I created this page in 2011 to record and share some questions I found interesting at the time. Many are still open; others are closed. In any event, the page hasn't been updated in quite a while. As part of my recent NSF CAREER award, I will be creating an Open Problems in Floer Homology Wiki with the help of several volunteers. The Wiki will be a much more expansive version of what I started below. It will, in addition to recording important open problems, provide background, context, and summaries of whatever progress has been made in solving these problems, and will hopefully be a boon to both grad students and more experienced researchers.
  1. Does contact -1 surgery on a Legendrian knot in a closed 3-manifold preserve tightness?

  2. Are there contact structures with support genus greater than one?

  3. Is a contact structure the perturbation of a taut foliation if it is compatible with a pseudo-Anosov open book whose fractional Dehn twist coefficients (FDTCs) are all at least one? Does the converse hold?

  4. Is there a contact structure that is not weakly symplectically fillable, but whose contact invariant is a non-zero element of HF+red?

  5. Consider a pseudo-Anosov open book with one binding component and orientable stable/unstable foliations. Is the corresponding contact structure tight whenever the FDTC is 1/n?

  6. Suppose Y is an irreducible 3-manifold which doesn't admit a taut foliation. Is Y an L-space?

  7. Which irreducible integer homology spheres are L-spaces?

  8. Is it the case that 2l-ν rk Khδ(L) ≥ rk HFKδ(L) in each δ-grading? Here, l is the number of components of L and ν refers to the rank of the first cohomology of Σ(L).

  9. Is rk HFK(L) ≥ 2l-ν rk HF(Σ(L))?

  10. Is the rank of HFK invariant under genus 2 mutation? Is HF(Y) invariant under genus 2 mutation? Is τ invariant under Conway mutation?

  11. Is there a spectral sequence from HOMFLY(K) to HFK(K)?

  12. Is there a symmetry in HOMFLY homology which recovers the symmetry of the HOMFLY polynomial?

  13. Is Szabo's homology theory an invariant of Σ(L)? Does his spectral sequence agree with the spectral sequence of Ozsvath-Szabo?

  14. How do the higher differentials in the Ozsvath-Szabo spectral sequence shift the δ-grading on Khovanov homology?

  15. Is the "Khovanov homology of open books" construction an invariant of factorizations up to Hurwitz equivalence?

  16. Is there a relationship between the Ozsvath-Szabo-Thurston invariant of a transverse link L in (S3, ξst) and the contact invariant of the naturally induced contact structure on Σ(L)?

  17. Is the O-Sz-T invariant of transverse links, defined using grid diagrams, the same as the L-O-Sz-S invariant, defined using open books?

  18. When does a transverse knot in (S3, ξst) have a Seifert surface with "protected boundary"?