Multi-Crossing Number for Knots and the Kauffman Bracket Polynomial

A 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〉 ⩽ 4c 2, 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. We also explore n-crossing additivity under composition, and find that for n ⩾ 4 there are examples of knots K 1and K 2 such that cn (K 1#K 2) = cn (K 1) + cn (K 2) − 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.

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