Krzysztof Putyra
Columbia University, New York
AMS Meeting, Newark
Special Session on Homology Theories for Knots and Skein Modules
22nd May 2010
What are link homologies?
 Cube of resolutions
 Even & odd link homologies
• via modules
• via chronological cobordisms
What are dotted cobordisms?
 chronology on dotted cobordisms
 neck-cutting relation and delooping
What is a chronological Frobenius algebra?
 dotted cobordisms as a baby-model
 universality of dotted cobordisms with NC
3
100
1
2
vertices are
smoothed
diagrams
000
010
001
110
101
011
edges are
cobordisms
Observation This is a commutative diagram in a category
of 1-mani-folds and cobordisms
111
Apply a graded functor
i.e.
Apply a graded pseudo-functor
FKh: 2Cob  Mod
FKh  X 
FKh g 
FKh  f  g 
FKh Y 
FKh Z 
FKh  f 
Result: a cube of
modules with
commutative faces
Mikhail
Khovanov
i.e.
FORS: 2Cob  Mod
FORS  X 
FORS g 
FORS  f  g 

FORS Y 
FORS Z 
FORS  f 
Result: a cube of
modules with both
commutative and
anticommutative faces
Peter
Ozsvath
Even: signs
given
explicitely
A 3
A 3
A
A 3
A
A 3
{+0+3}
C 3
{+1+3}
C 2
Odd: signs
given by
homological
properties
A 2
direct sums
create the
complex
A
{+2+3}
C 1
{+3+3}
C0
Theorem Homology groups of the complex C
are link invariants.
Mikhail
Khovanov
Peter
Ozsvath
100
110
1
3
2
000
010
001
101
111
011
edges are
cobordisms
with signs
Objects: sequences of smoothed diagrams
Morphisms: „matrices” of cobordisms
Theorem (2005) The complex is a link invariant under chain
homotopies and relations S/T/4Tu.
Dror Bar-Natan
Complexes for tangles in Cob
Dotted cobordisms:
Complexes for tangles in ChCob
?
Neck-cutting relation:
=
+
??
–
Delooping and Gauss elimination:
= {-1}
{+1}
Lee theory:
=1
???
????
=0
An arrow: choice of a in/outcoming
trajectory of a gradient flow of τ
Pick one
A chronology: a separative
Morse function τ.
An isotopy of chronologies: a smooth
homotopy H s.th. Ht is a chronology
Critical points cannot be permuted:
Critical points do not vanish:
Arrows cannot be reversed:
A change of a chronology is a smooth homotopy H. Changes H and
H’ are equivalent if H0 H’0 and H1 H’1.
Remark Ht might not be a chronology for some t (so called critical
moments).
Fact Every homotopy is equivalent to a homotopy with finitely
many critical moments of two types:
type I:
type II:
Theorem 2ChCob with changes of chronologies is a 2-category.
This category is weakly monoidal with a strict symmetry.
A solution in an R-additive extension for changes:
 type II: identity
Any coefficients can be replaced by 1’s due to scaling:
a
b
A solution in an R-additive extension for changes:
 type II: identity
 general type I:
MM = MB = BM = BB = X
SS = SD = DS = DD = Y
SM = MD = BS = DB = Z
-1
MS = DM = SB = BD = Z
Corollary Let bdeg(W) = (B-M, D-S). Then
AB = X Y Z
where bdeg(A) = ( , ) and bdeg(B) = ( , ).
X2 = 1
Y2 = 1
A solution in an R-additive extension for changes:
 type II: identity
 general type I:
MM = MB = BM = BB = X
SS = SD = DS = DD = Y
SM = MD = BS = DB = Z
-1
MS = DM = SB = BD = Z
AB =
X Y Z
-
X2 = 1
Y2 = 1
bdeg(A) = ( , )
bdeg(B) = ( , )
 exceptional type I:
1 / XY
X/Y
even odd
XYZ 1
-1
YXZ 1
-1
ZYX 1
-1
3
1
100
110
2
000
010
001
101
111
edges are
chronological
cobordisms with
coefficients in R
011
Fact The complex is independent of a choice of arrows and a sign
assignment used to make it commutative.
Theorem The complex C(D) is invariant under chain homotopies and
the following relations:
Dror Bar-Natan
where X, Y and Z are coefficients of chronology change relations.
Motivation Cutting a neck due to 4Tu:
Z(X+Y)
=
+
Add dots formally and assume the usual S/D/N relations:
(S)
=0
(D)
=1
bdeg( ) = (-1, -1)
=
(N)
+
–
A chronology takes care of dots, coefficients may be derived from (N):
M=
S
= XZ
= D = YZ-1
= XY
B
M
=
M
Motivation Cutting a neck due to 4Tu:
Z(X+Y)
=
+
Add dots formally and assume the usual S/D/N relations:
(S)
=0
(D)
=1
bdeg( ) = (-1, -1)
=
(N)
+
–
A chronology takes care of dots, coefficients may be derived from (N):
M=
B
= XZ
S
=
D
= YZ-1
Remark T and 4Tu can be derived from S/D/N.
Notice all coefficients are hidden!
= XY
Theorem (delooping) The following morphisms are mutually inverse:
{–1}
{+1}
–
Conjecture We can use it for Gauss elimination and a divide-conquer
algorithm.
Problem How to keep track on signs during Gauss elimination?
Theorem There are isomorphisms
Mor( , )
Mor( , )
given by
h
v+
XZ
[X, Y, Z 1, h, t]/(X2, Y2, (XY – 1)h, (XY – 1)t) =: R
v+R
t
v-R =: A
XZ
v-
bdeg(h) = (-1, -1)
bdeg(t) = (-2, -2)
bdeg(v+) = ( 1, 0)
bdeg(v- ) = ( 0, -1)
Corollary There is no odd Lee theory:
t=1
X=Y
Corollary There is only one dot in odd theory over a field:
X Y
XY 1
h=t=0
A chronological Frobenius system (R, A) in A is given by a monoidal
2-functor F: 2ChCob A:
R = F( )
A = F( )
Baby model: dotted algebra
R = Mor( , )
A = Mor( , )
Here, F(X) = Mor( , X).
Universality
A chronological Frobenius system (R, A) = (F( ), F( ))
Baby model: dotted algebra (R , A ): F(X) = Mor( , X)
 weak tensor product in ChCob
• product in R
• bimodule structure on A
=
A chronological Frobenius system (R, A) = (F( ), F( ))
Baby model: dotted algebra (R , A ): F(X) = Mor( , X)
 weak tensor product in ChCob
• product in R
• bimodule structure on A
left product
right product
A chronological Frobenius system (R, A) = (F( ), F( ))
Baby model: dotted algebra (R , A ): F(X) = Mor( , X)
 weak tensor product in ChCob
• product in R
• bimodule structure on A
left module:
=
right module:
=
A chronological Frobenius system (R, A) = (F( ), F( ))
Baby model: dotted algebra (R , A ): F(X) = Mor( , X)
 weak tensor product in ChCob
 changes of chronology
• torsion in R
• symmetry of A
AB =
X Y Z
-
bdeg(A) = ( , )
bdeg(B) = ( , )
cob:
bdeg:
(1, 1)
(0, 0)
= XY
(-1, -1) (-2, -2) (1, 0)
=
YZ-1
= XY
= XY
(1 – XY)a = 0,
= XZ-1
bdeg(a) < 0
bdeg(a) = 2n > 0
(0, -1)
no dots:
XZ / YZ
one dot:
1/1
two dots: XZ-1/ YZ-1
three dots: Z-2 / Z-2
A chronological Frobenius system (R, A) = (F( ), F( ))
Baby model: dotted algebra (R , A ): F(X) = Mor( , X)
 weak tensor product in ChCob
 changes of chronology
 algebra/coalgebra structure
=
= XZ
=
= XZ
=
= Z2


A chronological Frobenius system (R, A) = (F( ), F( ))
Baby model: dotted algebra (R , A ): F(X) = Mor( , X)
 weak tensor product in ChCob (right)
• product in R
• bimodule structure on A
 changes of chronology
• torsion in R:
0 = (1–XY)t = (1–XY)s02 = …
• symmetry of A:
tv+ = Z2v+t
hv- = XZv-h
 algebra/coalgebra structure
• right-linear, but not left
We further assume:
• R is graded, A = R1 Rα is bigraded
• bdeg(1) = (1, 0) and bdeg(α) = (0, -1)
…
A base change: (R, A)
(R', A') where A' := A
R R'
Theorem If (R', A') is obtained from (R, A) by a base change then
C(D; A') C(D; A) R'
for any diagram D.
Theorem (P, 2010) Any rank two chronological Frobenius system (R, A)
is a base change of (RU, AU), defined as follows:
RU = [X, Y, Z 1, h, t, a, c, e, f]/(ae–cf, 1–af+YZ-1 (cet–aeh))
AU = R[ ]/(
2
– h –t)
with
(1) = –c
( )=a
(1) = (et–fh) 1 1+ f (YZ 1
+
1) + e
( ) = ft 1 1+ et(1
+ YZ-1
1) + (f + YZ-1eh)
bdeg(c) = bdeg(e) = (1, 1)
bdeg(a) = bdeg(f) = (0, 0)
bdeg(h) = (-1, -1)
bdeg(t) = (-2, -2)
bdeg(1A) = (1, 0)
bdeg( ) = (0, -1)
A twisting: (R, A)
where y
(R', A')
' (w) = (yw)
' (w) = (y-1w)
A is invertible and deg(y) = (1, 0).
Theorem If (R', A') is a twisting of (R, A) then
C(D; A') C(D; A)
for any diagram D.
Theorem The dotted algebra (R , A ) is a twisting of (RU, AU).
Proof Twist (RU, AU) with y = f + e, where v+=1 and v– = .
Corollary (P, 2010) The dotted algebra (R , A ) gives a universal odd
link homology.
Complexes for tangles in Cob
Dotted chronological cobordisms
- universal
- only one dot over field, if X Y
Dotted cobordisms:
Neck-cutting relation:
=
+
Neck-cutting with no coefficients
–
Delooping and Gauss elimination:
= {-1}
{+1}
Lee theory:
=1
Complexes for tangles in ChCob
Delooping – yes
Gauss elimination – sign problem
Lee theory exists only for X = Y
=0
 Higher rank chronological Frobenius algebras may be given as
multi-graded systems with the number of degrees equal to the
rank
 For virtual links there still should be only two degrees, and a
punctured Mobius band must have a bidegree (–½, –½)
 Embedded chronological cobordisms form a (strictly) braided
monoidal 2-category; same for the dotted version unless (N) is
imposed
 The 2-category nChCob of chronological cobordisms of
dimension n can be defined in the same way. Each of them is a
universal extension of nCob in the sense of A.Beliakova