The higher-dimensional Heegaard Floer–Khovanov correspondence conjecture

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Let σ^\widehat{\sigma} be the link associated to a braid σ\sigma, and let Kh♯,k(σ^)Kh^{\sharp,k}(\widehat{\sigma}) denote the degree-kk part of the higher-dimensional Floer homology, while Khi,j(σ)Kh^{i,j}(\sigma) denotes the Khovanov homology in bidegree (i,j)(i,j). Higher-dimensional Heegaard Floer–Khovanov correspondence conjecture. There is an isomorphism

Kh♯,k(σ^)≃⨁i−j=k mod⁡ n−2Khi,j(σ).Kh^{\sharp,k}(\widehat{\sigma})\simeq\bigoplus_{i-j=k\,\operatorname{mod}\,n-2}Kh^{i,j}(\sigma).

This conjecture proposes that the invariant constructed in the paper recovers Khovanov homology after combining its grading according to the congruence i−j=k mod⁡ n−2i-j=k\,\operatorname{mod}\,n-2. The supplied text does not state whether the conjecture has been proved or disproved.

References

Primary source

Tianyu Yuan, “A link invariant from higher-dimensional Heegaard Floer homology”, arXiv:2309.13241 (2023).

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