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

Quadratic Bounds on the Quasiconvexity of Nested Train Track Sequences