Cubulated moves for 2-knots

TL;DR

This paper establishes that two 2-dimensional cubical links in four-dimensional space are isotopic if and only if they can be transformed into each other through a finite sequence of specific cubulated moves, analogous to classical knot moves.

Contribution

It introduces a set of cubulated moves for 2-knots and proves their sufficiency and necessity for isotopy, extending classical knot move concepts to higher dimensions.

Findings

Cubulated moves characterize isotopy of 2-knots in 4D.

The moves are analogous to Reidemeister and Roseman moves.

The main theorem provides a complete move-based classification.

Abstract

In this paper, we prove that given two cubical links of dimension two in ${\mathbb R}^4$, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister and Roseman moves for classical tame knots of dimension one and two, respectively.