| Meets: | Tue/Thu 9:30 AM - 10:50 AM in Wachman Hall, room 527 |
| Instructor: | David Futer |
| Office: | 1038 Wachman Hall |
| Office Hours: | by appointment |
| E-mail: | dfuter at temple.edu |
| Phone: | (215) 204-7854 |
Course content: This course will survey the modern theory of knots, coming at it from several very distinct points of view. We will start at the beginning with projection diagrams and the tabulation problem. We will proceed to several classical polynomial invariants, which can be constructed via the combinatorics of diagrams, via representation theory, or via the topology of the knot complement. We will touch on braid groups and mapping class groups, and use these groups to show that every (closed, orientable) 3-manifold can be constructed via knots. Finally, we will use these constructions to gain a glimpse of several skein-theoretic and quantum invariants of 3-manifolds.
Textbooks: We will draw material from the following sources. The selection of topics in Prasolov and Sossinsky is probably closest to the outline that we'll follow.
Grading: Grades will be assigned based on homework and a presentation toward the end of the semester.
| Day | Topic | Reference | Homework |
|---|---|---|---|
| 8/26 | Definitions, Reidemeister moves | PS, §1 | |
| 8/28 | Tri-colorability, fundamental group | L, p. 110-112 | Homework 1, due 9/4 |
| 9/2 | Seifert surfaces | L, p. 15-18 | |
| 9/4 | Prime factorization | L, p. 19-21 | |
| 9/9 | Alexander polynomial, part 1 | L, p. 49-51 | |
| 9/11 | Alexander polynomial, part 2 | L, p. 51-58 | Homework 2, due 9/18 |
| 9/16 | Skein relations, Kauffman bracket | PS, p. 23-28 | |
| 9/18 | Jones polynomial | PS, p. 29-32 | |
| 9/23 | Crossing number of alternating links | L, p. 41-45 | |
| 9/25 | Introduction to braids | PS, p. 47-52 | Homework 3, due 10/2 |
| 9/30 | Alexander and Markov theorems | PS, p. 54-60 | |
| 10/2 | Morton-Franks-Williams inequality | Article | |
| 10/7 | MFW inequality, finished | ||
| 10/9 | Braids and mappling class groups | PS, p. 61-65 | |
| 10/14 | Dehn-Lickorish theorem | PS, p. 90-93 | |
| 10/16 | Mapping class fundamentals | FM, p. 31-42, 55-57 | |
| 10/21 | Interactions between Dehn twists | FM, p. 72-78, 81-85 | Homework 4, due 10/30 |
| 10/23 | Heegaard splittings of 3-manifolds | PS, p. 67-71, 75-77 | |
| 10/28 | Lens spaces | PS, p. 77-80 | |
| 10/30 | Dehn surgery | PS, p. 84-86, 98-100 | |
| 11/4 | Handles, Morse theory | Wikipedia | |
| 11/6 | 4-manifolds, equivalent surgeries | PS, p. 88-90, 105-109 | |
| 11/11 | Kirby calculus | PS, p.117-122; Article | Homework 5, due 11/20 |
| 11/13 | Framed diagrams; skein algebras | PS, p. 122-124, 165-169 | |
| 11/18 | Temperley-Lieb algebra | PS, p. 170-171, 177 | |
| 11/20 | Jones-Wentzl idempotent | PS, p. 172-176 | |
| 12/2 | Presentation: Thomas | ||
| 12/4 | Presentation: Will | ||
| 12/9 | Presentation: Zach, Geoff | ||
| 12/11 | Presentation: Elif, Tim |