14
votes

### How to motivate the skein relations?

One of the earliest appearances of the ingredients for a skein relation can be found in Romilly Allen's 1904 book on Celtic Knotting. There he explains that designers of Celtic knot patterns first ...

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 ...

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 $q$ ...

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 ...

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 $...

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, ...

3
votes

### How to motivate the skein relations?

Here is a sketch of how the skein relations appear in the approach to knot invariants based on braided monoidal categories coming e.g. from representations of quantum groups.
Suppose $V$ is a ...

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
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. ...

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 ...

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