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Abstract:Most currently known kissing arrangements of size $840$ in $\mathbb R^{12}$ share a common structure. They consist of $60$ vectors supported on $\mathbb R^6\times\{\mathbf 0\}$, another $60$ vectors supported on $\{\mathbf 0\}\times\mathbb R^6$, and $720$ additional \emph{bridge vectors}. The bridge vectors encode the interaction between the two six-dimensional factors and are constructed from the unique $1$-factorization of the complete graph $K_6$.
In this paper we investigate kissing arrangements of this type while keeping the bridge vectors fixed. We show that each $60$-point block admits substantial flexibility: $12$ of its vectors may be chosen as the signed coordinate vectors $\pm e_i$, while the remaining $48$ vectors may vary within a positive-dimensional family of configurations, which we call $48$-systems. As a consequence, we obtain infinitely many pairwise non-isometric kissing arrangements of size $840$ in $\mathbb R^{12}$.
The geometric freedom revealed by these constructions provides new insight into the local structure of extremal configurations. Exploiting this structure, we develop a specialized initialization scheme for logarithmic Riesz energy optimization. Starting from such structurally informed initial configurations, we numerically construct a kissing arrangement of size $841$ in $\mathbb R^{12}$.

Submission history

From: Rustem Takhanov [view email]
[v1] Wed, 17 Jun 2026 12:05:28 UTC (1,088 KB)