Movie moves for singular link cobordisms in 4-dimensional space

TL;DR

This paper introduces a comprehensive set of movie moves that fully characterize isotopies between singular link cobordisms in 4-dimensional space, facilitating their classification and understanding.

Contribution

It provides a complete set of movie moves for singular link cobordisms in 4-space, extending the classical theory to include singular links and their cobordisms.

Findings

A finite set of movie moves suffices to connect isotopic singular link cobordisms.

The movie moves generalize classical link isotopy moves to singular links in 4D.

Framework aids in the classification of singular link cobordisms.

Abstract

Two singular links are cobordant if one can be obtained from the other by singular link isotopy together with a combination of births or deaths of simple unknotted curves, and saddle point transformations. A movie description of a singular link cobordism in 4-space is a sequence of singular link diagrams obtained from a projection of the cobordism into 3-space by taking 2-dimensional cross sections perpendicular to a fixed direction. We present a set of movie moves that are sufficient to connect any two movies of isotopic singular link cobordisms.