9
votes
Is there a known invariant for knotted surfaces defined by skein relations?
It is known that a knotted surface can be presented by a marked graph diagram, which is just a knot diagram while some crossing points are equipped with markers. On the other hand, two marked graph ...
- 517
6
votes
Is there a known invariant for knotted surfaces defined by skein relations?
The answer might depend a bit on exactly what you want; perhaps giving a precise formulation is the hard part!
There was an important first step in this direction for the Alexander polynomial, by ...
- 19.4k
6
votes
Accepted
Is the quantum $\mathfrak{sl}_3$ invariant stronger than the quantum $\mathfrak{sl}_2$ invariant?
The quantum $\mathfrak{sl}_3$ invariant is a special case of the HOMFLY-PT polynomial, which is essentially the $\mathfrak{sl}_N$ invariant. That polynomial has two variables $q$ and $t$. The $q$ ...
- 2,533
3
votes
Gap in Przytycki's computation of the skein module of links in a handlebody?
Here's something that isn't a complete rigorous proof, but maybe it can be completed to one (unless I'm missing something). Let's fix a non-positively curved metric on $F$ (which rules out the sphere, ...
- 2,869
3
votes
Accepted
Easy lemma for trivalent graphs in colored Jones polynomial
Rewrite each edge of the graph, labeled by $k$, as $k$ strands with the $k$-th JW idempotent in the middle. Make a similar modification at the vertices. Expand the sums appearing to one side of the $...
- 12.8k
3
votes
Dimension of the skein module of a closed manifold?
The finiteness conjecture for skein modules was recently proved by Gunningham, Jordan and Safronov in 2022.
Gunningham, S., Jordan, D. & Safronov, P. The finiteness conjecture for skein modules. ...
- 168
3
votes
Accepted
From braid representations to link invariants
Yes, but you need to check a normalization condition. (Thanks to Student for noting the error in my previous answer.)
An invariant $f$ of braids $b$ is an invariant of links (obtained as braid ...
- 2,533
2
votes
Actions of two types of Kauffman skein categories
The Dubrovnik normalization is the "right" one for quantum groups for the simple reason that it deforms the symmetric (over=under) skein relation which is what holds for ordinary groups. ...
- 28.1k
2
votes
Unusual skein relation in HOMFLY polynomial
First, doing a Reidemeister II move in the first diagram in your expression and then subtracting a multiple of the skein relation converts it to the change in the HOMFLY polynomial through a crossing ...
1
vote
Accepted
Basis for Annular Skein Algebra
The outlined proof almost works: only need to consider the weaker statement "$\sigma_{w}$ is conjugate to $\sigma_{w'}$ in $B_{n} \implies w$ is conjugate to $w'$ in $S_{n}$". (A useful discussion ...
- 318
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