TITLE: Knot Invariants from Twisted Quantum Doubles
QUESTION [12 upvotes]: In "Topological Gauge Theories and Group Cohomology", Dijkgraaf and Witten construct a 3-manifold invariant from a finite group $G$ and 3-cocycle $\omega$. I would think there is also an associated knot invariant, coming in particular from the twisted quantum double $D^\omega(G)$, but haven't been able to find a place where it is explicitly described or computed. 
If it exists, could someone point me to the right reference? I would be interested in both a general construction and a specific example, for instance computing it in the case of $\mathbb{Z}_n$.

REPLY [4 votes]: You may be interested in this paper by Chen.
One of the simplest nontrivial knot invariants is the number of "Fox n-colorings", which count homomorphisms of the knot group to a dihedral group $D_{2n}$ sending the meridians to reflections. This count can be interpreted as a value in a Dijkgraaf-Witten theory on the knot complement. Chen's paper computes those invariants in the nontwisted case in terms of the representation category of the quantum double $D(D_{2n})$ and gives a nice formula for the number of colorings of a rational link.
I tried computing values of the twisted invariant, but it turns out that these are equal to the untwisted invariants times a power of $i$ depending on the writhe.
It may be possible that something more intersting happens with a more complicated group: $D_{2n} = \mathbb{Z}_2 \ltimes \mathbb{Z}_n$ is the simplest possible case where you get interesting homomorphism counts, and I'd be curious what happens in the twisted (and untwisted) case for other semidirect products. In particular, the twisting that actually affects the DW invariants seems to mostly involve the $\mathbb{Z}_2$ part, which has few interesting cocycles.