Louis H. Kauffman

STATE MODELS AND THE JONES POLYNOMIAL

IN THIS PAPER I construct a state model for the (original) Jones polynomial [5]. (In [6] a state model was constructed for the Conway polynomial.) As we shall see, this model for the Jones polynomial arises as a normalization of a regular isotopy invariant of unoriented knots and links, called here the bracket polynomial, and denoted 〈K〉 for a link projectionK . The concept of regular isotopy will be explained below. The bracket polynomial has a very simple state model. In §2 (Theorem 2.10) I use the bracket polynomial to prove (via Proposition 2.9 and an observation of Kunio Murasugi) that the number of crossings in a connected, reduced alternating projection of a link L is a topological invariant of L. (A projection is reduced if it has no isthmus in the sense of Fig. 5.) In other words, any two connected, reduced alternating projections of the link L have the same number of crossings. This is a remarkable application of our technique. It solves affirmatively a conjecture going back to the knot tabulations of Tait, Kirkman and Little over a century ago (see [6], [9], [10]). Along with this application to alternating links, we also use the bracket polynomial to obtain a necessary condition for an alternating reduced link diagram to be ambient isotopic to its mirror image (Theorem 3.1). One consequence of this theorem is that a reduced alternating diagram with twist number greater than or equal to one-third the number of crossings is necessarily chiral. The paper is organized as follows. In §2 the bracket polynomial is developed, and its relationship with the Jones polynomial is explained. This provides a self-contained introduction to the Jones polynomial and to our techniques. The last part of §2 contains the applications to alternating knots, and to bounds on the minimal and maximal degrees of the polynomial. §3 contains the results about chirality of alternating knots. §4 discusses the structure of our state model in the case of braids. Here the states have an algebraic structure related to Jones’s representation of the braid group into a Von Neumann Algebra.

Virtual Knot Theory

This paper is an introduction to the theory of virtual knots. It is dedicated to the memory of Francois Jaeger.

Braiding operators are universal quantum gates

This paper explores the role of unitary braiding operators in quantum computing. We show that a single specific solution R (the Bell basis change matrix)oftheYang-Baxterequationisauniversalgateforquantumcomputing,in thepresenceoflocalunitarytransformations.Weshowthatthissame Rgeneratesa new non-trivial invariant of braids, knots and links. Other solutions of theYang- Baxter equation are also shown to be universal for quantum computation. The paperdiscussestheseresultsinthecontextofcomparingquantumandtopological points of view. In particular, we discuss quantum computation of link invariants, the relationship between quantum entanglement and topological entanglement, and the structure of braiding in a topological quantum field theory.

Self-reference and recursive forms

Introduction The purpose of this essay is to sketch a picture of the connections between the concept self-reference and important aspects of mathematical and cybernetic thinking . In order to accomplish this task, we begin with a very simple discussion of the meaning of self-reference and then let this unfold into many ideas . Not surprisingly, we encounter wholes and parts, distinctions, pointers and indications, local-global, circulation, feedback, recursion, invariance, self-similarity, re-entry of forms, paradox, and strange loops . But we also find topology, knots and weaves, fractal and recursive forms, infinity, curvature and imaginary numbers! A panoply of fundamental mathematical and physical ideas relating directly to the central turn of self-reference .

A Tutte polynomial for signed graphs

This paper introduces a generalization of the Tutte polynomial [14] that is defined for signed graphs. A signed graph is a graph whose edges are each labelled with a sign (+l or 1). The generalized polynomial will be denoted Q[G] = Q[G](A, B, d). Here G is the signed graph, and the letters A, B, d denote three independent polynomial variables. The polynomial Q[G] can be specialized to the Tutte polynomial, and it satisfies a spanning tree expansion analogous to the spanning tree expansion for the original Tutte polynomial. Planar signed graphs are, by a medial construction, in one-to-one correspondence with diagrams for knots and links. By this correspondence, the polynomial Q[G] specializes to the Kauffman bracket polynomial [5-S] and hence (with a normalization) to the Jones polynomial invariant [3] for knots and links. The Jones polynomial is an important invariant in knot theory. One purpose of this paper is to provide a link between knot theory and graph theory, and to explore a context embracing both subjects. Since the relationship with knots and knot diagrams is the primary motivation for our polynomial, we will explain this connection early in the paper. The first two sections provide graph theoretic and topological background. The reader may wish to begin reading directly in Section 4 and then turn to Section 2 and Section 3 for this background. On the other hand, a direct reading of the sections in order will give an account of the genesis of the polynomial Q[G]. Section 2 discusses chromatic, dichromatic and Tutte polynomials. Section 3 explains the medial graph construction and the relation to the bracket polynomial for unoriented link diagrams. Section 3 also contains a result of independent interest: a reformulation of the definitions of activities in maximal trees (if the graph is disconnected, one should properly refer to maximalforests to denote disjoint collections of trees; we shall speak of trees and ask the reader to read forest for tree when the graphs are disconnected) of a planar graph in terms of properties of Euler trails

The conway polynomial

§l. INTRODUCTION THE CLASSICAL Alexander polynomial[l] A(x) = A&) of a knot or link K C S3 is an invariant of ambient isotopy that is well-defined up to sign and multiplication by powers of the variable x. The Conway polynomial V(z) = V,(z) is a direct invariant of ambient isotopy. This polynomial, first introduced by Conway in[2], has remarkable properties that allow its computation from a knot diagram without recourse to matrices or determinants. It is related to the Alexander polynomial via the potential function V(x x-‘) which is (up to sign and powers of x) equivalent to A(x*). The purpose of this paper is to give an exposition of the Conway polynomial, and to explain the source of its properties by modelling it in analogy to the Alexander polynomial. There is a good geometric explanation for the potential function: this proceeds as follows: Given an oriented knot or link K C S3, let F C S3 be a connected, oriented spanning surface for K. Let 8: H,(F) x H,(F)+2 be the Seifert pairing (see 03). Let 0 also denote any matrix of this pairing with respect to a basis for H,(F). Define the potenfiaI function a,(x) by the formula 0,(x) = 0(x0 -x-‘(Y) where D denotes determinant and 8’ is the transpose of 8. (If H,(F) = 0, Let n,(x) = 1). Our main result is the following:

LINK POLYNOMIALS AND A GRAPHICAL CALCULUS

This paper constructs invariants of rigid vertex isotopy for graphs embedded in three dimensional space. For the Homfly and Dubrovnik polynomials, the skein formalism for these invariants is as shown below. Homfly. Dubrovnik.

Quantum entanglement and topological entanglement

This paper discusses relationships between topological entanglement and quantum entanglement. Specifically, we propose that it is more fundamental to view topological entanglements such as braids as entanglement operators and to associate with them unitary operators that are capable of creating quantum entanglement.

Visualizing quaternion rotation

Quaternions play a vital role in the representation of rotations in computer graphics, primarily for animation and user interfaces. Unfortunately, quaternion rotation is often left as an advanced topic in computer graphics education due to difficulties in portraying the four-dimensional space of the quaternions. One tool for overcoming these obstacles is the quaternion demonstrator, a physical visual aid consisting primarily of a belt. Every quaternion used to specify a rotation can be represented by fixing one end of the belt and rotating the other. Multiplication of quaternions is demonstrated by the composition of rotations, and the resulting twists in the belt depict visually how quaternions interpolate rotation. This article introduces to computer graphics the exponential notation that mathematicians have used to represent unit quaternions. Exponential notation combines the angle and axis of the rotation into concise quaternion expression. This notation allows the article to present more clearly a mechanical quaternion demonstrator consisting of a ribbon and a tag, and develop a computer simulation suitable for interactive educational packages. Local deformations and the belt trick are used to minimize the ribbons twisting and simulate a natural-appearing interactive quaternion demonstrator.

Ray tracing deterministic 3-D fractals

As shown in 1982, Julia sets of quadratic functions as well as many other deterministic fractals exist in spaces of higher dimensionality than the complex plane. Originally a boundary-tracking algorithm was used to view these structures but required a large amount of storage space to operate. By ray tracing these objects, the storage facilities of a graphics workstation frame buffer are sufficient. A short discussion of a specific set of 3-D deterministic fractals precedes a full description of a ray-tracing algorithm applied to these objects. A comparison with the boundary-tracking method and applications to other 3-D deterministic fractals are also included.