On the topological linkage of finite, entangled, spatial graphs in three-dimensions.

The observation, design and analysis of mesh-like networks in crystal chemistry, polymer physics and biological systems has brought forward an extensive catalog of fascinating structures of which a subgroup share a particular, yet critically under appreciated attribute: Intertwinedness.

Here, intertwinedness (or interpenetration, entanglement) describes the state of networks which are embedded in space such that one wouldn't be able to pull them apart without prior removal of a subset of edges.

One also refers to such networks as topologically linked, on the basis that their fundamental loop pairs have nonzero linking numbers (utilizing the Gaussian linking integral).

Though one may analyze such networks in terms of other topological invariants, e.g. Yamada polynomials or lattice symmetries (in case of order periodic systems), we would prefer an easily accessible and interpretable approach on the basis of Hopf-link identification.

Therefore, in this study we suggest a method to analyze the effective intertwinedness of any finite, amorphous, 2-component nets by characterization of their respective cycle spaces.

Doing so we are able to efficiently differentiate a variety of entangled spatial networks on the ground of their respective loop linkage and thereby offer a new tool for spatial network characterization with particular applications in developmental biology.