All Questions
10 questions
9
votes
1
answer
297
views
Is there a geometric interpretation of the second derivative of the Alexander polynomial at $1$?
For an (oriented) knot in $S^3$ the number $\Gamma(K) := \Delta_K’’(1)$ shows up in a number of places in knot theory, for example the Casson-Walker-Lescop invariant. Here $\Delta_K(t)$ is the ...
- 2,533
7
votes
0
answers
209
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 ...
- 40.2k
6
votes
0
answers
156
views
Relation between different versions of Bar-Natan homology
In Bar-Natan's paper: Khovanov’s homology for tangles and cobordisms, he defined a deformation of Khovanov homology. Namely, for any $m\geq 0$, Bar-Natan's homology $BN^{m}(K)$ is obtained by ...
- 1,069
5
votes
1
answer
348
views
Presentations of the monoidal categories of virtual tangles and of welded tangles by generators and relations
Reidemeister theorem implies, without too much fuss, that the monoidal categories of tangles, and of oriented tangles, can be presented by generators and relations. This is done for example in
a) ...
21
votes
1
answer
863
views
Why, in terms of quantum groups, does the knot determinant appear as an evaluation of both the Jones and Alexander polynomials?
The Jones polynomial can be computed from the representation theory of $\mathcal{U}_q(\mathfrak{sl}(2))$. The Alexander polynomial has an analogous description in terms of the representation theory of ...
- 1,474
5
votes
2
answers
554
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, $J_L(q)...
- 103
5
votes
3
answers
1k
views
Links with same Jones polynomial
Is there anything known about when two links have the same Jones polynomial (beyond a calculated list of small actual examples)? The first thing I would try is to compute the (formal - you would have ...
- 4,829
6
votes
1
answer
634
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 \...
- 8,723
7
votes
2
answers
497
views
Computations of the Link homology categorifying the second colored Jones polynomial
Has anybody done computations of such a theory? Is there a place I could look up and see what the answers are for low crossing knots?
- 3,474
8
votes
2
answers
834
views
What is the Alexander polynomial of a point?
According to the Baez-Dolan cobordism hypothesis, an extended TQFT is determined by its value on a single point. This value a fully dualizable object of a symmetric monoidal $n$ category (a fully ...
- 22.1k