Instantons, concordance, and Whitehead doubling

Abstract

We use moduli spaces of instantons and Chern-Simons invariants of flat connections to prove that the Whitehead doubles of (2;2 n − 1) torus knots are independent in the smooth knot concordance group; that is, they freely generate a subgroup of infinite rank. Call two knots in the 3-sphere concordant if they arise as the boundary of a smooth and properly embedded cylinder in the 3-sphere times an interval. Concordance is clearly an equivalence relation. Modulo this relation, the set of knots forms an abelian group C, with the role of addition played by connected sum and inverse given by considering the mirror image, with reversed orientation. This concordance group is a much studied object, well motivated by its role as a gateway into the mysterious world of 4-dimensional topology. Indeed, even in this relative situation of studying 3-dimensional manifolds in relation to the 4-dimensional manifolds they bound, one can observe the distinction between the smooth and topological categories in dimension four. Moreover, the group structure afforded by passing to concordance paves a clearer path through the often intractable field of knot theory. Our results focus on a particular satellite operation, (positive, untwisted) Whitehead doubling, and its effect on the concordance group, see Figure 1 and Section 3.1 for a definition. Our motivation comes from the following conjecture. To state it recall that a knot is slice if it is concordant to the unknot or, equivalently, if it bounds a smooth and properly embedded disk in the 4-ball.