Free Knots, Groups, and Finite-Type Invariants
arXiv:1004.4325
Abstract
Based on a recently introduced by the author notion of {\em parity}, in the present paper we construct a sequence of invariants (indexed by natural numbers ) of long virtual knots, valued in certain simply-defined group (the Cayley graphs of these groups are represented by grids in the -space); the conjugacy classes of elements of play the role of invariants of {\em compact} virtual knots. By construction, all invariants do not change under {\em virtualization}. Factoring the group algebra of the corresponding group by certain polynomial relations leads to finite order invariants of (long) knots which do not change under {\em virtualization
12 pages