Knot Theory - Spring 2017

**Instructor:** Louis H. Kauffman

**E-mail: ** kauffman@uic.edu

**Web page: **http://www.math.uic.edu/~kauffman

Course Number 39515. Meeting in Taft Hall Room 308. MWF 1:00PM to 1:50PM. Office Hours MWF 4:00PM on M, 3PM on WF in 533 SEO.

This is a course in knot theory, including virtual knot theory, knotoids, knotted surfaces in four-space and higher dimensional knots. We will study basic knot invariants such as the fundamental group and quandle, the Jones polynomial, quantum invariants and Vassiliev invariants and Khovanov homology. Other topics are available and may be explored depending on the interests of the class.

Assignment Number 1. Read Some Knot Theory. In this excerpt from Colin Adams' "The Knot Book", please do the following exercises: 1.1,1.3,1.4,1.5,1.6,1.7,1.10,1.11,1.12, 1.21,1.22,1.24 In problem 1.24 determine the modulus for coloring using a*b = 2b-a for each of the seven crossing knots. Note that Adams studies the three-coloring scheme without directly identifying it as a*b = 2b -a mod 3. Here are some other links that are relevant to this assignment. Kauffman Knot Notes. Tricolorability. Jozef Przytycki. Ben Webster. Knots and DNA. The problems in this assignment are due Wednesday, January 18, 2017.

A Sidetrack: Look at the knot game at Ayaka Shimizu's Game. Play the game! Read Ayaka Shimizu's Paper about the game. Read Ayaka Shimizu's Arxiv Paper proving that knots can be unknotted by region crossing change.

Assignment Number 2. Read Palette Graph and Coloring and Perko Paper. Do your best to decipher these two papers. We will discuss them in class. Read also Wirtinger Presentation Excerpt from the book "Classical Topology and Combinatorial Group Theory" by John Stillwell, on the generators and relations for the fundamental group of a knot complement. Try your hand at proving Ayaka Shimizu's Theorem that on a link diagram decorated with + and - at the crossings, region switching can change the labeling to all + or all -. See the reference to her research paper above. Read Introductory notes on Tempereley Lieb Algebra, and examine Notes from 2015 . Here are notes and problems from the course in 2015. Read the first 14 pages of these notes and you will find some of the material we have covered and a discussion of some fundamental groups and examples. For practice try some of the problems in these notes. Here is more reading. David Joyce Thesis on Quandles is worth reading. Excerpt from book by "A Survey of Knot Theory" A. Kawauchi. This contains an appendix about the differnt notions of isotopy for knots, an appendix on fundamental group and covering space and an extensive bibliography. Dehn Excerpt from "Formal Knot Theory" by Kauffman, explaining how the abelianization of the commutator subgroup of the knot group, leads to the Alexander Polynomial as defined by J. W. Alexander. Dehn Excerpt from "On Knots" by Kauffman, explaining how the abelianization of the commutator subgroup of the knot group, leads to the Alexander Polynomial as defined by J. W. Alexander. Alexander J. W. Alexander's original paper on the Alexander Polynomial. ProblemSetTwo Problems Set Number Two, due on February 22, 2017.

Perko. Here is another explanation by Ken Perko of his method for finding linking numbers in irregular branched coverings along knots.

Assignment Number 3. Please read: Bracket , Bracket on Doubled Strands , Statistical Mechanics and the Jones Polynomial , BaeMorton , Manchon . The last two examine more deeply the span and spread of the bracket polynomial. Homflypt Skein (These notes give an inductive proof of the validity of the regular isotopy version of the Homflypt polynomial.) More Bracket, Marker States Remarks on Marker States, Clock Theorem and Coloring . For Reference On Knots This monograph contains a discussion of the Seifert Pairing model for the Alexander-Conway Polynomial. Here is an excerpt from the book: Seifert Pairing . For Reference Notes on Fox Calculus . Read these notes. We will discuss them in class. Ralph Fox's Quick Trip Through Classical Knot Theory . Problem Set Three: (1) Rewrite the Reidemeister Moves in terms of signed graphs by using the Medial Construction. Explore moving knots around in the graphical language that you have produced. (2) In class we will explain how the moves on signed graphs correspond to moves in electrical circuits that do not disturb the conductance of the circuit and we will explain the Conductance Invariant that results from this observation. Explore this invariant with some examples of your own. (3) Choose a spanning surface (using bands) and redraw the boundary so that you know a concise representation of it and make sure that it is knotted. Then calculate the Seifert Pairing and use it to find the Alexander polynomial of the knotted boundary. (4) Calculate the bracket for some infinite classes of knots and links. For example you can obtain a recursive formula for the (2,n) torus links. Or you can examine other families of diagrams. (5) No one has yet found a non-trivial knot with unit Jones polynomial (i.e. normalized bracket polynomial equal to 1). Try your hand at thinking about this problem and working on it. Report on what you find.

Assignment Number 4. Choose some of the projects in these Project Notes and do the projects and write them up. You can also devise projects of your own. This assignment is due by Wednesday of the last week of class.

See Figure Eight Knot, excerpt from "A Topological Picturebook" by George Francis. See Thurston, excerpt from William Thurston's notes, showing how to see the hyperbolic structure on the complement of the figure eight knot. See Weeks, Jeff Week's notes on hyperbolic structures on knot complements.

See Combinatorial Heegard Floer Knot Homology. Read pages 49-52. The paper itself is about a combinatorial formulation of a homology theory that generalizes the Alexander polynomial. It is a 'categorification' of the Alexander polynomial. This theory is very powerful and can determine the minimal genus for spanning surfaces of any classical knot. I am asking you to look at the authors' formulation of the Alexander polynomial as a determinant associated with a grid diagram. The result is very startling and suggestive. You do not need to read the rest of the paper to understand this section, but will want to look at Notes on Fox Calculus for the way the result is proved.

See An Introduction to Khovanov Homology.

Khovanov Invariant Notes 1 -- Earlier notes. .

Khovanov Invariant Notes 2 -- Earlier notes. .

There is a great deal of material on Khovanov homology on the Arxiv. You should look at the original paper by Khovanov and at the papers by Bar-Natan. I will add links here to more papers shortly.

See Computations in Khovanov Homology.

See From Tangle Fractions to DNA for notes on rational tangles, rational knots and applications to DNA. This is a survey paper by LK and Sofia Lambropoulou. The paper contains a complete introduction to tangle fractions, using both coloring and the bracket polynomial. It also proves a "Knitting Theorem" showing how many recombinations are needed to determine the starting tangle in processive (rational) DNA recombination. See Knots and DNA -- pdf Download for an essay on topology and DNA by DeWitt Sumners.

See Electrical Graphs for an exposition of how to solve for conductance in a graphical network, and then see Matrix Tree Theorem for a proof that the Kirchoff matrix enumerates spanning trees in a graph. See Electric Matrix for matrix verifications of Kirchoff's Laws that state that the sum of the voltage drops around a closed loop is zero, and that the sum of the currents at a node is equal to zero. The local rules for conductance replacement follow from these principles. For more about electrical graphs and topology of knots and links see Electrical Knots for a graphical approach to knots that relates topology to conductance. See Wang Algebra for a clever approach to graphs and spanning trees. See Wang Algebra Progam for a Mathematica program that calculates the spanning trees in a graph via the Wang algebra. See Electricity for an exposition by Duffin of the application of Wang algegra to the calculation of electrical conductance. See Duffin for Duffin's long and intetersting article on this subject. See also Negative Resistance for a realistic discussion of devices that can exhibit negative resistance. We include this reference since the crossings in the knot diagram can be translated into positive and negative conductances.

Below this line are many references and some OLD assignments from previous courses. ##############################################################################################

Please read also: Eliahou, Kauffman, Thistlethwaite . Here we discuss how to make infinitely many links that are not detectable by the Jones polynomial.

Intro to Topological Quantum Field Theory.

Please read the following two papers: Conway , and ConwayGordon. The first paper is the pioneering paper by John Horton Conway, containing the skein formulation of Alexander polynomials, the tangle calculus and methods of enumerating knots and links. The second paper is the paper by Conway and Cameron Gordon, proving that any embedding of the complete graph on seven nodes has a knotted Hamiltonian cycle.

For Reference Formal Knot Theory This monograph contains the state model for the Alexander-Conway Polynomial, using marker states. See also, Alexander J. W. Alexander's original paper on the Alexander Polynomial.

For Reference On Knots This monograph contains a discussion of the Seifert Pairing model for the Alexander-Conway Polynomial.

For Reference Notes on Fox Calculus .

Ralph Fox's Quick Trip Through Classical Knot Theory .

Please also read the following background about electrical circuits and graph theory.

See Electrical Graphs for an exposition of how to solve for conductance in a graphical network, and then see Matrix Tree Theorem for a proof that the Kirchoff matrix enumerates spanning trees in a graph. For more about electrical graphs and topology of knots and links see Electrical Knots for a graphical approach to knots that relates topology to conductance. See Wang Algebra for a clever approach to graphs and spanning trees. See Wang Algebra Progam for a Mathematica program that calculates the spanning trees in a graph via the Wang algebra. See Electricity for an exposition by Duffin of the application of Wang algegra to the calculation of electrical conductance. See Duffin for Duffin's long and intetersting article on this subject.

See also Negative Resistance for a realistic discussion of devices that can exhibit negative resistance. We include this reference since the crossings in the knot diagram can be translated into positive and negative conductances.

See Knots and Hopf Algebras -- pdf Download for a paper on invariants of knots and links derived from Hopf algebras. Please read this paper. A related paper using state models and solutions to the Yang-Baxter equation is Oriented Quantum Algebras -- pdf Download, and a paper emphasizing the categorical algebra is Centrality and the KRH Invariant -- pdf Download.

Khovanov Invariant Notes 1 -- Earlier notes. .

Khovanov Invariant Notes 2 -- Earlier notes. .

Paul Turner's Notes on Khovanov Homology .

Below this line you will find various references that may be of use in the course.

----------------------------------------------------------------------------------------------A Short Course on Knots by LK. .

Joyce Thesis David Joyce's Thesis on Quandles.

Winker Thesis Steve Winker's Thesis on Quandles. The Knot Quandle, independently discovered by John Horton Conway, Gavin Wraith, David Joyce and Sergei Matveev, is a non-associative algebra structure that gives a complete invariant of knots up to reversal of orientation and mirror image. A complete invariant is obtained by adding longitude information to the quandle. There is a large knot theory literature devoted to the use of quandles. Quandle is David Joyce's terminology. Conway and Wraith discovered the quandle when they were undergraduates in the UK and they called it the Wrack. A version of this structure without dependence on the first Reidemeister move was call the Knot Crystal by Kauffman and the Rack by Fenn and Rourke. At the present time the term quandle continues and the term rack continues. Winker's thesis gives a generalization of Cayley diagram for groups that applies to quandles, and he proves that the Involutory Quandle (a weaker unoriented version of the quandle) of a knot detects the unknot.

You should download New Invariants -- pdf Download and possibly Functional Integral and Vassiliev Invariants -- pdf Download . The first is an introduction to the bracket polynomial, Jones polynomial and related invariants. The second is an introduction to Vassiliev invariants, Witten's quantum field theory approach to link invariants. We show in this paper how the Kontsevich integrals arise as Feynman integrals in the (1/k) expansion of Witten's functional integral.

Maps and Spin Nets Paper on bracket polynomial, spin networks and map coloring.

"From Knots to Quantum Groups and Back" by LK -- Download .

"Knot Diagrammatics" by LK -- Download .

See Bracket Calculations -- pdf Download for notes on skein calculations of bracket and Jones polynomial, and for a sketch of how to produce links that are not detectable by the Jones polynomial. This file is meant to be read in conjunction with the next file on Mathematica calculations.

Two Strand Bracket Notes on a state sum formulation for the two-strand bracket polynomial.

See Mathematica Calculates the Bracket Polynomial -- pdf Download for a Mathematica worksheet illustrating how to use a computer to calculate the bracket polynomial. The worksheet contains examples of calculations for the trefoil knot, the figure eight knot, the Hopf link, the knot 9_{42} (first example of a knot that is chiral but whose chirality cannot be detected by the Jones polynomial) and an example due to Morwen Thistlethwaite of a link of two components whose linking is invisible to the Jones polynomial. Read this worksheet in conjunction with the previous file on bracket calculations.

See Knots, Three Manifolds and Topological Quantum Field Theory -- pdf Download for an introduction to invariants of knots and three-manifolds from the point of view of Temperley-Lieb Recoupling Theory. These notes contain a sketch of the Witten-Reshetikhin- Turaev invariant, the Turaev-Viro invariant, and the Crane-Yetter invariant. See WRT Invariant and Virtual Knot Theory -- pdf Download for a paper by Heather Dye and L.K. about virtual three-manifolds.

See Braids and Quantum Gates -- pdf Download for a new paper on relationships between topology and quantum information theory. This paper contains an introduction to the Yang-Baxter Equation, Braid group, Markov Theorem, the construction of a new invariant of knots and links and applications to quantum computing.

Knot Quadratic Forms Notes on knots and quadratic forms, including Seifert's construction of the quadratic form of a knot from its two-fold branched covering space.

Ralph Fox's Quick Trip Through Classical Knot Theory .

Below this point are problems and other matters from previous courses.

Below this point are references relevant to other aspects of knot theory and to previous courses in combinatorial knot theory.

Virtual knot theory is a generalization of classical knot theory that utilizes a crossing that "is not there." This virtual crossing can be interpreted as a detour through a handle that is attached to the plane of projection, and the theory of virtual knots can be understood as a theory of (stabilized) knots and links in thickend surfaces. Here are three papers on virtual knot theory: Virtual Knots - pdf download , Detecting Virtual Knots - pdf download and A Self-Linking Invariant of Virtual Knots - pdf download. For more papers on virtual knot theory by Kauffman and others, please browse the arxiv on the web: www.arxiv.org, searching under mathematics.

ArxivPapers LK papers on the Arxiv.

Notes on Fox Calculus -- Download .

Ralph Fox's Quick Trip Through Classical Knot Theory -- Download .

J.W. Alexander's Paper on his polynomial -- Download .

"Remarks on Formal Knot Theory" by LK -- Download .

Wikipedia entry on Quandles. .

Gridlink -- a program by Marc Culler. .

Topological Quantum Information Theory -- Download .

There are a number of papers available from this website that are either notes about knot theory or explanations of research that are accessible to this course. I encourage you to browse around on the site and on the world wide web as well. Below you will find a few of such things.

It sometimes happens that pdf files print out upside down. Please note that there is a button on the Adobe Acrobat viewer that allows you to rotate all the pages by 90 or 180 degrees.