Triangulating Spun 2-Knot Complements

Abstract

A $2$-knot is an embedding of a $2$-sphere into the $4$-sphere. Similar to the case of embedding circles into the $3$-sphere, this allows the $2$-sphere to be knotted. In this short paper, we present an algorithm to generate triangulations of the exteriors of $2$-knots obtained by spinning $1$-knots. We give an implementation of the algorithm in \emph{Regina} and present triangulations of exteriors obtained from all $1$-knots with up to eight crossings.