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ISSN (electronic): 1472-2739
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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
References
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/