Vol. 8, No. 1, 2015

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Braid computations for the crossing number of Klein links

Michael Bush, Danielle Shepherd, Joseph Smith, Sarah Smith-Polderman, Jennifer Bowen and John Ramsay

Vol. 8 (2015), No. 1, 169–179
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 $n$.

Keywords
knots and links in S3, invariants of knots and 3-manifolds
Mathematical Subject Classification 2010
Primary: 57M25, 57M27