AI 中文总结
本文为Arnold不变量$J^-$和$J^+$构造同调提升，通过区域同调和圆同调细化Viro欧拉积分多项式与量子曲率多项式，并给出严格性证明和同调面积态和模型。
AI 中文摘要
Viro的欧拉积分多项式 $P_C(q)$ 和Lanzat--Polyak量子曲率多项式 $I_q(C)$ 细化了Arnold关于一般浸入单分量平面曲线的两个不变量 $J^-$ 和 $J^+$。我们构造了这两者的同调提升。双分次区域同调保留了每个连通Alexander指标区域的奇异同调；其分次欧拉示性数为 $P_C(q)$。三分次光滑圆同调由保持定向的光滑化产生的定向圆生成，并退范畴化为 $I_q(C)$ 中的光滑化项。保留每个光滑圆的实际区域求和项和边界区域，给出了定向光滑化构型（即Seifert态）的同调细化。一个无限族证明了严格性：多项式数据和普通同调提升均一致，而分量分次区域同调和分支分解圆同调能区分每一对。进一步构造通过顶点复形恢复了完整的 $I_q(C)$，通过边同调实现了其曲率积分的局部变化，并为无向曲线给出了典范的两态同调。Viro将其欧拉积分公式描述为量子结多项式面积态和公式的类比。通过本文发展的范畴化，我们获得了一个具体的同调面积态和模型，实现了这一类比。
英文摘要
Viro's Euler-integral polynomial $P_C(q)$ and the Lanzat--Polyak quantized-curvature polynomial $I_q(C)$ refine Arnold's invariants $J^-$ and $J^+$ for generic immersed one-component plane curves. We construct homological lifts of both. The bigraded region homology retains the singular homology of every connected Alexander-index region; its graded Euler characteristic is $P_C(q)$. The triply graded smoothing-circle homology is generated by the oriented circles of the orientation-preserving smoothing and decategorifies to the smoothing term in $I_q(C)$. Keeping the actual region summands and the boundary regions of every smoothing circle gives a homological refinement of the oriented smoothing configuration, or Seifert state. An infinite family proves strictness: both polynomial data and the ordinary homological lifts agree, while the component-graded region homology and the branch-decomposed circle homology distinguish every pair. Further constructions recover the full $I_q(C)$ by a vertex complex, realize the local change of its curvature integral by edge homology, and give a canonical two-state homology for unoriented curves. Viro described his Euler-integral formula as an analogue of face state-sum formulas for quantum knot polynomials. Through the categorifications developed here, we obtain one concrete homological face-state-sum model realizing that analogy.
Comments29 pages, 3 figures
