# Math 191 Introduction to Knot Theory

### Course information

This  is an introductory course in Knot Theory. There are no formal prerequisites, but some familiarity with linear and abstract algebra, as well as an ability to visualize objects in three dimensions is useful. The course  is  assessable to advanced undergraduate students.

One can imagine a knot as a continuous loop (e.g., made of very thin elastic rubber) in the three-dimensional space. Given a knot, one can ask: is it really knotted? I.e., can it be deformed into a ring without making any cuts? Given two knots, one wants to know whether  one of them can be deformed into the other. In order to answer such questions, we must introduce and be able to compute (e.g., numerical and polynomial) knot invariants. In this class, we will study many different invariants of knots and will see how they allow to distinguish knots.

Knot theory has many relations to topology, physics, and (more recently!) even the study of the structure of DNA. Some of these  connections  were explored in the second part of the class, and in some of the class projects.

Each student in the class will participate in a class-related project (see information below).

### Recommended books:

There are a lot of different  books on Knot Theory. I list below several books which are perhaps the closest to the topics we will study in class and are available at the UCLA library.

1. L. Kauffmann "Knots and Physics".
2. D. Rolfsen "Knots and Links".
3. V.V. Prasolov, A.B. Sossinsky "Knots, Links, Braids and 3-manifolds" .
4. C. Livingston "Knot theory".

### Lecture Notes

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

### Guest lecture

Tuesday, Feb. 22nd,

will speak on

### Knots and Graphs

Abstract:  Graphs are used to model many kinds of networks, from social networks to circuit diagrams.  A common problem is finding the nicest way to draw the graph to present the information most clearly.  In two dimensions, this means having as few crossings as possible.  In three dimensions, we can generalize this to ask whether a graph has a knotted cycle or a pair of disjoint linked cycles.  I will review the question of graphs in the plane, and then show that there are graphs that are "intrinsically linked" - they cannot be drawn in three dimensions without linked cycles!  I will also mention some other results and open problems in this area.

### Class Projects

Throughout the quarter, students in the class worked on class projects, which involved studying a class-related topic, writing a term paper on this topic, and making an in-class presentation.
The students  read original scientific papers as well as review papers and books, and discussed  the material with me on office hours. The goal was  for each student to master a selected topic,  write a term paper on it and make  an in-class presentation.

The titles and abstracts of the presentations, as well as the complete texts of the projects are available below. Each project was read and reviewed by at least two other students in the class. Their reviews are also available at the bottom of the page.

Possible Project Topics
Term Paper Information

Class Projects

 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

### Student Presentations

 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 polynomialinvariants for links, and show that they are nontrivial extensions of the Jones polynomial and are distinct. Using the Kauffman polynomial, we provethe invariance of the writhe for reduced alternating diagrams. Finally,we define an invariant called the arc index and use the Kauffmanpolynomial to produce a lower bound (sharp for alternating links).` 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 toCryptography` `We shall be dicussing the word problem and the conjugacy problemin the braid group B_n. After a quick glance at the terminology of thetheory of computing and complexity, we shall introduce the word andconjugacy problems. We shall then discuss Dehornoy's solution to the wordproblem and the NP-completeness of the NON-MINIMAL BRAIDS problem and endthe talk with the description of two cryptographic protocols based on thebraid group.` March 10th Matthew Krauel The Mathematical Theory of Hitches `The mathematical theory of hitches is the analyzing of theforces involved in the creation and holding of a hitch. By formulatingmathematical representations of tension forces, turnings, crossings, andfriction, the theory of hitches is able to express the fundamentals ofhow hitches work, and provide criteria for whether a hitch will hold ornot and under what conditions.` March 15th Janet Tong Surgery of 3-manifolds March 15th John Weng The fundamental group of a knot