On biquandle-based invariant of immersed surface-links and on Yoshikawa oriented fifth move

Abstract: We resolve an open problem showing that the Yoshikawa's fifth oriented move in his list cannot be reproduced by any finite sequence of the other nine moves and planar isotopies. Our proof introduces a link-type semi-invariant that remains unchanged under all moves except the fifth; contrasting values on two equivalent diagrams force the move's independence. Second, we extend the algebraic toolkit for immersed surface-links. After revisiting the banded-unlink description of immersed surfaces and the twelve local moves that relate their diagrams, we develop a coloring theory based on biquandles. By assigning elements of a biquandle to diagram arcs according to local rules, we obtain a counting invariant of immersed surfaces up to isotopy.