Genus two Goeritz equivalence in lens spaces $L(p,1)$

The Berge conjecture has been a motivating goal for low-dimensional topology for the last 50 years, using special positions of a knot on a genus 2 Heegaard splitting and the induced surface slopes to characterize the kinds of manifolds that can arise from Dehn surgery on the knot. This open conjecture has driven a great deal of research to answer the question (e.g., [8], [10], [13]), as well as variations and generalizations (e.g., [2], [5], [11]).

In exploring some of these variations, some surprising results have been found, including [1], [7], and [9], which find infinite families of knots, each with inequivalent positions on a genus 2 Heegaard splitting in S 3 but with equal induced surface slopes and corresponding surgeries giving rise to the same non-hyperbolic manifolds. The latter two authors of [1] endeavored to build a larger framework for simplifying the question of when two positions of a knot on a genus 2 Heegaard splitting are inequivalent, via the action of the Goeritz group.

The Goeritz group of a Heegaard splitting of a 3-manifold is the group of isotopy classes of orientation-preserving automorphisms of the manifold preserving the Heegaard splitting, set-wise. In [6], we provide homological obstructions for two curves on the (unique) genus 2 Heegaard surface in S 3 to be related by Goeritz group elements. The current work continues this enterprise. Building on work of Cho [3], which provides a presentation for each of the Goeritz groups for the (unique) genus 2 Heegaard splitting of lens spaces, L(p, 1), we are able to emulate the methods of [6], and provide homology obstructions in the same spirit, albeit more byzantine in some cases.

In Section 2, we present background on the Goeritz group for L(p, 1) and the induced action on the homology of a genus 2 Heegaard surface. In Section 3, we explore the images and kernels of these induced actions, and in Section 4 we provide obstructions to the existence of such actions. We prove Theorem 4.3, which provides a collection of relatively straightforward necessary arithmetic conditions for a Goeritz action between two curves in terms of their homology vectors. Theorem 4.4 spells out seven explicit relationships between the homology vectors’ curves, at least one of which must be satisfied in order for the curves to be Goeritz equivalent. Finally, in Theorem 4.5, we provide an obstruction to two curves being related by a Goeritz element in terms of their fundamental group representatives in each handlebody.

2. The Goeritz group of L(p, 1)

Observe that every Heegaard splitting of a lens space is standard: a stabilization of the unique genus 1 Heegaard splitting. We will exclusively examine genus 2 Heegaard splittings, so we will often simply refer to the Goeritz group of the lens space L(p, 1), for p ≥ 2. Cho uses the complex of primitive disks to provide a presentation for each of the Goeritz groups of L(p, 1).

Theorem 2.1 (Cho [3]). For p ≥ 2, a presentation for the genus-2 Goeritz group G p for the lens space L(p, 1) is given by:

(1) ⟨ β, ρ, γ | ρ 4 = γ 2 = (γρ

In Theorem 2.1, the maps β, ρ, γ, α, δ, and σ refer to specific homeomorphisms. Let (F ′ , V ′ , W ′ ) be the genus one Heegaard splitting of L(p, 1), with the (p, 1)torus knot bounding a disk in V ′ , and let (F, V, W ) be the genus 2 stabilization of (F ′ , V ′ , W ′ ). We consider the action G p on H 1 (F ; Z) by considering how the elements affect a basis for homology. As most of the homeomorphisms in [3] are described in terms of the handlebody W , we will choose a standard basis for the homology of F from this perspective. Let the homology vectors a, x, b, and y be the standard basis vectors, represented by the oriented curves a, x, b, and y shown in Figure 1, with a and x bounding disks in W , y bounding a disk in V , and the cross products a × b and x × y (of the tangent vectors) pointing into V at the the points of intersection. The actions then produce a map ⋆ ∶ G p → GL(4, Z). We denote by α * , β * , γ * , δ * , ρ * , and σ * the automorphisms of H 1 (F ; Z) induced by α, β, γ, δ, ρ, and σ, respectively. Lemma 2.2. With respect to the ordered basis {a, x, b, y}, the matrices induced by the generators of G p are as follows.

, and

Proof. The proof will consist of tracking the images of the basis curves under each of the homeomorphisms in the generating set for G p . Let ι(⋅, ⋅) denote the oriented intersection number between two curves, with the positive surface orientation pointing into V . Observe, then, that for a curve z representing vector z, and an element g ∈ G p , When p = 2, there is a 4-fold rotation, ρ, on the Heegaard surface. There exists a primitive pair of disks D ′ and E ′ in W , and a unique primitive pair of disks D and E in V that are commonly dual to D ′ and E ′ . Cutting W along D ′ and E ′ results in a 3-ball, with four distinguished disks on the boundary sphere, which we can think of as a 4-holed sphere. The curves ∂D and ∂E on this sphere are sym

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