MT 855: Applications of homology-type invariants in low-dimensional topology

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: Khovanov homology references: Heegaard Floer homology references:
Topic 1: 4-ball genus of knots and the topological Milnor conjecture

Historical background:
Homology-type invariants and the four-ball genus:

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