Finite Type Invariants and Virtual Twist Moves of Virtual Knots

Generalizing twist moves of classical knots, we introduce t(a(1), ... ,a(m))-moves of virtual knots for an m-tuple t(a(1), ... ,a(m)) of nonzero integers. In [4], M. Goussarov, M. Polyak and O. Viro introduced fi nite type invariants of virtual knots and Gauss diagram formulae giving combinatorial presentations of finite type invariants. By using the Gauss diagram formulae for the finite type invariants of degree 2, we give a necessary condition for a virtual long knot K to be transformed to a virtual long knot K' by a finite sequence of t(a(1), ... ,a(m))-moves for an m-tuple (a(1), ... ,a(m))of nonzero integers with the same sign.