Quantum Invariants of Links and 3-Manifolds with Boundary defined via Virtual Links

Abstract: We introduce new topological quantum invariants of compact oriented 3-manifolds with boundary where the boundary is a disjoint union of two identical surfaces. The invariants are constructed via surgery on manifolds of the form $F \times I$ where $I$ denotes the unit interval. Since virtual knots and links are represented as links in such thickened surfaces, we are able also to construct invariants in terms of virtual link diagrams (planar diagrams with virtual crossings). These invariants are the first meaningful, nontrivial, and calculable examples of quantum invariants of 3-manifolds with non-vacuous boundary. We give a new invariant of classical links in the 3-sphere in the following sense: Consider a link $L$ in $S^3$ of two components. The complement of a tubular neighborhood of $L$ is a manifold whose boundary consists in two copies of a torus. Our invariants apply to this case of bounded manifold and give new invariants of the given link of two components. Invariants of knots are also obtained.