All Questions
Tagged with knot-theory skein-relation
16 questions
4
votes
1
answer
336
views
From braid representations to link invariants
If one has a $\mathbb{C}$-linear representation of the braid algebra into e.g. the Temperley-Lieb algebra i.e. $\rho:\mathbb{C}[B_{n}]\to TL_{n}(\delta)$, we can deduce a skein relation $\mathcal{S}$....
- 309
5
votes
1
answer
136
views
Actions of two types of Kauffman skein categories
Consider the quotient of the monoidal category of framed tangles by one of the two skein relations
together with the twist and dimension relations
Here $1_\mathbb{1}$ denotes the identity morphism ...
- 618
4
votes
1
answer
264
views
Unusual skein relation in HOMFLY polynomial
If I take the HOMFLY(PT) polynomial defined by
$$l \,P(L_+) + l^{-1}\,P(L_-) + m\,P(L_0) = 0,$$
I have looked at expressions of the form
(knots that are the same except inside a small disk, where ...
- 1,465
3
votes
1
answer
239
views
Easy lemma for trivalent graphs in colored Jones polynomial
In his 2008 paper,
Tanaka, Toshifumi, The colored Jones polynomials of doubles of knots, J. Knot Theory Ramifications 17, No. 8, 925-937 (2008). ZBL1149.57023.
Tanaka stated a lemma (Lemma 3.3) ...
- 219
5
votes
1
answer
143
views
Dimension of the skein module of a closed manifold?
I'm looking for a reference to Witten's conjecture that the free part of the (Kauffman bracket) skein module of a closed 3-manifold is finitely generated, i.e. the dimension of $K(M)$, where $M$ is a ...
- 1,465
10
votes
1
answer
324
views
Is the quantum $\mathfrak{sl}_3$ invariant stronger than the quantum $\mathfrak{sl}_2$ invariant?
Both the $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$ quantum framed link invariants can be computed using linear skeins. The first being computed using the Kauffman bracket and the second using a similar ...
- 103
11
votes
2
answers
450
views
Is there a known invariant for knotted surfaces defined by skein relations?
Is there a known invariant for knotted surfaces in $\mathbb{R}^4$ (possibly with additional structures, e.g. colored, framed, etc.) which can be defined using skein relations? By skein relations for ...
- 1,430
3
votes
1
answer
211
views
Basis for Annular Skein Algebra
Background/Notation:
Given the Iwahori-Hecke algebra $H_{n}$ (over some ring commutative ring $R$ with identity) with generators $\{T_{1},\ldots T_{n-1}\}$, we know it has basis
$\{T_{w}\}_{w\in S_{...
- 318
13
votes
2
answers
1k
views
Gap in Przytycki's computation of the skein module of links in a handlebody?
I am reading the paper [1], where the author proves that the skein module of links in a handlebody $F\times I$ has a free basis given by products $D_1 \cdots D_n$ where each $D_i$ is the closure of $...
- 3,752
6
votes
1
answer
396
views
What vector space does the Kauffman bracket skein algebra of FxI act on?
The Kauffman bracket skein module $K_t(F\times I)$ (where $t$ is an indeterminant and $F$ is a closed surface) is an associative algebra (the operation being "stacking" links in the $I$ direction). ...
- 18.7k
7
votes
1
answer
853
views
Trace identities and the Kauffman Bracket skein module
Let's consider $K_t(M)$, the Kauffman bracket skein module (see this and this papers) of a three-manifold $M$. When $t=-1$, $K_t(M)$ is easily seen to be isomorphic to the ring of functions on the ...
- 18.7k
0
votes
1
answer
280
views
"Skein" equations sets that can reduce any graph
Consider for example the approach to the Kuperberg G2 or the Yamada polynomial. I don't know whether relations of graph (in contrast to knot) theory are also usually called "skein" but
I simply carry ...
- 4,829
4
votes
1
answer
500
views
Cubic skein relations
please note that this question deals with undirected knots/links!
The most generic cubic skein relation for a knot polynome would be
$$S^2=wvS+w/S+w^2(u-v)I-u\cdot\infty$$
where $w^3$ is one ...
- 4,829
7
votes
2
answers
774
views
3-manifold with torus boundary with trivial "peripheral ideal"?
Given a 3-manifold $M$, one can define the Kauffman bracket skein module $K_t(M)$ as the $C$-vector space with basis "links (including the empty link) in $M$ up to ambient isotopy," modulo the skein ...
- 2,869
16
votes
9
answers
4k
views
How to motivate the skein relations?
I am teaching an advanced undergraduate class on topology. We are doing introductory knot theory at the moment. One of my students asked how do we know to use this skein relation to compute all these ...
- 30.5k
18
votes
4
answers
2k
views
Who thought that the Alexander polynomial was the only knot invariant of its kind?
I apologize that this is vague, but I'm trying to understand a little bit of the historical context in which the zoo of quantum invariants emerged.
For some reason, I have in my head the folklore:
...
- 1,756