Stein-fillable open books of genus one that do not admit positive factorisations

Abstract

We construct an infinite family of genus one open book decompositions supporting Stein-fillable contact structures and show that their monodromies do not admit positive factorisations. This extends a line of counterexamples in higher genera and establishes that a correspondence between Stein fillings and positive factorisations only exists for planar open book decompositions.

Figures (6)

• Figure 1: An open book decomposition $(\Sigma_{1,2}, \tau_\alpha \tau_\beta \tau_\gamma^{-1} \tau_{\delta_1} \tau_{\delta_2}^{4 + n})$ with $n \geqslant 0$.
• Figure 2: On the left: an open book $(\Sigma_{1,1}, \tau_\alpha \tau_\beta)$ supporting $(S^3, \xi_{\textrm{std}})$. On the right: an open book $(\Sigma_{1,2}, \tau_\alpha \tau_\beta \tau_\gamma^{-1} \tau_{\delta_1} \tau_{\delta_2}^4)$ supporting $(L(5,1), \xi)$, the result of transverse (+5)-surgery on a right-handed trefoil in $(S^3, \xi_{\textrm{std}})$.
• Figure 3: A surgery diagram showing that, topologically, the $+5$-surgery on a trefoil and the $-5$-surgery on the unknot give diffeomorphic 3-manifolds.
• Figure 4: The lantern relation on $\Sigma_{0,4}$ is $\tau_{\delta_1} \tau_{\delta_2} \tau_{\delta_3} \tau_{\delta_4} = \tau_{\sigma_1} \tau_{\sigma_2} \tau_{\sigma_3}$, up to cyclic permutation of $\tau_{\sigma_i}$ and reordering of $\tau_{\delta_i}$.
• Figure 5: An open book decomposition $(\Sigma_{1,2}, \tau_\alpha \tau_\beta \tau_\gamma^{-1} \tau_{\delta_1} \tau_{\delta_2})$ supporting the result of inadmissible transverse $(+2)$-surgery on a right-handed trefoil in $(S^3, \xi_{\textrm{std}})$.

Theorems & Definitions (8)

• Theorem 1.1
• Remark 1.2
• Proposition 3.1: Conway2019
• Proposition 3.2: Conway2019
• Lemma 4.1
• proof
• Theorem 4.2
• proof