Lower bounds for the warping degree of a knot projection

TL;DR

This paper establishes lower bounds for the warping degree of knot projections by analyzing the maximum number of regions sharing no crossings, contributing to understanding knot diagram complexities.

Contribution

It introduces new lower bounds for the warping degree based on the number of regions without crossings in a fixed-crossing knot projection.

Findings

Lower bounds depend on the maximum number of crossing-free regions.

Provides a method to estimate minimal crossing sequences in knot diagrams.

Enhances understanding of the relationship between regions and warping degree.

Abstract

The warping degree of an oriented knot diagram is the minimal number of crossings which we meet as an under-crossing first when we travel along the diagram from a fixed point. The warping degree of a knot projection is the minimal value of the warping degree for all oriented alternating diagrams obtained from the knot projection. In this paper, we consider the maximal number of regions which share no crossings for a knot projection with a fixed crossing, and give lower bounds for the warping degree.