Triple Crossing Number of Knots and Links

TL;DR

This paper introduces the concept of triple crossing projections for knots and links, proves their universal existence, and explores their properties, bounds, and relationships with other knot invariants and hyperbolic volume.

Contribution

It establishes the existence of triple crossing projections for all knots and links, and relates triple crossing number to traditional crossing number, polynomial span, and hyperbolic volume.

Findings

Every knot and link admits a triple crossing projection.

Bounds on triple crossing number are tight and achievable.

Triple crossing number relates to the span of the bracket polynomial and hyperbolic volume.

Abstract

A triple crossing is a crossing in a projection of a knot or link that has three strands of the knot passing straight through it. A triple crossing projection is a projection such that all of the crossings are triple crossings. We prove that every knot and link has a triple crossing projection and then investigate c_3(K), which is the minimum number of triple crossings in a projection of K. We obtain upper and lower bounds on c_3(K) in terms of the traditional crossing number and show that both are realized. We also relate triple crossing number to the span of the bracket polynomial and use this to determine c_3(K) for a variety of knots and links. We then use c_3(K) to obtain bounds on the volume of a hyperbolic knot or link. We also consider extensions to c_n(K).