Multi-crossing number for knots and the Kauffman bracket polynomial

Abstract

AbstractA multi-crossing (or n-crossing) is a singular point in a projection of a knot or link at which n strands cross so that each strand bisects the crossing. We generalise the classic result of Kauffman, Murasugi and Thistlethwaite relating the span of the bracket polynomial to the double-crossing number of a link, span〈K〉 ⩽ 4c2, to the n-crossing number. We find the following lower bound on the n-crossing number in terms of the span of the bracket polynomial for any n ⩾ 3: $$\text{span} \langle K \rangle \leq \left(\left\lfloor\frac{n^2}{2}\right\rfloor + 4n - 8\right) c_n(K).$$ We also explore n-crossing additivity under composition, and find that for n ⩾ 4 there are examples of knots K1 and K2 such that cn(K1#K2) = cn(K1) + cn(K2) − 1. Further, we present the the first extensive list of calculations of n-crossing numbers of knots. Finally, we explore the monotonicity of the sequence of n-crossings of a knot, which we call the crossing spectrum.