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

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10.1063/1.4774172

### Abstract

To each three-component link in the 3-sphere 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. We view this as a natural extension of the familiar situation for two-component links in 3-space, where the linking number is the degree of the classical Gauss map from the 2-torus to the 2-sphere. The generalized Gauss map, like its prototype, is geometrically natural in the sense that it is equivariant with respect to orientation-preserving isometries of the ambient space, thus positioning it for application to physical situations. When the pairwise linking numbers of a three-component link are all zero, we give an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. This new integral is also geometrically natural, like its prototype, in the sense that the integrand is invariant under orientation-preserving isometries of the ambient space. Versions of this integral have been applied by Komendarczyk in special cases to problems of higher order helicity and derivation of lower bounds for the energy of magnetic fields. We have set this entire paper in the 3-sphere because our generalized Gauss map is easiest to present here, but in a subsequent paper we will give the corresponding maps and integral formulas in Euclidean 3-space.

© 2013 American Institute of Physics

Received 05 July 2012
Accepted 05 December 2012
Published online 25 January 2013

Acknowledgments: We are grateful to Fred Cohen and Jim Stasheff for their valued input during the preparation of this paper, and to Toshitake Kohno, whose 2002 paper^{54} provided one of the inspirations for this work. R. Komendarczyk and C. Shonkwiler also acknowledge support from DARPA (Grant No. FA9550-08-1-0386), and Vela-Vick from a NSF postdoctoral fellowship.

The Borromean rings shown on the first page are from a panel in the carved walnut doors of the Church of San Sigismondo in Cremona, Italy. The photograph is courtesy of P. Cromwell.^{55}

Article outline:

I. INTRODUCTION

A. The generalized Gauss map of a three-component link in the 3-sphere

B. Statement of results

C. Organization of the paper

II. THE MILNOR μ-INVARIANT

III. THE PONTRYAGIN ν-INVARIANT

A. Definition and properties

B. Converting the relative Pontryagin ν-invariant into an absolute invariant

C. Computing the absolute Pontryagin ν-invariant

IV. EXPLICIT FORMULAS FOR THE GENERALIZED GAUSS MAP

A. The Grassmann manifold

B. The Gauss map *g* _{ L }

C. The asymmetric Gauss map *h* _{ L }

D. A Gaussian view of the asymmetric Gauss map

E. Proof of the first statement in Theorem A

V. THE PONTRYAGIN ν-INVARIANT OF THE GENERALIZED GAUSS MAP

A. Outline of the procedure for computing ν(*h* _{ L })

B. The standard open books in and *S* ^{3}

C. Generic links

D. Bicycles and icycles

E. Some examples of bicycles and their associated icycles

F. The Bicycle theorem

G. Computing ν(*h* _{ L }) for a generic link *L* using the bicycle theorem

H. Double crossing changes

VI. PROOF OF THEOREM A

A. Proof of the base case of Theorem A

B. Proof of the inductive step of Theorem A

VII. PROOF OF THEOREM B, FORMULAS (1) AND (2)

A. Explicit formula for the 2-form ω_{ L } and the vector field on *T* ^{3}

B. Proof of Theorem B, formula (1)

C. Proof of Theorem B, formula (2)

VIII. FOURIER SERIES AND THE PROOF OF THEOREM B, FORMULA (3)

A. Fourier series and the fundamental solution of the Laplacian

B. Proof of Proposition 8.1

C. Fourier series and the calculus of differential forms on the 3-torus

D. Proof of Proposition 8.2

E. Proof of Theorem B, formula (3)

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