AI 中文总结
本文通过上链构造建立结三角剖分中面矩阵与粘合矩阵的对偶关系，证明零化度加倍及FAMED恒等式，并给出Wong猜想的拟议证明。
AI 中文摘要
对于$S^3$中结补空间的有序理想三角剖分，我们给出一个上链构造，将其面矩阵$C$与优选经线粘合矩阵$A,B$联系起来。唯一性的障碍是理想紧化的一阶上同调。其消失导致零化度加倍和FAMED矩阵恒等式，包括对奇异面矩阵的约束表述。在严格角度结构下的一个独立秩论证，为Wong的广义FAMED猜想提供了拟议证明。在整数上，该构造给出$\text{ker}_{\mathbb{Z}}B$的直和分解，以及从两个面余核到粘合余核的精确序列，其商为$\mathbb{Z}/2\mathbb{Z}$。在未减半经线约定下，这意味着$|\det B|=2|\det C|^2$。我们还获得循环面挠率、Smith标准形的逆序不变性，以及固定提升粘合系统的仿射面坐标。行列式为一的断言和完整的Andersen--Kashaev体积猜想在此仍未解决。
英文摘要
For an ordered ideal triangulation of a knot exterior in $S^3$, we give a cochain construction relating its face matrix $C$ to its preferred-longitude gluing matrices $A,B$. The obstruction to uniqueness is the first cohomology of the ideal compactification. Its vanishing yields nullity doubling and the FAMED matrix identity, including a constrained formulation for singular face matrices. A separate rank argument under a strict angle structure supplies a proposed proof of Wong's generalized FAMED conjecture. Over the integers, the construction gives a direct-sum decomposition of $\ker_{\mathbb{Z}}B$ and an exact sequence from the two face cokernels to the gluing cokernel, with quotient $\mathbb{Z}/2\mathbb{Z}$. In the unhalved longitude convention this implies $|\det B|=2|\det C|^2$. We also obtain cyclic face torsion, order-reversal invariance of Smith normal form, and affine face coordinates for fixed lifted gluing systems. The determinant-one assertion and the full Andersen--Kashaev volume conjecture remain open here.
Comments26 pages, 1 figure; exact-arithmetic verification scripts and data included as ancillary files
