Similarity indices and the Miyazawa polynomials of virtual links

Abstract

Y. Miyazawa introduced a two-variable polynomial invariant of virtual knots in 2006 [Magnetic graphs and an invariant for virtual links, J. Knot Theory Ramifications 15 (2006) 1319-1334] and then generalized it to give a multi-variable one via decorated virtual magnetic graph diagrams in 2008. A. Ishii gave a simple state model for the two-variable Miyazawa polynomial by using pole diagrams in 2008 [A multi-variable polynomial invariant for virtual knots and links, J. Knot Theory Ramifications 17 (2008) 1311-1326]. H. A. Dye and L. H. Kauffman constructed an arrow polynomial of a virtual link in 2009 which is equivalent to the multi-variable Miyazawa polynomial [Virtual crossing number and the arrow polynomial, preprint (2008), arXiv: 0810.3858v3, http://front.math.ucdavis.edu.]. We give a bracket model for the multi-variable Miyazawa polynomial via pole diagrams and polar tangles similarly to the Ishii's state model for the two-variable polynomial. By normalizing the bracket polynomial we get the multi-variable Miyazawa polynomial f(K) is an element of Z[A, A(-1), K-1, K-2,...] of a virtual link K. n-similar knots take the same value for any Vassiliev invariant of degree < n. We show that f(K1) = f(K2) mod (A(4) -1)(n) if two virtual links K-1 and K-2 are n-similar. Also we give a necessary condition for a virtual link to be periodic by using n-similarity of virtual tangles and the Miyazawa polynomial.