Questions tagged [skein-relation]
The skein-relation tag has no usage guidance.
19 questions
4
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1
answer
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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
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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
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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
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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
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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
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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
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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
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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
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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
2
answers
322
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On trivalent spines of surfaces
Let $\Sigma$ be a surface with non-empty boundary labelled by finitely many points $x_i$.
For our purposes a spine $s$ for $\Sigma$ is a uni-trivalent graph embedded in $\Sigma$ whose 1-vertices are ...
- 365
8
votes
1
answer
510
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Reference request: the "Kauffman bracket skein category"?
There should be a category $3\text{CobTang}$ whose
objects are some kind of surfaces with a finite set of marked points
morphisms $M : S \to T$ are some kind of $3$-dimensional cobordisms $...
- 118k
2
votes
1
answer
361
views
Skein Modules on closed 3-manifolds
Hi all! Why are skein modules 1-dimensional on closed 3-manifolds? The result seems clear on closed manifolds with vanishing first Betti number (e.g. $S^3$), but I don't see how to prove it for, say, $...
- 43
6
votes
1
answer
396
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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
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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
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280
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"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
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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
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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
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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
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4
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2k
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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