Knots having the same Seifert form and primary decomposition of knot concordance

TL;DR

This paper constructs infinite families of algebraically slice knots sharing the same Seifert form, which are linearly independent in the concordance group and cannot be concordant to knots with coprime Alexander polynomials, using advanced rho-invariants.

Contribution

It introduces a method to generate infinite families of knots with identical Seifert forms that are distinct in the concordance group and not concordant to knots with coprime Alexander polynomials.

Findings

Existence of infinite linearly independent knots with same Seifert form

Knots are not concordant to any with coprime Alexander polynomial

Utilizes Cheeger-Gromov-von Neumann rho-invariants for proofs

Abstract

We show that for each Seifert form of an algebraically slice knot with nontrivial Alexander polynomial, there exists an infinite family of knots having the Seifert form such that the knots are linearly independent in the knot concordance group and not concordant to any knot with coprime Alexander polynomial. Key ingredients for the proof are Cheeger-Gromov-von Neumann rho-invariants for amenable groups developed by Cha and Orr and polynomial splittings of metabelian rho-invariants.