Connected Primitive Disk Complexes and Genus Two Goeritz Groups of Lens Spaces

Abstract

Given a stabilized Heegaard splitting of a 3-manifold, the primitive disk complex for the splitting is the subcomplex of the disk complex for a handlebody in the splitting spanned by the vertices of the primitive disks. In this work, we study the structure of the primitive disk complex for the genus two Heegaard splitting of each lens space. In particular, we show that the complex for the genus two splitting for the lens space L(p, q) with 1 ≤ q ≤ p/2 is connected if and only if p ≡ ±1 (mod q), and describe the combinatorial structure of each of those complexes. As an application, we obtain a finite presentation of the genus two Goeritz group of each of those lens spaces, the group of isotopy classes of orientation preserving homeomorphisms of the lens space that preserve the genus two Heegaard splitting of it.

Genus one fibered knots in 3-manifolds with reducible genus two Heegaard splittings

“…Lemma 1 (Cho, Koda, [7] [8] [9]) In the above setting, if the word determined by a simple closed curve l contains a subword of the form xy n x −1 for some n ∈ N, then this word is reduced.…”

Disk surgery and the primitive disk complexes of the 3-sphere

“…Recall that when a handlebody V is one of the handlebodies of the genus-g Heegaard splitting (V, W ; Σ), with g ≥ 2, of the 3-sphere, S 2 × S 1 or a lens space L(p, q), the primitive disk complex P(V ) is the full subcomplex of K(V ) spanned by the vertices of primitive disks in V . The followings are known results on the primitive disk complexes P(V ) for the genus-2 splittings (see [1,2,3,4]).…”

“…For the genus-2 Heegaard splitting (V, W ; Σ) of each of the 3-sphere, S 2 × S 1 and lens spaces L(p, q), the structure of the primitive disk complex P(V ) is fully studied in [1], [2], [3], [4] and [5]. Understanding the structure of the primitive disk complexes enables us to obtain finite presentations of the mapping class groups of the splittings by investigating the simplicial action of the group on the primitive disk complex.…”

The disk complex and 2-bridge knots

“…The combinatorial structure of P(V ) for each lens space was fully studied in [3], [4] and [5]. We describe it as follows.…”

“…When V is one of the handlebodies of a genus-2 Heegaard splitting for a lens space, the non-separating disk complex for V admits a special subcomplex, called the primitive disk complex. In Section 3, the combinatorial structure of the primitive disk complex for each lens space is described, which was done in the previous works [3], [4] and [5]. From the primitive disk complex, we construct the primitive tree and introduce some properties of the simplicial automorphisms of the complex that we need.…”