The warping degree of a knot diagram

TL;DR

This paper introduces the warping degree of a knot diagram, a measure of how many crossing changes are needed to make it monotone, and establishes a key inequality relating it to the crossing number, with equality characterizing alternating diagrams.

Contribution

The paper defines the warping degree for knot diagrams and proves a fundamental inequality linking it to the crossing number, characterizing alternating diagrams.

Findings

d(D) + d(-D) + 1 ≤ crossing number of D

Equality holds if and only if D is an alternating diagram

Provides a new perspective on knot diagram complexity

Abstract

For an oriented knot diagram D, the warping degree d(D) is the smallest number of crossing changes which are needed to obtain the monotone diagram from D in the usual way. We show that d(D) + d(-D) + 1 is less than or equal to the crossing number of D. Moreover the equality holds if and only if D is an alternating diagram.