VIRTUAL KNOT DIAGRAMS AND THE WITTEN–RESHETIKHIN–TURAEV INVARIANT

Abstract

The Witten–Reshetikhin–Turaev invariant of classical link diagrams is generalized to virtual link diagrams. This invariant is unchanged by the framed Reidemeister moves and the Kirby calculus. As a result, it is also an invariant of the 3-manifolds represented by the classical link diagrams. This generalization is used to demonstrate that there are virtual knot diagrams with a non-trivial Witten–Reshetikhin–Turaev invariant and trivial 3-manifold fundamental group.