In the nuclei of eukaryotic cells, DNA is packaged through several levels of compaction in an orderly retrievable way that enables the correct regulation of gene expression. The functional dynamics of this assembly involves the unwinding of the so-called 30-nm chromatin fiber and accordingly imposes strong topological constraints. We present a general method for computing both the twist and the writhe of any winding pattern. An explicit derivation is implemented for the chromatin fiber which provides the linking number of DNA in eukaryotic chromosomes. We show that there exists one and only one unwinding path which satisfies both topological and mechanical constraints that DNA has to deal with during condensation/decondensation processes.