RIBBON-MOVES OF 2-KNOTS: THE TORSION LINKING PAIRING AND THE $\tilde{\eta}$-INVARIANTS OF 2-KNOTS

Abstract

We discuss the ribbon-move for 2-knots, which is a local move. Let K and K' be 2-knots. Then we have the following: Suppose that K and K' are ribbon-move equivalent. (1) Let [Formula: see text] (respectively, [Formula: see text]) be the ℤ-torsion submodule of the Alexander module [Formula: see text] (respectively, [Formula: see text]). Then [Formula: see text] is isomorphic to [Formula: see text] not only as ℤ-modules but also as ℤ[t, t-1]-modules. (2) The Farber–Levine pairing for K is equivalent to that for K'. (3) The set of the values of the ℚ/ℤ-valued [Formula: see text] invariants for K is equivalent to that for K'.