QUESTIONS

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.

- Does contact -1 surgery on a Legendrian knot in a closed 3-manifold preserve tightness?

- Are there contact structures with support genus greater than one?

- 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?

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

- 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?

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

- Which irreducible integer homology spheres are L-spaces?

- Is it the case that 2
^{l-ν}rk Kh^{δ}(L) ≥ rk HFK_{δ}(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).

- Is rk HFK(L) ≥ 2
^{l-ν}rk HF(Σ(L))?

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

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

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

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

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

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

- Is there a relationship between the Ozsvath-Szabo-Thurston invariant of a transverse link L in (S
^{3}, ξ_{st}) and the contact invariant of the naturally induced contact structure on Σ(L)?

- 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?

- When does a transverse knot in (S
^{3}, ξ_{st}) have a Seifert surface with "protected boundary"?