Ribbon-moves of 2-links preserve the μ-invariant of 2-links

Abstract

We introduce ribbon-moves of 2-knots, which are operations to make 2-knots into new 2-knots by local operations in B^4. (We do not assume the new knots is not equivalent to the old ones.) Let L_1 and L_2 be 2-links. Then the following hold. (1) If L_1 is ribbon-move equivalent to L_2, then we have μ(L_1)=μ(L_2). (2) Suppose that L_1 is ribbon-move equivalent to L_2. Let W_i be arbitrary Seifert hypersurfaces for L_i. Then the torsion part of H_1(W_1)+H_1(W_2) is congruent to G+G for a finite abelian group G. (3) Not all 2-knots are ribbon-move equivalent to the trivial 2-knot. (4) The inverse of (1) is not true. (5) The inverse of (2) is not true. Let L=(L_1,L_2) be a sublink of homology boundary link. Then we have: (i) L is ribbon-move equivalent to a boundary link. (ii) μ(L)= μ(L_1) + μ(L_2). We would point out the following facts by analogy of the discussions of finite type invariants of 1-knots although they are very easy observations. By the above result (1), we have: the μ-invariant of 2-links is an order zero finite type invariant associated with ribbon-moves and there is a 2-knot whose μ-invariant is not zero. The mod 2 alinking number of (S^2, T^2)-links is an order one finite type invariant associated with the ribbon-moves and there is an (S^2, T^2)-link whose mod 2 alinking number is not zero.