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 movesPS, §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
9/11Alexander polynomial, part 2 L, p. 51-58 Homework 2, due 9/18
9/16Skein relations, Kauffman bracket PS, p. 23-28
9/18Jones polynomial PS, p. 29-32
9/23Crossing number of alternating links L, p. 41-45
9/25Introduction to braids PS, p. 47-52 Homework 3, due 10/2
9/30Alexander and Markov theorems PS, p. 54-60
10/2Morton-Franks-Williams inequality Article
10/7MFW inequality, finished
10/9Braids and mappling class groups PS, p. 61-65
10/14Dehn-Lickorish theorem PS, p. 90-93
10/16Mapping class fundamentals FM, p. 31-42, 55-57
10/21Interactions between Dehn twists FM, p. 72-78, 81-85 Homework 4, due 10/30
10/23Heegaard splittings of 3-manifolds PS, p. 67-71, 75-77
10/28Lens spaces PS, p. 77-80
10/30Dehn surgery PS, p. 84-86, 98-100
11/4Handles, Morse theory Wikipedia
11/6 4-manifolds, equivalent surgeries PS, p. 88-90, 105-109
11/11Kirby calculus PS, p.117-122; Article Homework 5, due 11/20
11/13Framed diagrams; skein algebras PS, p. 122-124, 165-169
11/18Temperley-Lieb algebra PS, p. 170-171, 177
11/20Jones-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