The knot invariant Upsilon, defined by Ozsváth, Stipsicz, and Szabó, induces a homomorphism from the smooth knot concordance group to the group of piecewise linear functions on the interval [0,2]. Here we define a set of related secondary invariants, each of which assigns to a knot a piecewise linear function on [0,2]. These secondary invariants provide bounds on the genus and concordance genus of knots. Examples of knots for which Upsilon vanishes but which are detected by these secondary invariants are presented.

Abstract. We show that if the connected sum of two knots with coprime Alexander polynomials has vanishing von Neumann ρ-invariants associated with certain metabelian representations, then so do both knots. As an application, we give a new example of an infinite family of knots which are linearly independent in the knot concordance group.

Abstract. Let Z[1/p] denote the ring of integers with the prime p inverted. There is a canonical homomorphism Ψ :Q , where Θ 3 R denotes the three-dimensional smooth R-homology cobordism group of R-homology spheres and the direct sum is over all prime integers. Gauge theoretic methods prove the kernel is infinitely generated. Here we prove that Ψ is not surjective, with cokernel infinitely generated. As a basic example we show that for p and q distinct primes, there is no rational homology cobordism from the lens space L(pq, 1) to any Mp # Mq, where H 1 (Mp) = Zp and H 1 (Mq) = Zq. More subtle examples include cases in which a cobordism to such a connected sum exists topologically but not smoothly. (Conjecturally, such a splitting always exists topologically.) Further examples can be chosen to represent 2-torsion in Θ 3 Q . Let K denote the kernel of Θ 3 Q → Θ 3 Q , where Θ 3 Q denotes the topological homology cobordism group. Freedman proved that Θ 3 Z ⊂ K. A corollary of results here is that K/Θ 3 Z is infinitely generated. We also demonstrate the failure in dimension three of splitting theorems that apply to higher dimensional knot concordance groups.