Knot Theory - Fall 2018 in Novosibirsk, Summer 2019 in Xiamen, Fall 2019 in Beijing and in Novosibirsk.

Instructor: Louis H. Kauffman

E-mail: kauffman@uic.edu

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

Knot theory in Novosibirsk, Akademgorodok, Fall of 2018,2019 and Xiamen and Beijing in 2019.

Here are notes from the course LK gave in Xiamen, China in June 2016: Xiamen Notes.

Here is some background material about fundamental group. Fundamental Group, notes on fundamental group and Van Kampen Theorem. Torus Knots, an excerpt from the book "introduction to Algebraic Topology" by W. Massey. 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.

Here is more background material. 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.

Some guidance. Important background for this course is some familiarity with fundamental group and covering spaces. Quandles are a topic closely related to fundamental group, and we will use the idea of the quandle from time to time. Most of the work in the course will concentrate on combinatorially definied invariants such as the bracket polynomial and relations with statistical mechanics and graph theory. Thus you should not get bogged down in reading the background material. Get on with the job of working with the combinatorial invariants as introduced below.

For tangles and tangle fractions and relations with DNA, please read: Tangle Fractions and DNA , Tangles, Hard Unknots and DNA . We can use bracket tangle calculus to locate infinitely many non-trivial links that cannot be detected by the bracket (or the Jones polynomial). See John Horton Conway paper on tangles, skein identities and enumeration of knots. See also Kawauchi Knot Table, a knot table indicating those knots that are rational and those that are insertions into Conway polyhedra. See research paper by Eliahou, Kauffman and Thistlethwaite and also a survey paper on knots and diagrams by Kauffman. See Section 9 of the latter for a quick introduction to this construction of distinct links with the same Jones polynomial.

For classical knot theory, read Notes on Fox Calculus . Read these notes. We will discuss them in class. Here is a great source paper: Ralph Fox's Quick Trip Through Classical Knot Theory . We will be using these readings for a while and also notes that I shall write and post here.

Read Perko Paper. Do your best to decipher this paper. We will discuss it in class.

See also Combinatorial Knot Theory - a first draft LaTeX version of a Book by L.K. -- This book is an introduction to knot theory and to Witten's approach to knot theory via his functional integral.. I will be modifying these notes during the time of the course.

Here are references for knots made from straight sticks. Colin Adams , Seiya Negami .

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. 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.

Here is some information about hyperbolic structures on knot complements. We will not directly work with this in the course, but it is worth your while to look at it! You may wish to start with Not Knot Film and then 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 and Purcell, Deraux, Jacquemet, for other information about hyperbolic structures on knot complements.

Below you will find the links and assigments of previous courses.

#####################################

2017 Spring 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.

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

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.

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 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.

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 .

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.

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

----------------------------------------------------------------------------------------------

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.

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.

• 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.
• Tietze Proof of Tietze's Theorem for isomorphisms of group presentations.
• Torus Knots An excerpt from the book "introduction to Algebraic Topology" by W. Massey.
• Simplicial Paper by Lickorish on simplicial moves on manifolds.
• Kirby Calculus Excerpt from article by Richard Mandelbaum on Four Manifolds. The article is in BAMS,Vol. 1, January 1980. This is a good survey of four manifold theory before the breakethrough work of Freedman and Donaldson. We will add more information about Four Manifolds after F&D and also more about Kirby Calculus.
• Cancel Knot Conway proof that you cannot cancel knots.

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

• Problem Set #1 First problem set. Note that Problem 2 is erroneous. I should have said: Prove that any K_{m,n} curve on the torus can be mapped to a K_{1,1} curve on the torus. The embedding of the K_{1,1} curve in three space via the standard embedding in the torus is unknotted. But there is no way to throw the K_{n,m} curve to a trivial curve ON the torus by a homeomorphism of the torus (since the curve is non-trivial in the fundamental group of the torus).
• Problem Set #2 These are problems about the quandle and showing how to obtain representations of knot groups to dihedral groups. Correction: In Problem 1, prove that you generate the dihedral group for n odd and that you obtain a subgroup of the dihedral group for n even.
• Problem Set #3 Third problem set.
• Problem Set #4 Fourth problem set.
• Seifert Pairing An excerpt from the book "On Knots" by LK. These sections are on the Seifert pairing and the Arf invariant related to the mod-2 quadratic form defined via the Seifert pairing.
• Cyclic Branched Coverings An excerpt from the book "On Knots" by LK. These sections are on the stucture of cyclic branched covering spaces of knots and of surfaces in four-balls, and the relationship with signatures of knots and four-manifolds.
• LinkSing Excerpt from "On Knots" by LK, about knots as links of singularities, Brieskorn manifolds, exotic spheres, knot products and Arf invariants via band-passing -- to classify exotic shpheres.
• Exotic Spheres Article by LK giving an introduction to characteristic classes, cobordism, Hirzebruch Index Theorem and the existence and construction of spheres with exotic differentiable structures.
• Milnor Spheres John Milnor's original breakthrough article constructing exotic spheres.
• Characteristic Classes Notes by LK giving an introduction to characteristic classes, cobordism, Hirzebruch Index Theorem and the existence and construction of spheres with exotic differentiable structures.
• LHK Thesis 1 Scan of PhD Thesis of LK, "Cyclic Branched Covers, O(n) Actions and Hypersurface Singularities," Princeton University (1972). Pages 37 to 52 are in a separate scan.
• LHK Thesis 2 Scan of PhD Thesis of LK, "Cyclic Branched Covers, O(n) Actions and Hypersurface Singularities," Princeton University (1972). Pages 37 to 52 are in this scan. These pages treat the Seifert pairing for a knot or link in the three-sphere.
• Seifert Scan of the 1930's classic paper "Uber das geschlect von knoten" by H. Seifert.
• Branched Covers Paper by LK on iterative construction of branched covers with applications to Brieskorn manifolds and knot periodicity.
• Link Manifolds Paper by LK on differentiable classification of high dimensional manifolds with O(n) actions and orbit spaces the four-ball, fixed point set corresponding to a link in the three-sphere. Example given of a link where the reversal of an orientation of one component corresponds to the addition of an exotic sphere to the O(n) manifold.
• KNOT PRODUCTS 1 Short paper by LK on a product construction of knots in all dimensions that constructs exotic differentiable structures, Brieskorn manifolds and knot periodicity of the knot cobordism groups in high dimensions.
• KNOT PRODUCTS 2 Notes by LK on the knot product construction. These notes are more informal then the paper with Neumann, and they contain some geometric material that is not in the paper. See also the paper below on generalized twist spinning.
• KNOT PRODUCTS 3 Paper by LK and Walter Neumann about knot products and links of singularities.
• Zeeman's Twist Spun Knots Paper on twist spinning fibration theorem of Christopher Zeeman.
• KNOT PRODUCTS 4 Generalized Twist Spinning - Paper by LK on a generalization of the twist spinning fibration theorem of Christopher Zeeman, using the knot product construction.
• Schoenflies Morton Brown's proof of the Schoenflies Theorem -- A bicollored (n-1)-sphere in an n-sphere divides it into two n-balls.
• Horned Sphere The Alexander Horned Sphere as depicted in "Topology" by Hocking and Young.
• Alexander J. W. Alexander's original paper on the Alexander Polynomial.
• Fundamental Group Notes on fundamental group and Van Kampen Theorem.
• Waldhausen Paper of Waldhausen, from which it follows that the pair (Knot Group, Peripheral Subgroup) classifies knots.
• Conway-Gordon Group A generalization of the fundamental group that completely classifies classical knots.

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.

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.