AI 中文总结
本文研究霍万诺夫拉普拉斯算子的谱几何，将其与加权图无符号拉普拉斯算子关联，推导了间隙界与挠率对应，还给出李谱序列的调和模型。
AI 中文摘要
对于定向纽结图 $D$，霍万诺夫上链复形带有一个典范埃尔米特内积，该内积定义了一个组合霍奇拉普拉斯算子。其核自然同构于有理霍万诺夫上同调，而其正谱依赖于所选取的图。在琼斯和魏的数值工作基础上，我们对这种依赖于图的高阶谱开展了结构研究。在最小和最大立方体次数下，我们将霍万诺夫拉普拉斯算子与显式加权图的无符号拉普拉斯算子等同起来，最大次数下相差一个对角符号共轭。该图模型通过其二分分量描述了有理霍万诺夫类与调和代表元，且当对应有理上同调消失时，在极值 $q$ 次数附近的固定宽度带中给出了逆多项式间隙界。它还对扭曲 unknot、奇数 $(2,N)$-环面纽结和偶数扭结给出了精确的 $\u0398(N^{-2})$ 双次数间隙。这些精确间隙计算部分受霍万诺夫同调的近期量子算法中谱分辨率要求的驱动。我们还证明了霍万诺夫复形的有限维分析-组合挠率对应，表明非零拉普拉斯伪行列式的交替乘积在有理无环 $q$ 次数中恢复了整数霍万诺夫挠率，而在一般情况下与它相差一个显式调节器。最后，我们将李的变形转移到调和霍万诺夫子空间上的滤过微分，给出了李谱序列的典范调和模型。
英文摘要
For an oriented link diagram $D$, the Khovanov cochain complex carries a canonical Hermitian inner product that defines a combinatorial Hodge Laplacian. Its kernel is naturally isomorphic to rational Khovanov cohomology, while its positive spectrum depends on the chosen diagram. Building on the numerical work of Jones and Wei, we develop a structural study of this diagram-dependent higher spectrum. In the minimal and maximal cube degrees, we identify the Khovanov Laplacian with signless Laplacians of explicit weighted graphs, up to diagonal sign conjugation in maximal degree. The graph model describes rational Khovanov classes and harmonic representatives through its bipartite components and gives inverse-polynomial gap bounds in fixed-width bands near the extremal $q$-degrees when the corresponding rational cohomology vanishes. It also yields exact $Θ(N^{-2})$ bidegree gaps for twisted unknots, odd $(2,N)$-torus knots, and even twist knots. The exact gap calculations are motivated in part by the spectral-resolution requirement in a recent quantum algorithm for Khovanov homology. We also prove a finite-dimensional analytic--combinatorial torsion correspondence for the Khovanov complex showing that an alternating product of nonzero Laplacian pseudodeterminants recovers integral Khovanov torsion in rationally acyclic $q$-degrees and differs from it by an explicit regulator in general. Finally, we transfer Lee's deformation to a filtered differential on the harmonic Khovanov subspace, giving a canonical harmonic model for the Lee spectral sequence.
Comments35 pages
