Knot Concordance and Homology Sphere Groups. (29th May 2017)

• Record Type: Journal Article
• Authors: Aceto, Paolo; Larson, Kyle
• Is Part Of: International mathematics research notices. Volume 2018:Number 23(2018)
• Page Start: 7318
• Page End: 7334
• Publication Date: 2017-05-29
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnx091
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: We study two homomorphisms to the rational homology sphere group $\Theta^3_\mathbb{Q}$ . If $\psi$ denotes the inclusion homomorphism from the integral homology sphere group $\Theta^3_\mathbb{Z}$, then using work of Lisca we show that the image of $\psi$ intersects trivially with the subgroup of $\Theta^3_\mathbb{Q}$ generated by lens spaces. As corollaries this gives a new proof that the cokernel of $\psi$ 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 $\beta$ denotes the homomorphism from the knot concordance group $\mathcal{C}$ defined by taking double branched covers of knots, we prove that the kernel of $\beta$ contains a $\mathbb{Z}^{\infty}$ summand by analyzing the Tristram–Levine signatures of a family of knots whose double branched covers all bound rational homology balls.