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4. Knot homology theories and categorification

    1.     Cristina Anghel has several papers on interpretations of quantum invariants (Alexander, Jones, colored versions of these, WRT) in terms of intersections in covers of symmetric products of Riemann surfaces. It is natural to ask whether these constructions are shadows of certain Lagrangian Floer theories. See, e.g., [MR4418386] and followup works.

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      Problem 4.1.

      [Ciprian Manolescu] Can Cristina Anghel’s approach to knot polynomials (via intersections of Lagrangians in configuration spaces) be categorified in instances via Lagrangian Floer cohomology?
        1. Remark. [Mina Aganagic] For colored Jones polynomials, the Lagrangians written down in Anghel’s work may not be graded in general.
            • Remark. [Ivan Smith] If Lagrangian Floer cohomology could be defined, is it a link invariant? One needs to show invariance under moves that do not change the underlying link.


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                  Problem 4.2.

                  [Alexei Oblomkov] Is it true that $\mathfrak{gl}_{n|m}$ and $\mathfrak{gl}_{m|n}$ knot homologies are isomorphic if one switches the role of $t$ and $q$ variable? In particular, what is happening for the $\mathfrak{gl}_{1|1}$ theory (= knot Floer homology)?
                    1. Remark. [Robert Lipshitz] For $\mathfrak{gl}_{1|1}$, it asks for switching Alexander and Maslov grading.
                        • Remark. [Mina Aganagic] The answer should be negative for general links but may be positive for knots.


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                              Problem 4.3.

                              [Alexei Oblomkov] Compare different approaches to $\mathfrak{gl}_{n|m}$ knot homologies.


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                                  Problem 4.4.

                                  [Alexei Oblomkov] Compute knot homology of a ‘local’ infinite twist in any of these cases.


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                                      Problem 4.5.

                                      [Mohammed Abouzaid] What is a B-side construction of Heegaard knot Floer homology?
                                        1. Remark. [Robert Lipshitz] The same question has been asked since circa 2003.


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                                              Problem 4.6.

                                              [Sheel Ganatra] Is there a symplectic way to understand the entire $2$-categorical representation theory framework from which, e.g., KLRW or nil Hecke algebra arises?


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                                                  Problem 4.7.

                                                  [Ivan Smith] How does Ekholm–Shende’s “Skeins on Branes" [arXiv:1901.08027] story fit into a symplectic production of, e.g., (nil) Hecke algebras, and how can we better use it?
                                                    1. Remark. Cf. Honda–Tian–Yuan’s work [arXiv:2202.05593].

                                                        •     Aganagic’s proposal on defining $\mathfrak{gl}_{1|1}$ knot homology proves invariance of the relevant Lagrangian Floer homology under Markov moves and skein relations by resolving the Lagrangians using a specific choice of generators, which reduces the proofs to algebraic arguments. The following problem arises from an inquiry to see if there is a more direct approach.

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                                                          Problem 4.8.

                                                          [Robert Lipshitz] Is there a direct proof that $HF^*(I \ \text{brane}, E \ \text{brane})$ is a knot invariant which does not pass through resolving $E$ branes by $T$ branes?
                                                            1. Remark. It is tempting to see if the desired relations and invariance could be proved by directly looking at some moduli spaces.


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                                                                  Problem 4.9.

                                                                  [Lipshitz-Smith] Does Aganagic’s Khovanov homology construction have a Lee/Bar-Natan deformation? Does this theory provide a geometric explanation for odd Khovanov homology?
                                                                    1. Remark. There is a proposal which involves compactification of Lee deformation in Seidel–Smith’s symplectic Khovanov homology, but it is unclear how to translate this proposal to the case of Floer theory with potentials.

                                                                          Cite this as: AimPL: Floer theory of symmetric products and Hilbert schemes , available at http://aimpl.org/floerhilbert.