Generalized Gauss maps and integrals for threecomponent links: Toward higher helicities for magnetic fields and fluid flows
Abstract
To each threecomponent link in the 3sphere we associate a generalized Gauss map from the 3torus to the 2sphere, 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 twocomponent links in 3space, where the linking number is the degree of the classical Gauss map from the 2torus to the 2sphere. The generalized Gauss map, like its prototype, is geometrically natural in the sense that it is equivariant with respect to orientationpreserving isometries of the ambient space, thus positioning it for application to physical situations. When the pairwise linking numbers of a threecomponent 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 orientationpreserving 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 3sphere 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 3space.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 January 2013
 DOI:
 10.1063/1.4774172
 arXiv:
 arXiv:1101.3374
 Bibcode:
 2013JMP....54a3515D
 Keywords:

 fluid dynamics;
 geometry;
 integral equations;
 02.30.Rz;
 02.40.Pc;
 Integral equations;
 General topology;
 Mathematics  Geometric Topology;
 Mathematical Physics;
 Mathematics  Algebraic Topology;
 Mathematics  Differential Geometry;
 Mathematics  Symplectic Geometry;
 57M25;
 55S30;
 55Q25
 EPrint:
 60 pages, 37 figures