Patterns of the V₂-polynomial of knots

2024/09/05 by Garoufalidis, Stavros, Li, Shana Yunsheng · 2 citations
#FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #High Energy Physics - Theory (hep-th)

paper · doi:10.48550/arxiv.2409.03557

Abstract

Recently, Kashaev and the first author defined a sequence V_n of 2-variable knot polynomials with integer coefficients, coming from the R-matrix of a rank 2 Nichols algebra, the first polynomial been identified with the Links--Gould polynomial. In this note we present the results of the computation of the V_n-polynomials for n=1,2,3,4. This leads to the discovery of emerging patterns, including the genus bound for V₂ being an equality for all 58 million knots with at most 18 crossings, as well as unexpected Conway mutations that seem undetected by the V_n-polynomials as well as by Heegaard Floer Homology and Khovanov Homology.

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• On the colored Links--Gould polynomial
• Skein theory for the Links-Gould polynomial
• Extending knot polynomials of braided Hopf algebras to links

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