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 tracefree element in SU (2) are binary dihedral. This is a generalization of being a 2-bridge knot. Pretzel knots with bridge number ≥ 3 are not SU (2)-simple. We provide an infinite family of knots K with bridge number ≥ 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 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. 2 RAPHAEL ZENTNER homology to the Alexander polynomial, established independently by Kronheimer and Mrowka in [20] and Lim in [24].Proposition 7.3. If a knot K is SU (2)-simple and satisfies the genericity hypothesis 7.2, then its instanton Floer chain complex CI π (K) has no non-zero differentials and is of total rank det(K). In particular, the total rank of reduced instanton knot Floer homology I (K) is also equal to det(K).Denoting Y (T (p, q)) the complement of a tubular neighborhood of the torus knot T (p, q), we may glue Y (T (p, q)) and Y (T (r, s)) together along their boundary torus in such a way that a meridian of the first torus knot is mapped to a Seifert fibre of the second and vice versa. It is a result of Motegi [30] that these 3-manifolds Y = Y (T (p, q), T (r, s)) are SO(3)-cyclic, i.e. admit only cyclic SO(3) representations of their fundamental group. Using the concept of strongly invertible knots, we obtain Theorem 4.14. The 3-manifold Y (T (p, q), T (r, s)) comes with an involution with quotient S 3 . It is a branched double cover of some knot or 2-component link L(T (p, q), T (r, s)) in S 3 , well defined up to mutation by the involution on either side of the essential torus in Y (T (p, q), T (r, s)). If pqrs − 1 is odd then L(T (p, q), T (r, s) is a knot. If in addition both T (p, q) and T (r, s) are non-trivial torus knots, then the knot L(T (p, q), T (r, s) is SU (2)-simple, but is not a 2-bridge knot.We give an explicit description of the knots L(T (p, q), T (r, s)) as a decomposition of two tangles in Section 6 below. In fact, each tangle is explicitly described in Theorem 6.1. Somewhat to our surprise, we have obtained Theorem 6.5. The knots L(T (p, q), T (r, s)) are alternating. The Conway sphere giving rise to the essen...

Khovanov homology and binary dihedral representations for marked links

“…In [17], Zentner studied knots with the property that all of its SU (2) representations are binary dihedral and called such knots "SU (2)-simple". He showed that if a knot K is SU (2)-simple and satisfies a certain genericity hypothesis, then the higher differentials on the instanton complex vanish.…”

Surgery obstructions and character varieties

“…Example 2.6. The manifold Y (T 2,3 , T 2,−3 ) is the branched double cover of 8 17 (see [Zen17,§5]), which has unknotting number 1. One can check that this SU(2)-cyclic manifold is 37 2 -surgery on the (−2, 3, 7)-pretzel knot, which will appear in Theorem 3.1 as the mirror of the knot k(2, 2, 0, 0).…”

A menagerie of SU(2)-cyclic 3-manifolds

“…We know very little about the remaining case, namely hyperbolic manifolds, but we can prove that some closed hyperbolic 3-manifolds are SU(2)-cyclic by the following, which was claimed without proof in [Zen17] and builds on work of Cornwell [Cor15].…”