A Graph-Theoretic Approach to a Partial Order of Knots and Links

TL;DR

This paper introduces a graph-theoretic framework to establish a partial order among prime alternating knots and links based on crossing modifications and smoothings, and determines this order for all such links with up to six crossings.

Contribution

It defines a new partial order relation among prime alternating links and characterizes this order for links with up to six crossings using graph-theoretic methods.

Findings

Established the partial order for all prime alternating links with ≤6 crossings.

Provided graph-theoretic proofs for the partial order relations.

Mapped the structure of prime alternating links within this partial order.

Abstract

We say that a link $L_1$ is an s-major of a link $L_2$ if any diagram of $L_1$ can be transformed into a diagram of $L_2$ by changing some crossings and smoothing some crossings. This relation is a partial ordering on the set of all prime alternating links. We determine this partial order for all prime alternating knots and links with the crossing number less than or equal to six. The proofs are given by graph-theoretic methods.