Minimal area of the spun trefoil knot on the canonical cubulation of $\mathbb{R}^4$

TL;DR

This paper investigates the minimal cubical area required to realize the spun trefoil knot within the canonical cubulation of four-dimensional space, establishing an invariant measure for this knot type.

Contribution

It introduces the concept of minimal cubical area for cubical 2-knots and provides an explicit analysis for the spun trefoil knot case.

Findings

Defines minimal cubical area as a knot invariant.

Calculates minimal area for the spun trefoil knot.

Answers the natural question for this specific knot type.

Abstract

We say that a \emph{cubical 2-knot} $K^{2}$ is an embedding of the 2-sphere in the 2-skeleton of the canonical cubulation of $\mathbb{R}^4$; in particular, $K^{2}$ is the union of $m(K^{2})$ unit squares, hence $m(K^{2})$ is its area. We define the minimal area of $K^{2}$ as the minimum over all the areas of cubical 2-knots isotopic to the given knot type. The minimal area of a cubical 2-knot is an invariant, and the following natural question arose: Given a knot type, what area is needed for a cubical 2-knot in the canonical cubulation of $\mathbb{R}^4$ to realise that type with minimal area? In this paper, we answer this question for the spun trefoil knot in the weakly minimal case.