Welcome to the REU! We'll spend ten weeks (May 29-August 3) working on a research project in knot Floer homology.
Contact:
Akram, alishahi@math.columbia.edu
Linh, ltruong@math.columbia.edu
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.
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.
Date | Topics | References | Problems |
---|---|---|---|
Introduction. Knots, diagrams, Reidemeister moves, tangles. Overview. | Murasugi: Knot Theory and its applications
Lickorish: An Introduction to Knot Theory |
pset1 | |
Chain complexes and homology | Marcus: notes
Weibel: An introduction to homological algebra |
pset2 | |
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 |
pset5 | |
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 |