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- 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. AMS Classification 57M27; 68Q15

- 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)

A knot in S 3 whose complement contains an essential** torus is called a satellite knot. In this paper we discuss algebraic invariants of satellite knots, giving short proofs of some known results as well as new results. To each essential torus in the complement of an oriented satellite knot S , one may associate two oriented knots C and E (the companion… (More)

- Blake Mellor, Paul Melvin
- 2002

Milnor’s 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. AMS Classification 57M25; 57M27

- DENNIS DETURCK, HERMAN GLUCK, +4 authors DAVID SHEA VELA-VICK
- 2013

To each three-component link in Euclidean 3–space, we associate a 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 generalized Gauss… (More)

HEEGAARD FLOER INVARIANTS AND CABLING Jennifer Hom Paul Melvin, Advisor 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 Floertheoretic properties. In particular, we will give a formula for the Ozsváth-Szabó concordance… (More)

- Dennis Deturck, Herman Gluck, +7 authors Vela-Vick
- 2009

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 further integer μ, the “triple linking number” of the title, which is well-defined… (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)

- 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 τM 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 ε1 ⊕ τM (where… (More)