On Descriptional Complexity of the Planarity Problem for Gauss Words

TL;DR

This paper explores the complexity of determining planarity in knot diagrams represented by Gauss words, using automata models over infinite alphabets to establish upper bounds on recognition problems.

Contribution

It introduces upper bounds for the planarity problem of Gauss words in the context of automata over infinite alphabets, advancing understanding of knot diagram recognition complexity.

Findings

Upper bounds for planarity recognition of Gauss words

Application of automata over infinite alphabets to knot problems

Analysis of descriptional complexity in knot theory

Abstract

In this paper we investigate the descriptional complexity of knot theoretic problems and show upper bounds for planarity problem of signed and unsigned knot diagrams represented by Gauss words. Since a topological equivalence of knots can involve knot diagrams with arbitrarily many crossings then Gauss words will be considered as strings over an infinite (unbounded) alphabet. For establishing the upper bounds on recognition of knot properties, we study these problems in a context of automata models over an infinite alphabet.