The warping degree of a link diagram

TL;DR

This paper investigates the warping degree of link diagrams, establishing bounds related to crossing numbers and linking properties, and characterizes conditions for equality, contributing to knot theory understanding.

Contribution

It provides a new inequality involving warping degree, inverse diagram, and self-crossings, along with conditions for equality and relations to classical knot invariants.

Findings

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

Characterization of when equality holds

Relations between warping degree and linking/unknotting/splitting numbers

Abstract

For an oriented link diagram D, the warping degree d(D) is the smallest number of crossing changes which are needed to obtain a monotone diagram from D. We show that d(D)+d(-D)+sr(D) is less than or equal to the crossing number of D, where -D denotes the inverse of D and sr(D) denotes the number of components which have at least one self-crossing. Moreover, we give a necessary and sufficient condition for the equality. We also consider the minimal d(D)+d(-D)+sr(D) for all diagrams D. For the warping degree and linking warping degree, we show some relations to the linking number, unknotting number, and the splitting number.