
Math
9023: Knot Theory and LowDimensional 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 
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. Finally, we will use these constructions to gain a glimpse of
several skeintheoretic and quantum
invariants of 3manifolds.
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, FM for FarbMargalit.
Day 
Topic 
Reference 
Homework 
8/26  Definitions, Reidemeister moves  PS, §1 
 8/28  Tricolorability, fundamental group  L,
p. 110112  Homework 1, due 9/4
 9/2  Seifert surfaces  L, p. 1518 
 9/4  Prime factorization  L, p. 1921 
 9/9  Alexander polynomial, part 1  L,
p. 4951 
 9/11  Alexander polynomial, part 2  L, p. 5158 
Homework 2, due 9/18
 9/16  Skein relations, Kauffman bracket  PS, p. 2328 
 9/18  Jones polynomial  PS, p. 2932 
 9/23  Crossing number of alternating links  L, p. 4145 
 9/25  Introduction to braids  PS,
p. 4752  Homework 3, due 10/2
 9/30  Alexander and Markov theorems  PS, p. 5460 
 10/2  MortonFranksWilliams inequality  Article 
 10/7  MFW inequality, finished  
 10/9  Braids and mappling class groups  PS, p. 6165 
 10/14  DehnLickorish theorem  PS, p. 9093 
 10/16  Mapping class fundamentals  FM, p. 3142, 5557 
 10/21  Interactions between Dehn twists  FM, p. 7278,
8185  Homework 4, due 10/30
 10/23  Heegaard splittings of 3manifolds  PS, p. 6771, 7577 
 10/28  Lens spaces  PS, p. 7780 
 10/30  Dehn surgery  PS, p. 8486, 98100 
 11/4  Handles, Morse
theory  Wikipedia 
 11/6  4manifolds, equivalent surgeries  PS, p. 8890, 105109 
 11/11  Kirby calculus  PS,
p.117122; Article  Homework
5, due 11/20
 11/13  Framed diagrams; skein algebras  PS, p. 122124,
165169 
 11/18  TemperleyLieb algebra  PS, p. 170171, 177 
 11/20  JonesWentzl idempotent  PS, p. 172176 
 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
