Local knots and the prime factorization of links
Sergey A. Melikhov
TL;DR
The paper provides a shorter, conceptually transparent proof of Hashizume's theorem that every non-split link in $S^3$ admits a unique prime factorization under Hashizume connected sum, using multiply punctured $3$-spheres. It then extends the framework to string links, establishing a string-link version of Rolfsen's theorem and showing that a string link can lack local knots even when its closure contains one. Central to the approach are the notions of beaded links inside punctured $3$-spheres, genus additivity under connected sums, and a non-cancellation lemma that yields a unique, sphere-based decomposition into prime factors. The results clarify the relationship between local knots and ambient isotopy for string links and provide a robust tool for comparing links and string links through their prime decompositions, with implications for how link tables and related invariants are organized.
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.