Structure and unique factorization in concordance groups of links

TL;DR

This paper fully characterizes the structure of certain link concordance groups introduced by Donald and Owens, revealing they decompose into infinite direct sums and establishing a unique prime factorization theory.

Contribution

It proves the complements are isomorphic to infinite direct sums of integers and 2-torsion, and introduces a prime element concept with a unique factorization theorem.

Findings

Complements are isomorphic to ^} \u2295 (\u001d/2)^

Established a prime decomposition theorem for the groups

Provided a canonical normal form for the group structure

Abstract

Donald and Owens introduced two link concordance groups with a marked component and showed that they contain the knot concordance group as a direct summand with infinitely generated complements. While not explicitly posed by Donald and Owens, the problem of determining the structure of these complements arises naturally from their work. In this paper, we completely resolve this problem by proving that both complements are isomorphic to $\mathbb{Z}^{\infty} \oplus (\mathbb{Z}/2\mathbb{Z})^{\infty}$. Moreover, we introduce a notion of prime element and establish a unique prime decomposition theorem. This yields a canonical normal form, providing a complete description of the group structure.