7
votes
HOMFLYPT vs. Jones vs. Alexander polynomial?
I will give an example that is likely not the simplest one. The example comes from the paper Behavior of knot invariants under genus 2 mutation by Dunfield, Garoufalidis, Shumakovitch, and ...
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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 ...
2
votes
Proving knot polynomial dependencies and skein relations
For 1, see Lickorish's "An Introduction to Knot Theory" Proposition 3.7 (page 28).
For 2, the substitution $l=i\alpha$ and $m=-i z$ gives the $\alpha P(L_+)-\alpha^{-1}P(L_-)=z P(L_0)$ version of the ...
- 413
2
votes
Accepted
Categorifying skein algebras?
Much of the research in knot homology has been about categorifying these algebras!
Khovanov's paper math/0103190 is devoted to defining and studying the Temperley-Lieb 2-category which is a ...
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