The warping degree of a link diagram

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 $d(D) + d(-D) + \mathit{sr}(D)$ is less than or equal to the crossing number of $D$, where $-D$ denotes the inverse of $D$ and $\mathit{sr}(D)$ 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 $d(D) + d(-D) + \mathit{sr}(D)$ 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.