  This note explores two questions: (1) Which bigraded groups arise as the knot
Floer homology of a knot in the three-sphere? (2) Given a knot, how many
distinct knots share its Floer homology? Regarding the first, we show there
exist bigraded groups satisfying all previously known constraints of knot Floer
homology which do not arise as the invariant of a knot. This leads to a new
constraint for knots admitting lens space surgeries, as well as a proof that
the rank of knot Floer homology detects the trefoil knot. For the second, we
show that any non-trivial band sum of two unknots gives rise to an infinite
family of distinct knots with isomorphic knot Floer homology. We also prove
that the fibered knot with identity monodromy is strongly detected by its knot
Floer homology, implying that Floer homology solves the word problem for
mapping class groups of surfaces with non-empty boundary. Finally, we survey
some conjectures and questions and, based on the results described above,
formulate some new ones.
