Primary decompositions of knot concordance

TL;DR

This paper investigates the structure of the smooth concordance group of knots, demonstrating that certain primary decompositions cannot exist for topologically slice knots, thus revealing limitations in algebraic approaches to knot concordance.

Contribution

It proves that strong primary decompositions of the smooth concordance group for topologically slice knots are impossible, providing new insights into the algebraic structure of knot concordance.

Findings

Existence of a topologically slice knot with a specific Alexander polynomial factorization

Demonstration that such knots are not smoothly concordant to connected sums of knots with factors

Limitations on primary decompositions in the smooth concordance group

Abstract

For all n > 0 there is a homomorphism from the smooth concordance group of knots in dimension 2n + 1 to an algebraically defined group called the rational algebraic concordance group. This algebraic concordance group splits as a direct sum of groups indexed by polynomials. For n > 1 the homomorphism is injective. This leads to what is called a primary decomposition theorem for knot concordance. In the classical dimension, the kernel of this homomorphism includes the smooth concordance group of topologically slice knots, and Jae Choon Cha has begun studying possible primary decompositions of this subgroup. Here we will show that primary decompositions of a strong type cannot exist. In more detail, it is shown that there exists a topologically slice knot K for which there is a factorization of its Alexander polynomial as f(t)g(t), where f(t) and g(t) are relatively prime and each is the Alexander polynomial of a topologically slice knot, but K is not smoothly concordant to any connected sum of a pair of knots with Alexander polynomials f(t) and g(t).