Triangulations of the 3-sphere with knotted edge

TL;DR

This paper demonstrates that any knot can be embedded as an edge in a one-vertex triangulation of the 3-sphere, providing a constructive method based on augmented links and analyzing minimal triangulations.

Contribution

It introduces a constructive approach to embed any knot as an edge in a 3-sphere triangulation and identifies minimal triangulations up to a constant factor.

Findings

Existence of one-vertex triangulations containing any knot as an edge

Construction method based on fully augmented links

Identification of smallest possible triangulations up to a constant factor

Abstract

We prove that for any knot $K$, there exists a one-vertex triangulation of the $3$-sphere containing an edge forming $K$. The proof is constructive, and based on fully augmented links. We use our method to produce ``complicated'' simplicial triangulations of the $3$-sphere that we show are smallest possible, up to a constant multiplicative factor.