A refined 1-cocycle for regular isotopies and the refined tangle equations

TL;DR

This paper introduces a refined combinatorial 1-cocycle for regular isotopies of long knots, leading to Laurent polynomial coefficients in tangle equations that help distinguish knot diagrams and analyze knot isotopies.

Contribution

It develops a refined 1-cocycle with values in a Laurent polynomial module, enabling more detailed tangle equations and knot distinction methods.

Findings

Refined 1-cocycle with Laurent polynomial values for regular isotopies.

Tangle equations with Laurent polynomial coefficients provide quantitative knot information.

Solutions to tangle equations indicate isotopy relations between knot diagrams.

Abstract

We refine the combinatorial 1-cocycle $\mathbb{L}R_{reg}$ for regular isotopies of long knots to a 1-cocycle with values in the free $\mathbb{Z}[x,x^{-1}]$-module generated by regular isotopy classes of oriented tangles with exactly one signed ordinary double point. We use it to define the refined tangle equations for couples of knot diagrams, where the coefficients are now Laurent polynomials instead of integers. A solution of the tangle equations gives quantitative information about any knot isotopy which relates two given knot diagrams. If the tangle equations have no solution, then the diagrams represent different knots.