Remarks on the invariants valued in the generalization of Conway algebra

TL;DR

This paper explores a generalized Conway algebra invariant, demonstrating it can distinguish links that the Homflypt polynomial cannot, and investigates its properties related to Vassiliev invariants.

Contribution

It introduces an example showing the generalized Conway type invariant can differentiate links with identical Homflypt polynomial values, advancing understanding of link invariants.

Findings

Generalized Conway type invariant can distinguish certain links

It relates to properties of Vassiliev invariants

Provides directions for future research

Abstract

In~\cite{Kim} the author generalized the Conway algebra and constructed the invariant valued in the generalized Conway algebra defined by applying two skein relations to crossings, which is called a generalized Conway type invariant. The generalized Conway type invariant is a generalization of Homflypt polynomial. In this paper we show that an example of links, which have the same value of Homflypt polynomial, but have different values of the generalized Conway type invariant. We study a properties of Conway type invariant related to Vassiliev invariant. In section 3 we discuss about further researches.