MT855

MT 855: Surfaces, braids, and homology-type invariants

Spring 2016

Time/Location: WF 3:00-4:15, Maloney 560

Instructor: Eli Grigsby

Office hours: Th 2-4, Maloney 522 or by appointment

Course description: I will discuss some 2-, 3-, and 4-dimensional aspects of braid theory from both an algebraic and topological point of view, emphasizing connections to contact/symplectic geometry. Along the way, I will introduce tools from Khovanov homology and Heegaard Floer homology that have proven to be quite useful in studying smoothly properly imbedded surfaces in the 4-ball (e.g.: the knot Floer complex as a (Z+Z)-filtered object, the tau invariant, the s invariant, and their younger cousins: Hom's epsilon invariant and Ozsvath-Stipsicz-Szabo's Upsilon invariant). We'll also probably spend a reasonable amount of time talking about the slice-ribbon conjecture, since you should all be working on it...once you get tenure. (J/K. Sort of.)

What follows is an incomplete list of books/articles to keep on your bedside table for reference.

Obviously, we will be able to recite each one by heart by end-of-semester.

Braids:

• Joan Birman and Tara Brendle, Braids: a survey

• John Milnor, Singular points of complex hypersurfaces

• Lee Rudolph, Knot theory of complex plane curves

Slice-Ribbon conjecture:

• Chuck Livingston and Swatee Naik, Introduction to knot concordance (work in progress)

• A nice math overflow discussion

• Andrew Casson and Cameron Gordon, A loop theorem for duality spaces and fibred ribbon knots

• Francis Bonahon, Ribbon fibred knots, cobordism of surface diffeomorphisms, and pseudo-Anosov diffeomorphisms

• Katura Miyazaki, Nonsimple, ribbon fibered knots

• Ken Baker, A note on the concordance of fibered knots

• Tetsuya Abe and Keiji Tagami, Fibred knots with the same 0-surgery and the slice-ribbon conjecture

Khovanov-Lee homology:

• Mikhail Khovanov, A categorification of the Jones polynomial

• Dror Bar-Natan, On Khovanov's categorification of the Jones polynomial

• Eun Soo Lee, An endomorphism of the Khovanov invariant

• Jacob Rasmussen, Khovanov homology and the slice genus

• Dror Bar-Natan and Scott Morrison The Karoubi envelope and Lee's degeneration of Khovanov homology

• Dror Bar-Natan Fast Khovanov homology computations

• Olga Plamenevskaya, Transverse knots and Khovanov homology

• John Baldwin and Eli Grigsby, Categorified invariants and the braid group

• Eli Grigsby and Tony Licata and Stephan Wehrli, Annular Khovanov homology and knotted Schur-Weyl representations

Heegaard Floer homology:

• Peter Ozsváth and Zoltan Szabó, Holomorphic disks and knot invariants

• Jacob Rasmussen, Floer homology and knot complements (PhD thesis)

• Ozsváth and Szabó, Knot Floer homology and the four-ball genus

• Matt Hedden, Notions of positivity and the Ozsvath-Szabo concordance invariant

• Jen Hom, A survey on Heegaard Floer homology and concordance

• Peter Ozsváth and Andras Stipsicz and Zoltan Szabó, Concordance homomorphisms from knot Floer homology

• Chuck Livingston, Notes on the concordance invariant Upsilon

Other useful (for me) background references:

• Glen Bredon, Topology and Geometry

• Robert Gompf and András Stipsicz, 4-manifolds and Kirby Calculus

• Victor Guillemin and Alan Pollack, Differential Topology

• Allen Hatcher, Algebraic Topology*

• Burak Ozbagci and András Stipsicz, Surgery on Contact 3-manifolds and Stein Surfaces*

*These may or may not be available in electronic form! Email me for assistance.

Problems