Local knots and the prime factorization of links

TL;DR

This paper presents a shorter proof of the prime factorization theorem for links in $S^3$, and extends it to string links, highlighting subtle differences in local knot properties between links and their closures.

Contribution

It offers a new, concise proof of Hashizume's prime decomposition theorem and applies it to establish a string link version of Rolfsen's theorem, revealing nuanced local knot phenomena.

Findings

New shorter proof of prime link factorization

String link version of isotopy classification

Existence of string links without local knots whose closures have local knots

Abstract

The present note contains a new proof of Y. Hashizume's 1958 theorem that every non-split link in $S^3$ admits a unique factorization into prime links. While the new proof does not go far beyond standard techniques, it is considerably shorter than the original proof and avoids most of its case exhaustion. We apply this proof to obtain a string link version (and also an alternative proof) of a 1972 theorem of D. Rolfsen: two PL links in $S^3$ are ambient isotopic if and only if they are PL isotopic and their respective components are ambient isotopic. It is tempting to dismiss this string link version as obvious by deriving it directly either from Rolfsen's or Hashizume's theorem. But this does not seem to be possible, as it turns out that there exists a string link that has no local knots, while its closure has a local knot.