Paul Melvin

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.

Dedekind sums, μ-invariants and the signature cocycle

0 Introduction The Dedekind eta function, defined by o~ ~(z) = e ~z/12 l-I(1-e 2€ n=l for z in the upper half plane H, plays a central role in number theory. It is a modular form of fractional weight whose 24 th power is proportional to the fundamental discriminant cusp form of weight 12. In particular ~?(z)24dz 6, is invariant under the modular group PSL(2, Z), and so there is a function r PSL(2, Z)-* Z given by (0.1) r) for A= 7r~ c where #(A) = ~ log if c =~ 0, and #(A) = 0 if c = 0. Dedekind [D] gave a formula \ i sign c~ (0.2) r = 2 if c = 0 a+d 12sign(c)s(a,c) if c~0 C in terms of certain arithmetic sums s(a, c) defined for coprime integers a and c by Lcl-I k=l where ((x)) = x-[x]-1/2. These sums, now called Dedekind sums, arise in many contexts and have been intensively studied during the past hundred years. Many of their fundamental properties were discovered by Rademacher, and the function r is

Finite type invariants of 3-manifolds

Contents 1 Introduction 2 2 Finiteness 13 3 The Conway polynomial 20 4 Finite type invariants from quantum invariants 30 5 Combinatorial structure of finite type invariants 47 6 Finite type invariants for spin manifolds 58 7 Finite type invariants for bounded manifolds 61 8 Finite type invariants for marked manifolds 62 9 Further generalizations 65 10 Relationships with other theories and other results 68

Quantum cyclotomic orders of 3-manifolds☆

Abstract This paper provides a topological interpretation for number theoretic properties of quantum invariants of 3-manifolds. In particular, it is shown that the p-adic valuation of the quantum SO(3)-invariant of a 3-manifold M, for odd primes p, is bounded below by a linear function of the mod p first betti number of M. Sharper bounds using more delicate topological invariants are given as well.

A geometric interpretation of Milnor's triple linking numbers

Milnors triple linking numbers of a link in the 3-sphere are interpreted geometrically in terms of the pattern of intersections of the Seifert surfaces of the components of the link. This generalizes the well known formula as an algebraic count of triple points when the pairwise linking numbers vanish.

2-SPHERE BUNDLES OVER COMPACT SURFACES

Closed 4-manifolds which fiber over a compact surface with fiber a sphere are classified, and the fibration is shown to be unique (up to diffeomorphism). It is well known that there are at most two orientable 4-manifolds which fiber over a given compact surface with fiber the 2-sphere S2. (There is exactly one if the surface has nonempty boundary, and two if it is closed.) If the orientability condition is dropped, then the situation becomes more involved. In particular the (mod 2) intersection pairing is no longer sufficient to distinguish among the mani- folds that arise. One must also consider the ?Tl-action on g2 and the peripheral structure. The purpose of this note is to classify all 4-manifolds (orientable or not) which are total spaces of S2-bundles over compact surfaces. We shall work in the smooth category. Since Diff(S2) deformation retracts to 0(3), we may assume that all bundles that arise have 0(3) as structure group. Along the way it is shown that the bundle structures are unique. That is, if any two 4-manifolds, fibered as above, are diffeomorphic, then there is a fiber preserving diffeomorphism between them which is orthogonal on fibers. Our interest in S2-bundles arose in the study of Lie group actions (in particular of S0(3)) on 4-manifolds. The results obtained here are used in the equivariant classification of such actions (MP).

On 4-Manifolds with Singular Torus Actions

This paper studies the homeomorphism classification problem for closed oriented 4-manifolds which support effective codimension-two torus actions. With suitable restrictions on the actions a classification is obtained in terms of numerical invariants computable from orbit data. These invariants are related to the fundamental groups and the cohomology rings of the underlying manifolds. They do not distinguish among all 4-manifolds with actions as above, and so the general classification problem remains open.

Generalized Gauss maps and integrals for three-component links: Toward higher helicities for magnetic fields and fluid flows

We describe a new approach to triple linking invariants and integrals, aiming for a simpler, wider and more natural applicability to the search for higher order helicities of fluid flows and magnetic fields. To each three-component link in Euclidean 3-space, we associate a geometrically natural generalized Gauss map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its generalized Gauss map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two-component link in 3-space is the degree of its associated Gauss map from the 2-torus to the 2-sphere. When the pairwise linking numbers are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers, but patterned after J.H.C. Whiteheads integral formula for the Hopf invariant. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of 3-space, while the integral itself can be viewed as the helicity of a related vector field on the 3-torus. In the first paper of this series [math.GT 1101.3374] we did this for three-component links in the 3-sphere. Komendarczyk has applied this approach in special cases to derive a higher order helicity for magnetic fields whose ordinary helicity is zero, and to obtain from this nonzero lower bounds for the field energy.To each three-component link in the 3-sphere, we associate a geometrically natural characteristic map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its characteristic map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two-component link in 3-space is the degree of its associated Gauss map from the 2-torus to the 2-sphere. When the pairwise linking numbers are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of the 3-sphere, while the integral itself can be viewed as the helicity of a related vector field on the 3-torus.

Tori in the diffeomorphism groups of simply-connected 4-manifolds

Let M be a closed simply-connected 4-manifold. All manifolds will be assumed smooth and oriented. The purpose of this paper is to classify up to conjugacy the topological subgroups of Diff( M ) isomorphic to the 2-dimensional torus T 2 (Theorem 1), and to give an explicit formula for the number of such conjugacy classes (Theorem 2). Such a conjugacy class corresponds uniquely to a weak equivalence class of effective T 2 -actions on M . Thus the classification problem is trivial unless M supports an effective T 2 -action. Orlik and Raymond showed that this happens if and only if M is a connected sum of copies of ± ; P 2 and S 2 × S 2 (2), and so this paper is really a study of the different T 2 -actions on these manifolds.

Stable isotopy in four dimensions

We construct infinite families of topologically isotopic but smoothly distinct knotted spheres in many simply connected 4-manifolds that become smoothly isotopic after stabilizing by connected summing with