Boltzmann Enhancements of Biquasile Counting Invariants

TL;DR

This paper introduces Boltzmann weights for biquasile invariants, enhancing knot coloring invariants and distinguishing links that previous invariants could not differentiate.

Contribution

It presents biquasile Boltzmann weights that improve the biquasile counting invariant and establishes conditions for their linear functions.

Findings

Demonstrated that the enhancement distinguishes links with identical counting invariants.

Provided conditions for linear functions to serve as Boltzmann weights for Alexander biquasiles.

Extended the framework of biquasile invariants with new enhancement techniques.

Abstract

In this paper, we build on the biquasiles and dual graph diagrams introduced in arXiv:1610.06969. We introduce \textit{biquasile Boltzmann weights} that enhance the previous knot coloring invariant defined in terms of finite biquasiles and provide examples differentiating links with the same counting invariant, demonstrating that the enhancement is proper. We identify conditions for a linear function $\phi:\mathbb{Z}_n[X^3]\to\mathbb{Z}_n$ to be a Boltzmann weight for an Alexander biquasile $X$.