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

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 a non-trivial integer value, demonstrating the nontriviality of the invariant in the immersed setting.

These results unify and extend biquandle cocycle invariants from embedded to immersed surface-links.