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; 3-colorings; the number of 3-colorings as a simple
invariant. |
Operations |
3. The number of 3
colorings |
3-colorings as an F3
vector space; computing the number of 3-colorings from a diagram; |
3-colorings |
4. The unknotting number, etc. |
p-colorings; 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 |
Oriented-1 |
9. The oriented state model |
The oriented state model of the
Jones' polynomial - II |
Oriented-2 |
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 3-manifolds |
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 non-isotopic knots , and, therefore, is not a complete invariant. The knot quandle allows to distinguish some knots that we could not distinguish using the 3-coloring 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 3-manifolds |
||
March 15th |
John Weng |
The fundamental group of a knot |