Knot Theory

Instructor: Louis H. Kauffman


Web page:

Course Number 35672. Meeting in Taft Hall Room 305. MWF 1:00PM to 1:50PM. Office Hours MWF 2:00PM to 3:00PM in 533 SEO.

This is a course in knot theory. We will study basic knot invariants such as the fundamental group and quandle, the Jones polynomial, quantum invariants and Vassiliev invariants and then Khovanov homology and the relationships between quantum groups (Hopf algebras) and link homology theories. We will also discuss relationships with combinatorics and graph theory and with physics, as time permits.

Specific assignments and notes will appear on these pages.

Notes and Homework . Here are notes and homework assignments.

Please read: Bracket , More Bracket, Marker States Remarks on Marker States, Clock Theorem and Coloring and Intro to Topological Quantum Field Theory.

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 .

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 0 -- An introduction to Khovanov homology motivating the algebraic constructions from the idea of categorification. .

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.

Notes on Fox Calculus .

Ralph Fox's Quick Trip Through Classical Knot Theory .

  • 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.

    Dennis Roseman's generalization of the Reidemeister moves to handle immerision diagrams for surfaces embedded in four-dimensional space. .

  • 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:, searching under mathematics.

    ArxivPapers LK papers on the Arxiv.

    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..

    UnKnots and DNA -- Download .

    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 .

    See From Tangle Fractions to DNA -- pdf Download for notes on rational tangles, rational knots and applications to DNA. This is a survey paper by LK and Sofia Lambropoulou. See Knots and DNA -- pdf Download for an essay on topology and DNA by DeWitt Sumners.

    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.