Work

Patterns of the V2-Polynomial of Knots

Year: 2026

Type: article

Abstract: Recently, Kashaev and the first author constructed an R-matrix from a Nichols algebra with an automorphism, that leads, via the Reshetikhin–Turaev functor, to a multivariable polynomial invariant of knots. Applying this to a rank 2 Nichols algebra, results in a sequence Vn of 2-variable knot polynomials with integer coefficients, the first polynomial been identified with the Links–Gould polynomial. In this note we present the results of the computation of the Vn-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 352.2 million knots with at most 19 crossings, as well as unexpected Conway mutations that seem undetected by the Vn-polynomials as well as by Heegaard Floer Homology and Khovanov Homology.

Source: Experimental Mathematics

DOI: 10.1080/10586458.2026.2651081

Authors: Stavros Garoufalidis, Shana Yunsheng Li

Institutions: Southern University of Science and Technology, University of Illinois Urbana-Champaign, University of Illinois System

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: National Science Foundation

Award: Interactions between geometry, topology, number theory, and dynamics