Questions tagged [quantum-topology]
Finite-type (Vassiliev) invariants, quantum invariants, and perturbative invariants of knotted objects and of manifolds.
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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 ...
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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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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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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 +\...
- 1,410
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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 ...
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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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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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S-matrix for the BMW category
This question concerns the Birman-Murakami-Wenzl category, or equivalently the tangle category associated to the 2-variable Kauffman polynomial. (See here for example.)
The minimal idempotents of ...
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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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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. ...
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generators for Hecke algebra quotients
What are generators for the kernel of the (k,r)-quotient of the Hecke algebras of type A? Are just the two projections onto the reps. corresponding to Young digarams with 1-row of length r-k+1 and ...
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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 ...
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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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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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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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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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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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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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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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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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