Spring 2014

Time/Location: MW 12:00-1:20, McGuinn 526

Instructor: Eli Grigsby

Office: Carney 357

Course description: We will survey the many exciting applications of homology-type invariants-in particular, Heegaard-Floer homology and Khovanov homology-to questions in knot theory and low-dimensional topology. These will probably include (and may not be limited to) bounds on the 4-ball genus of knots, detection of the Seifert genus and fiberedness of knots (along with unknot detection), trivial mapping class detection for surfaces with boundary, and obstructions to the existence of exceptional Dehn surgeries. We will review the constructions of Khovanov and Heegaard-Floer homology, as well as historical background on the problems discussed, as needed.

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

• John Milnor and James Stasheff, Characteristic classes

Khovanov homology references:

• Mikhail Khovanov, A categorification of the Jones polynomial

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

Heegaard Floer homology references:

• Peter Ozsváth and Zoltan Szabó, Holomorphic disks and topological invariants for closed three-manifolds

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

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

• Peter Ozsváth and Zoltan Szabó, On Heegaard diagrams and holomorphic disks

• Manolescu, Ciprian, An introduction to knot Floer homology

Topic 1: 4-ball genus of knots and the topological Milnor conjecture

Historical background:

• Peter Kronheimer and Tomasz Mrowka, Gauge theory for embedded surfaces, I (Introduction)

• Lee Rudolph, Some knot theory of complex plane curves

• John Milnor, Singular points of complex hypersurfaces

• Charles Livingston, A survey of classical knot concordance (Sec. 1-3)

Homology-type invariants and the four-ball genus:

• Eun Soo Lee, An endomorphism of the Khovanov invariant

• Jacob Rasmussen, Khovanov homology and the slice genus

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

• Stanislav Jabuka, Heegaard Floer homology and knot concordance: a survey of recent advances

• Ozsváth and Szabó, On knot Floer homology and lens space surgeries

• Louise Moser, Elementary surgery along a torus knot

Lecture 1, 1/13

Lecture 2, 1/15

Lecture 3, 1/17

Lecture 4, 1/22

Lecture 5, 1/24

Lecture 6, 2/24

Lecture 7, 2/26

Lecture 8, 2/28

Lecture 9, 3/12

Lecture 10, 3/17

Lecture 11, 3/19

Lecture 12, 3/24

Lecture 13, 3/26

Lecture 14, 3/31

Lecture 15, 4/2

Lecture 16, 4/9

Lecture 17, 4/11

Lecture 18, 4/16

Lecture 19, 4/23

Lecture 20, 4/28

Lecture 21, 4/30

Problems