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- PAUL MELVIN
- 2008

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)

- Paul Melvin, Sumana Shrestha
- 2005

Examples are given of prime Legendrian knots in the standard contact 3–space that have arbitrarily many distinct Chekanov polynomials, refuting a conjecture of Lenny Ng. These are constructed using a new " Legendrian tangle replacement " technique. This technique is then used to show that the phenomenon of multiple Chekanov polynomials is in fact quite… (More)

- Dennis Deturck, Herman Gluck, Rafal Komendarczyk, Paul Melvin, Clayton Shonkwiler, David Shea Vela-Vick +4 others
- 2009

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 natural question in knot theory is to ask how certain properties of a knot behave under satellite operations. We will focus on the satellite operation of cabling, and on Heegaard Floer-theoretic properties. In particular, we will give a formula for the Ozsvath-Szabo concordance invariant tau of iterated cables of a knot K in terms of the cabling… (More)

- Robion Kirby, Paul Melvin
- 2004

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)

- PAUL MELVIN
- 1999

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)

A general topological formula is given for the 517(2) quantum invariant of a 3-manifold M at the sixth root of unity. It is expressed in terms of the homology, Witt invariants and signature defects of the various 2-fold covers of M, and thus ties in with basic 4-dimensional invariants. A discussion of the range of values of these quantum invariants is… (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)

- Robion Kirby, Paul Melvin
- 1999

The E 8 –manifold has several natural framed link descriptions, and we give an efficient method (via " grapes ") for showing that they are indeed the same 4–manifold. This leads to explicit handle pictures for the perturbation of singular fibers in an elliptic surface to a collection of fishtails. In the same vein, we show how the degeneration of a regular… (More)