We present a conjecture for the crossing symmetry rules for Chern-Simons
gauge theories interacting with massive matter in $2+1$ dimensions. Our
crossing rules are given in terms of the expectation values of particular
tangles of Wilson lines, and reduce to the standard rules at large Chern-Simons
level. We present completely explicit results for the special case of two
fundamental and two antifundamental insertions in $SU(N)_k$ and $U(N)_k$
theories. These formulae are consistent with the conjectured level-rank,
Bose-Fermi duality between these theories and take the form of a $q=e^{\frac{ 2
\pi i }{\kappa}}$ deformation of their large $k$ counterparts. In the ‘t Hooft
large $N$ limit our results reduce to standard rules with one twist: the
$S$-matrix in the singlet channel is reduced by the factor $\frac{\sin \pi
\lambda}{\pi \lambda} $ (where $\lambda$ is the ‘t Hooft coupling), explaining
`anomalous’ crossing properties observed in earlier direct large $N$
computations.
