Knot Concordance and Homology Sphere Groups. (29th May 2017) - Aceto, Paolo; Larson, Kyle
Abstract: We study two homomorphisms to the rational homology sphere group Θ^3_Q . If ψ denotes the inclusion homomorphism from the integral homology sphere group Θ^3_Z, then using work of Lisca we show that the image of ψ intersects trivially with the subgroup of Θ^3_Q generated by lens spaces. As corollaries this gives a new proof that the cokernel of ψ is infinitely generated, and implies that a connected sum K of two-bridge knots is concordant to a knot with determinant 1 if and only if K is smoothly slice. Furthermore, if β denotes the homomorphism from the knot concordance group C defined by taking double branched covers of knots, we prove that the kernel of β contains a Z^∞ summand by analyzing the Tristram–Levine signatures of a family of knots whose double branched covers all bound rational homology balls.