Braid computations for the crossing number of Klein links

Abstract

Klein links are a nonorientable counterpart to torus knots and links. It is shown that braids representing a subset of Klein links take on the form of a very positive braid after manipulation. Once the braid has reached this form, its number of crossings is the crossing number of the link it represents. Two formulas are proven to calculate the crossing number of $K\left(m,n\right)$ Klein links, where $m\ge n\ge 1$. In combination with previous results, these formulas can be used to calculate the crossing number for any Klein link with given values of $m$ and $n$.