
Math
9023: Knot Theory and LowDimensional 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 2:304:00 PM 
Email:  dfuter
at
temple.edu 
Phone:  (215)
2047854 
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) 3manifold 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 806162 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 PrasolovSossinsky, P for Purcell.
Day 
Topic 
Reference 
Homework 
8/27  Definitions, Reidemeister moves  PS, §1 
 8/29  Tricolorability and the fundamental group  L,
p. 11, p. 110112  Homework 1, due 9/5
 9/5  Seifert surfaces  L, p. 1518 
 9/10  The linking number  Rolfsen; Epple article 
 9/12  Prime factorization  L,
p. 1921; Hedegard,
p. 2229  Homework 2, due 9/19
 9/17  Alexander polynomial, part 1  L,
p. 4951 
 9/19  Alexander polynomial, part 2  L, p. 5158 
 9/24  Skein relations, Kauffman bracket  PS, p. 2328 
 9/26  Jones polynomial  PS,
p. 2932  Homework 3, due 10/3
 10/1  Crossing number of alternating links  L, p. 4145 
 10/3  Introduction to braids  PS,
p. 4752 
 10/8  Alexander and Markov theorems  PS, p. 5460 
 10/10  MortonFranksWilliams
inequality  Article  Homework
4, due 10/17
 10/15  Braids and mappling class groups  PS, p. 6165 
 10/17  DehnLickorish theorem  PS, p. 9093 
 10/22  Heegaard splittings of 3manifolds  PS, p. 6771, 7577 
 10/24  Lens spaces, Dehn surgery  PS, p. 7780,
8486  Homework 5, due 10/31
 10/29  Introduction to hyperbolic
knots  P,
Chapter 1 
 10/31  Hyperbolic structure on the figure8 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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Last modified: Fri Aug 21 13:41:22 PDT 2009
