A virtual link diagram is even if the virtual crossings divide each component into an even number of arcs. The set of even virtual link diagrams is closed under classical and virtual Reidemeister moves, and it contains the set of classical link diagrams. For an even virtual link diagram, we define a certain linking invariant which is similar to the linking number. In contrast to the usual linking number, our linking invariant is not preserved under the forbidden moves. In particular, for two fused isotopic even virtual link diagrams, the difference between the linking invariants of them gives a lower bound of the minimal number of forbidden moves needed to deform one into the other. Moreover, we give an example which shows that the lower bound is best possible.

Two string links are equivalent up to 2n-moves and link-homotopy if and only if their all Milnor link-homotopy invariants are congruent modulo n. Moreover, the set of the equivalence classes forms a finite group generated by elements of order n. The classification induces that if two string links are equivalent up to 2n-moves for every n > 0, then they are link-homotopic.

We give a classification of n-component links up to C n -move. In order to prove this classification, we characterize Brunnian links, and have that a Brunnian link is ambient isotopic to a band sum of a trivial link and Milnor's links.

We introduce the multiplexing of a crossing, replacing a classical crossing of a virtual link diagram with multiple crossings which is a mixture of classical and virtual. For integers m i (i = 1, . . . , n) and an ordered ncomponent virtual link diagram D, a new virtual link diagram D(m 1 , . . . , mn) is obtained from D by the multiplexing of all crossings. For welded isotopic virtual link diagrams D and D ′ , D(m 1 , . . . , mn) and D ′ (m 1 , . . . , mn) are welded isotopic. From the point of view of classical link theory, it seems very interesting that D(m 1 , . . . , mn) could not be welded isotopic to a classical link diagram even if D is a classical one, and new classical link invariants are expected from known welded link invariants via the multiplexing of crossings.2010 Mathematics Subject Classification. 57M25, 57M27.

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