Knot Floer homology and bordered invariants


Instructor: Peter Ozsváth
phone: 609-258-4222
office: 1106
TA: Cole Hugelmeyer

The course:

This course is a continuation of Zoltán Szabó's lecture course Bordered algebras and a bigraded knot invariant. The course will start with a quick overview of Heegaard Floer homology, and its associated knot invariant. After describing this construction, we describe computational techniques, focusing on one inspired by "bordered Floer homology", proving that the invariant constructed in Szabo's course in fact agrees with a version of knot Floer homology.

The course will give details to a program outlined in An overview of knot Floer homology (P.S.O. and Z. Szabó); aiming to identify the holomorphic knot invariant with the bordered knot invariant. I have attempted to give a list of background and further reading for each lecture below. Much of the math draws upon available preprints and papers, but some of it is not yet available; so I have attempted to suggest similar materials instead.

This page is a work in progress. Please keep checking it for updates. In particular, the precise syllabus from lecture to lecture is subject to change. For example, the exact content of Lecture 3 will depend a little on what is covered in the first course.

Announcements:

Relevant announcements may well appear here.

Lecture 1: An introduction to Heegaard Floer homology.

Heegaard Floer homology is an invariant for three-manifolds constructed using Lagrangian Floer homology. In this first lecture, I will introduce this invariant, starting with topological background material material. For background reading, see:
  • An introduction to Heegaard Floer homology. (P.S.O. and Z. Szabó)

    For further reading:
  • Lectures on Heegaard Floer homology (P.S.O. and Z. Szabó) are notes from a lecture course, giving applications of the theory.
  • Holomorphic disks and topological invariants for closed three-manifolds (P.S.O. and Z. Szabó) is the original paper defining Heegaard Floer homology.
  • Holomorphic disks and three-manifold invariants: properties and applications (P.S.O. and Z. Szabó) fundamental properties of Heegaard Floer homology.

    Lecture 2: An introduction to knot Floer homology.

    Knot Floer homology is an adaptation of Heegaard Floer homology to an invariant for knots in three-manifolds; and indeed in this lecture we will specialize to knots in the three-sphere. Formally, this knot invariant is a bigraded vector space whose Euler characteristic, suitably interpreted, is the Alexander polynomial. I will sketch the construction of knot Floer homology and outline some of its properties.

    The original references for knot Floer homology are:
  • Holomorphic disks and knot invariants (P.S.O. and Z. Szabó)
  • Floer homology and knot complements (J. A. Rasmussen).

    The bordered knot invariant was defined in Zoltán's course; it is described in the following two papers:
  • Kauffman states, bordered algebras, and a bigraded knot invariant (P.S.O. and Z. Szabó) gives an algebraic definition of a simplified version of the bordered invariant, and a proof of its invariance properties.
  • Bordered knot algebras with matchings(P.S.O. and Z. Szabó) generalizes the constructions from the previous paper.

    Further remarks. The bordered knot invariant can be viewed as an alternative combinatorial definition of knot Floer homology. There is another easier to define combinatorial definition, using ``grid diagrams'' (which, however, is much more computationally intensive).
  • A combinatorial description of knot Floer homology (C. Manolescu, P.S.O., and S. Sarkar), for the original paper
  • On combinatorial link Floer homology (C. Manolescu, P.S.O., Z. Szabó, and D. P. Thurston), for a development of ``grid homology'' as a knot invariant.
  • Grid homology for knots and links (P.S.O., A. Stipsicz, Z. Szabó): for a book developing the grid homology approach to knot Floer homology. (This also contains an introduction to knot theory, in Chapter 2).

    Lecture 3: Decomposing Heegaard diagrams: algebraic aspects.

    Bordered Floer homology is an invariant for three-manifolds with boundary, defined in collaboration with Robert Lipshitz and Dylan Thurston, as a tool for computing the U=0 specalization of Heegaard Floer homology. It comes as an algebraic package, which associates to an oriented, parameterized surface a differential graded algebra; to a three-manifold with parameterized boundary a (suitable) module over that algebra. In fact, there are two possible types of modules one obtains, according to whether or not the orientations agree. Finally, to a three-manifold decomposed along a separating surface, the Heegaard Floer homology is computed as a pairing between the two modules. The knot invariants discussed here is inspired by that bordered theory.

    In this lecture, I will describe whatever algebraic background is needed from bordered Floer homology which has not already been covered in Zoltán's course. Topics might include:
  • A-infinity algebras and modules over them
  • ``Type D'' structures
  • The box product pairing between A-infinity modules and type D structures
  • A generalization to bimodules (either here, or in Lecture 6)
    References:
  • See Bernhard Keller's Introduction to A-infinity algebras and modules for a nice introduction to A-infinity algebras
  • Bordered Heegaard Floer homology (R. Lipshitz, P.S.O., and D. P. Thurston) is a reference on bordered Floer homology; Chapter 2 focuses on the algebraic aspects which will be discussed in this lecture (including type D structures and the box product).
  • Bimodules in bordered Heegaard Floer homology (R. Lipshitz, P.S.O. and D. P. Thurston) is mostly beyond the scope of this lecture; but Chapter 2 gives a further description of the necessary algebra, including the case of bimodules.

    Lecture 4: Decomposing Heegaard diagrams and holomorphic curve counting.

    I will describe geometric aspects of the curve counting used for bordered invariants.

    References and further reading:
  • A cylindrical reformulation of Heegaard Floer homology (R. Lipshitz) is used to simplify the holomorphic curves counted in the Heegaard Floer differential.
  • Chapter 5 Bordered Heegaard Floer homology discusses the holomorphic curves considered in bordered Floer homology. Although the precise complications that arise here are slightly different from the ones that arise in the study of the bordered knot invariant, that chapter should serve as a useful reference.

    Lecture 5: Gluing Heegaard diagrams and holomorphic curve counting.

    I will explain how to reconstruct the pseudo-holomorphic curve counting in a Heegaard diagram from the algebraic counts coming from the two sides obtained by splitting the diagram. The result is a pairing theorem, analogous to the pairing theorem for bordered Floer homology; compare Chapter 9 of Bordered Heegaard Floer homology.

    Lecture 6: Chopping up and reassembling Heegaard diagrams.

    I will sketch the remaining material needed to identify knot Floer homology with the bordered knot invariant from Zoltán's lecture course. Key points are the following:
  • decomposing the Heegaard diagram further into smaller pieces, corresponding to crossings, cups, and caps
  • defining type DA bimodules associated to smaller pieces
  • generalizing the pairing theorem to the case of bimoudles
  • counting pseudo-holomorphic curves in the basic pieces

    Further reading:
  • Computing HF^ by factoring mapping classes (R. Lipshitz, P.S.O., and D. P. Thurston) gives an analogous algebraic description of the U=0 specialization of Heegaard Floer homology.
  • Combinatorial Proofs in Bordered Heegaard Floer homology (B. Zhan) gives a combinatorial verification of the invariance of the algebraic description.