Enhanced Kauffman bracket

TL;DR

This paper generalizes an existing Kauffman bracket invariant to a polynomial ring and introduces a new invariant that provides bounds on crossings in tricolorable link diagrams.

Contribution

It extends the invariant $\Phi^{eta}_X$ to a polynomial ring and introduces a crossing bound invariant for tricolorable links.

Findings

Invariant generalized to polynomial ring

Provides lower bounds on crossings in tricolorable diagrams

Enhances understanding of link invariants

Abstract

S. Nelson, M. Orrison, V. Rivera {\cite{S}} modified Kauffman's construction of bracket. Their invariant $\Phi^{\beta}_X$ takes value in a finite ring $Z_2[t]/(1+t+t^3)$. In this paper, the author generalizes this invariant. The new invariant takes value in a polynomial ring. Furthermore, for a tricolorable link diagram, the author gives a bracket invariant which gives lower bound on number of crossings with different (same) colors.