Oriented local moves and divisibility of the Jones–Kauffman polynomial

Abstract

For any virtual link [Formula: see text] that may be decomposed into a pair of oriented [Formula: see text]-tangles [Formula: see text] and [Formula: see text], an oriented local move of type [Formula: see text] is a replacement of [Formula: see text] with the [Formula: see text]-tangle [Formula: see text] in a way that preserves the orientation of [Formula: see text]. After developing a general decomposition for the Jones polynomial of the virtual link [Formula: see text] in terms of various (modified) closures of [Formula: see text], we analyze the Jones polynomials of virtual links [Formula: see text] that differ via a local move of type [Formula: see text]. Succinct divisibility conditions on [Formula: see text] are derived for broad classes of local moves that include the [Formula: see text]-move and the double-[Formula: see text]-move as special cases. As a consequence of our divisibility result for the double-[Formula: see text]-move, we introduce a necessary condition for any pair of classical knots to be [Formula: see text]-equivalent.