Non-convex polytopes realizing kissing contact graphs in dimensions 5-7

@misc{self2026nonconvex,
  title = {{Non-convex polytopes realizing kissing contact graphs in dimensions 5-7}},
  author = {Matthew Self},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.12640},
  eprint = {2609.12640},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12640},
}

TY  - UNPB
TI  - Non-convex polytopes realizing kissing contact graphs in dimensions 5-7
AU  - Matthew Self
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.12640
UR  - https://arxiv.org/abs/2609.12640
AB  - Each of the 12 known conjecturally optimal kissing configurations in dimensions 5, 6, and 7 carries a contact-graph polytope: a polytope whose vertices are the configuration points and whose edges are exactly its contacts. Eight of these polytopes are non-convex and distinct from their convex hulls, which include longer edges. Each polytope is radial about the sphere's center: its boundary meets every ray from the center exactly once. It is the unique radial polytope with those vertices and edges. Seven of the nine non-lattice configurations carry polytopes constructed from the lattice contact polytope in the same dimension by splitting some facets and reassembling the pieces. This split-and-reassemble construction describes the configurations by facets rather than by layers. The fold angles of the reassembled facets explain the inner products 1/(10−n), giving 1/5, 1/4, and 1/3 in dimensions 5, 6, and 7, respectively. The contact graphs of all 12 configurations are infinitesimally rigid. We also prove that Q_5 and R_5 are infinitesimally jammed, extending the 10 previously established cases to all 12 configurations. All mathematical computations use exact arithmetic.
ER  -