Triangulating Spun 2-Knot Complements

Rhuaidi Antonio Burke; Benjamin Burton; Arunima Ray; Jonathan Spreer; Finn Thompson; Orion Zymaris

Triangulating Spun 2-Knot Complements

Geometric Topology 2026-07-07 v1

Authors: Rhuaidi Antonio Burke , Benjamin Burton , Arunima Ray , Jonathan Spreer , Finn Thompson , Orion Zymaris

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.

Cite

@article{arxiv.2607.05923,
  title  = {Triangulating Spun 2-Knot Complements},
  author = {Rhuaidi Antonio Burke and Benjamin Burton and Arunima Ray and Jonathan Spreer and Finn Thompson and Orion Zymaris},
  journal= {arXiv preprint arXiv:2607.05923},
  year   = {2026}
}

Comments

15 pages, 4 figures, comments welcome