Rafal Komendarczyk

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.

Homotopy Brunnian links and the -invariant

We provide an alternative proof that Koschorkes kappa-invariant is injective on the set of link homotopy classes of n-component homotopy Brunnian links. The existing proof (by Koschorke) is based on the Pontryagin--Thom theory of framed cobordisms, whereas ours is closer in spirit to techniques based on Habegger and Lins string links. We frame the result in the language of the rational homotopy Lie algebra of the configuration space of n points in R^3, which allows us to express Milnors higher linking numbers as homotopy periods of the configuration space.

Tightness in contact metric 3-manifolds

This paper begins the study of relations between Riemannian geometry and global properties of contact structures on 3-manifolds. In particular we prove an analog of the sphere theorem from Riemannian geometry in the setting of contact geometry. Specifically, if a given three dimensional contact manifold (M,ξ) admits a complete compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure ξ is tight; in particular, the contact structure pulled back to the universal cover is the standard contact structure on S3. We also describe geometric conditions in dimension three for ξ to be universally tight in the nonpositive curvature setting.

Homotopy string links and the κ-invariant

Koschorke introduced a map from the space of closed n-component links to the space of maps from the n-torus to the ordered configuration space of n-tuples of points in R3. He conjectured that this map separates homotopy links. The purpose of this paper is to construct an analogous map for string links, and to prove (1) this map in fact separates homotopy string links, and (2) Koschorkes original map factors through the map constructed here together with an analog of Markovs closure map defined on the level of certain function spaces.

Homotopy string links and the

Koschorke introduced a map from the space of closed n-component links to the space of maps from the n-torus to the ordered configuration space of n-tuples of points in R3. He conjectured that this map separates homotopy links. The purpose of this paper is to construct an analogous map for string links, and to prove (1) this map in fact separates homotopy string links, and (2) Koschorkes original map factors through the map constructed here together with an analog of Markovs closure map defined on the level of certain function spaces.

\kappa

Koschorke introduced a map from the space of closed n-component links to the space of maps from the n-torus to the ordered configuration space of n-tuples of points in R3. He conjectured that this map separates homotopy links. The purpose of this paper is to construct an analogous map for string links, and to prove (1) this map in fact separates homotopy string links, and (2) Koschorkes original map factors through the map constructed here together with an analog of Markovs closure map defined on the level of certain function spaces.

-invariant

This paper begins the study of relations between Riemannian geometry and contact topology in any dimension and continues this study in dimension 3. Specifically we provide a lower bound for the radius of a geodesic ball in a contact manifold that can be embedded in the standard contact structure on Euclidean space, that is on the size of a Darboux ball. The bound is established with respect to a Riemannian metric compatible with an associated contact form. In dimension three, it further leads us to an estimate of the size for a standard neighborhood of a closed Reeb orbit. The main tools are classical comparison theorems in Riemannian geometry. In the same context, we also use holomorphic curves techniques to provide a lower bound for the radius of a PS-tight ball.

Homotopy string links and the κ-invariant: HOMOTOPY STRING LINKS AND THE κ-INVARIANT

This article presents an algebraic topology perspective on the problem of finding a complete coverage probability of a domain by a random cover. In particular we obtain a general formula for the chance that a collection of finitely many compact connected random sets placed on a 1-complex

Quantitative Darboux theorems in contact geometry

X

Finite random coverings of one-complexes and the Euler characteristic

has a union equal to