直棍结与链的判别簇
Discriminant Varieties for Stick Knots and Links
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中文总结 AI 辅助
该研究解决固定数量直棍可构造结型数量的增长阶问题，通过将多边形自相交转化为稀疏实代数腔问题，结合辫结构造得到阶乘级最优上界，建立了多个数学领域间的直接联系。
中文摘要 AI 辅助
用固定数量的直棍可以构造出多少种结型？我们证明该答案具有阶乘级增长，首次确定了其增长阶。此前已发表的一般上界均未超过由交叉数枚举得到的平方指数级估计；我们将其替换为阶乘级上界，该上界在增长阶层面是最优的。证明过程将多边形自相交问题转化为仅需线性维数的稀疏实代数腔问题，同时互补的辫结构造可得到阶乘级数量的不同结。该结果在纽结拓扑、实代数几何、少项式结构与置换组合学之间建立了直接联系。
英文摘要
How many knot types can be built from a fixed budget of straight sticks? We prove that the answer has factorial-scale growth, settling its order for the first time. No previously published general upper bound improves on the exponential-in-the-square estimate obtained from crossing-number enumeration; we replace it with a factorial-scale upper bound, which is optimal at the level of growth order. The proof turns polygonal self-intersection into a sparse real-algebraic chamber problem in only linearly many dimensions, while a complementary braid construction supplies factorially many distinct knots. The result creates a direct bridge between knot topology, real algebraic geometry, fewnomial structure, and permutation combinatorics.
