A saturated 1-system of curves on the surface of genus 3

TL;DR

The paper addresses $N(1,3)$, the maximal size of a $1$-system on the genus-$3$ surface $\Sigma_3$, where prior bounds ranged broadly. It constructs a saturated $1$-system $X_0$ with $33$ curves by encoding curves as multitwists relative to a fixed pants decomposition with dual graph $K_4$ and organizing them into $24$ type $\triangle$ curves and $3$ type $\square$ curves via $A_4$-orbits. The authors prove saturation for the $\triangle$ family and extend it to include the $\square$ curves, establishing $N(1,3)\ge 33$ and providing a constructive framework toward maximality, though the exact value remains open. This work blends pants decompositions, Dehn twist encodings, and subsurface projection techniques to advance understanding of maximal $1$-systems on genus-$3$ surfaces and suggests concrete avenues for proving maximality.

Abstract

We construct a system of 33 essential simple closed curves that are pairwise non-homotopic and intersect at most once on the oriented, closed surface of genus 3. Moreover, we show that our construction is saturated, in the sense that it is not properly contained in any other such system of curves.

Figures (17)

• Figure 2.1: The pants decomposition $\mathcal{P}_0$ of $\Sigma$, whose pants curves $\alpha_1,\dots,\alpha_6$ are marked in red, and its dual graph $K_4$.
• Figure 2.2: The induced pants decomposition of $\Sigma_{1,3}\subset\Sigma$, whose dual graph is a $3$-cycle in $K_4$. For a $4$-cycle in $K_4$, its corresponding pants decomposition of $\Sigma_{1,4}\subset\Sigma$ can be obtained by cutting $\Sigma$ along the curves corresponding to the two edges of $K_4$ not in the $4$-cycle.
• Figure 2.3: An embedding $K_4\hookrightarrow\Sigma$ mapping the vertices to the 'extremities' of $\Sigma$. Every cycle in $K_4$ is associated with a curve $\gamma_0$ under this embedding, which we choose to be the reference curves for which every other curve $\gamma\in X_0$ will be a multitwist thereof.
• Figure 2.4: A reference curve $\gamma_0$ and its multitwist $\gamma=T_{\alpha_1}^1T_{\alpha_2}^1(\gamma_0)$, encoded by $\left(1,1,0\right)$.
• Figure 2.5: A reference curve $\gamma_0$ and its multitwist $\gamma=T_{\alpha_1}^1T_{\alpha_6}^1(\gamma_0)$, encoded by $\left(1,-,0,-,0,1\right)$.

Theorems & Definitions (14)

• Definition 2.1
• Theorem 2.2
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5
• proof
• Lemma 2.6
• proof