The Goeritz groups of $(1,1)$-decompositions

TL;DR

The paper determines the Goeritz groups for all $(1,1)$-decompositions of knots in closed orientable 3-manifolds. It develops a piecewise-linear framework for $(g,n)$-decompositions and analyzes the mapping class group of a trivial $1$-tangle in a solid torus to obtain explicit presentations of the relevant Goeritz groups. The main result classifies $\,\mathcal{G}(M,K;\Sigma)$ across ambient manifolds, including lens spaces $L(p,q)$ with even $p$, showing the groups are finite precisely when the associated distance $d(M,K;\Sigma)\ge 2$. This yields concrete, case-wise descriptions such as $\,\mathcal{G}(M,K;\Sigma)\cong \mathbb{Z}/2\mathbb{Z}\langle \alpha\rangle \times \mathbb{Z}\langle \tau\rangle \times \langle \beta,\gamma \mid \gamma^2=1\rangle$ in several scenarios and simplifies to $\,\mathbb{Z}/2\mathbb{Z}\langle \alpha\rangle$ in others, clarifying how Goeritz groups behave for low-comistance bridge decompositions and enabling further study of knot-3-manifold symmetries.

Abstract

A $(g, n)$-decomposition of a link $L$ in a closed orientable $3$-manifold $M$ is a decomposition of $M$ by a closed orientable surface of genus $g$ into two handebodies each intersecting the link $L$ in $n$ trivial arcs. The Goeritz group of that decomposition is then defined to be the group of isotopy classes of orientation-preserving homeomorphisms of the pair $(M, L)$ that preserve the decomposition. We compute the Goeritz groups of all $(1,1)$-decompositions.

Figures (14)

• Figure 1: From a $(0,2)$-decomposition of the figure-eight knot to its $(1, 1)$-decomposition.
• Figure 2: Every torus knot admits a $(1, 1)$-decomposition.
• Figure 3: Canceling disks of $C_{T_1}$.
• Figure 4: The quotient graph $\mathcal{T} / \mathrm{MCG}_+(\widetilde{V}_1, C_{T_1})$.
• Figure 5: The four elements $\alpha, \beta_D, \tau, \gamma_{\{D, E\}} \in \mathrm{MCG}_+(\widetilde{V}_1, C_{T_1})$.

Theorems & Definitions (27)

• Theorem 1: Iguchi--Koda Iguchi-Koda
• Theorem 2
• Corollary 3
• Theorem 1.1
• Theorem 1.2: Saito Saito
• Lemma 2.1
• proof
• Theorem 2.2: Cho Cho
• Lemma 2.3
• proof