Volume 13, issue 5 (2013)

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

Dennis DeTurck, Herman Gluck, Rafal Komendarczyk, Paul Melvin, Haggai Nuchi, Clayton Shonkwiler and David Shea Vela-Vick

Algebraic & Geometric Topology 13 (2013) 2897–2923
Abstract

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.

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 map is a natural successor to Gauss’s original map from the $2$–torus to the $2$–sphere. Like its prototype, it is equivariant with respect to orientation-preserving isometries of the ambient space, attesting to its naturality and positioning it for application to physical situations.

When the pairwise linking numbers are all zero, we give an integral formula for the triple linking number which is a natural successor to the classical Gauss integral for the pairwise linking numbers, with an integrand invariant under orientation-preserving isometries of the ambient space. This new integral is patterned after J H C Whitehead’s integral formula for the Hopf invariant, and hence interpretable as the ordinary helicity of a related vector field on the $3$–torus.

Keywords
Gauss integral, triple linking, helicity
Mathematical Subject Classification 2010
Primary: 57M25, 76B99, 78A25
Publication
Received: 18 November 2012
Accepted: 3 March 2013
Published: 23 July 2013
Authors
 Dennis DeTurck Department of Mathematics University of Pennsylvania David Rittenhouse Lab 209 South 33rd Street Philadelphia, PA 19104-6395 USA http://www.math.upenn.edu/~deturck/ Herman Gluck Department of Mathematics University of Pennsylvania David Rittenhouse Lab 209 South 33rd Street Philadelphia, PA 19104-6395 USA Rafal Komendarczyk Department of Mathematics Tulane University New Orleans, LA 70118 USA http://dauns.math.tulane.edu/~rako/ Paul Melvin Department of Mathematics Bryn Mawr College Bryn Mawr, PA 19010 USA http://www.brynmawr.edu/math/people/melvin Haggai Nuchi Department of Mathematics University of Pennsylvania David Rittenhouse Lab 209 South 33rd Street Philadelphia, PA 19104-6395 USA http://www.math.upenn.edu/~hnuchi/ Clayton Shonkwiler Department of Mathematics University of Georgia Athens, GA 30602 USA http://www.math.uga.edu/~clayton David Shea Vela-Vick Department of Mathematics Louisiana State University Baton Rouge, LA 70803 USA https://www.math.lsu.edu/~shea/