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 B4. (We do not assume the new knots is not equivalent to the old ones.) Let L1 and L2 be 2-links. Then the following hold. (1) If L1 is ribbon-move equivalent to L2, then we have μ(L1)=μ(L2). (2) Suppose that L1 is ribbon-move equivalent to L2. Let Wi be arbitrary Seifert hypersurfaces for Li. Then the torsion part of H1(W1)+H1(W2) 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=(L1,L2) be a sublink of homology boundary link. Then we have: (i) L is ribbon-move equivalent to a boundary link. (ii) μ(L)= μ(L1) + μ(L2). 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 (S2, T2)-links is an order one finite type invariant associated with the ribbon-moves and there is an (S2, T2)-link whose mod 2 alinking number is not zero.
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