Efficient geodesics in the curve complex and their dot graphs

TL;DR

This paper studies the geometric structure of efficient geodesics in the curve complex of a surface, revealing that their dot graphs are contained within a spindle shape, which influences the coordinate behavior of associated curves.

Contribution

It provides a detailed analysis of the shape of dot graphs for efficient geodesics, showing they are contained within a spindle shape, enhancing understanding of curve intersection patterns.

Findings

Dot graphs of efficient geodesics are contained within a spindle shape.

Shape of dot graphs influences the coordinate behavior of curves.

Provides a geometric characterization of efficient geodesics in the curve complex.

Abstract

For the complex of curves of a closed orientable surface of genus $g$, $\mathcal{C}(S_{g>1})$, the notion of efficient geodesic in was introduced in arXiv:1408.4133. There it was established that there always exists (finitely many) efficient geodesics between any two vertices, $ v_{\alpha} , v_{\beta} \in \mathcal{C}(S_g)$, representing homotopy classes of simple closed curves, $\alpha , \beta \subset S_g$. The main tool for used in establishing the existence of efficient geodesic was a dot graph, a booking scheme for recording the intersection pattern of a reference arc, $\gamma \subset S_g$, with the simple closed curves associated with the vertices of geodesic path in the zero skeleton, $\mathcal{C}^0(S_g)$. In particular, for an efficient geodesic between $v_\alpha $ and $v_\beta$ of length $d \geq 3$, it was shown that any curve corresponding to the vertex that is distance one from $v_\alpha$ intersects any $\gamma$ at most $d -2$ times. In this note we make a more expansive study of the characterizing "shape" of the dot graphs over the entire set of vertices in an efficient geodesic edge-path. The key take away of this study is that the shape of a dot graph for any efficient geodesic is contained within a spindle shape region. Since the Nielson-Thurston coordinates of any curve on $S_g$ are directly derived from its intersection number with finitely many reference arcs, spindle shaped dot graphs control the coordinate behavior of curves associated with the vertices of an efficient geodesic.