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

Michal Jablonowski

Published 2022-08-17Version 1

For immersed surfaces in the four-space, we have a generating set of the Swenton-Hughes-Kim-Miller spatial moves that relate banded singular 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. This gives a potential room for a stronger immersed surface invariant than the diffeomorphism type of its complement. We also discuss the fundamental group of the immersed surface-link complement in the four-space.