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Math
9023: Knot Theory and Low-Dimensional Topology
Fall 2014
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.
Prerequisites: Math 8061-62 or permission of the instructor.
Grading: Grades will be assigned based on homework and a presentation
toward the end of the semester.
Class Schedule and Homework
This table will be gradually filled in as the course progresses. L stands for
Lickorish, PS for Prasolov-Sossinsky, FM for Farb-Margalit.
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 | |
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Last modified: Fri Aug 21 13:41:22 PDT 2009
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