Multi-Crossing Braids

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, each strand bisecting all the other strands at 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 then find relationships between the n-crossing braid indices.