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 generalisation 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 $I^{♮}(K)$ of a $SU(2)$-simple knot to be as small as it can be -- of rank equal to the knot determinant $\det (K)$. In fact, the complex underlying $I^{♮}(K)$ is of rank equal to $\det (K)$, 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 show that an integer homology 3-sphere which is a graph manifold always admits irreducible representations of its fundamental group.

22 pages, 10 figures, to appear in Selecta Mathematica

References in corpus (2)

• Traceless SU(2) representations of 2-stranded tangles
• Strong Heegaard diagrams and strong L-spaces

Cited by in corpus (6)

• Integer homology 3-spheres admit irreducible representations in SL(2,C)
• Stein fillings and SU(2) representations
• A menagerie of SU(2)-cyclic 3-manifolds
• Small Dehn surgery and SU(2)
• Small Heegaard genus and SU(2)
• Surgery obstructions and character varieties