AI 中文总结
该研究证明了(-2,3,q)扭结的约化奇异瞬子同调秩，在枕形空间模型中计算了其朴素拉格朗日-Floer同调，得到秩差规律并计算了首批非零边界上同调，区分了证明、计算与猜想内容。
AI 中文摘要
我们证明，对于所有奇数 q≥3，pretzel 扭结 P(-2,3,q) 的约化奇异瞬子同调秩为 q+2：通过链环递归（Hironaka 类莱默多项式）闭式计算得到的该族的亚历山大多项式给出了下界 q+2，而 Manion 的闭式约化 Khovanov 同调给出了匹配的上界。随后我们转向扭结 Atiyah-Floer 程序的枕形空间（辛）侧，在 Herald-Kirk 和 Smith 的浸入曲线组合模型中，我们重构了该族自然缠结分解的枕形空间拉格朗日量，并计算了其中 q=13 时成员的朴素拉格朗日-Floer 同调。结果得到了一个精确的实验规律：朴素秩与 I^natural 秩恰好相差一个微分，差值为 2 sgn(det K - 3)，对于环面成员 q=3 该差值为零，且当行列式 det=|q-6| 穿过 3 时（等价于二元二面体无迹特征出现时）差值方向发生改变。最后我们计算了 Cazassus-Herald-Kirk-Kotelskiy 所猜想的用于修正缺陷的边界上同调：对于 q=5，是一个通过浸入四边形作用的唯一双交叉上同调；对于 q=7，是一个通过三角形作用的唯一单交叉上同调（每个均抵消一个二边形并使秩提升 2）；对于 q=11，是沿相反方向作用的单交叉上同调（产生一个微分并使秩降低 2）。据我们所知，这些是在康威和缠结上首次计算得到的非零边界上同调，也是首次通过抵消作用的边界上同调；它们在同一族内实现了猜想修正的两个方向，且具有刚性不对称性：抵消作用存在唯一的最小上同调，而产生作用则有多个。我们全程区分了无条件证明的内容、模型内计算的内容以及仍为猜想的内容。
英文摘要
Hedden, Herald and Kirk conjectured that every knot admits a decomposition along a Conway sphere for which the Lagrangian Floer homology of the two associated immersed curves in the pillowcase recovers Kronheimer and Mrowka's reduced singular instanton homology. We prove the conjecture for the hyperbolic pretzel knots $P(-2,3,q)$; it was previously known for two-bridge knots and for some torus knots. The decomposition is obtained from Hedden, Herald and Kirk's decomposition of the torus knot $T(3,5)$ by Dehn twists along the Conway sphere, which act on the pillowcase by a linear shear. Building on our analysis of the torus knots $T(3,n)$, we compute the Floer complex and its $\mathbb{Z}/4$ grading by hand; its differential is nonzero for every member of the family except $P(-2,3,7)$. The same method proves the conjecture for a family of twisted torus knots, under a hypothesis on the chirality of the torus-knot decomposition. On the instanton side, we show that the instanton homology of $P(-2,3,q)$ is free abelian of rank $q+2$ for every odd $q\ge3$, extending Lobb and Zentner's rational computation to the integers by means of Manion's integral Khovanov homology; for the hyperbolic members this also follows from work of Daemi and Scaduto.
Comments37 pages, 1 figure
