Questions tagged [quantum-topology]
Finite-type (Vassiliev) invariants, quantum invariants, and perturbative invariants of knotted objects and of manifolds.
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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 $...
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0
answers
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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. ...
6
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1
answer
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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, ...
2
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1
answer
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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, $...
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2
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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 ...
6
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0
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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 ...
13
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1
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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 $n^{k-1}$? ...
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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}...
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How to find Colin Day's PhD Thesis
A week or so ago, I was saddened to read Jim Stasheff's post on the AlgTop mailing list, announcing the passing of Colin Day, after a long bout with cancer.
I was thinking of reading Colin Day's PhD ...
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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 ...
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Why is the quantum Lorentz group not connected?
Podles and Woronowicz' construct the quantum Lorentz group, by which they mean $SL_q(2,\mathbb{C})$, as a quantum double of the compact quantum group $SU_q(2)$. More precisely, it is a bicrossed ...
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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 ...
7
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Quantum coordinate ring at root of unity
Noah Snyder gave a great answer to this question about different versions of a quantum group $U_q(\mathfrak g)$ when $q$ is a root of unity. I want to ask about forms of the deformed coordinate ring $...
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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 ...
6
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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). ...
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vote
1
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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 ...
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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 $\...
3
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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 ...
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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 ...
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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 ...
3
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1
answer
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What if I change field in a Topological Quantum Field Theory?
Of course I'm talking about the algebraic notion of field.
In a few words, if a TQFT consists of a functor $Z\colon Cob(n)\to \mathbf{Vec}_k$, I'm wondering if there are sensible relations among ...
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Quantum mathematics?
"Quantum" as a term/prefix used to be genuinely physical: what was supposed to be physically continuous turned out to be physically quantized.
What sense does this distinction make inside ...
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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 "...
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Why is the volume conjecture important?
The volume conjecture, a formula relating hyperbolic volume of a knot complement with the semiclassical limit of a family of coloured Jones polynomials, is widely considered the biggest open problem ...
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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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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 ...
5
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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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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 ...
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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 ...
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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 ...
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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 ...
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Is there a canonical Hopf structure on the center of a universal enveloping algebra?
Let $\mathfrak g$ be a finite-dimensional Lie algebra over $\mathbb C$. Define $\mathcal Z(\mathfrak g)$ to be the center of the universal enveloping algebra $\mathcal U\mathfrak g$, and define $(\...
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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 ...
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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 ...