Ciprian Manolescu

On combinatorial link Floer homology

Link Floer homology is an invariant for links defined using a suitable version of Lagrangian Floer homology. In an earlier paper, this invariant was given a combinatorial description with mod 2 coefficients. In the present paper, we give a self-contained presentation of the basic properties of link Floer homology, including an elementary proof of its invariance. We also fix signs for the differentials, so that the theory is defined with integer coefficients.

Nilpotent slices, Hilbert schemes, and the Jones polynomial

Seidel and Smith have constructed an invariant of links as the Floer cohomology for two Lagrangians inside a complex affine variety Y. This variety is the intersection of a semisimple orbit with a transverse slice at a nilpotent in the Lie algebra

Involutive Heegaard Floer homology

sl_{2m}.

Combinatorial cobordism maps in hat Heegaard Floer theory

We exhibit bijections between a set of generators for the Seidel-Smith cochain complex, the generators in Bigelows picture of the Jones polynomial, and the generators of the Heegaard Floer cochain complex for the double branched cover. This is done by presenting Y as an open subset of the Hilbert scheme of a Milnor fiber.

Floer Homology on the Extended Moduli Space

Using the conjugation symmetry on Heegaard Floer complexes, we define a three-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to

A generalized asynchronous computability theorem

\mathbb{Z}_4

An untwisted cube of resolutions for knot Floer homology

-equivariant Seiberg-Witten Floer homology. Further, we obtain two new invariants of homology cobordism,

On the algebra of cornered Floer homology

\underline{d}

INVOLUTIVE HEEGAARD FLOER HOMOLOGY AND PLUMBED THREE-MANIFOLDS

and

A combinatorial description of knot Floer homology

\bar{d}