The knot Floer homology and embedded contact homology correspondence conjecture

Let KMK\subset M be a null-homologous knot of genus gg. Let NN be the complement of an open tubular neighbourhood of KK, and let α\alpha be a contact form on NN for which N\partial N is a negative Morse-Bott torus foliated by meridians of KK. For each integer ii, let ECHi(N,α)ECH^\sharp_i(N,\alpha) denote the homology of the subcomplex of ECH(N,α)ECH^\sharp(N,\alpha) generated by orbit sets in NN with total linking number ii with KK.

ECH–knot Floer correspondence conjecture. There is an isomorphism

ECHi(N,α)HFK^(M,K,ig).ECH^\sharp_i(N,\alpha)\cong \widehat{HFK}(M,K,i-g).

This conjecture is obtained by combining the preceding sutured ECH–SFH conjecture with a theorem relating the relevant ECH groups and the relation between sutured Floer homology and knot Floer homology. Its resolution is not specified in the source.

Sources & referencesView supporting material

Primary source

Paolo Ghiggini, Vincent Colin and Ko Honda, “An exposition of the equivalence of Heegaard Floer homology and embedded contact homology”, arXiv:2004.09626 (2020).

Additional references

2 papers in this index state this conjecture (2006–2020). The statement above is taken from the most recent of them; the others are arXiv:math/0601443.

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