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

TL;DR

This paper links Milnor's triple linking number for three-component links to Pontryagin's homotopy invariants via characteristic maps, providing explicit integral formulas and clarifying their geometric and algebraic relationships.

Contribution

It establishes a geometric interpretation of Milnor's triple linking number using characteristic maps and integral formulas, connecting link invariants to homotopy invariants of maps from the 3-torus to the 2-sphere.

Findings

Milnor's mu-invariant equals Pontryagin's nu-invariant divided by two.

Pairwise linking numbers correspond to degrees of characteristic map restrictions.

Explicit integral formulas for the invariants are derived using Fourier series and Laplacian solutions.

Abstract

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.