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Abstract: Three-component links in the 3-dimensional sphere were classified up to linkhomotopy by John Milnor in his senior thesis, published in 1954. A complete setof invariants is given by the pairwise linking numbers p, q and r of thecomponents, and by the residue class of one further integer mu, the -triplelinking number- of the title, which is well-defined modulo the greatest commondivisor of p, q and r.To each such link L we associate a geometrically natural characteristic mapg L from the 3-torus to the 2-sphere in such a way that link homotopies of Lbecome homotopies of g L. Maps of the 3-torus to the 2-sphere were classifiedup to homotopy by Lev Pontryagin in 1941. A complete set of invariants is givenby the degrees p, q and r of their restrictions to the 2-dimensional coordinatesubtori, and by the residue class of one further integer nu, an -ambiguous Hopfinvariant- 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 Lare equal to the degrees of its characteristic map g L restricted to the2-dimensional subtori, and that twice Milnor-s mu-invariant for L is equal toPontryagin-s nu-invariant for g L.When p, q and r are all zero, the mu- and nu-invariants are ordinaryintegers. In this case we use J. H. C. Whitehead-s integral formula for theHopf invariant, adapted to maps of the 3-torus to the 2-sphere, together with aformula for the fundamental solution of the scalar Laplacian on the 3-torus asa Fourier series in three variables, to provide an explicit integral formulafor nu, and hence for mu.



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

Source: https://arxiv.org/







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