On the generalization of Conway algebra

TL;DR

This paper generalizes Conway algebras to create a new invariant for oriented links, extending previous polynomial invariants by incorporating non-linear skein relations and two binary operations.

Contribution

It introduces a generalized Conway algebra with two binary operations and constructs a novel link invariant satisfying non-linear skein relations.

Findings

Defines a new algebraic structure with two binary operations.

Constructs a link invariant valued in the generalized algebra.

The invariant satisfies non-linear skein relations.

Abstract

In \cite{PrzytyskiTraczyk} J.H.Przytyski and P.Traczyk introduced an algebraic structure, called {\it a Conway algebra,} and constructed an invariant of oriented links, which is a generalization of the Homflypt polynomial invariant. On the other hand, in \cite{KauffmanLambropoulou} L. H. Kauffman and S. Lambropoulou introduced new 4-variable invariants of oriented links, which are obtained by two computational steps: in the first step we apply a skein relation on every mixed crossing to produce unions of unlinked knots. In the second step, we apply the skein relation on crossings of the unions of unlinked knots, which introduces a new variable. In this paper, we will introduce a generalization of the Conway algebra $\widehat{A}$ with two binary operations and we construct an invariant valued in $\widehat{A}$ by applying those two binary operations to mixed crossings and self crossings respectively. Moreover, the generalized Conway algebra gives us an invariant of oriented links, which satisfies non-linear skein relations.