All Questions
Tagged with knot-theory quantum-topology
35 questions
4
votes
0
answers
164
views
Coloured Jones polynomial at 4th root of unity and Arf invariant
Looking at the link invariants of $\operatorname{SU}(2)$ Chern-Simons theory, if we take the coloured Jones polynomial of a knot K, say $J_N^K$ at fundamental representation $N=2$, then we get the ...
- 81
3
votes
0
answers
187
views
Reshuffling power series (aka Melvin–Morton expansion in knot theory)
I am struggling to understand a statement which follows from some change of variables in a power series. I think that the context does not really matter here, so I will put it at the end of the ...
- 601
8
votes
1
answer
735
views
Inverse Kirby knot
Given an (oriented framed) knot $K$ in the 3-sphere $S^3$, we can perform a surgery along $K$ to get another 3-manifold $M$. From $M$, we can perform the inverse surgery back to $S^3$.
However, the ...
- 5,230
4
votes
1
answer
168
views
Link invariants from modular categories (strictification and computation)
By the theory of Reshetikhin and Turaev, a modular tensor category $C$ gives rise to a link invariant. While $C$ is strict as a monoidal category (e.g. $\mathbb{Fib}$), calculating the link can be ...
- 5,230
1
vote
1
answer
200
views
Is there a combinatorial way to determine the coefficients of the universal finite-type invariant on a given knot?
There are various (equivalent?) descriptions of a universal finite-type knot invariant, e.g. https://arxiv.org/abs/q-alg/9603010. They take the form of formal power series valued in Feynman diagrams (...
- 2,533
5
votes
1
answer
379
views
Computation of \tau invariant
I am trying to understand the following inequality, $$0 \leq \tau (K_{+}) - \tau(K_{-}) \leq 1$$ from the following paper by Livingston. \ https://arxiv.org/pdf/math/0311036.pdf . At page 737 , he ...
- 341
4
votes
0
answers
187
views
Are Turaev-Viro invariants holonomic?
Consider a 3-manifold $M$ with a boundary, which is a genus $g\geq 1$ surface $\Sigma$. Fix a triangulation $T$ of $\Sigma$. Then Turaev-Viro invariants $TV_q(M)$ are functions, assigning to integer ...
- 4,316
7
votes
0
answers
209
views
IH-moves on trivalent graphs, and a complex that might be known to low-dimensional topologists
Here is a combinatorial problem which is hard to Google but seems like it might have a solution well known to people who study finite type invariants etc.
Let $G_{g,b}$ denote the set of finite ...
- 40.2k
4
votes
1
answer
265
views
Framing dependence of HOMFLY polynomial
I want to understand the framing dependence of the Khovanov-Rozansky homology, and as its first step, I am trying to understand the framing dependence of the HOMFLY polynomial (i.e. quantum $sl(n)$ ...
- 1,430
6
votes
3
answers
651
views
Polynomial invariants for unoriented links
I have seen that usually one finds polynomial invariants for oriented links (for example the Jones polynomial, the Hompfly polynomial). Does anyone know what polynomial invariants exist for non-...
- 743
4
votes
1
answer
958
views
Jones polynomial of tangles using Temperley-Lieb algbra?
The definition of the Jones polynomial of tangles (a la Reshetikhin and Turaev) uses the space of invariants for $U_q sl_2$ and R-matrices. It seems to me the same thing cane be done in terms of the ...
- 803
2
votes
0
answers
104
views
Mutant pairs distinguished by [2,1]-colored HOMFLY polynomials
Morton and Cromwell showed that the famous mutant pair, Kinoshita-Terasaka and Conway knots, can be distinguished by HOMFLY polynomials colored by [2,1] Young diagram. Are there any other mutant pairs ...
- 2,273
8
votes
1
answer
413
views
Does the limit in the Volume conjecture converge?
The Volume conjecture says that if $J_n(q)$ are the colored Jones polynomials of a knot $K \subset S^3$, then
$$\lim_{N \to \infty} \frac{ 2 \pi}N \left\vert J_N(e^{2\pi i / N})\right\vert = vol(K)$$
...
- 2,869
8
votes
1
answer
681
views
Proving that the Jones polynomial is q-holonomic
The Jones polynomial is known to have many different interpretations or definitions, by now. There are connections with QFT, quantum groups, Hilbert schemes, Cherednik algebras, etc.
My question is ...
- 85.5k
38
votes
3
answers
3k
views
The Jones polynomial at specific values of $t$
I've been calculating some Jones polynomials lately and I was just curious if there was a "physical" (or, rather, geometric) meaning to evaluating the Jones polynomial at a particular value of $t$.
...
- 503
14
votes
1
answer
2k
views
Kontsevich integral : state of the art
The Kontsevich integral is known to be a universal Vassiliev invariant.
It is still an open question whether it is a complete knot invariant, i.e. whether it distinguishes a given knot from all other ...
- 1,609
4
votes
0
answers
236
views
Is a generic link diagram semi-adequate?
Each crossing in a link diagram of a link $L$ has an A-resolution and a B-resolution.
Resolving all crossings gives a collection of circles in the plane, connected by dotted lines. A state of a link ...
- 22.1k
9
votes
1
answer
1k
views
Diagrammatic proof of unique prime decomposition of knots
Consider a knot to be a diagram in a plane--- i.e. a drawing of a finite connected planar graph (loops and multiple edges allowed) whose vertices are 4-valent with cyclic ordering for the incident ...
- 22.1k
7
votes
2
answers
417
views
Vassilliev invariants of knots and their cables
The following is perhaps a standard question, but i could not find a plain enough answer
by simply searching online.
Q: Given a knot $K$ and its $(p,q)$-cable $K_{p,q}$ what is a relation
between ...
- 73
10
votes
3
answers
562
views
Is a knotted trivalent graph determined by its set of unzips?
A knotted trivalent graph (KTG) is a framed embedding of a trivalent graph in $\mathbb{R}^3$, modulo ambient isotopy. I believe they were first considered (in the quantum topological context, at least)...
- 22.1k
5
votes
0
answers
346
views
On finding A-polynomials
I have two questions to obtain the explicit forms of A-polynomials.
Takata used the mathematica pacage qMultisum.m to obtain the recursion relation of the colored Jones polynomials for twist knots. ...
- 2,273
6
votes
1
answer
483
views
Computations and applications of Khovanov's functor valued invariant of tangles
Soon after his famous paper "A categorification of the Jones polynomial", Khovanov
introduced a "bordered" version. His theory assingns to every oriented even tangle
a complex of (H_n,H_m)-bimodules, ...
- 587
13
votes
1
answer
702
views
Integer matrices with a strange divisibility property
Fix an integer $n$. What can you say about a (not necessarily square) matrix $A$ with integer entries that has the property that for any $k$, every $k\times k$ minor of $A$ is divisible by $n^{k-1}$? ...
- 1,960
8
votes
1
answer
1k
views
Closed formula for colored Jones polynomial of the trefoil? (reference request)
(EDIT: Powers of $q$ in the formula corrected.)
I've been doing some computations with skein modules, and I found the following formula for the N-th colored Jones polynomial of the trefoil:
$\frac{1}...
- 2,869
7
votes
2
answers
693
views
Quantum E6/E7 knot polynomials
Has anybody seen seen quantum knot invariants associated to (E6, 27) or (E7, 56) worked out in the literature? Even for just simple knots like the trefoil or figure-8?
I suspect these haven't been ...
- 141
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
1
vote
1
answer
695
views
Recovering the Alexander Polynomial from Ocneanu's HOMFLYPT
Let $H_q(n)$ denote the Hecke algebra associated to the symmetric group $S(n)$: this is the $\mathbb{Z}[q^{\pm 1/2}]$ algebra generated by $T_1, \ldots, T_{n-1}$ satisfying the braid relations along ...
- 952
9
votes
2
answers
838
views
Torus knots in Euclidean space -- a symmetry argument
Consider a $(p,q)$ torus knot $K$ in 3-dimensional Euclidean space $\mathbb R^3$ where $p,q \geq 2$ and $\operatorname{GCD}(p,q)=1$.
Let $\operatorname{Isom}(\mathbb R^3,K)$ be the isometries of $\...
- 44.3k
3
votes
1
answer
289
views
Invariants of unframed, oriented links from Reshetikhin Turaev construction
Hello!
I have a few questions on Reshetikhin Turaev invariants.
By RT any ribbon category ${\mathcal C}$ yields an invariant of oriented, framed links labelled with objects of ${\mathcal C}$.
Is ...
- 2,756
2
votes
1
answer
540
views
Now that I got a mutant-discriminating invariant...
...what can I do with the darn thing?
Background: I read that still no Vassiliev Invariant with mutant-discriminating
power is known (correct me if this is outdated). Now, my research lead to a
whole ...
- 4,829
0
votes
1
answer
378
views
Some Vassiliev Invariant Questions
The V.I. definition goes doublepoint=overpass-underpass (or was it the other
way around? If it's 50:50, I score 0 always :-). Would it lead anywhere
to define doublepoint=overpass+underpass? (Even if ...
- 4,829
8
votes
0
answers
381
views
Which presentations of (non)planar algebras give rise to knots?
Reidermeister's theorem states that the set of knots, modulo ambient isotopy, is isomorphic to the planar algebra generated by crossings, modulo Reidemeister moves. This planar algebra presentation is ...
- 22.1k
21
votes
1
answer
2k
views
How are the Conway polynomial and the Alexander polynomial different?
Background story:
I have just come out from a talk by Misha Polyak on generalizations of an arrow-diagram formula for coefficients of the Conway polynomial by Chmutov, Khoury, and Rossi. In it, he ...
- 22.1k
1
vote
3
answers
995
views
SO(3) knot polynomials
Can one use the real lie algebra so(3) to get knot polynomials? If so, do they have a skein relation (I presume they would, if they come from R-matrices in some standard way. If so, is the R-matrix ...
- 1,129
28
votes
2
answers
3k
views
Why is the Alexander polynomial a quantum invariant?
When we think of quantum invariants, we usually think of the Jones polynomial or of the coloured HOMFLYPT. But (arguably) the simplest example of a quantum invariant of a knot or link is its Alexander ...
- 22.1k