I am looking for non-trivial examples of flat $U(2)$ connections over the complement of a torus link $\mathcal{S}^3-L$ i.e.


$\mathcal{A}:\mathcal{S}^3-L \longrightarrow \mathfrak{U}(2)$ such that $F\_{\mathcal{A}} = 0$ and $Hol\_{\gamma}(\mathcal{A}) \neq 0$ with $\gamma$ a non-trivial element in $\pi\_1(\mathit{S}^3-L)$


Are there any known examples? Perhaps for $SU(2)$ it is known? I guess these connections would be classical solutions to a Euclidean $U(2)$ Chern-Simons theory on $\mathit{S}^3-L$ but when looking for Chern-Simons and knots all one usually finds is the quantum case...


Thus an explicit example of a classical solution with non-trivial holonomy is what I am looking for as well as the references where one may find how to construct explicit examples of this type.


Thanks



Explicit examples of Classical, Flat $U(2)$-connections on a torus link complement with non-trivial holonomy
For torus *knots*, all of the representations into $SU(2)$ were rather explicitly worked out by Eric Klassen (Representations of knot groups in $SU(2)$. Trans. Amer. Math. Soc. 326 (1991), no. 2, 795–828). The starting point is a rather simple presentation of the fundamental group of the complement of a torus knot; since torus links also have similar presentations it seems likely that the same technique would work for more than one component.


If you just want one example, consider $L$, a $(2,2n)$ torus link. Its double branched cover is a lens space $L(2n,1)$ with $\pi\_1 = Z/2n$. Hence there is a dihedral representation of the group $S^3 -L$, which you can think of as an SO(3) representation. I think this lifts to a representation of $\pi\_1(S^3 -L)$ to $SU(2)$ with image a binary dihedral group.


(I wasn't sure if you really meant multi-component links or if examples with knots would be satisfactory.)


