Multi-crossing number for knots and the Kauffman bracket polynomial

作者：Adams Colin^*; Capovilla Searle Orsola; Freeman Jesse; Irvine Daniel; Petti Samantha; Vitek Daniel; Weber Ashley; Zhang Sicong

来源：Mathematical Proceedings of the Cambridge Philosophical Society, 2018, 164(1): 147-178.

DOI：10.1017/S0305004116000906

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:
span < K > <= (left perpendicular n(2)/2 right perpendicular + 4n - 8 ) c(n)( K).
We also explore n-crossing additivity under composition, and find that for n >= 4 there are examples of knots K-1 and K-2 such that c(n)(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.

• 出版日期2018-1
• 单位MIT