Volume 18, issue 7 (2018)

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Knot Floer homology and Khovanov–Rozansky homology for singular links

Nathan Dowlin

Algebraic & Geometric Topology 18 (2018) 3839–3885
Abstract

The (untwisted) oriented cube of resolutions for knot Floer homology assigns a complex ${C}_{F}\left(S\right)$ to a singular resolution $S$ of a knot $K$. Manolescu conjectured that when $S$ is in braid position, the homology ${H}_{\ast }\left({C}_{F}\left(S\right)\right)$ is isomorphic to the homfly-pt homology of $S\phantom{\rule{0.3em}{0ex}}$. Together with a naturality condition on the induced edge maps, this conjecture would prove the existence of a spectral sequence from homfly-pt homology to knot Floer homology. Using a basepoint filtration on ${C}_{F}\left(S\right)$, a recursion formula for homfly-pt homology and additional ${\mathfrak{s}\mathfrak{l}}_{n}$–like differentials on ${C}_{F}\left(S\right)$, we prove Manolescu’s conjecture. The naturality condition remains open.

Keywords
knot theory, knot Floer, Khovanov–Rozansky, HOMFLY-PT, homology
Primary: 57M27
Publication
Received: 29 June 2017
Revised: 14 April 2018
Accepted: 23 April 2018
Published: 11 December 2018
Authors
 Nathan Dowlin Department of Mathematics Columbia University New York, NY United States