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Abstract
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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
–space,
we associate a generalized Gauss map from the
–torus to the
–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
–torus to the
–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
–torus.
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Keywords
Gauss integral, triple linking, helicity
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Mathematical Subject Classification 2010
Primary: 57M25, 76B99, 78A25
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Publication
Received: 18 November 2012
Accepted: 3 March 2013
Published: 23 July 2013
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