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1. Link invariants from quivers


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      Problem 1.1.

      Is there an association between the mutation equivalence of plabic graphs or the mutation equivalence of quivers to the invariants of isotopy classes of their associated links? Can these be used to produce invariants of quivers up to mutation equivalence?


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          Problem 1.2.

          To a quiver, we can associate a point count polynomial defined recursively (Galashin-Lam). This is sometimes equal to the HOMFLY-PT polynomial of the related link. For any simple plabic graphs, are these polynomials always equivalent?


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              Problem 1.3.

              In high levels of generality, varieties pick up a mixed Hodge structure (Lam-Speyer). When the cluster variety is adequately nice, the mixed Hodge structure has only two dimensions, and the point count polynomial corresponds to a certain polynomial, but not the HOMFLY-PT.

              What is the relationship between the mixed Hodge structure on the cohomology of a cluster variety coming from a plabic graph and knot invariants (eg. the Khovanov-Rozansky homology)?


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                  Problem 1.4.

                  When is the plabic cluster algebra locally acyclic? If it is not locally acyclic, is it still a polynomial count and can we say anything about a mixed Hodge structure?

                      Cite this as: AimPL: Cluster algebras and braid varieties, available at http://aimpl.org/clusterbraid.