Multi-crossing Braids

Abstract

Traditionally, knot theorists have considered projections of knots where there are two strands meeting at every crossing. A multi-crossing is a crossing where more than two strands meet at a single point, such that each strand bisects the crossing. In this paper we generalize ideas in traditional braid theory to multi-crossing braids. Our main result is an extension of Alexander's Theorem. We prove that every link can be put into an $n$-crossing braid form for any even $n$, and that every link with two or more components can be put into an $n$-crossing braid form for any $n$. We find relationships between the $n$-crossing braid indices, or the number of strings necessary to represent a link in an $n$-crossing braid.