摘要
Genuine multi-entropy is the part of the𝚚 -partite multi-entropy that captures entanglement genuinely shared among all𝚚parties, rather than entanglement already present among fewer subsystems. It was recently proposed that this genuine part – so-called a multipartite entanglement signal – can be studied via the graph-encoded manifold (GEM) built from the underlying𝚚 -partite multi-entropy permutation family: reading its𝚚 -colored contraction graphΓ_(𝚚,n)as a triangulation by(𝚚-1) -simplices. Focusing on𝚚=4 , we classify the topology of every vertex link ofΓ_(4,n)and findχ_(\rm link)(n)=n(3-n) : the graph satisfies the GEM condition, namely every vertex link is a two-sphere, if and only ifn=2 , while for everyn≥3the vertex links are closed surfaces of strictly positive genusg_(n)=\tfrac{1}{2}{(}n-1)(n-2) , so that the graph instead defines a simplicial complex with a conical singularity at every vertex.