Integer homology 3-spheres admit irreducible representations in SL(2,C)

Computing Heegaard Genus is NP-Hard

“…Schleimer showed that 3-Sphere Recognition is in NP [Sch11]. And, using Kuperberg's work [Kup14], Zentner showed that 3-Sphere Recognition is also in co-NP if we assume that the Generalized Riemann Hypothesis is true [Zen16]. Thus, without disproving a major conjecture, we do not expect the special case Heegaard Genus ≤ 0 to be NP-hard.…”

Stein fillings and SU(2) representations

“…Remark 1.3. Zentner [Zen16] has recently proven a version of this conjecture with SL 2 (C) in place of SU (2). He relies on geometrization and the fact that every hyperbolic 3-manifold automatically admits a nontrivial SL 2 (C) representation, which is far from clear for SU (2).…”

On the complexity of torus knot recognition

“…We now fix an integer r = 0 and construct 3-manifolds Y n for any n ∈ Z by Dehn filling the exterior of P along a pair of curves: we fill ∂N (P ) along the slope µ + rλ, and then we fill S 1 × ∂D 2 along nM + (nrw 2 + 1)L. The filling curves belong to the homology classes [µ] + rw[L] and nw[µ] + (nrw 2 + 1) [L], which span all of H 1 (E p ), so Y n is a homology sphere. If Y n is not homeomorphic to S 3 , then Zentner [Zen16] proved that π 1 (Y n ) admits an irreducible SL 2 (C) representation, and since π 1 (Y n ) is a quotient of π 1 (E P ), the composition…”

“…Theorem 1.6 is analogous to the theorem of Kronheimer and Mrowka [KM04] that nontrivial knots in S 3 admit nonabelian SU (2) representations, which Kuperberg used to certify knottedness. (Our proof relies on recent work of Zentner [Zen16], which in turn depends on [KM04].) In both cases, these representations are complex points of algebraic varieties defined over Z, and the use of GRH allows Kuperberg and us to assert that these varieties also have F p -points where p is a reasonably small prime.…”