Unknot Recognition Through Quantifier Elimination

Abstract

Unknot recognition is one of the fundamental questions in low dimensional topology. In this work, we show that this problem can be encoded as a validity problem in the existential fragment of the first-order theory of real closed fields. This encoding is derived using a well-known result on SU(2) representations of knot groups by Kronheimer-Mrowka. We further show that applying existential quantifier elimination to the encoding enables an UnKnot Recogntion algorithm with a complexity of the order $2^{\mathcal{O}(n)}$, where $n$ is the number of crossings in the given knot diagram. Our algorithm is simple to describe and has the same runtime as the currently best known unknot recognition algorithms.