# Questions tagged [quantum-topology]

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

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### 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 ...

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### 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 $\...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 (...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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"
...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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}(\...

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### 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 ...

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### 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 ...

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### 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 $\Lambda$ ...

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### 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 ...

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### 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 +\...

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### 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 $sl(n)$ ...

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### 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.

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 \...

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### 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 ...

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### 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-...

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### 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 ...

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### 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 ...

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### 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
$$\lim_{N \to \infty} \frac{ 2 \pi}N \left\vert J_N(e^{2\pi i / N})\right\vert = vol(K)$$
...

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### 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 ...

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### Why are Witten-Reshetikhin-Turaev invariants expected to be integral?

A Witten-Reshetikhin-Turaev (WRT) Invariant $\tau_{M,L}^G(\xi)\in\mathbb{C}$ is an invariant of closed oriented 3-manifold $M$ containing a framed link $L$, where $G$ is a simple Lie group, and $\xi$ ...

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### 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 $t$.
...

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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 ...

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### Construction of 3D Topological Quantum Field Theory from a 2D Modular Functor

I have been reading Bakalov and Kirillov's Lectures on Tensor Categories and Modular Functors in which the authors state that a direct construction of a $C$-extended 3D TQFT from a $C$-extended 2D ...

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### Quantization of symplectic vector space and choice of lagrangian subspaces

My question is related to Geometric Quantization.
I don't undrestand the philosophy of following assertion
If $(V,\omega)$ be a symplectic vector space then the quantizations of
$V$ corresponds ...

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### The Maslov triple product is alternating in its entries

Let $(V,\omega)$ be a $2g$-dimensional symplectic vector space. I'm trying to understand the Maslov triple product. I know that it can be defined in a variety of ways, but for the applications I'm ...

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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 ...

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### Provide a citation for the "spine lemma"?

I'm looking for a citable reference for the following (perhaps folkloric?) result on topological field theories.
(There are obviously generalizations to other dimensions; I'm happy with just the 2-...

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### 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 ...

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### Are Turaev--Viro invariants secretly a discretized path integral?

Turaev--Viro http://www.ams.org/mathscinet-getitem?mr=1191386 defined an invariant of three-manifolds $M$ denoted $TV(M)$, which was subsequently shown by Kevin Walker to coincide with $\left|WRT(M)\...

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### 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 ...

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### 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 ...

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### 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)...

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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 $...