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Classification of links up to self pass-move. (English) Zbl 1046.57009

In the paper under review, three kinds of equivalence relations on oriented links are considered. Namely, the link-homotopy introduced by J. Milnor, the pass-equivalence defined by L. Kauffman and the \(\sharp\)-equivalence defined by H. Murakami. These relations can be easily described using local moves on link diagrams. The aim of the paper is to classify oriented links up to self pass-move (resp. self \(\sharp\)-move).
The main result is a necessary and sufficient condition for two oriented links to be self pass-equivalent (self \(\sharp\)-equivalent). These conditions are given in terms of link-homotopy and the Arf invariant of proper sublinks.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
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