Enhanced Hantzsche Theorem

A closed 3-manifold $M$ may be described up to some indeterminacy by a Heegaard diagram $\mathcal{D}$. The question "Does $M$ smoothly embed in ${\mathbb{R}}^4$?‘’ is equivalent to a property of $\mathcal{D}$ which we call $\mathit{\text{doubly}}\mathit{\text{unlinked}}$ (DU). This perspective leads to an enhancement of Hantzsche’s embedding obstruction.