A state-sum invariant for immersed surface-links is defined using singular biquandle 3-cocycles and proven invariant under all generating moves including the singular move (h).
Carrell, The surface biquandle, Pomona College, 2009
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Biquandle cocycle condition for invariants of immersed surface-links in the four-space
A state-sum invariant for immersed surface-links is defined using singular biquandle 3-cocycles and proven invariant under all generating moves including the singular move (h).