Spherical braids with the same knot Floer homology
Question: are there infinitely many spherical braids in $S^1\times S^2$ with the same knot Floer homology? There is a simple way of constructing infinitely many knots in $S^3$ with the same knot Floer homology: take a nontrivial band sum of two unknots; adding twists to the band preserves knot Floer homology (see Theorem 1 of Hedden--Watson ). The family of genus-2 fibered hyperbolic pretzel knots $P(-3,3,2n)$ can be described in this way. The existence of this family of pretzels shows that the knot Floer homology of a hyperbolic fibered knot in $S^3$ cannot in general bound the dilatation of its pseudo-Anosov monodromy. Indeed, a hyperbolic fibered knot is determined by the conjugacy class of the pA representative of its monodromy, and there are only finitely many conjugacy classes of pA homeomorphisms of a given surface with dilatation less than some fixed constant. (One can also conclude that infinitely many of the pretzels in this family must have monodromies which are not ...