New link invariants and Polynomials (II), unoriented case

TL;DR

This paper introduces new unoriented link invariants derived from generalized skein relations, expanding the framework of polynomial invariants through algebraic structures and homomorphisms.

Contribution

It constructs a family of new link invariants satisfying generalized skein relations, unifying and extending known invariants like the Kauffman bracket and Q-polynomial.

Findings

New invariants satisfy a generalized 4-term skein relation

Coefficients are from a commutative ring, enabling algebraic generalizations

Recover known invariants such as the Kauffman bracket and Q-polynomial

Abstract

Given any unoriented link diagram, a group of new knot invariants are constructed. Each of them satisfies a generalized 4 term skein relation. The coefficients of each invariant is from a commutative ring. Homomorphisms and representations of such a ring defines new link invariants. In this sense, they produce the well-known Kauffman bracket, the Kauffman 2-variable polynomial, and the $Q$-polynomial.