Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Patterns of the $V_2$-polynomial of knots

Stavros Garoufalidis, Shana Yunsheng Li

TL;DR

This work develops and studies the $V_n(t,q)$ invariants arising from rank-2 Nichols algebras, with $V_1=\mathrm{LG}$ and a genus bound $\deg_t V_{K,n}\le 4g(K)$. Through expansive computation up to 18 crossings, $V_2$ is shown to detect genus for all knots in the dataset ($58{,}021{,}794$ knots) and to behave predictively for the Kinoshita–Terasaka family; it also reveals numerous Conway mutant pairs that are $V_2$-equivalent, sometimes coinciding with other invariants. The study demonstrates that $V_2$ is not governed by the 2-loop Kontsevich invariant nor by any single homology theory, but exhibits concrete relations to $V_1$ via explicit linear formulas. Finally, a scalable tensor-network computation framework is developed, enabling efficient evaluation of $V_n$ and providing data and code to the community, thereby enabling broader verification and exploration of genus-detection and mutation phenomena in knot theory.

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 $V_2$ 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.

Patterns of the $V_2$-polynomial of knots

TL;DR

This work develops and studies the invariants arising from rank-2 Nichols algebras, with and a genus bound . Through expansive computation up to 18 crossings, is shown to detect genus for all knots in the dataset ( knots) and to behave predictively for the Kinoshita–Terasaka family; it also reveals numerous Conway mutant pairs that are -equivalent, sometimes coinciding with other invariants. The study demonstrates that is not governed by the 2-loop Kontsevich invariant nor by any single homology theory, but exhibits concrete relations to via explicit linear formulas. Finally, a scalable tensor-network computation framework is developed, enabling efficient evaluation of and providing data and code to the community, thereby enabling broader verification and exploration of genus-detection and mutation phenomena in knot theory.

Abstract

Recently, Kashaev and the first author defined a sequence of 2-variable knot polynomials with integer coefficients, coming from the -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 -polynomials for . This leads to the discovery of emerging patterns, including the genus bound for being an equality for all 58 million knots with at most crossings, as well as unexpected Conway mutations that seem undetected by the -polynomials as well as by Heegaard Floer Homology and Khovanov Homology.
Paper Structure (19 sections, 8 theorems, 57 equations, 4 figures, 6 tables)

This paper contains 19 sections, 8 theorems, 57 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. A sequence of 2-variable knot polynomials
  3. Does $V_2$ detect the genus of a knot?
  4. $V_2$ detects the genus for the Kinoshita--Terasaka family
  5. $V_2$-trivial Conway mutations
  6. Independence of the $V_2$ and the 2-loop polynomials
  7. A relation between $V_1$ and $V_2$
  8. $V_n$ detects the genus of torus knots with two strings
  9. Positivity of the $V_1$ and $V_2$-polynomials of alternating knots?
  10. Computing the $V_n$-polynomials
  11. Overview
  12. Terminology
  13. Long knots and rotation numbers
  14. Tensor contraction
  15. Process of computation
  16. ...and 4 more sections

Key Result

Proposition 1.1

$V_2$ detects the genus of all 58,021,794 knots with at most $18$ crossings.

Figures (4)

  • Figure 1: The 3 pairs of knots from \ref{['pairs12a']}.
  • Figure 2: The long knot diagram corresponding to the $4_1$ knot, with nonzero rotation numbers labeled.
  • Figure 3: The tensor network resulted from an oriented long knot diagram of the $4_1$ knot.
  • Figure 4: A bigon in a knot diagram.

Theorems & Definitions (16)

  • Proposition 1.1
  • Remark 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Remark 1.8
  • Proposition 1.10
  • Proposition 1.11
  • ...and 6 more