Paul Melvin

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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(More)
To Manfredo do Carmo in friendship and admiration, on his 80 th birthday Abstract. Three-component links in the 3-dimensional sphere were classified up to link homotopy by John Milnor in his senior thesis, published in 1954. A complete set of invariants is given by the pairwise linking numbers p, q and r of the components, and by the residue class of one(More)
In [11], two of us constructed a closed oriented 4-dimensional manifold with fundamental group Z that does not split off S 1 × S 3. In this note we show that this 4-manifold, and various others derived from it, do not admit smooth structures. Moreover, we find an infinite family of 4-manifolds with exactly the same properties. As a corollary, we obtain(More)
A formula for the Arf invariant of a link is given in terms of the singularities of an immersed surface bounded by the link. This is applied to study the computational complexity of quantum invariants of 3–manifolds. The quantum 3–manifold invariant of Witten [38] and Reshetikhin–Turaev [28] with gauge group SU (2) at the fourth root of unity is given by(More)
A framing of an oriented trivial bundle is a homotopy class of sections of the associated oriented frame bundle. This paper is a study of the framings of the tangent bundle τ M of a smooth closed oriented 3-manifold M , often referred to simply as framings of M. 1 We shall also discuss stable framings and 2-framings of M , that is framings of ε 1 ⊕ τ M(More)
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 S 2 ×S 2 , and as a consequence, analogous families of diffeomorphisms and metrics of positive scalar curvature for such 4-manifolds. We also construct(More)