Relationships between quantum and Heegaard Floer Invariants

Nathan Dowlin

Knots in Washington XLVI: 70th Birthday of Oleg Viro;
May 4-6, 2018
George Washington University,
Washington, DC, USA

Organizers
Valentina Harizanov (GWU), Jozef H. Przytycki (GWU), Yongwu Rong (GWU), Radmila Sazdanovic (NCSU), Alexander Shumakovitch (GWU)

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Relationships between quantum and Heegaard Floer Invariants
by
Nathan Dowlin
Columbia University
Coauthors: Akram Alishahi

There are two main conjectures relating Khovanov-type invariants with knot Floer homology. The first is the existence of a spectral sequence from Khovanov homology to delta-graded knot Floer homology, and the second is the existence of a spectral sequence from HOMFLY-PT homology to (bigraded) knot Floer homology. We will define a family of invariants on the knot Floer side which are analogous to the sl(n) homology of Khovanov and Rozansky which shed some light on these conjectures. In the n=2 case, we construct an algebraically defined filtered homology theory such that the E₂ page is isomorphic to Khovanov homology, and the higher pages are link invariants. The construction is inspired by counting holomorphic discs in a particular Heegaard diagram, so we expect the E_∞ page of the spectral sequence to recover delta-graded knot Floer homology.

Date received: March 30, 2018