Parity, free knots, groups, and invariants of finite type

In this paper, on the basis of the notion of parity introduced recently by the author, for each positive integer m we construct invariants of long virtual knots with values in some simply defined group Gm; conjugacy classes of this group play a role as invariants of compact virtual knots. By construction, each of the invariants is unaltered by the move of virtualization. Factorization of the group algebra of the corresponding group leads to invariants of finite order of (long) virtual knots that do not change under virtualization. The central notion used in the construction of the invariants is parity: the crossings of diagrams of free knots is endowed with an additional structure — each crossing is declared to be even or odd, where even crossings behave regularly under Reidemeister moves. © 2012 American Mathematical Society.