many distinct ways, the bound is to be interpreted as stated: regardless of the choice of fiber, the twist coefficient of the resulting monodromy is bounded by a number depending only on $Y$. To the best of our knowledge, the only situation prior to our theorem in which such a bound was known is for knots in the 3-sphere, in which case work of Gabai [@Gabai5] and Kazez-Roberts [@kazezroberts] shows that $|\tau(\phi_K)|\le 1/2$. Their proof relies on the application of thin position, among other things, and does not extend to other manifolds in an obvious way. Our proof exploits the connection between twist coefficients and contact topology, and a connection between contact topology and Heegaard Floer homology. Recall that by a construction of Thurston-Winkelnkemper [@Thurston], a fibered knot $K\subset Y$, regarded as an open book decomposition, uniquely determines a contact structure $\xi_K$ on $Y$ (see [@Torisu] for uniqueness). It was shown by Honda, Kazez, and Matic that if $\xi_K$ is tight, then $\tau(\phi_K)\geq 0$ [@HKMVeer1 Theorem 1.1 and Propositions 3.1, 3.2]. Using property $(3)$ of $\tau$, we see that to obtain a lower bound on $\tau(\phi_K)$ it suffices to show that there is a number $N_Y$ such that the monodromy 