2018/02/27 by Meesum, Syed Mohammed, Prathamesh, T. V. H
#03C10 #FOS: Mathematics #Geometric Topology (math.GT) #Primary 57M25 #Secondary 11Y16
paper · doi:10.48550/arxiv.1803.00413
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 2O(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.