 10-crossings. We can find
epimorphisms explicitely, but not find all epimorphisms if there exist.
For the epimorpshism we could find, the following partial order
relations can be realized by epimorphisms induced from degree zero
maps.
}
810 , 820 , 924 , 1062 , 1065 , 1077 ,
≥ 31
1082 , 1087 , 1099 , 10140 , 10143
1059 , 10157 ≥ 41
In this list, Montesinos knots appear as above.
Remark
The other knots are given by Conway’s notation as follows:
1082 = 6 ∗ ∗4.2,
1087 = 6 ∗ ∗22.20,
1099 = 6 ∗ ∗2.2.20.20
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
177 / 226
As another application of Kawauchi’s theory, we can see the following.
Proposition
For any knot K , there exists a hyperbolic knot K ′ s.t. there exist two
epimorphisms from G (K ′ ) onto G (K ). Further the one is induced by
degree one map and another one induced by degree zero map.
We feel that the above epimorphism induced from a degree zero map
is standard. Then finally we put open problem.
Problem
If an epimorphism φ : G (K ) → G (K ′ ) is induced by a degree zero
map, then there exists a knot K1 s.t.
φ can be the composite of epimorphisms
G (K ) → G (K1 ) → G (K ′ ♯K̄ ′ ) → G (K ′ ).
The above epimorphism G (K ) → G (K ′ ♯K̄ ′ ) is induced by a
reflection with degree zero.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
178 / 226
Problem
How strong is Twisted Alexander polynomial for a represenation
over a finite field ?
Can we detect the fiberedness by using only representations over
finite fields ?
Can we also do the existence of epimorphisms ?
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
179 / 226
D. Johnson’s theory of Reidemeister torsion
Let M be a compact 3-manifold.
Let ρ : π1 M → SL(2; C) be a represenation.
SL(2; C) acts on V = C2 .
Let ρ : π1 (M) → SL(2; C) be an acyclic representation. Then the
Reidemeister torsion of M with Vρ - coefficients is defined by
τρ (M) = τ (C∗ (M; Vρ )) ∈ C \ {0}.
Remark
We define the τ (M; Vρ ) to be zero for a non-acyclic representation ρ.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
180 / 226
Definition
ρ : π1 (X ) → SL(2; C) is parabolic if ρ(x) is trivial or parabolic for
any x ∈ π1 (X ).
We recall torsions for T 2 and S 1 × D 2 .
Proposition
If ρ : π1 (T 2 ) → SL(2; C) is not parabolic, then ρ is acyclic and
τρ (T 2 ) = 1.
Proposition
If ρ : π1 (S 1 × D 2 ) → SL(2; C) is not parabolic, then ρ is acyclic and
τρ (S 1 × D 2 ) =
1
det(ρ(x) − E )
for the generator x ∈ π1 (S 1 × D 2 ).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
181 / 226
Let M be a closed 3-manifold with a torus decomposition
M = A ∪T 2 B.
For simplicity, we write the same symbol ρ for restricted
representation to subspaces, for example, ρ|π1 A or ρ|π1 B .
By this decomposition, we have the following exact sequence:
0 → C∗ (T 2 ; Vρ ) → C∗ (A; Vρ ) ⊕ C∗ (B; Vρ ) → C∗ (M; Vρ ) → 0.
Proposition
Let ρ : π1 (M) → SL(2; C) a representation which restricted to T 2 is
not parabolic. Then H∗ (M; Vρ ) = 0 if and only if
H∗ (A; Vρ ) = H∗ (B; Vρ ) = 0. In this case it holds
τρ (M) = τρ (A)τρ (B).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
182 / 226
We apply this proposition to a 3-manifold obtained by Dehn-surgery
along a knot.
K ⊂ S 3 : a knot in S 3
N(K ):open tubular neighborhood of K .
N̄:its closure of N(K ) ∼
= S 1 × D 2.
E (K ) = S 3 \ N(K ):its knot exterior
1
Mn : a 3-manifold obtained by the -surgery along K
n
m, l ∈ π1 E (K ) a meridian and a longitude
We can decompose Mn = E (K ) ∪ N̄.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
183 / 226
Let ρ : π1 (E (K )) → SL(2; C) a representation.
Lemma
A given representation ρ can be extended to π1 (Mn ) → SL(2; C) as a
representation if and only if ρ(ml n ) = E .
Proof.
π1 E (K ) = ⟨x1 , · · · , xn | r1 , · · · , rn−1 ⟩
π1 Mn = ⟨x1 , · · · , xn | r1 , · · · , rn−1 , ml n ⟩
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
184 / 226
Proposition
τρ (Mn ) =
τρ (E (K ))
.
det(ρ(l) − E )
Proof.
τρ (Mn ) = τρ (E (K ))τρ (N̄)
1
det(ρ(l) − E )
τρ (E (K ))
=
.
det(ρ(l) − E )
= τρ (E (K ))
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
185 / 226
K ⊂ S 3 : a torus (p, q)-knot T (p, q)
1
Mn : a 3-manifold obtained by the -surgery along K
n
π1 Mn = ⟨x, y | x P = y q , ml n = 1⟩.
Remark
In this case we write M0 to S 3 , not 0-suregeried manifold.
Remark
Mn is Brieskorn homology 3-sphere Σ(p, q, N) where
N = pqn + 1 (n > 0), or pq|n| − 1 (n < 0).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
186 / 226
Brieskorn manfold
Σ(p, q, r ) = {(z1 , z2 , z3 ) ∈ C3 |z1p + z2q + z3r = 0} ∩ S 5 ⊂ C3 ∼
= R6 .
Example
K = T (2, 3) is the trefoil knot.
M−2 = Σ(2, 3, 11)
M−1 = Σ(2, 3, 5):Poincaré homology 3-sphere
M0 = S 3
M1 = Σ(2, 3, 7)
M2 = Σ(2, 3, 13)
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
187 / 226
Now we consider only irreducible representations. The set of the
distinct conjugacy classes of the SL(2; C)-irreducible representations
R n = {ρ : π1 Mn → SL(2, C) : irreducible}/conjugate
is finite. Any element of R n is represented by the representation
{ρ(a,b,k) } such that
1
0 < a < p, 0 < b < q, a ≡ b mod 2,
2
0 < k < N = pq|n| ± 1, k ≡ na mod 2,
3
tr ρ(a,b,k) (x) = 2 cos πa/p,
4
tr ρ(a,b,k) )(y ) = 2 cos πb/q,
5
tr ρ(a,b,k) (m) = 2 cos πk/N.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
188 / 226
Proposition
ρ(a,b,k) is acylic if and only if a ≡ b ≡ 1, k ≡ n mod 2.
Johnson computed τρ(a,b,k) (Mn ) as follows.
Theorem (Johnson)
For ρ(a,b,k) with a ≡ b = 1, k = n mod 2(namely ρ(a,b,k) is acyclic),
then
τρ(a,b,k) (Mn ) =
Teruaki Kitano (Soka University)
(
2 1 − cos
πa
p
)(
1
1 − cos
Reidemeister torsion
πb
q
)(
1 + cos πkpq
N
).
Nov. 03-07, 2014 in SNU
189 / 226
Proof.
τρ(a,b,k) (Mn ) = τρ(a,b,k) (E (T (p, q))τρ(a,b,k) (N̄)
det(Aρ,1 |t=1 )
1
det(X1 − E )) det(L − E )
1
1
(
)(
)(
)
πb
1 + cos πkpq
2 1 − cos πa
1
−
cos
N
p
q
=
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
190 / 226
Example
M−1 = Σ(2, 3, 5).
In this case, a = b = 1 and k = 1, 3 < 5.
τρ(a,b,k) (M−1 ) =
(
)(
π
1
)
1 + cos 6πk
5
1 − cos 3
1
=
1
2(1 − 0)(1 − 2 )(1 + cos 6πk
)
5
1
=
1 + cos 6πk
5
{
√
2√
3− 5
=
(k = 1),
2√
3+ 5
=
2√
= 3+2 5 (k = 3).
3− 5
Teruaki Kitano (Soka University)
2 1 − cos
1
)(
π
2
Reidemeister torsion
Nov. 03-07, 2014 in SNU
191 / 226
Torsion polynomial
τρ (M) is an ivariant of (M, ρ). We want to get an invariant of M
itself.
By considering the set of all nontrivial values of Reidemeister torsion
for irreducible representations, Johnson proposed and defined the
torsion polynomial
∏
σM (t) =
(t − τρ ) .
ρ:acyclic
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
192 / 226
In the case of the n1 -surgeried manifold along the trefoil knot,
Johnson modified the semi-torsion polynomial as
∏ (
τρ )
t−
σ̄Mn (t) = ±
.
2
ρ:acyclic
normalized this polynomial with
σ̄Mn (0) = (−1)n
for Mn = Σ(2, 3, 6|n| ± 1) along the trefoil knot T (2, 3). We write
simply
σ̄(2,3,n) (t) for σ̄Mn (t).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
193 / 226
Theorem (Johnson)
The semi-torsion polynomial is given by
(√ )
(√ )
T6n+2 2t − T6n 2t
σ̄(2,3,n) (t) = 2
t −4
where TN is the N-th Tchebychev polynomial.
Recursive formula:
σ̄(2,3,0) (t) := 1
σ̄(2,3,n+1) (t) = (t 3 − 6t 2 − 9t − 2)σ̄(2,3,n) (t) − σ̄(2,3,n−1) (t)
where
(
)
1√
t 3 − 6t 2 − 9t − 2 = 2T6
t .
2
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
194 / 226
Example
σ̄(2,3,−1) (t) = −t 2 + 3s − 1
σ̄(2,3,0) (t) = 1
σ̄(2,3,1) (t) = t 3 − 5t 2 + 5t − 1
σ̄(2,3,2) (t) = t 6 − 11t 5 + 26t 4 + 12t 3 − 29t 2 − t + 1
σ̄(2,3,3) (t) =
t 9 − 17t 8 + 83t 7 − 47t 6 − 313t 5 + 13t 4 + 243t 3 + 66t 2 − 12t − 1
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
195 / 226
Tchebychev polynomials
By using
cos 2θ = 2 cos2 θ − 1, sin2 θ = 1 − cos2 θ,
cos nθ = Tn (cos θ) + sin θUn (cos θ)
= Tn (cos θ)
for some polynomial Tn (t) and Un (t) = 0.
This
Tn (t) = cos nθ = Tn (t)
is the n-th Tchebychev polynomial.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
196 / 226
By definition, T0 (t) = 1, T1 (t) = t. The Tchebychev polynomials
have following properties.
Proposition
1
2
3
4
5
T−n = Tn .
Tn (1) = 0, Tn (−1) = (−1)n .
{
0 n is odd,
Tn (0) =
n
(−1) 2 n is even.
Tn+1 = 2tTn − Tn−1 .
The degree of Tn is n.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
197 / 226
He we put a short list of Tn .
T0 = 1,
T1 = t,
T2 = 2t 2 − 1,
T3 = 4t 3 − 3t,
T4 = 8t 4 − 8t 2 + 1,
T5 = 16t 5 − 20t 3 + 5t,
T6 = 32t 6 − 48t 4 + 18t 2 − 1.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
198 / 226
We give a formula of the normalized semi-torsion polynomial
σ̄(p,q,n) (t) for Mn = Σ(p, q, pq|n| ± 1) obtained by the n1 -Dehn
surgery along the (p, q)-torus knot.
Main result is the following.
Theorem
The normalized semi-torsion polynomial is given by
( √ )
( √ )
Tpqn+2 12 t − Tpqn 12 t
σ̄(p,q,n) (t) = 2
.
t −4
(
σ̄(p,q,n+1) (t) = 2Tpq
Teruaki Kitano (Soka University)
)
1√
t σ̄(p,q,n) (t) − σ̄(p,q,n−1) (t)
2
Reidemeister torsion
Nov. 03-07, 2014 in SNU
199 / 226
To devide τρ by 2, what meaning it has ? Directly we can say
nothing. However when Johnson was working, he wanted to
study the relation between Casson’s invariant and Reidemeister
torsion.
The special normalization of σ(0) = (−1)n has also some
meaning ? Or it is useful only for Brieskorn manifolds ?
Now we only consider the case that a sapce of the conjugacy
classes of the irreducible representations is finite. How can we
treat the case of the positive dimennsion.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
200 / 226
Ray-Singer analytic torsion
D. Ray and I. M. Singer, R-torsion and the Laplacian on
Riemannian manifolds, Advances in Math. 7 (1971), 145–210.
J. Cheeger, Analytic torsion and the heat equation. Ann. of
Math. (2) 109 (1979), no. 2, 259–322.
W. Müller, Analytic torsion and R-torsion of Riemannian
manifolds, Adv. in Math. 28 (1978), no. 3, 233–305.
W. Müller, Analytic torsion and R-torsion for unimodular
representations, J. Amer. Math. Soc. 6 (1993), no. 3, 721–753.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
201 / 226
Combinatorial Laplacian
Here let F = R. By using a basis cq = {cq1 , · · · , cqn } we identify Cq
with Rn . We consider the inner product on Cq s. t. cq is an
orhonormal basis. Then we can take the adjoint matrix
∂q∗ : Cq−1 → Cq of ∂q : Cq → Cq−1 .
Definition
(c)
The combinatorial Laplacian ∆∗ is defined by
∗
∗
∆(c)
q = −(∂q ∂q + ∂q+1 ∂q+1 ) : Cq → Cq .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
202 / 226
Lemma
(c)
∆q is a symmetric matrix with the nonpositive eigenvalues.
Proof.
We take an eigenvalue λ and a corresponding eigenvector u.
λ(u, u) = (λu, u)
= (∆(c)
q u, u)
∗
= −(∂q∗ ∂q u, u) − (∂q+1 ∂q+1
u, u)
∗
∗
= −(∂q u, ∂q u) − (∂q+1
u, ∂q+1
u)
∗
= −||∂q u||2 − ||∂q+1
u||2
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
203 / 226
∗
From the above arguments, If ∆q u = 0, then ∂q u = ∂q+1
u = 0.
(c)
Proposition
1∑
log τ (C∗ ) =
(−1)q+1 q log det(−∆(c)
q ).
2 q=0
m
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
204 / 226
Proof
(c)
Bq = ∂q+1 (Cq+1 ) ⊂ Cq is invariant subspace under ∆q . Because
∗
∆q(c) (∂q+1 u) = −(∂q∗ ∂q + ∂q+1 ∂q+1
)(∂q+1 u)
∗
= −∂q+1 (∂q+1
∂q+1 u) ∈ Bq .
We can take
bq = {bq1 , · · · , bqrq }
to be an orthonormal basis of Bq consisting of eigenvectors
j
∗
∗
j
∆(c)
q bq = −(∂q ∂q + ∂q+1 ∂q+1 )bq
∗
bqj
= −∂q+1 ∂q+1
= λq,j bqj .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
205 / 226
We set
1
∂ ∗ bj
λq−1,j q q−1
1
j
=−
(−∆q−1 bq−1
λq−1,j
b̃jq−1 = −
so that
∂q b̃jq−1 = −
1
λq−1,j
Teruaki Kitano (Soka University)
j
∂q ∂q∗ bq−1
=−
1
λq−1,j
Reidemeister torsion
j
(−λq−1,j )bq−1
= bqj .
Nov. 03-07, 2014 in SNU
206 / 226
Note that b̃jq−1 are orthogonal, and
∥ b̃jq−1 ∥2 =
=
=
=
=
1
λ2q−1,j
1
λ2q−1,j
1
λ2q−1,j
1
λ2q−1,j
1
λ2q−1,j
=−
Teruaki Kitano (Soka University)
j
j
(∂q∗ bq−1
, ∂q∗ bq−1
)
j
j
(bq−1
, ∂q ∂q∗ bq−1
)
j
j
(bq−1
, −∆q−1 bq−1
)
j
j
(bq−1
, −λq−1,j bq−1
)
j
(−λq−1,j ) ∥ bq−1
∥2
1
λq−1,j
Reidemeister torsion
Nov. 03-07, 2014 in SNU
207 / 226
Hence
∥ b̃jq−1 ∥= (−λq−1,j )− 2 .
1
m
∑
log τ (C∗ ) =
(−1)q log[bq , bq−1 /cq ]
=
q=0
m
∑
(−1)q log[bq , b̃q−1 /cq ]
q=0
rq −1
m
∑
∏
1
q
=
(−1) log
(−λq−1,j )− 2
q=0
Teruaki Kitano (Soka University)
j=1
Reidemeister torsion
Nov. 03-07, 2014 in SNU
208 / 226
On the other hand, The orthonomal base consisting of bq and
1
1 rq−1
{(−λq−1,1 )− 2 b̃1q−1 , · · · , (−λq−1,rq−1 )− 2 b̃q−1
} clearly diagonalizes
(c)
∆q so that
det(−∆(c)
q )
=
rq
∏
(−λq,j )
1
Teruaki Kitano (Soka University)
∏
rq−1
Reidemeister torsion
(−λq−1,j ).
1
Nov. 03-07, 2014 in SNU
209 / 226
This implies
∏
rq−1
log
(−λq−1,j ) =
1
n
∑
(−1)k−q log det(−∆(c)
q )
k=q
Therefore we have
1∑
log τ (C∗ ) =
(−1)q+1 q log det(−∆(c)
q ).
2 q=0
m
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
210 / 226
Here we can replace combinatorial Laplacians to analytic Laplacians.
One problem
( is how
) to treat determinants for analytic Laplacians.
a 0
Let A = 1
where a1 , a2 > 0. Then
0 a2
log det A = log(a1 a2 )
= log(a1 ) + log(a2 ).
Here we consider the following function:
ζA (x) = a1−x + a2−x .
ζA′ (x) = −a1−x log a1 − a2 log a2 .
Hnece
Teruaki Kitano (Soka University)
ζA′ (0) = − log a1 − log a2 .
Reidemeister torsion
Nov. 03-07, 2014 in SNU
211 / 226
(c)
We consider the zeta function of ∆q :
∑
ζ−∆q(c) (s) =
λ−s
q,j .
j
By using this zeta function, we can describe
1∑
log τ (C∗ ) =
(−1)q+1 q log det(−∆(c)
q )
2 q=0
m
1∑
=
(−1)q q(− log det(−∆(c)
q ))
2 q=0
m
1∑
′
=
(−1)q qζ−∆
(c) (0).
q
2 q=0
m
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
212 / 226
Definition of Ray-Singer torsion
M is a closed oriented smooth m-manifold.
ρ : π1 M → SL(n; F) is a representation (F = R, C).
Eρ → M is its associated flat vector bundle.
We take a metric on Eρ .
If ρ is an orthognal representaion, or a unitary representation,
we take the metric associtaed to the flat bundle structure.
q
∞
Ω(M; Eρ ) = ⊕m
q=0 Ω (M; Eρ ) is the linear space of C -forms.
d : Ωq → Ωq+1 is the exterior differential with d 2 = 0.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
213 / 226
We assume M has a Riemannian metric. Then we can define
A Hodge ∗-operator: ∗ : Ωq → Ωm−q .
∫
an inner product: (f , g ) = M f ∧ ∗g .
This inner product gives a Hilbert space structure on Ω.
We can define a formal adjoint δ = (−1)mq+m+q ∗ d∗ of d on Ωq .
We define the Laplacian
∆ = (δd + dδ) : Ω → Ω.
We write ∆q for the restriction of ∆ to Ωq .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
214 / 226
∆q is a second order defferential order and it is symmetric and
positive on Ω.
The eigenvalues:
0 ≤ λ1 ≤ λ2 ≤ · · · ≤ λk ≤ · · · → +∞.
Definition
ζq,ρ (s) = ζ∆q (s)
∑
=
λ−s
j
λj
is the operator zeta function of ∆q .
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
215 / 226
Definition
1∑
′
log Tρ (M) =
(−1)n qζq,ρ
(0).
2 q=0
m
Here we assume any λj > 0. For orthogonal or unitary
representations, the hollowing holds.
Theorem (Ray-Singer)
log Tρ (M) is independent of choices of Riemannian metric. Nemly
log Tρ (M) is a invariant of a smooth manifold.
Further Ray proved.
Theorem (Ray)
If M is a lens space, then
log Tρ (M) = log τρ (M).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
216 / 226
For a general manifold, independently Cheeger and Müller proved.
Theorem (Cheeger, Müller)
log Tρ (M) = log τρ (M).
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
217 / 226
More generally, for any SL(n; F)-represenations, the following holds.
Theorem (Müller)
log Tρ (M) = log τρ (M).
Remark
Ray-singer torsion is defined analytically. However it is essentially
combinatorial invariant.
Howver by taking a some kind of limit we can get the hyperbolic
volume.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
218 / 226
W. Müller, The asymptotics of the Ray-Singer analytic torsion of
hyperbolic 3-manifolds, in the book “Metric and differential
geometry”, p317–352, Progr. Math., 297, Birkhäuser/Springer,
Basel, 2012.
P. Menal-Ferrer and J. Porti, Higher-dimensional Reidemeister
torsion invariants for cusped hyperbolic 3-manifolds, J. Topol. 7
(2014), no. 1, 69–119.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
219 / 226
Let M be a closed hyperbolic 3-manifold.
Let ρ1 : π1 M → SL(2; C) be a lift of the holonomy
representation.
Let V = C2 = ⟨e1 , e2 ⟩.
In this situation we can consider τρ1 (M) ∈ C.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
220 / 226
Here we take an irreducibel representation SL(2; C) → SL(n + 1; C)
defined by the action of SL(2; C) on the n-th symmetric tensor of C 2 .
Then we get a family of representation {ρn : π1 M → SL(n + 1; C)}.
Example
n = 1:ρ1 is the holonomy representation.
n = 2: Sym2 (C2 ) ∼
= C3 = ⟨e1 ⊗ e1 , e2 ⊗ e2 , e1 ⊗ e2 + e2 ⊗ e1 ⟩
where V = ⟨e1 , e2 ⟩.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
221 / 226
Theorem (Müller)
lim −4π
n→+∞
log τρn (M)
= vol(M).
n2
Remark
The proof of this theorem is done for the same statement for
Ray-Singer torsion by using analytic methods.
It is known that there are finitely many closed oriented
hyperbolic 3-manifolds with the same volume. Then we can see
that a closed oriented hyperbolic 3-manifold M is determined up
to finitely many possibilities by the set {τρn (M)} of Reidemeister
torsions.
It is generalized to the case of a complete hyperbolic 3-manifold
with finite volume by Menal-Ferrer and Porti.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
222 / 226
L2-torsion
W. Lück, L2 -invariants: theory and applications to geometry and
K-theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3.
Folge. A Series of Modern Surveys in Mathematics, 44.
Springer-Verlag, Berlin, 2002.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
223 / 226
L2 -torsion is defined analytically or combinatorially. It is essentially
the hyperbolic volume for a complete hyperbolic odd-dimensional
manifold. The idea to define combinatorial L2 -torsion is the following.
Let ρ : π1 (M) → N(l 2 (π1 M)) be a left regular representation
where N(l 2 (π1 M)) is soem von-Neumann algebra.
Let l 2 (π1 M) be the l 2 -completion of Cπ1 M.
We get Alexander matrix is a matrix over Cπ1 M.
The point is how to define a determinant for such a matrix.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
224 / 226
Fundamental
( idea)is the following.
a 0
For A = 1
where a1 , a2 > 0,
0 a2
log det(A + E ) = log((a1 + 1)(a2 + 1))
= log(a1 + 1) + log(a2 + 1)
= tr (log(A + E ))
Formally we can define the log for a matrix over Cπ1 (M) as a
infinite sequense of a matrix.
For such a matrix there exists a “good” trace.
The trace comes essentially from
∑
Cπ1 M ∋
ng g 7→ n1 ∈ C.
The determinant is called Fuglede-Kadison determinant.
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
225 / 226
We have no direct proof that a sequence of Reidemeister
torsions or combinatorial L2 -torsin detect a hyperbolic volume.
How can we prove it by more topological or combinatorial
arguments ?
Teruaki Kitano (Soka University)
Reidemeister torsion
Nov. 03-07, 2014 in SNU
226 / 226