Embedded contact knot homology conjecture for links

About 12 years old · traced to

Let LL be a link in a contact three-manifold YY, and let α\alpha be any contact form on YY adapted to LL. Denote by ECK(L,Y,α)ECK(L,Y,\alpha) and ECK^(L,Y,α)\widehat{ECK}(L,Y,\alpha) the full and hat versions of embedded contact knot homology. Embedded contact knot homology conjecture for links. One should have

ECK(L,Y,α)≅HFK−(L,Y),ECK(L,Y,\alpha)\cong HFK^{-}(L,Y),

and

ECK^(L,Y,α)≅HFK^(L,Y).\widehat{ECK}(L,Y,\alpha)\cong\widehat{HFK}(L,Y).

These are the link version of the proposed correspondence between embedded contact knot homology and knot Floer homology. The source does not state whether the conjecture has been resolved.

References

Primary source

Gilberto Spano, “A categorification of the Alexander polynomial in embedded contact homology”, arXiv:1410.5081 (2014).

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