Questions tagged [homflypt-polynomial]

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4
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1answer
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Unusual skein relation in HOMFLY polynomial

If I take the HOMFLY(PT) polynomial defined by $$l \,P(L_+) + l^{-1}\,P(L_-) + m\,P(L_0) = 0,$$ I have looked at expressions of the form (knots that are the same except inside a small disk, where ...
0
votes
1answer
140 views

Proving knot polynomial dependencies and skein relations

I have two questions: From the definition of the Jones polynomial as the normalization of the Kauffman bracket $(-A^3)^{-w(D)} \langle D\rangle$ and substituting $A\rightarrow t^{-1/4}$, how does one ...
4
votes
1answer
213 views

Categorifying skein algebras?

We can obtain the Jones polynomial by the Temperly-Lieb algebra and the HOMFLYPT polynomial from the Hecke algebra. Were there attempts to categorify the algebras itself and obtain the Khovanov ...
6
votes
1answer
429 views

HOMFLYPT vs. Jones vs. Alexander polynomial?

I'm searching for examples (perhaps the simplest one?) to show that the HOMFLYPT polynomial is stronger than the Jones and Alexander polynomial, respectively. Any ideas what is the 1st knot in the ...