Menasco showed that a non-split, prime, alternating link that is not a 2-braid is hyperbolic in S 3 . We prove a similar result for links in closed thickened surfaces S × I. We define a link to be fully alternating if it has an alternating projection from S × I to S where the interior of every complementary region is an open disk. We show that a prime, fully alternating link in S × I is hyperbolic. Similar to Menasco, we also give an easy way to determine primeness in S ×I. A fully alternating link is prime in S ×I if and only if it is "obviously prime". Furthermore, we extend our result to show that a prime link with fully alternating projection to an essential surface embedded in an orientable, hyperbolic 3-manifold has a hyperbolic complement.

We show that the pseudo-Anosov representative of any genus-two, hyperbolic, fibered knot in S 3 (with nonzero fractional Dehn twist coefficient) has a fixed point. Combined with previous work of Baldwin-Hu-Sivek [BHS21], this proves that the cinquefoil knot T (2, 5) is the only genus-two L-space knot. Our results have applications to the Floer homology of cyclic branched covers over knots in S 3 , to SU (2)-abelian Dehn surgeries, and to Khovanov and annular Khovanov homology. Along the way to proving our fixed point result, we describe for each stratum Q on a punctured disk (satisfying a very mild condition), a small list of train tracks carrying all pseudo-Anosovs in Q. As a consequence, we find a canonical track τ carrying all pseudo-Anosovs in a particular stratum Q0 on the genus-two surface with one boundary component, and describe every fixed point-free pseudo-Anosov in Q0. The main technical tools in our proofs are a restrictive variant of splitting for pseudo-Anosov braids, and a careful analysis of train track maps on the canonical track τ .

We investigate when a Legendrian knot in standard contact R 3 has a non-orientable exact Lagrangian filling. We prove analogs of several results in the orientable setting, develop new combinatorial obstructions to fillability, and determine when several families of knots have such fillings. In particular, we determine completely when an alternating knot (and more generally a plus-adequate knot) is decomposably non-orientably fillable, and classify the fillability of most torus and 3strand pretzel knots. We also describe rigidity phenomena of decomposable non-orientable fillings, including finiteness of the possible normal Euler numbers of fillings, and the minimization of crosscap numbers of fillings, obtaining results which contrast in interesting ways with the smooth setting.

For an open book decomposition (S, φ), the fractional Dehn twist coefficients are rational numbers measuring the amount that the monodromy φ twists the surface S near each boundary component. In general, the twist coefficients do not behave nicely under the operation of capping off a boundary component. The goal of this paper is to use Heegaard Floer homology to constrain the behavior of the fractional Dehn twist coefficients after capping off. We also use our results about fractional Dehn twists to study the Floer homology of cyclic branched covers over fibered two-component links.