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

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.

34 pages, 16 figures