The rank of a warping matrix

TL;DR

This paper establishes a precise relationship between the rank of the warping matrix and the crossing number of a knot diagram, providing insights into knot diagram invariants and their linear independence.

Contribution

It proves that the rank of the warping matrix is exactly one greater than the crossing number and explores linear independence of knot diagrams via the warping incidence matrix.

Findings

Rank of warping matrix equals crossing number plus one

Warping matrix uniquely represents knot diagrams

Analysis of linear independence of knot diagrams

Abstract

The warping matrix has been defined for knot projections and knot diagrams by using warping degrees. In particular, the warping matrix of a knot diagram represents the knot diagram uniquely. In this paper we show that the rank of the warping matrix is one greater than the crossing number. We also discuss the linearly independence of knot diagrams by considering the warping incidence matrix.