Knotted Polyhedral Tori

TL;DR

This paper establishes a relationship between the stick number of knots and the minimal vertices needed for knotted polyhedral tori, providing bounds on the complexity of such structures.

Contribution

It proves that for any knot with stick number k, a knotted polyhedral torus of that knot type can be constructed with 3k vertices, and at least 3k-2 vertices are necessary.

Findings

Constructed knotted tori with 3k vertices for knots with stick number k

Proved lower bound of 3k-2 vertices for such tori

Established a tight relationship between knot complexity and polyhedral torus vertices

Abstract

For every knot K with stick number k there is a knotted polyhedral torus of knot type K with 3k vertices. We prove that at least 3k-2 vertices are necessary.