Multi-crossing Braids

TL;DR

This paper extends classical braid theory to multi-crossing braids, proving that any link can be represented as an n-crossing braid for even n, and exploring the relationships between n-crossing braid indices.

Contribution

It generalizes Alexander's Theorem to multi-crossing braids, establishing that all links can be represented in n-crossing braid form for various n.

Findings

Every link can be represented as an n-crossing braid for any even n.

Links with two or more components can be represented as n-crossing braids for any n.

Relationships between n-crossing braid indices are established.

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.