Triple Crossing Number and Double Crossing Braid Index

TL;DR

This paper explores the relationship between triple crossing number and double crossing braid index in knot theory, establishing bounds and identifying specific knots where these invariants are equal.

Contribution

It introduces a relationship between triple crossing number and double crossing braid index, and identifies an infinite family of knots where equality holds.

Findings

Established that β₂(L) ≤ c₃(L) + 1 for knots.

Identified an infinite family of knots achieving equality.

Determined both invariants for these knots.

Abstract

Traditionally, knot theorists have considered projections of knots where there are two strands meeting at every crossing. A triple crossing is a crossing where three strands meet at a single point, such that each strand bisects the crossing. In this paper we find a relationship between the triple crossing number and the double crossing braid index, namely $\beta_2(L) \le c_3(L) + 1$. We find an infinite family of knots that achieve equality, which allows us to determine both the double crossing braid index and the triple crossing number of these knots.