Bordered algebras and a bigraded knot invariant


Instructor: Zoltán Szabó
Office: Fine 805
TA: Konstantinos Varvarezos

The course:

The aim of this mini-course is to present some recent advances in knot Floer homology. The main focus will be the use of algebraic techniques and in particular a version of bordered Floer homology. In the course we will discuss certain algebraic methods (differential graded algebras, D-structures and curved bimodules, box tensor product) and use them to construct a Floer homology invariant for knots in the three-dimensional sphere.

Suggested readings:

  • Knot polynomials and knot homologies (J. Rasmussen).
  • An overview of knot Floer homology (P. Ozsváth. and Z. Sz.)
  • Heegaard Floer homology and alternating knots (P. Ozsváth. and Z. Sz.)

    Reference materials and possible further readings:

  • Kauffman states, bordered algebras, and a bigraded knot invariant (P. Ozsváth. and Z. Sz.) gives an algebraic definition of a simplified version of the bordered invariant.
  • Bordered knot algebras with matchings(P. Ozsváth. and Z. Sz.) generalizes the constructions from the previous paper.
    Note that in the class we are going to work with a simplified curved version of the construction, (compare Chapter 13 of the 'Bordered algebras with matchings' paper).

    Lecture 1: A short introduction to knot invariants and knot homologies.

  • Exercises for the first lecture

    Lecture 2: Differential graded algebras.

    Lecture 3: D-structures

    Lecture 4: Bimodules and A-infinity relations.

    Lecture 5: DA- bimodules for crossings and minimums

    Lecture 6: DD-bimodules and proof of invariance.