Minimal Surface Representations of Virtual Knots and Links

Equivalence classes of virtual knot diagrams are in a one to one correspondence with knot diagrams (decorated immersions of S1) in orientable, closed surfaces modulo stable handle equivalence and Reidemeister moves. Each virtual knot diagram corresponds to a diagram in a unique minimal surface. If a virtual knot diagram is equivalent to a classical knot diagram then this minimal surface is a sphere. We use minimal surfaces and a generalized version of the bracket polynomial for surfaces to determine when a virtual knot diagram is non-trivial and non-classical.