loop braid group
and the invariant
for the welded link
Bingjin Liu,
Beihang University
& Texas A&M
University
loop braid group and the invariant for the
welded link
Bingjin Liu, Beihang University & Texas A&M University
05/11/2015
loop braid group
background
and the invariant
for the welded link
Bingjin Liu,
Beihang University
& Texas A&M
University
I
braid group Bn , motion group of n points in a plane,
relations between links and Bn , Turaev’s invariant for
links
I
loop braid group LBn , motion group of n circles in R3
relations between welded links and LBn ,
I
question? generalized Turaev’s invariant for welded
links?
loop braid group
the definition of LBn
LBn : generated by σi and si for 1 ≤ i ≤ (n − 1) satisfying :
the braid relations
f or
for the welded link
Bingjin Liu,
Beihang University
& Texas A&M
University
1. σi σi+1 σi = σi+1 σi σi+1
2. σi σj = σj σi
and the invariant
|i − j| > 1
the symmetric group relations
1. si si+1 si = si+1 si si+1
2. si sj = sj si for |i − j| > 1
3. s2i = 1
the mixed relations
1. σi sj = sj σi for |i − j| > 1
2. si si+1 σi = σi+1 si si+1
3. σi σi+1 si = si+1 σi σi+1
loop braid group
the relations of loop braid group and welded links
and the invariant
for the welded link
Bingjin Liu,
Beihang University
Theorem (Algebraic Markov Theorem for welded links)
Two welded links are isotopic if and only if any two
corresponding loop braids differ by the group relations and
by a finite sequence of the following moves or their inverses:
1. congugation: α ∼ βαβ −1
2. stabilization : ασn±1 ∼ α ∼ αsn
where α, β ∈ LBn and σn , sn ∈ LBn+1 .
I
To find a welded invariant is to find a function
`
F : n≥1 LBn → K which is invariant under the
relations stated in Markov Theorem.
& Texas A&M
University
loop braid group
the relations of loop braid group and welded links
and the invariant
for the welded link
Bingjin Liu,
Beihang University
Theorem (Algebraic Markov Theorem for welded links)
Two welded links are isotopic if and only if any two
corresponding loop braids differ by the group relations and
by a finite sequence of the following moves or their inverses:
1. congugation: α ∼ βαβ −1
2. stabilization : ασn±1 ∼ α ∼ αsn
where α, β ∈ LBn and σn , sn ∈ LBn+1 .
I
To find a welded invariant is to find a function
`
F : n≥1 LBn → K which is invariant under the
relations stated in Markov Theorem.
& Texas A&M
University
loop braid group
the representation for LBn
and the invariant
for the welded link
Bingjin Liu,
Beihang University
& Texas A&M
University
I
Yang-Baxter operator R
V is a m-dim v.s
R ∈ GL(V ⊗2 ) satisfying the Yang-Baxter equation:
(R⊗IdV )(IdV ⊗R)(R⊗IdV ) = (IdV ⊗R)(R⊗IdV )(IdV ⊗R)
For any n ∈ N, any 1 ≤ i ≤ n − 1, define
Ri : V ⊗n → V ⊗n
Ri = IdV ⊗i−1 ⊗ R ⊗ IdV ⊗n−i−1
easy to see Ri satisfies the braid relations.
loop braid group
the representation for LBn
and the invariant
for the welded link
Bingjin Liu,
Beihang University
& Texas A&M
I
If R and S are both Yang-Baxter operators, and they
satisfies the equations:
1. S ◦ S = IdV ⊗2
2. (S ⊗ IdV ) ◦ (IdV ⊗ S) ◦ (R ⊗ IdV ) =
(IdV ⊗ R) ◦ (S ⊗ IdV ) ◦ (IdV ⊗ S)
3. (R ⊗ IdV ) ◦ (IdV ⊗ R) ◦ (S ⊗ IdV ) =
(IdV ⊗ S) ◦ (R ⊗ IdV ) ◦ (IdV ⊗ R)
(V,R,S) is called a loop braid vector space(LBVS).
I
define a representation ρn for the loop braid group LBn
via ρn (σi ) = Ri , ρn (si ) = Si ρn (αβ) = ρn (α) ◦ ρn (β).
University
loop braid group
an example of LBV S– affine group type
and the invariant
for the welded link
Bingjin Liu,
Beihang University
& Texas A&M
University
I
{e1 , e2 , ..., em } basis of V
I
define R via R(ei ⊗ ej ) := ej ⊗ hj (ei )
S via S(ei ⊗ ej ) := ej ⊗ ei (the usual flip)
where hj an automophism of V, hj (ei ) = eαi+βj for
some α, β ∈ N
several equations are satisfied to make it a LBVS


gcd(α, m) = 1
 β(α + β − 1) = 0
(mod
m)
loop braid group
an example of LBV S– affine group type
and the invariant
for the welded link
Bingjin Liu,
Beihang University
& Texas A&M
University
I
{e1 , e2 , ..., em } basis of V
I
define R via R(ei ⊗ ej ) := ej ⊗ hj (ei )
S via S(ei ⊗ ej ) := ej ⊗ ei (the usual flip)
where hj an automophism of V, hj (ei ) = eαi+βj for
some α, β ∈ N
several equations are satisfied to make it a LBVS


gcd(α, m) = 1
 β(α + β − 1) = 0
(mod
m)
loop braid group
link invariant and welded link invariant
and the invariant
for the welded link
Bingjin Liu,
In Turaev’s paper, enhanced YB operator is a collection {a
YB operator R, a hom µ : V → V , a, b in K} satisfying
some conditions:
1. The homomorphism µ ⊗ µ : V ⊗2 → V ⊗2 commutes
with R
2. Sp2 (R ◦ (µ ⊗ µ)) = abµ
Sp2 (R−1 ◦ (µ ⊗ µ)) = a−1 bµ
`
Then define a map Ts : n≥1 Bn → K to be an invariant of
links.
n
Tp (ξ) = a−w(ξ) b−n Sp(ρn (ξ) ◦ µ⊗ : V ⊗n → V ⊗n )
Beihang University
& Texas A&M
University
loop braid group
link invariant and welded link invariant
I
we can generalize enhanced YB operator to
enhanced LBVS : a collection { LBVS (V, R, S), a
K-homomorphism µ : V → V and a and b of K }
satisfying
The homomorphism µ ⊗ µ : V ⊗2 → V ⊗2 commutes
with R and S
Sp2 (R±1 ◦ (µ ⊗ µ)) = a±1 bµ
Sp2 (S ◦ (µ ⊗ µ)) = bµ
Then generalize the link invariant to
`
Tp : n≥1 LBn → K as follows.
n
Tp (ξ) = a−w(ξ) b−n Sp(ρn (ξ) ◦ µ⊗ : V ⊗n → V ⊗n )
and the invariant
for the welded link
Bingjin Liu,
Beihang University
& Texas A&M
University
loop braid group
the verification of invariant
I
and the invariant
for the welded link
To verify the map we defined is a welded link invariant,
Bingjin Liu,
Beihang University
& Texas A&M
we only need to show:
University
For any ξ and η ∈ LBn
Tp (ξ) = Tp (η −1 ξη) = Tp (ξσn ) = Tp (ξσn−1 ) = Tp (ξsn )
I
operator trace Spn (f ) is a homomorphism
V ⊗n−1 → V ⊗n−1 given as follows:
X
Spn (f )(ei1 ⊗· · ·⊗ein−1 ) =
j ,··· ,j
,j
n−1
fi11,··· ,in−1
,j ej1 ⊗· · ·⊗ejn−1
1≤j1 ,··· ,jn−1 ,j≤m
where
f (ei1 ⊗ · · · ⊗ ein ) =
X
1≤j1 ,··· ,jn ≤m
,··· ,jn
fij11,···
,in ej1 ⊗ · · · ⊗ ejn
loop braid group
an example
For any affine group type (V,R,S), we can always set
µ = bIdV to get a enhanced LBVS. Furthermore, if we
assume V is a 3-dim space, then α = β = 2, we can compute
the welded link invariant. Here is the invariant table:
and the invariant
for the welded link
Bingjin Liu,
Beihang University
& Texas A&M
University
loop braid group
evaluation of the invariant and other questions
and the invariant
for the welded link
Bingjin Liu,
Beihang University
& Texas A&M
University
I
From the table above, we can see that the invariant
does distinguish certain different welded links.
I
But we still do not know the relations of our invariant
with other known invariants. Maybe we can study these
later.
loop braid group
evaluation of the invariant and other questions
and the invariant
for the welded link
Bingjin Liu,
Beihang University
& Texas A&M
University
I
From the table above, we can see that the invariant
does distinguish certain different welded links.
I
But we still do not know the relations of our invariant
with other known invariants. Maybe we can study these
later.
loop braid group
acknowledgement
and the invariant
for the welded link
Bingjin Liu,
Beihang University
& Texas A&M
University
I
Thanks for my instructor Eric Rowell for his patience
and advice.
I
Thanks for my classmate Zheming Gao for his help on
the softwares.
I
Thank you!
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Any queations?