Train tracks of pseudo-Anosov 3-braids and non-degenerate flype

In this thesis, we prove two results concerning properties of knots and braids. We focus on 3-braids which are isomorphic to the mapping class group of a 3-punctured disk, and their closures which form knots.

The first part of the thesis focuses on examining the slice-Bennequin inequality, which gives an upper bound for the self-linking number of a knot in terms of its four-ball genus. The s-Bennequin and τ-Bennequin inequalities provide upper bounds on the self-linking number of a knot in terms of the Rasmussen s invariant and the Ozsváth-Szabó τ invariant. We exhibit examples in which the difference between self-linking number and four-ball genus grows arbitrarily large, whereas the s-Bennequin inequality and the τ -Bennequin inequality are both sharp. This is the first known infinite sequence of braids with this property in the literature.

In the second part of the thesis, we study Agol cycles of pseudo-Anosov 3-braids using Farey sequences. An Agol cycle is a conjugacy class invariant of a pseudo-Anosov mapping class. We give a sufficient condition for two 3-braids to have equivalent or mirror equivalent Agol cycles and give infinitely many examples. Since non-degenerate flypes play a significant role in knot theory, we also study behavior of Agol cycles of pseudo-Anosov 3-braids admitting flype moves.