A prime decomposition theorem for string links in a thickened surface

Abstract

We prove a prime decomposition theorem for string links in a thickened surface. Namely, we prove that any non-braid string link $\ell \subset \hat{I}\pounds \times I$, where $\hat{I}\pounds$ is a compact orientable (not necessarily closed) surface other than $S^2$, can be written in the form $\ell ={\ell }_1\#\dots \#{\ell }_m$, where ${\ell }_j,j=1,\dots ,m,$ is prime string link defined up to braid equivalence, and the decomposition is unique up to possibly permuting the order of factors in its right-hand side.

Minor revisions in accordance with reviewer suggestions