Adequacy of Link Families

TL;DR

This paper develops criteria for assessing the adequacy of link families in knot theory, introduces a new invariant called the adequacy polynomial, and demonstrates its effectiveness in distinguishing alternating link families up to 12 crossings.

Contribution

It introduces the adequacy polynomial as a new invariant for alternating links and establishes criteria for adequacy using computer calculations and pretzel tangle representatives.

Findings

Adequacy polynomial distinguishes all alternating link families up to 12 crossings.

Established general adequacy criteria for different classes of knots and links.

Defined the adequacy number based on adequate graphs from Kauffman states.

Abstract

Using computer calculations and working with representatives of pretzel tangles we established general adequacy criteria for different classes of knots and links. Based on adequate graphs obtained from all Kauffman states of an alternating link we defined a new numerical invariant: adequacy number, and computed adequacy polynomial which is the invariant of alternating link families. Adequacy polynomial distinguishes (up to mutation) all families of alternating knots and links whose generating link has at most $n=12$ crossings.