Minimal generating sets of moves for surfaces immersed in the four-space

TL;DR

This paper identifies a minimal set of spatial and planar moves that generate all isotopies of immersed surfaces in four-space, simplifying the understanding of their diagrammatic transformations.

Contribution

It derives a minimal generating set of moves for immersed surfaces in four-space and relates these to Kirby calculus and invariants.

Findings

Established a minimal generating set of spatial moves.

Connected the moves to Kirby calculus without handle slides.

Discussed invariants like the fundamental group and quandle colorings.

Abstract

For immersed surfaces in the four-space, we have a generating set of the Swenton--Hughes--Kim--Miller spatial moves that relate singular banded diagrams of ambient isotopic immersions of those surfaces. We also have Yoshikawa--Kamada--Kawauchi--Kim--Lee planar moves that relate marked graph diagrams of ambient isotopic immersions of those surfaces. One can ask if the former moves form a minimal set and if the latter moves form a generating set. In this paper, we derive a minimal generating set of spatial moves for diagrams of surfaces immersed in the four-space, which translates into a generating set of planar moves. We also show that the complements of two equivalent immersed surfaces can be transformed one another by a Kirby calculus not requiring the 1-1-handle or 2-1-handle slides. We also discuss the fundamental group of the immersed surface-link complement in the four-space and a quandle coloring invariant of an oriented immersed surface-link.