Questions tagged [quantum-topology]

Finite-type (Vassiliev) invariants, quantum invariants, and perturbative invariants of knotted objects and of manifolds.

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14 votes
1 answer
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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 ...
4 votes
0 answers
136 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 ...
6 votes
1 answer
163 views

Does $U_q (\mathfrak{sl}_2)$ have a universal $R$-matrix?

Consider the standard quantum group $U_q (\mathfrak{sl}_2)$ over the field $\mathbb{C}(q)$ of rational functions (or over $\mathbb{C}$ if $q \in \mathbb{C}$ is not a root of unity), with the usual ...
17 votes
1 answer
1k views

Is there an "arithmetic cobordism category"?

This question is a clumsy attempt to apply a certain analogy. I hope that if the answer is negative it comes with a clarification of the scope and limitations of the analogy. Arithmetic topology is ...
3 votes
1 answer
177 views

Squier's conjecture on Burau at roots of unity

In Squier's short, yet influential, paper about the Burau representation, he made two conjectures that might have provided a proof for the faithfulness of the Burau representation (which we now know ...
5 votes
0 answers
77 views

References for completions of finite group tensor categories

Let $G$ be a finite group and $\operatorname{Vec}_G$ be the tensor category of $G$-graded vector spaces (or, if you prefer, $\pi_{\le 2}(BG)$). The completion $\overline{\operatorname{Vec}_G}$ of $\...
3 votes
0 answers
181 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 ...
3 votes
1 answer
124 views

Closed-form expressions for the Kashaev invariant via surgery

For a knot $K$, let $J_N(K)$ denote the $N$th Kashaev invariant of $K$. This is the same as the $N$th colored Jones polynomial evaluated at an $N$th root of unity (or $2N$th depending on your ...
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 ...
28 votes
2 answers
2k 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 ...
20 votes
4 answers
2k views

Is there a volume conjecture for closed 3-manifolds?

A typical statement of the volume conjecture, for instance in Murakami's survey 1002.0126, is Conjecture: For $K$ a knot in $S^3$, the N-th colored Jones polynomials are related to the volume of ...
7 votes
1 answer
173 views

Quantum homology of $(S^2 \times S^2,\omega_{FS}\oplus \omega_{FS})$ and Poincare duality

I am having some issues computing Poincare duality for the quantum homology $QH(M)$ when $(M,\omega)=(S^2 \times S^2,\omega_{FS}\oplus \omega_{FS})$. I am using the simple novikov ring $\Lambda$ ...
5 votes
2 answers
369 views

Does WRT invariant detect hyperelliptic involution on the genus 2 surface?

The Witten-Reshetikhin-Turaev invariant cannot detect the hyperelliptic involution on the genus 1 surface, and that if $M_U$ is the mapping torus for a mapping class group element $U\in \mathrm{Mod}(\...
9 votes
1 answer
1k views

Casson's invariant and the trivial connection contribution to witten's 3-manifold invariant

This question might turn out not to make any sense, but here it is: Witten's (and Reshetikhin and Turaev's) 3-manifold invariant can be "defined" as an integral over the space of connections on the ...
9 votes
1 answer
717 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 ...
13 votes
1 answer
1k views

Why Lagrangian cobordism?

There are a good number of quantum topology papers in which a TQFT-like set-up is constructed as a functor to the category of vector spaces from some category of cobordisms which satisfy some "...
2 votes
0 answers
141 views

A spectral sequence in Khovanov Homology

Szabo equipped the mod $2$ Khovanov complex with a family of differentials $\{d_{i} \}_{i=1}^{\infty}$ such that each $d_{i}$ has bigrading $(i,2i-2)$ where $d_1$ is the mod $2$ Khovanov differential ...
48 votes
5 answers
7k views

Algebraic proof of 4-colour theorem?

4-colour Theorem. Every planar graph is 4-colourable. This theorem of course has a well-known history. It was first proven by Appel and Haken in 1976, but their proof was met with skepticism because ...
4 votes
1 answer
152 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 ...
2 votes
1 answer
204 views

Non-extendable 3D TQFTs

Non-extendable 2D TQFTs correspond to finite dimensional Frobenius algebras [1]. How about 3D TQFTs? While the answer is clear for the extended ones (e.g. (3,2,1) TQFTs almost correspond to modular ...
1 vote
1 answer
161 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 (...
9 votes
0 answers
202 views

Algebraic context for Mednykh's formula?

Let $S$ be a closed orientable surface and let $G$ be a finite group, then Mednykh's formula says that $$ \sum_{V}d(V)^{\chi(S)} = |G|^{\chi(S) - 1} |\text{Hom}(\pi_1 S, G)| $$ where the sum is over ...
14 votes
2 answers
458 views

Lagrangian of Reshetikhin-Turaev TFT's

One of the results from the Reshetikhin-Turaev package is that given a modular tensor category $\mathscr{C}$ one can construct a TFT $Z$. In the case where $\mathscr{C}$ is the category of positive ...
1 vote
0 answers
77 views

Ordering in Cobordism Category

Let $Cob^{3}$ denote the cobordism category of $1$ dimensional manifolds i.e the objects are finite disjoint union of circles and morphisms are represented by surfaces. Is it possible to treat the ...
5 votes
1 answer
323 views

How does the scalar TV invariant of a 3-manifold with boundary fit into the TQFT picture?

Chen and Yang have a more general version of the volume conjecture that they state for all hyperbolic $3$-manifolds (Conjecture 1.1 of [2]) including those with boundary. To do this, they have to ...
12 votes
2 answers
714 views

Relations between quantum groups at roots of unity, modular representation theory, and physics

I understand that quantum groups at roots of unity are related to physics because they are used in the construction of Reshetikhin-Turaev invariants, conjectured by Witten. Are there other relations ...
3 votes
0 answers
89 views

Are the tangle functors based off Khovanov homology braided monoidal functors?

I was wondering if the tangle functors constructed in "A functor-valued invariant of tangles" https://arxiv.org/pdf/math/0103190.pdf "An invariant of tangle cobordisms via subquotients of arc rings" ...
9 votes
1 answer
389 views

Generators and relations for the 2-dimensional unoriented cobordism category

It is very well known in the field of TQFT that the 2-dimensional oriented cobordism category is generated by the disk and the pair of pants (each going in both directions), subject to a finite set of ...
6 votes
1 answer
183 views

Applications of quantum representations of the mapping class group to quantum computers

Quantum representations of the mapping class group of a surface are certain representations constructed from the data of a TQFT and described, for example, in and 1 and 2. The following sources 3 ...
5 votes
1 answer
339 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 ...
4 votes
0 answers
183 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 ...
15 votes
1 answer
741 views

P-adic Volume Conjecture

Let $M$ be a closed hyperbolic 3-manifold. One can use hyperbolic structure on $M$ to define hyperbolic volume $Vol(M)$. Thanks to Mostow's rigidity theorem the volume depends only on the topology of ...
67 votes
4 answers
9k views

Why hasn't anyone proved that the two standard approaches to quantizing Chern-Simons theory are equivalent?

The two standard approaches to the quantization of Chern-Simons theory are geometric quantization of character varieties, and quantum groups plus skein theory. These two approaches were both first ...
7 votes
0 answers
206 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 ...
9 votes
0 answers
223 views

What is the "classical limit" of Khovanov homology?

Let me first explain what I mean by the "classical limit". For quantum group invariants of links and webs (such as colored Jones polynomials), the "classical limit" means the limit $k\rightarrow +\...
4 votes
1 answer
260 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 $sl(n)$ ...
32 votes
2 answers
2k 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 $t$. ...
3 votes
0 answers
179 views

Example of a non-extendable TQFT?

All 3D TQFTs I know of are of Reshetikhin-Turaev type. These are fully extended and I wondered if there are known examples of TQFTs such that there is no once-extended TQFT extending it.
13 votes
1 answer
1k views

Classification of unitary modular tensor categories (UMTCs)

Context/background: I'm approaching this topic from the perspective of anyonic systems. In the study of anyons, one works with fusion categories. Of course, for physicality, we demand that i) The ...
18 votes
2 answers
1k views

Hyperbolic Volume and Chern-Simons

In the paper ``Analytic Continuation Of Chern-Simons Theory'' (arXiv:1001.2933) Witten postulates that hyperbolic volume of 3-dimensional manifold coincides with the value of the Chern-Simons ...
15 votes
0 answers
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Representations of quantum groups at roots of unity

I'm interested in the semisimplified category of representations of a quantum group at a root of unity. I've heard that simple objects in this category correspond to certain "integral" conjugacy ...
31 votes
4 answers
2k views

Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?

Varieties of representations and characters of $3$-manifold groups in $SL(2,\mathbb{C})$ have been intensively studied. They have provided tools to identify geometric structures on manifolds, and are ...
6 votes
3 answers
618 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-...
12 votes
2 answers
537 views

S-matrix for the HOMFLY/Hecke category

This question concerns the HOMFLY-PT category, closely related to Hecke algebras. (See here for example.) The minimal idempotents of this category are indexed by pairs $(\lambda_+, \lambda_-)$ of ...
8 votes
1 answer
397 views

Brauer-Picard for a fusion category coming from a quantum group

In Fusion Categories and Homotopy Theory, ENO attatch a 3-groupoid to a fusion category. In the case of A graded vector spaces they further compute it's truncation as an orthogonal group $O(A \...
15 votes
1 answer
520 views

Examples of n+1D TQFT with 1 dimensional Hilbert spaces on n-torus and n-sphere but higher dimensional Hilbert spaces on other n-manifolds

Are there simple examples of $n+1$D TQFT that assign 1-dimensional Hilbert spaces to both $n$-torus and $n$-sphere but higher dimensional Hilbert spaces to some other $n$-manifolds? Here I am assuming ...
4 votes
1 answer
864 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 ...
2 votes
0 answers
101 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 ...
7 votes
2 answers
402 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 ...
8 votes
1 answer
665 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 ...