与等效面心立方（FCC）格纽结相比，顶点数更少的珍珠项链纽结
Pearl necklace knots with fewer vertices than an equivalent FCC lattice knot
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中文总结 AI 辅助
该研究通过大语言模型寻找满足珍珠项链数小于等效面心立方格纽结格点数的构型，发现5-7交叉数纽结等可实现$N_P=N_L-1$，$8_1$纽结可实现$N_P=N_L-2$，未找到三叶结的14顶点构型。
中文摘要 AI 辅助
纽结的珍珠项链数$N_P$是构造纽结所需单位球的最小数量，要求每个球与两个相邻球相切且无重叠。此前推测珍珠项链数等于在面心立方（FCC）格上嵌入纽结所需的最小格点数$N_L$，即三叶结无法用少于15个球构造。本文通过大语言模型（LLM）寻找满足$N_P<N_L$的纽结构型，报告相关发现、验证尝试及意义：未找到三叶结的14顶点构型，但在5到7交叉数的纽结、$8_{19}$和$10_{124}$中发现$N_P=N_L-1$的构型，$8_1$纽结可实现$N_P=N_L-2$的构造。
英文摘要
The pearl necklace number, $N_P$, of a knot is the smallest number of unit spheres required to construct a knot if each sphere is tangent to two neighbors and no sphere overlaps with another. It has been speculated that the pearl necklace number is equal to the minimum number of lattice sites required to embed a knot on a face centered cubic (FCC) lattice $N_L$, implying that a trefoil knot cannot be constructed from fewer than 15 spheres. A large language model (LLM) was prompted to find configurations of knot for which $N_P<N_L$, and this manuscript describes its findings, attempts at validating them, and their implications. No 14-vertex example was found for the trefoil knot, but examples with $N_P=N_L-1$ were found for all knots from 5 to 7 crossings as well as $8_{19}$, and $10_{124}$. One example, $8_1$, could be constructed with $N_P=N_L-2$.
