Flag algebras and the stable coefficients of the Jones polynomial

TL;DR

This paper explores the stable coefficients of the Jones polynomial for alternating links, linking them to graph invariants, and introduces a polynomial algebra that refines these coefficients, supported by experimental evidence.

Contribution

It introduces a polynomial algebra of graph invariants that refines the first four stable coefficients of the Jones polynomial for alternating links.

Findings

First four stable coefficients correspond to graph invariants.

Proposed algebra potentially contains all stable coefficients.

Experimental evidence supports the conjecture for coefficients five and six.

Abstract

We study the structure of the stable coefficients of the Jones polynomial of an alternating link. We start by identifying the first four stable coefficients with polynomial invariants of a (reduced) Tait graph of the link projection. This leads us to introduce a free polynomial algebra of invariants of graphs whose elements give invariants of alternating links which strictly refine the first four stable coefficients. We conjecture that all stable coefficients are elements of this algebra, and give experimental evidence for the fifth and sixth stable coefficient. We illustrate our results in tables of all alternating links with at most 10 crossings and all irreducible planar graphs with at most 6 vertices.