Small Dehn surgery and SU(2)

Abstract

We prove that the fundamental group of 3-surgery on a nontrivial knot in S 3 always admits an irreducible SU (2)-representation. This answers a question of Kronheimer and Mrowka dating from their work on the Property P conjecture. An important ingredient in our proof is a relationship between instanton Floer homology and the symplectic Floer homology of genus-2 surface diffeomorphisms, due to Ivan Smith. We use similar arguments at the end to extend our main result to infinitely many surgery slopes in the interval [3, 5).JAB was supported by NSF FRG Grant DMS-1952707.1 If b1(Y ) = 0 then ρ has abelian image if and only if it has cyclic image. 2 Though see [LPCZ21, BS21] and [SZ19] for progress on this problem in the cases of toroidal homology spheres and non-hyperbolic geometric manifolds, respectively.

An instanton take on some knot detection results

“…Such knots are fibered, and in [BLSY21, §2] we gave a partial characterization of their possible monodromies; more recently, Farber, Reinoso, and Wang [FRW22, Corollary 1.8] used this to show that K is necessarily T (2, 5). We remark that if we wanted to work over Z/2Z rather than Q, then we could finish the proof that K = T (2, 5) without recourse to [FRW22], using instead the arguments in [BLSY21]. Now if Kh(K; Z/2Z) is 5-dimensional and supported in the δ-grading σ = −2, then we apply the above to its mirror K to conclude that K = T (2, 5), and hence that K = T (−2, 5).…”