Elementary combinatorics of the HOMFLYPT polynomial
Sergei Chmutov, Michael Polyak
TL;DR
This paper reformulates Jaeger's state model for the HOMFLYPT polynomial using Gauss diagrams and derives new formulas for related Vassiliev invariants, including those of degree 3.
Contribution
It introduces Gauss diagram formulas for a family of Vassiliev invariants from the HOMFLYPT polynomial, advancing the understanding of their combinatorial structure.
Findings
Gauss diagram formulas for Vassiliev invariants of the HOMFLYPT polynomial
New formulas for invariants of degree 3
Enhanced combinatorial understanding of knot invariants
Abstract
We explore Jaeger's state model for the HOMFLYPT polynomial. We reformulate this model in the language of Gauss diagrams and use it to obtain Gauss diagram formulas for a two-parameter family of Vassiliev invariants coming from the HOMFLYPT polynomial. These formulas are new already for invariants of degree 3.
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