Jones polynomial invariants

Abstract

The Jones polynomial is a well-defined invariant of virtual links. We observe the effect of a generalised mutation M of a link on the Jones polynomial. Using this, we describe a method for obtaining invariants of links which are also invariant under M . The Jones polynomial of welded links is not well-defined in Z[q 1/4 , q −1/4 ]. Taking M = Fo allows us to pass to a quotient of Z[q 1/4 , q −1/4 ] in which the Jones polynomial is well-defined. We get the same result for M = Fu , so in fact, the Jones polynomial in this ring defines a fused isotopy invariant. We show it is non-trivial and compute it for links with one or two components.