Department of Mathematics
Columbia University

Summer Undergraduate Research Program

Knot Floer homology, bordered algebras and double points

Advisors: Akram Alishahi and Linh Truong

Summer 2018

Welcome to the REU! We'll spend ten weeks (May 29-August 3) working on a research project in knot Floer homology.


Location: 528 Mathematics


Knot theory is a classical area of low-dimensional topology, directly connected with the theory of three-manifolds and smooth four-manifold topology. In order to tell knots apart and study their topological properties, we use algebraic invariants that can distinguish them and give us more information.

Below is a reading list. We encourage you to complete these readings before the REU begins.

Pre-REU Reading List

Project Description

The knot Floer chain complex is an invariant for knots defined by Ozsváth and Szabó, and independently, Rasmussen. This invariant can be combinatorially computed from a grid diagram representation of a knot. This definition involves counting "empty rectangles" in the grid diagram. It is natural to elaborate this complex by counting more rectangles. Lipshitz defined a modified version of the knot Floer chain complex which counts rectangles with "double points" in addition to the "empty rectangles". It is an open question whether the modified invariant is equivalent to original knot Floer chain complex. This project involves investigating the relationship between the two invariants.


Tentative Schedule

Date Topics References Problems
Introduction. Knots, diagrams, Reidemeister moves, tangles. Overview. Murasugi: Knot Theory and its applications
Lickorish: An Introduction to Knot Theory
Chain complexes and homology Marcus: notes
Weibel: An introduction to homological algebra
Grid diagrams and grid moves Ozsváth, Stipsicz, Szabó: Grid Homology for Knots and Links, p. 43-49. pset3
Grid homology and double pointed version Ozsváth, Stipsicz, Szabó: Grid Homology for Knots and Links pset4
Tangles and grid diagrams for tangles Petkova - Vértesi: An introduction to tangle Floer homology
Akram: notes on tangles
Differential graded (dg) algebra Lipshitz - Ozsváth - Thurston: Slicing planar grid diagrams
Type D structures for tangles Petkova - Vértesi: An introduction to tangle Floer homology pset6
Type A modules for tangles. Boxed tensor product. pset7

Group meetings will be held at 11am on the following dates in June: 6/14, 6/19, 6/25, 6/27.
Individual meetings will be held on Wednesday 6/27.

Group meetings in July: TBA.

Final presentations will be held on August 1, 2018.

Other resources: