5
votes
1answer
240 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 ...
4
votes
2answers
336 views

Jones polynomial of the concatenation of two braids

Let $\sigma_1$ and $\sigma_2$ be two braids with $n$-strings. Are there any formulas relating $J_{\widehat{\sigma_1\sigma_2}}(q)$, $J_{\hat{\sigma_1}}(q)$, and $J_{\hat{\sigma_2}}(q)$? Here, ...
9
votes
0answers
174 views

The Markov trace via Bott-Samelson fibers?

Let $H_n$ be the Hecke algebra of GL(n), i.e., the algebra over $\mathbb{Q}(q)$ with generators $T_1, \ldots, T_{n-1}$ which satisfy the braid relations and also $T^2 = (q-1) T + q$. Recall the ...
12
votes
3answers
593 views

Representing SU(3) with 3 ropes in 3 dimensions

The short question is: how exactly is SU(3) realized with ropes? The long question: There is this idea that deformations of a configuration of three infinitely long, flexible ropes that cross each ...
8
votes
0answers
311 views

What is (explicitly) known about the SL(n,C) character varieties of 3-manifolds ?

The $SL(2,{\bf C})$ character variety of a 3-manifold with 1-cusp $M$ (like a knot complement in the 3-sphere) essentially coincide with the variety defined by the A-polynomial. Those polynomials are ...
3
votes
0answers
197 views

Knot polynomials of non-crystallographic Coxeter groups?

I learnt that the Coxeter groups have a few members more than the classic simple Lie groups: $H_3, H_4$ and $I_2(p)$. Is there a Reshetikhin-Turaev invariant for those, too? If not, where does the ...
0
votes
0answers
450 views

Knot Numerology

EDIT: Ok, I condense it to only that what is needed. Assume it's possible to use the method described here Matrix decomposition the other way to decompose a $S$ matrix from knot theory. Then each ...
7
votes
2answers
303 views

Is there a (hope for) formula for a colored HOMFLYPT invariant in terms of HOMFLYPT invariants colored by the fundamental represenations?

Any finite dimensional representation of $SL(n,\mathbb{C})$ can be found as a direct summand inside of some tensor product of the fundamental representations. Since we use finite dimensional ...
6
votes
1answer
386 views

What is the image of the half/full twist in the Hecke algebra, in the Kazhdan-Lusztig basis? What is the corresponding complex of Soergel bimodules?

Let $B_n$ be the braid group on $n$ strands. It has generators $\tau_i$ for $i = 1,\ldots,n-1$ which exchange the $i$th and $(i+1)$st strands, and which satisfy the relations $\tau_i \tau_j = \tau_j ...
17
votes
2answers
571 views

Lie algebra automorphisms and detecting knot orientation by Vassiliev invariants

Recall that there are knots in $\mathbf{R}^3$ that are not invertible, i.e. not isotopic to themselves with the orientation reversed. However, it is not easy to tell whether or not a given knot is ...
6
votes
5answers
491 views

Is there a version of Temperley-Lieb using sl(3) rather than sl(2)?

This question is a spin-off from Sammy Black's question on super Temperley-Lieb. Please see there for the background. The short version is that Sammy defines the Temperley-Lieb at index d as the ...
15
votes
2answers
902 views

Does the super Temperley-Lieb algebra have a Z-form?

Background Let V denote the standard (2-dimensional) module for the Lie algebra sl2(C), or equivalently for the universal envelope U = U(sl2(C)). The Temperley-Lieb algebra TLd is the algebra of ...
15
votes
4answers
852 views

HOMFLY and homology; also superalgebras

My understanding is that an analogy along the following lines is (roughly) true: "The Alexander polynomial is to knot Floer homology is to gl(1|1) as the Jones polynomial is to Khovanov homology is ...