A class of knots with simple SU(2)-representations

Abstract

We call a knot in the 3-sphere SU(2)-simple if all representations of the fundamental group of its complement which map a meridian to a trace-free element in SU(2) are binary dihedral. This is a generalization of being a 2-bridge knot. Pretzel knots with bridge number ${\ge }3$ are not SU(2)-simple. We provide an infinite family of knots K with bridge number ${\ge }3$ which are SU(2)-simple. One expects the instanton knot Floer homology $I^\natural (K)$ of a SU(2)-simple knot to be as small as it can be—of rank equal to the knot determinant $\text {det}(K)$. In fact, the complex underlying $I^\natural (K)$ is of rank equal to $\text {det}(K)$, provided a genericity assumption holds that is reasonable to expect. Thus formally there is a resemblance to strong L-spaces in Heegaard Floer homology. For the class of SU(2)-simple knots that we introduce this formal resemblance is reflected topologically: The branched double covers of these knots are strong L-spaces. In fact, somewhat surprisingly, these knots are alternating. However, the Conway spheres are hidden in any alternating diagram. With the methods we use, we obtain the result that an integer homology 3-sphere which is a graph manifold always admits irreducible representations of its fundamental group. This makes use of a non-vanishing result of Kronheimer–Mrowka.