On the complexity of meander-like diagrams of knots

2023/12/08 by Yury Belousov, Belousov, Yury
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Advanced Combinatorial Mathematics

paper · pdf · doi:10.48550/arxiv.2312.05014

Abstract

It is known that each knot has a semimeander diagram (i. e. a diagram composed of two smooth simple arcs), however the number of crossings in such a diagram can only be roughly estimated. In the present paper we provide a new estimate of the complexity of the semimeander diagrams. We prove that for each knot K with more than 10 crossings, there exists a semimeander diagram with no more than 0.31 ⋅ 1.558^cr(K) crossings, where cr(K) is the crossing number of K. As a corollary, we provide new estimates of the complexity of other meander-like types of knot diagrams, such as meander diagrams and potholders. We also describe an efficient algorithm for constructing a semimeander diagram from a given one.

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