- Louis H. Kauffman
- Department of Mathematics, Statistics, and Computer Science
- University of Illinois at Chicago
- Chicago, IL 60607-7045
- Phone: (312)-996-3066
- E-Mail: kauffman AT uic.edu

- .

- I am a topologist working in knot theory and its relationships with statistical mechanics, quantum theory, algebra, combinatorics and foundations. This material is based upon work supported by the National Science Foundation under Grant No. DMS - 0245588 (and previous grants) and by a grant to study quantum computation and information theory under the auspices of the Defense Advanced Research Projects Agency (DARPA).

See BLOG.

I teach Math 313 - Mathematical Analysis I - in Fall 2014. See Analysis

I teach Math 442 - Differential Geometry - in Fall 2014. See Geometry

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FORM Papers on Form. FORM = FORM(DISTINCTION). "The form we take to exist arises from framing nothing." "We take the form of distinction for the form."

KNOTS. See MATH 569. (History: From September 1, 2003 to August 31, 2004 at the University of Waterloo and the Perimeter Institute in Waterloo, Canada. In July 2004, a workshop on knots and applications at the MSRI workshop at University of British Columbia in Vancouver, BC. Knot Theory in Trieste at the Abdus Salam International Centre for Theoretical Physics in May 2007. Jyvaskyla, Finland in August 2007. Trieste in May 2009. This link also applies to Math 569 for Spring 2007, Math 568 for Spring 2009 and Math 569 for Spring 2011.)

I taught Math 435 - Abstract Algebra - in Spring 2014. See Algebra

I taught Math 300 - Writing Mathematics - in Fall 2013. See Writing! See also the Box Algebra Exercise. This is an exercise in the writing course and it is an introduction to the mathematics of G. Spencer-Brown and Charles Sanders Peirce.

I taught Applied Linear Algebra, Math 310 in Fall 2013. See Vector Algebra.

I taught 215 - Introduction to Advanced Mathematics - in Spring 2013. See Proofs! for the course webpage. If you google "Eccles, An Introduction to Mathematical Reasoning", you will obtain information about the book for the course,

I taught Math 548 - Algebraic Topology - in Spring 2013. See Homology! for the course webpage.

I taught Calculus, Math 181 in Fall 2010. See Calculus.

I taught Math 423 - Graph Theory - in Spring 2010. See Graph Theory

I taught a course in knot theory and topological visualization. For some notes about knots in the the seven color map on the torus see the pdf file The Knot in the Seven Color Map.

See KNOTS IN CHICAGO, Sept 10-12, 2010.

Here is a collection of internal and external links.

- Just for a beginning, before the knots begin to appear. Let's consider:
- What is a knot? If it is in the plane, then it is not. If it is not then it is a knot.
- Along with knots and nots, I am fascinated by PARADOX.
- Let G be defined by the equation
- Gx = F(xx).
- Then
- GG=F(GG).
- Thus, if J=GG then J=F(J) for any F! (This is the fixed point theorem of Church and Curry in the untyped lambda calculus). The fixed point theorem gets you very quickly to paradox. For example, let AB mean "A is a member of B". Let Rx = not(xx). (That is, x is a member of R only if x is not a member of x.) Then RR = not(RR). R is a member of R only if it is not a member of R! R's Self-Membership is in a state of doubt.
- Now imagine a simple loop of rope. Allow that when a bit of line passes underneath another bit of line, we shall say that the underpassing bit "belongs" to the overpassing bit. Membership by underpassage.
- The simple loop is then an empty "knot set". Put a twist in the loop and it underpasses itself. The singly twisted loop is a member of itself. Loop and twisted loop are topologically equivalent. Hence, speaking {topo}logically, the simple loop is both a member of itself and not a member of itself. By this simple twist of logic, the paradox becomes a phenomenon of three dimensional space.

- Paradox is not all there is to knots. The problem of finding invariants of knots has led to extraordinary connections of knot theory with many different fields. In the next few paragraphs, I will comment on some of these connections.

- Here is a brief introduction to the subject of knots and statistical mechanics. The motivating idea is that a physical system has many different configurations that it assumes over time. These are the "states" of the system. Significant physical quantities are obtained by averaging over all the states of the system. By analogy, a topological space may have a natural collection of states associated with it. Significant topological quantities may be obtained by averaging over the states of the topological space!
- We shall describe the state summation model for the bracket polynomial and its relation to the original Jones polynomial. Given a diagram for a knot, it is possible to reduce it to a collection of Jordan curves in the plane by "smoothing" each crossing in one of the two possible ways shown in the diagram below.

- A "state" S of a link diagram is a choice of smoothing for each of its crossings.
- It is of historical interest to realize that the idea of smoothing crossings in a knot diagram was used by the designers of Celtic knots. Starting from a highly regular diagram, the designer smooths collections of crossings to obtain the desired design. To see a demonstration of this, try the following link for celtic knot design.
- We encode the type of smoothing by labelling it "A" or "B" according as the regions that are joined are labelled A or B. This labelling is illustrated above. The four local regions incident at a crossing are labelled A and B with the two A's occupying vertical angles as are the two B's. In this labelling, the two A regions are swept out when the overcrossing line is swept counterclockwise. (This convention pinpoints the assignment of A's and B's.) Thus a state is decorated with the labels at the sites of its smoothings. We call these labels the "vertex weights" of the state. If K is a link diagram and S is a state of that diagram, let [K|S] denote the product of all of the vertex weights (labels A or B) for that state. Note that [K|S] depends upon the structure of over and under crossings in the link. Let d be a third algebraic variable commuting with A and B (A and B commute with each other). Let ||S|| denote the number of Jordan curves in the state S. The Bracket Polynomial is defined to be the summation
- [K] = SUM Over States S {[K|S]d^||S||}.
- By adjusting the variables A,B and d correctly, the bracket polynomial [K] reads out deep topological information about the link K.
- The correct adjustment turns out to be B=1/A and d=-A^2 -A^(-2) where X^Y denotes X raised to the Yth power. With this adjustment the bracket polynomial is invariant under the basically flat Reidemeister II and III moves and multiplies by -A^3 or -A^(-3) under a type I Reidemeister move. What is a Reidemeister move? In the next incarnation of this page, there will be a hyperlink to a discussion of the Reidemeister moves. They are a simple set of basic moves on link diagrams that generate topological equivalence. The unnormalized bracket polynomial is an invariant of what is called "regular isotopy."
- We normalize the bracket polynomial to create a full invariant of ambient isotopy for knots and links, denoted by f[K](A) with the formula
- f[K](A) = (-A^3)^(-w(K)) [K]
- where w(K) is the sum of the signs of the crossings of the oriented diagram K. The sign of a crossing is plus one if a counterclockwise rotation of the overcrossing line puts it in parallel orientation to the undercrossing line.
- The Jones polynomial (discovered by Vaughan Jones in 1984) can be expressed in terms of the bracket polynomial. The Jones polynomial is modelled by the bracket through the formula
- V[K](t) = f[K](t^(-1/4)).
- It is an open problem whether the Jones polynomial detects knots! That is we can conjecture that V[K](t) =1 implies that K is unknotted for a knot diagram K. So far there is no counterexample to this conjecture.
- On the other hand, there are many pairs of knots K, K' such that K and K' have the same Jones polynomial, but K and K' are topologically distinct. One such pair is shown below. They are KT, the Kinoshita-Terasaka knot, and C the Conway knot. C and KT are MUTANTS of one another; each can be obtained from the other by removing a box with four strands (a "2-tangle") and replacing the box after turning it around by 180 degrees. The KT and the C are both 11 crossing knots, with non-trivial Jones polynomial. They can be distinguished from one another by subtler means. It is also noteworthy that KT and C are the smallest knots with Alexander polynomial equal to one. There are many knots undetectable by the Alexander polynomial, and no classification of them is known.
- Below the picture of KT and C, you will see a twelve crossing diagram of KT. Contemplation of this picture reveals that KT is a ribbon knot, a special form of knot that is "slice" ( i.e. it bounds a smooth disk in the four dimensional ball). A ribbon knot bounds a disk immersed in the three sphere with so-called "ribbon singularities". In a ribbon singularity two arcs from the disk cross transversely. One arc is in the interior of the disk. One arc has its boundary points in the boundary of the disk. It is unknown whether every slice knot is ribbon.