Topics 
Descriptions 
Lecture Notes (PDF) 

1. Introduction (What is
knot theory all about?) 
Knots and links;
diagrams; simple examples; ambient and planar isotopy; Reidemeister
moves and theorem; number of link components as a simple invariant;
signs of oriented crossings; the writhe of a knot. 
Introduction 
2. Basic operations and
simplest invariants 
Operations on knots: mirror
image, reverse, inverse, connected sum, Whitehead double; the
linking number; the Whitehead link; the linking number of parallel
twisted strands; 3colorings; the number of 3colorings as a simple
invariant. 
Operations 
3. The number of 3
colorings 
3colorings as an F_{3}
vector space; computing the number of 3colorings from a diagram; 
3colorings 
4. The unknotting number, etc. 
pcolorings; the unknotting
number; alternating knots; states of a knot; fun part: the rope trick;
unknotting vs. unbraiding. 
more_invariants 
5. The bracket polynomial 
The bracket polynomial and the
normalized bracket polynomial; 
bracket 
6. The Jones polynomial 
The Jones polynomial via the
bracket; Tait's conjecture on minimality of alternating diagrams. 
Jones 
7. Tait's conjectures 
The proof of Tait's conjecture
via bracket (Jones) polynomial; 
Tait 
8. The oriented state model 
The oriented state model of the
Jones' polynomial  I 
Oriented1 
9. The oriented state model 
The oriented state model of the
Jones' polynomial  II 
Oriented2 
10. Alexander's polynomial 
Conway's approach to Alexander's
polynomial; 
Alexander 
11. Finite type invariants 
Introduction to Vassiliev
invariants 
Vassiliev 
Tuesday, Feb.
22nd,
Prof. Blake Mellor (Loyola Marymount University)will speak on
Knots and Graphs 
Student 
Title 
Reviewed
by 
Erin Colberg 
The
history of Knot
Theory 
Matt and
John 
Jordan Fassler 
Braids, the
Artin group and the
Jones polynomial 
Greg
and Erin 
Kyle Graehl 
The enhanced
linking number 
Marcos
and Steven 
Umut Isik 
Computational
problems in the Braid Group with applications to Cryptography 
Jordan
and Sean 
Matthew Krauel 
The
mathematical theory of
Hitches 
Janet and Erin 
Gregory McNulty 
Generalized
knot polynomials and an application 
Janet and Kyle 
Marcos Morinigo 
Prime
factorization of knots 
John
and Sean 
Steven Read 
The knot
quandle 
Marcos
and Jordan 
Sean Rostami 
Using Braids
to unravel Knots 
Matt and Steven 
Janet Tong 
The surgery of 3manifolds 
Umut and Kyle 
John Weng 
Funudamental
groups and knots 
Umut and
Greg 
Date 
Speaker 
Title 
Abstract 
Handout 
Feb. 1st 
Steven Read 
A discussion of the Knot Quandle 
A quandle is a set with two
operations that satisfy three conditions. For example, there is a
quandle naturally associated to any group. It turns out that one can associate a quandle to any knot. The knot quandle is invariant under Reidemeister moves (and , thus, an invariant of ambient isotopy). However, it fails to distinguish some nonisotopic knots , and, therefore, is not a complete invariant. The knot quandle allows to distinguish some knots that we could not distinguish using the 3coloring invaraint. 
Quandles 
Feb. 10th 
Kyle Graehl 
The enhanced linking number 
TBA 

Feb. 17th 
Marcos Morinigo 
Knot factorization 
A knot is called prime if
it can not be represented as a connected sum of two knots such
that both of these are knotted. Using the notion of a Seifert
surface of a knot , we define a knot's genus, an
additive invariant which allows to prove the
existence of prime knots. Then, after defining an
equivalence relation on all possible ways of factoring a knot, we will
show that there is only one equivalence class. Hence we show that
every knot has a unique (up to order) factorization into prime
components.

Factorization 
Feb. 24th 
Gregory McNulty 
Generalized polynomials and some
applications 
We introduce the two variable Kauffman and HOMFLY polynomial 
Generalizations 
March 3rd 
Sean Rostami 
Using Braids to unravel knots 

March 3rd 
Jordan Fassler 
Bsic properties of Braids 

March 8th 
Umut Isik 
Computational Problems in the Braid Group and Applications to 
We shall be dicussing the word problem and the conjugacy problem 

March 10th 
Matthew Krauel 
The Mathematical Theory of
Hitches 
The mathematical theory of hitches is the analyzing of the 

March 15th 
Janet Tong 
Surgery of 3manifolds 

March 15th 
John Weng 
The fundamental group of a knot 