带符号图的符号GLMY同调：通过双覆盖实现
Signed GLMY Homology of Signed Graphs via Double Covers
浏览论文内容
中文总结 AI 辅助
本文通过双覆盖定义带符号图的符号GLMY同调，证明其切换不变性、与符号拉普拉斯核的联系及函子性，并分类五顶点九箭头有向图的512种符号得到四个Betti向量。
中文摘要 AI 辅助
我们利用片标记正则路径，为带符号有向图定义了在实数域上的符号GLMY链复形。该复形自然同构于带符号双覆盖上普通GLMY复形的对跖反不变子复形。双覆盖实现导出了切换不变性，并恢复了切换平衡符号情形的普通GLMY同调。双向完备化给出了带符号图的与定向无关的同调理论。对于带符号图，零维同调与符号拉普拉斯算子的核一致，其维数等于平衡连通分量的个数。符号GLMY同调在符号弱态射下具有函子性，符号弱态射将顶点映射与切换函数结合，并允许相容的箭头收缩。对于带符号有向图，全正约简保留了普通GLMY同调的定向敏感性，而显式计算显示了对箭头符号的额外敏感性。对于具有五个顶点和九条箭头的固定有向图，我们分类了全部512种箭头符号，并恰好得到四个符号Betti向量。精确地说，有16种符号具有非零的第二符号GLMY同调。
英文摘要
We define a signed GLMY chain complex over $\mathbb{R}$ for signed digraphs using sheet-labelled regular paths. The complex is naturally isomorphic to the deck anti-invariant subcomplex of the ordinary GLMY complex on the signed double cover. The double-cover realization yields switching invariance and recovers ordinary GLMY homology for switching-balanced signings. Bidirected completion gives an orientation-independent homology theory for signed graphs. For a signed graph, the zero-dimensional homology identifies with the kernel of the signed Laplacian and has dimension equal to the number of balanced connected components. Signed GLMY homology is functorial under signed weak morphisms, which combine vertex maps with switching functions and allow compatible arrow contractions. For signed digraphs, the all-positive reduction retains the orientation sensitivity of ordinary GLMY homology, while explicit computations show additional sensitivity to the arrow signs. For a fixed digraph with five vertices and nine arrows, we classify all 512 arrow signings and obtain exactly four signed Betti vectors. Precisely 16 signings have nonzero second signed GLMY homology.
发表机构
- Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）
- Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
机构由 AI 辅助整理，请以论文原文为准。
