# Triple linking numbers, ambiguous Hopf invariants and integral formulas for three-component links

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

Arxiv ID: 0901.1612•Last updated: 8/12/2021

Three-component links in the 3-dimensional sphere were classified up to link
homotopy by John Milnor in his senior thesis, published in 1954. A complete set
of invariants is given by the pairwise linking numbers p, q and r of the
components, and by the residue class of one further integer mu, the "triple
linking number" of the title, which is well-defined modulo the greatest common
divisor of p, q and r.
To each such link L we associate a geometrically natural characteristic map
g_L from the 3-torus to the 2-sphere in such a way that link homotopies of L
become homotopies of g_L. Maps of the 3-torus to the 2-sphere were classified
up to homotopy by Lev Pontryagin in 1941. A complete set of invariants is given
by the degrees p, q and r of their restrictions to the 2-dimensional coordinate
subtori, and by the residue class of one further integer nu, an "ambiguous Hopf
invariant" which is well-defined modulo twice the greatest common divisor of p,
q and r.
We show that the pairwise linking numbers p, q and r of the components of L
are equal to the degrees of its characteristic map g_L restricted to the
2-dimensional subtori, and that twice Milnor's mu-invariant for L is equal to
Pontryagin's nu-invariant for g_L.
When p, q and r are all zero, the mu- and nu-invariants are ordinary
integers. In this case we use J. H. C. Whitehead's integral formula for the
Hopf invariant, adapted to maps of the 3-torus to the 2-sphere, together with a
formula for the fundamental solution of the scalar Laplacian on the 3-torus as
a Fourier series in three variables, to provide an explicit integral formula
for nu, and hence for mu.

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