A method of encoding generalized link diagrams

TL;DR

This paper introduces a novel encoding method for various link diagrams, enabling efficient representation and analysis of their equivalence classes, with applications to classical and virtual knots.

Contribution

The paper presents a new encoding technique for diverse link diagrams, including virtual and welded types, with a cubic-length notation based on riser marks and moves for diagram equivalence.

Findings

Encoding length is cubic in the number of riser marks.

Classical knots with minimal bridge form can be encoded in space proportional to the square of the bridge index.

The method provides a new proof of the non-classicality of the Kishino virtual knot.

Abstract

We describe a method of encoding various types of link diagrams, including those with classical, flat, rigid, welded, and virtual crossings. We show that this method may be used to encode link diagrams, up to equivalence, in a notation whose length is a cubic function of the number of 'riser marks'. For classical knots, the minimal number of such marks is twice the bridge index, and a classical knot diagram in minimal bridge form with bridge index $b$ may be encoded in space $\mathcal{O}(b^2)$. A set of moves on the notation is defined. As a demonstration of the utility of the notation we give another proof that the Kishino virtual knot is non-classical.