H. R. Morton

Seifert circles and knot polynomials

In this paper I shall show how certain bounds on the possible diagrams presenting a given oriented knot or link K can be found from its two-variable polynomial P K defined in [3]. The inequalities regarding exponent sum and braid index of possible representations of K by a closed braid which are proved in [5] and [2] follow as a special case.

The Coloured Jones Function

The invariantsJK,k of a framed knotK coloured by the irreducibleSU(2)q-module of dimensionk are studied as a function ofk by means of the universalR-matrix. It is shown that whenJK,k is written as a power series inh withq=eh, the coefficient ofhd is an odd polynomial ink of degree at most 2d+1. This coefficient is a Vassiliev invariant ofK. In the second part of the paper it is shown that ask varies, these invariants span ad-dimensional subspace of the space of all Vassiliev invariants of degreed for framed knots. The analogous questions for unframed knots are also studied.

The coloured Jones function and Alexander polynomial for torus knots

In [2] it was conjectured that the coloured Jones function of a framed knot K, or equivalently the Jones polynomials of all parallels of K, is sufficient to determine the Alexander polynomial of K. An explicit formula was proposed in terms of the power series expansion JKk(h) = Ti < d°=oad(k)h d , where JKk(h) is the SU(2)q quantum invariant of K when coloured by the irreducible module of dimension k, and q = e h is the quantum group parameter. In this paper I show that the explicit formula does give the Alexander polynomial when K is any torus knot.

DISTINGUISHING MUTANTS BY KNOT POLYNOMIALS

We consider the problem of distinguishing mutant knots using invariants of their satellites. We show, by explicit calculation, that the Homfly polynomial of the 3-parallel (and hence the related quantum invariants) will distinguish some mutant pairs. Having established a condition on the colouring module which forces a quantum invariant to agree on mutants, we explain several features of the difference between the Homfly polynomials of satellites constructed from mutants using more general patterns. We illustrate this by our calculations; from these we isolate some simple quantum invariants, and a framed Vassiliev invariant of type 11, which distinguish certain mutants, including the Conway and Kinoshita-Teresaka pair.

Threading knot diagrams

Alexander [ 1 ] showed that an oriented link K in S 3 can always be represented as a closed braid. Later Markov [ 5 ] described (without full details) how any two such representations of K are related. In her book [ 3 ], Birman gives an extensive description, with a detailed combinatorial proof of both these results.

THE HOMFLY POLYNOMIAL OF THE DECORATED HOPF LINK

The main goal is to find the Homfly polynomial of a link formed by decorating each component of the Hopf link with the closure of a directly oriented tangle. Such decorations are spanned in the Homfly skein of the annulus by elements Qλ, depending on partitions λ. We show how the 2-variable Homfly invariant of the Hopf link arising from decorations Qλ and Qμ can be found from the Schur symmetric function sμ of an explicit power series depending on λ. We show also that the quantum invariant of the Hopf link coloured by irreducible sl(N)q modules Vλ and Vμ, which is a 1-variable specialisation of , can be expressed in terms of an N × N minor of the Vandermonde matrix (qij).

The 2-variable polynomial of cable knots

The 2-variable polynomial PK of a satellite K is shown not to satisfy any formula, relating it to the polynomial of its companion and of the pattern, which is at all similar to the formulae for Alexander polynomials. Examples are given of various pairs of knots which can be distinguished by calculating P for 2-strand cables about them even though the knots themselves share the same P. Properties of a given knot such as braid index and amphicheirality, which may not be apparent from the knots polynomial P, are shown in certain cases to be detectable from the polynomial of a 2-cable about the knot.

An irreducible 4-string braid with unknotted closure

Every oriented knot or link in S 3 can be represented in many ways as the closure of a braid β ∈ B n , the braid group on n strings, for some n . Braids β ∈ B n , γ ∈ B m are called closure-equivalent if and are equivalent as oriented knots. It is a well-known result of Markov, see, for example, (l), that β and γ are closure equivalent if and only if there is a sequence of elementary (Markov) moves in which β ∈ B n is replaced by (a) a conjugate in B n , or , or (c) β 1 ∈ B n−1 , where , and the process repeated until γ is reached.

Invariants of Links and 3-Manifolds From Skein Theory and From Quantum Groups

Starting with Kauffman’s bracket polynomial the techniques of linear skein theory are used to present and package a family of polynomial invariants for a framed link. An equivalent family of invariants is derived from representations of the quantum group SU(2) q Specialisation of the variable q leads to invariants of a 3-manifold defined by surgery on a framed link, in terms of the invariants of the link. A similar programme is outlined relating the invariants constructed from the Homfly polynomial to those derived from the quantum groups SU(k) q

Symmetric products of the circle

The n th symmetric product of a topological space, X , is defined to be the quotient of the Cartesian product X n by the action of the symmetric group which permutes the factors. Even if X is a manifold, this product is, in general, not a manifold. The purpose of this note is to determine these products when X is the circle, S 1 , and to show that they are manifolds with boundary.