Multi-crossing Number for Knots and the Kauffman Bracket Polynomial

TL;DR

This paper extends classical bounds on the bracket polynomial to n-crossing numbers, investigates their properties under knot composition, and provides extensive calculations and analysis of the crossing spectrum.

Contribution

It generalizes the span bound for the bracket polynomial to n-crossings, explores n-crossing additivity, and presents the first comprehensive list of n-crossing numbers for knots.

Findings

Upper bound on span of bracket polynomial for n-crossings

Existence of sub-additive n-crossing numbers for n ≥ 4

Extensive calculations of n-crossing numbers for various knots

Abstract

A multi-crossing (or n-crossing) is a singular point in a projection at which n strands cross so that each strand bisects the crossing. We generalize the classic result of Kauffman, Murasugi, and Thistlethwaite, which gives the upper bound on the span of the bracket polynomial of K as 4c_2(K), to the n-crossing number: span<K> is bounded above by ([n^2/2] + 4n-8) c_n(K) for all integers n at least 3. We also explore n-crossing additivity under composition, and find that for n at least 4, there are examples of knots such that the n-crossing number is sub-additive. Further, we present the first extensive list of calculations of n-crossing numbers for knots. Finally, we explore the monotonicity of the sequence of n-crossings of a knot, which we call the crossing spectrum.