Multi-crossing Number for Knots and the Kauffman Bracket Polynomial

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 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.