Knot tabulation

Abstract: Hyperbolic Knots - Colin Adams. Braids: A Survey - Joan S. Birman and Tara E. Brendle. Legendrian and Transversal Knots - John B. Etnyre. Knot Spinning - Greg Friedman. The Enumeration and Classification of Knots and Links - Jim Hoste. Knot Diagrammatics - Louis H. Kauffman. A Survey of Classical Knot Concordance - Charles Livingston. Knot Theory of Complex Plane Curves - Lee Rudolph. Thin Position in the Theory of Classical Knots - Martin Scharlemann. Computation of Hyperbolic Structures in Knot Theory - Jeff Weeks .

Abstract: We give a brief historical overview of the Tait conjectures, made 120 years ago in the course of his pioneering work in tabulating the simplest knots, and solved a century later using the Jones polynomial. We announce the solution, again based on a substantial study of the Jones polynomial, of one (possibly his last remaining) problem of Tait, with the construction of amphicheiral knots of almost all odd crossing numbers. An application to the non-triviality problem for the Jones polynomial is also outlined. 1. The first knot tables Knot theory originated in the late 19th century. At that time, W. Thomson (“Lord Kelvin”), P. G. Tait and J. Maxwell propagated the vortex-atom theory in an attempt to explain the structure of the universe. They believed that a supersubstance, ether, makes up all of matter, and atoms are knotted tubes of ether. Knotting is hereby understood as tying a piece of rope and then identifying both its ends so that the tying cannot be any more undone. Thus, in the realm of constructing a periodic table of elements, Tait began the catalogization of the simplest knots. He depicted knots (as we still do today) by diagrams consisting of a (smooth) plane curve with transverse self-intersections, or crossings. At each crossing one of the two strands passes over the other. The term “simplest” refers to the number of crossings of the diagram. We can define the crossing number of a knot as the minimal crossing number of all its diagrams, and say that Tait sought the list of knots with given (small) crossing number. The list meant to present each knot by exactly one diagram. This entails that knots from different diagrams should be inequivalent, in the sense that one cannot turn a (closed) piece of rope knotted one way into one knotted the other way without cutting the rope. The simplest knots are shown in figure 1. The leftmost one, of crossing number 0, is the trivial knot or unknot. It has some special importance, much like the unit element in a group. Tait completed the list up to 7 crossings. Little, Kirkman, later Conway [Co] and others took over and continued his work. In the modern computer age, tables have reached the knots of 17 crossings, with millions of entries, even though Tait’s vortex-atom theory has long been dismissed. An account on knot tabulation is Received by the editors May 30, 2007. 2000 Mathematics Subject Classification. Primary 57M25; Secondary 01A55, 01A60.

Abstract: A study is made of the factorization of prime knots into tangles. Several infinite families of knots which do not factor into prime tangles are examined, and a new characterization of knot primality is developed. This paper examines the factorization of knots into tangles. The notion of tangle was first introduced by J. H. Conway (3) in order to classify knots of small crossing number and to provide a streamlined method of computing certain algebraic knot invariants. F. Bonahon and L. Siebenmann (2) have given a factorization of simple knots into pieces which are either "algebraic" in a certain sense or are "7-hyper- bolic". Further, R. Kirby and W. B. R. Lickorish (6, 7) have offered the notion of prime tangle; intuitively, tangles from which one can construct prime knots. We present here several infinite families of prime knots which do not factor into prime tangles. In the first section we review the definitions and theorems of tangle theory and put forward the notion of string primality. Although our main concern is the classical situation of two strings, our definitions are phrased to apply to higher string numbers. In the second section several families of knots with high degrees of string primality are given; in ?3 it is shown that for certain forms of satellite knots, the string primality of the satellite is characterized by the primality of the companion. The author is indebted to C. McA. Gordon and W. B. R. Lickorish for many helpful discussions and to D. Seal for the explanation of his construction of 2-fold branched covers for the K( p/q) knots of ?2.. The reader is referred to Rolfsen's book (13) for the standard defintions and results of knot theory. Throughout this work a knot is a link of one component and all work is done in the PL category. 1. Definitions and enabling lemmas. A tangle (B, t) is a 3-ball B with a finite

Abstract: There have been many attempts to settle the question whether there exist nontrivial knots with trivial Jonespolynomial. In this paper we show that such a knot must have crossing number at least 18. Furthermore we give the number of prime alternating knots and an upper bound for the number of prime knots up to 17 crossings. We also compute the number of different HOMFLY, Jonesand Alexander polynomials for knots up to 15 crossings.