Morwen Thistlethwaite

A SPANNING TREE EXPANSION OF THE JONES POLYNOMIAL

A NEW combinatorial formulation of the Jones polynomial of a link is used to establish some basic properties of this polynomial. A striking consequence of these properties is the result that a link admitting an alternating diagram with m crossings and with no “nugatory” crossing cannot be projected with fewer than m crossings.

Infinite families of links with trivial Jones polynomial

Abstract For each k⩾2, we exhibit infinite families of prime k-component links with Jones polynomial equal to that of the k-component unlink.

Computing Varieties of Representations of Hyperbolic 3-Manifolds into SL(4, ℝ)

The geometric structure on a closed orientable hyperbolic 3- manifold determines a discrete faithful representation ρ of its fundamental group into SO+(3, 1), unique up to conjugacy. Although Mostow rigidity prohibits us from deforming ρ, we can try to deform the composition of ρ with inclusion of SO+(3, 1) into a larger group. In this sense, we have found by exact computation a small number of closed manifolds in the Hodgson- Weeks census for which ρ deforms into SL(4,ℝ), thus showing that the hyperbolic structure can be deformed in these cases to a real projective structure. In this paper we describe the method for computing these deformations, particular attention being given to the manifold Vol3.

Kauffman's polynomial and alternating links

These theorems result in quick tests which will often distinguish between links having alternating diagrams with the same number of crossings; links having reduced alternating diagrams with different numbers of crossings are automatically distinguished by Corollary 1 of [12] (proved also in [4] and [9]). Also, as L. H. Kauffman has pointed out, it follows from Theorem 1 that any reduced alternating diagram of an amphicheiral alternating knot must have writhe 0. K. Murasugi, in [lo], has independently obtained a different proof of Theorem 1; he has discovered a formula relating the extreme powers of t in the Jones polynomial VL(t) with the writhe of the reduced alternating diagram of L and the signature of L. I am grateful to W. B. R. Lickorish for making the following important observation: since F,(u, z) determines VL(t) (see [7]), the combination of these two proofs of Theorem 1 yields the result that, for alternating links L, F,(a, z) determines the signature. That the polynomial F,(a, z) does not determine the signature in general is evidenced by the knot 9,, and its obverse: these knots share the same Kauffman polynomial, yet have different signatures. Theorem 2 may be regarded as a tentative first step towards settling the famous “flyping conjecture” (see [ 111,

LINKS WITH TRIVIAL JONES POLYNOMIAL

2). In proving the results of this article, substantial use is made of the ideas in M. E. Kidwell’s paper [6]. We use the same terminology as in [ 121, except that in deference to custom we now say that a diagram with no nugatory crossing m is reduced (rather than “irreducible”). As in [12], a diagram is prime if it is not decomposable as a non-trivial diagrammatic connected sum .Of course, a prime diagram might represent a composite link, or indeed a split link; moreover, a connected, non-prime diagram can represent a prime link. The reader is also referred to [12] for graph-theoretical definitions and background information on the Tutte polynomial. The original paper introducing the

ON THE STRUCTURE AND SCARCITY OF ALTERNATING LINKS AND TANGLES

Examples of non-trivial 2- and 3-component links are given that cannot be distinguished from the corresponding unlink by means of the Jones polynomial.

Character Varieties For : The Figure Eight Knot

It is shown that the proportion of alternating n-crossing, prime link types amongst all n-crossing prime link types tends to zero exponentially with increasing n. Also, a characterization is established for essential annuli in alternating tangles, and a simple criterion is given for equivalence of alternating tangles.

THE LINEAR GROWTH IN THE LENGTHS OF A FAMILY OF THICK KNOTS

ABSTRACT We give a description of several representation varieties of the fundamental group of the complement of the figure eight knot in PGL (3, ℂ) or PSL (3, ℂ). We obtain a description of the projection of the representation variety into the character variety of the boundary torus into SL (3, ℂ).

Zariski Dense Surface Subgroups in SL(4

For any given knot K, a thick realization K0 of K is a knot of unit thickness which is of the same knot type as K. In this paper, we show that there exists a family of prime knots {Kn} with the property that Cr(Kn)→∞(as n→∞) such that the arc-length of any thick realization of Kn will grow at least linearly with respect to Cr(Kn).

Constructing non-congruence subgroups of flexible hyperbolic 3-manifold groups

ABSTRACT An infinite family of Zariski dense surface groups of fixed genus is exhibited inside , and an account is given of the computational method.