Oriented Local Moves and Divisibility of the Jones Polynomial

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For any virtual link L = S ∪ T that may be decomposed into a pair of oriented n-tangles S and T, an oriented local move of type T 7→ T ′ is a replacement of T with the n-tangle T ′ in a way that preserves the orientation of L. After developing a general decomposition for the Jones polynomial of the virtual link L = S ∪ T in terms of various (modified) closures of T, we analyze the Jones polynomials of virtual links L1, L2 that differ via a local move of type T 7→ T ′ . Succinct divisibility conditions on V (L1) − V (L2) are derived for broad classes of local moves that include the ∆-move and the double-∆-move as special cases. As a consequence of our divisibility result for the double-∆-move, we introduce a necessary condition for any pair of classical knots to be S-equivalent.