On biquandle-based invariant of immersed surface-links, Yoshikawa oriented fifth move, and ribbon 2-knots

TL;DR

This paper proves the necessity of a specific move in surface-link theory, introduces a biquandle coloring invariant for immersed surfaces, and demonstrates the existence of ribbon 2-knots with identical groups but distinct quandles.

Contribution

It establishes the independence of the fifth Yoshikawa move, develops a biquandle-based invariant for immersed surface-links, and shows the existence of ribbon 2-knots with isomorphic groups but different quandles.

Findings

The fifth Yoshikawa move cannot be generated by other moves.

A new biquandle coloring invariant for immersed surface-links is introduced.

Infinitely many ribbon 2-knots have the same group but different quandles.

Abstract

We resolve an open problem by 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, highlighting its necessity in generating the full move set. 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 biquandle-based coloring theory. By assigning elements of a biquandle to diagram arcs according to local rules, we obtain a counting invariant of immersed surfaces up to isotopy. Third, we show that there are infinitely many pairs of ribbon $2$-knots with isomorphic groups but different knot quandles.