On Descriptional Complexity of the Planarity Problem for Gauss Words

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.