Biquandle cocycle condition for invariants of immersed surface-links in the four-space

TL;DR

This paper develops a biquandle cocycle framework to define and prove invariance of new surface-link invariants in four-space, extending previous embedded-case invariants to immersed surfaces with singularities.

Contribution

It introduces singular biquandle 3-cocycles and a state-sum invariant for immersed surface-links, proving invariance under all Roseman move types, including the singular move.

Findings

Established minimal generating set of Roseman moves for immersed surfaces.

Defined a bijection of coloring sets under these moves, leading to a coloring number invariant.

Computed a non-trivial example demonstrating the invariant's effectiveness.

Abstract

We consider a biquandle-cohomological framework for invariants of oriented immersed surface-links in the four-space. After reviewing projections and Roseman moves for immersed surfaces, we prove that the move types (a,b,c,e,f,g,h) form a minimal generating set, showing in particular that the singular move (h) is independent of the embedded-case set (a,b,c,e,f,g). We extend biquandle colorings to broken surface diagrams with singular points and establish that coloring sets are in bijection for diagrams related by these moves, yielding a coloring number invariant for immersed surface-links. We introduce singular biquandle 3-cocycles: biquandle 3-cocycles satisfying an additional antisymmetry when the singular relations hold. Using such cocycles, we define a triple-point state-sum with Boltzmann weights and prove its invariance under all generating moves, including (h), thereby obtaining a state-sum invariant for immersed surface-links. The theory is illustrated on the Fenn-Rolfsen link example, where a computation yields non-trivial value, demonstrating nontriviality of the invariant in the immersed setting. These results unify and extend biquandle cocycle invariants from embedded to immersed surface-links.