Quadratic Bounds on the Quasiconvexity of Nested Train Track Sequences

Let $S_{g,p}$ denote the genus $g$ orientable surface with $p$ punctures. We show that nested train track sequences constitute $O((g,p)^{2})$-quasiconvex subsets of the curve graph, effectivizing a theorem of Masur and Minsky. As a consequence, the genus $g$ disk set is $O(g^{2})$-quasiconvex. We also show that splitting and sliding sequences of birecurrent train tracks project to $O((g,p)^{2})$-unparameterized quasi-geodesics in the curve graph of any essential subsurface, an effective version of a theorem of Masur, Mosher, and Schleimer.