A TQFT for Wormhole cobordisms over the field of rational functions

We consider a cobordism category whose morphisms are punctured connected sums of ${\displaystyle S^1\times S^2}$'s (wormhole spaces) with embedded admissibly colored banded trivalent graphs. We define a TQFT on this cobordism category over the field of rational functions in an indeterminant A. For r large, we recover, by specializing A to a primitive 4rth root of unity, the Witten-Reshetikhin-Turaev TQFT restricted to links in wormhole spaces. Thus, for r large, the rth Witten-Reshetikhin-Turaev invariant of a link in some wormhole space, properly normalized, is the value of a certain rational function at ${\displaystyle e^{\frac{\pi i}{2r}}}$. We relate our work to Hoste and Przytycki's calculation of the Kauffman bracket skein module of ${\displaystyle S^1\times S^2}$.