Triple linking numbers, ambiguous Hopf invariants and integral formulas for three-component links. (English) Zbl 1194.57007

Milnor showed that link homotopy classes of three-component links in the \(3\)-sphere are completely determined by a set of invariants: the three linking numbers of pairs of components of the link, together with the triple linking number, which is a residue class modulo the greatest common divisor of the three linking numbers.
The authors show that these invariants can be interpreted as Pontryagin invariants for homotopy classes of maps from the \(3\)-torus to the \(2\)-sphere. Indeed, to each three component link, one can associate a map from the \(3\)-torus, which parametrises triplets of points in the \(3\)-sphere, one on each component of the link, to the \(2\)-sphere as follows: any (generic) triple defines an oriented linear \(2\)-plane in \(4\)-space, i.e. a point on the Grassmannian \(G_2(4)\). This Grassmannian is isometric (up to scaling) to the product of two \(2\)-spheres. To obtain the desired map, it is then sufficient to project onto any of the two factors.
The set of Pontryagin invariants of the (homotopy class of) the given map are the degrees of the restrictions of the map to the three coordinate \(2\)-subtori of the \(3\)-torus together with an ambiguous Hopf invariant. In the case of the above map, the degrees coincide with the three linking numbers of the original link, while twice Milnor’s triple linking number coincides with the ambiguous Hopf invariant, modulo twice the greatest common divisor of the three linking numbers.
When the linking numbers are all zero, the authors also provide an integral formula for triple linking number.


57M25 Knots and links in the \(3\)-sphere (MSC2010)
53C65 Integral geometry
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