Math 9023: Knot Theory and Low-Dimensional Topology

Fall 2018

Meets: Mon/Wed 9:00 - 10:20 AM in Wachman Hall, room 527
Instructor: David Futer
Office: 1026 Wachman Hall
Office Hours: Mon 10:30 - 12:00, Tue 2:30-4:00 PM
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. We will conclude by looking at knot complements via the tools of hyperbolic geometry.

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, P for Purcell.

Day Topic Reference Homework
8/27 Definitions, Reidemeister movesPS, §1
8/29 Tri-colorability and the fundamental group L, p. 11, p. 110-112 Homework 1, due 9/5
9/5 Seifert surfaces L, p. 15-18
9/10The linking number Rolfsen; Epple article
9/12 Prime factorization L, p. 19-21; Hedegard, p. 22-29 Homework 2, due 9/19
9/17 Alexander polynomial, part 1 L, p. 49-51
9/19Alexander polynomial, part 2 L, p. 51-58
9/24Skein relations, Kauffman bracket PS, p. 23-28
9/26Jones polynomial PS, p. 29-32Homework 3, due 10/3
10/1Crossing number of alternating links L, p. 41-45
10/3Introduction to braids PS, p. 47-52
10/8Alexander and Markov theorems PS, p. 54-60
10/10Morton-Franks-Williams inequality ArticleHomework 4, due 10/17
10/15Braids and mappling class groups PS, p. 61-65
10/17Dehn-Lickorish theorem PS, p. 90-93
10/22Heegaard splittings of 3-manifolds PS, p. 67-71, 75-77
10/24Lens spaces, Dehn surgery PS, p. 77-80, 84-86 Homework 5, due 10/31
10/29Introduction to hyperbolic knots P, Chapter 1
10/31Hyperbolic structure on the figure-8 knot P, Chapter 2
11/5 Hyperbolic structures on surfaces P, Chapter 3
11/7 Developing map and completeness P, Chapter 3 Homework 6, due 11/16
11/12 Gluing and completeness equations P, Chapter 4
11/14 Gluing and completeness equations P, Chapter 4
11/26 Completion and Dehn filling P, Chapter 6
11/28 Presentation: Khanh, Rebekah
12/3 Presentation: Abeer, Dong Bin
12/5 Presentation: Rosie
12/10 Presentation: Kyle, Ben

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dfuter at temple edu
Last modified: Fri Aug 21 13:41:22 PDT 2009