Algebra of Families of Alternating Knots and Links

TL;DR

This paper develops an algebraic framework for families of alternating knots and links using Conway's tangles, representing rational knots with continued fractions and matrices, and identifies 65 prime families with unique behaviors.

Contribution

It introduces an algebraic structure for alternating knots and links, generalizing from rational knots, and catalogs 65 prime families with novel properties.

Findings

65 families of prime alternating knots identified

11 families with six conways exhibit unique behaviors

Representation of rational knots via continued fractions and matrices

Abstract

Families of alternating knots (links) and tangles are studied using as building block the conway defined as the twisting of two strands. The regular representation of knots assumes the projection has the minimal number of overpassings, and the minimal number of conways. The continued fraction associated to rational knots is represented by gaussian brackets and products of 2-dimensional matrices. This gives birth to an algebra of rational knots and tangles which is easily generalized to alternating knots. A collection of 65 families of prime alternating knots with one to six conways is found. Eleven families with six conways show peculiar behavior not present in families with a lower or equal number of conways.