Odd knot invariants from quantum covering groups

Abstract

We show that the quantum covering group associated to $\mathfrak{o}\mathfrak{s}\mathfrak{p}\left(1|2n\right)$ has an associated colored quantum knot invariant à la Reshetikhin–Turaev, which specializes to a quantum knot invariant for $\mathfrak{o}\mathfrak{s}\mathfrak{p}\left(1|2n\right)$, and to the usual quantum knot invariant for $\mathfrak{s}\mathfrak{o}\left(1+2n\right)$. In particular, this furnishes an “odd” variant of $\mathfrak{s}\mathfrak{o}\left(1+2n\right)$ quantum invariants, even for knots labeled by spin representations. We then show that these knot invariants are essentially the same, up to a change of variables and a constant factor depending on the knot and weight.