Realizing homology boundary links with arbitrary patterns

Homology boundary links have become an increasingly important class of links, largely due to their significance in the ongoing concordance classification of links. Tim Cochran and Jerome Levine defined an algebraic object called a pattern associated to a homology boundary link which can be used to study the deviance of a homology boundary link from being a boundary link. Since a pattern is a set of $m$ elements which normally generates the free group of rank $m$, any invariants which detect non-trivial patterns can be applied to the purely algebraic question of when such a set is a set of conjugates of a generating set for the free group. We will give a constructive geometric proof that all patterns are realized by some homology boundary link $L^n$ in $S^{n+2}$. We shall also prove an analogous existence theorem for calibrations of $\mathbb{E}$-links, a more general and less understood class of links tha homology boundary links.