arXiv subjects

Jernej Grlj

Publications and source records attributed to Jernej Grlj.

4 recordsLinked to original sources

arXiv2026

Spectral geometry of Khovanov Laplacians

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.

math.GT↗

arXiv2026

6-valent vertex in the $\mathfrak{gl}_N$ web category and its categorification

We define a $2π/3$-rotationally invariant 6-valent vertex in the $\mathfrak{gl}_N$ web category. When $N = 4, 5$, we provide a categorification of the 6-valent vertex using $\mathfrak{gl}_N$ foams and decompose the hexagon web into a direct sum of indecomposables. A similar decomposition is conjectured for $N \geq 6$.

math.QA↗

arXiv2026

Analytic Torsion and Spectral Gap Capture Persistent-Laplacian Performance

While persistent Laplacians (PL) offer a richer geometric representation of data than persistent homology, utilizing their full eigenspectrum for learning tasks is often hampered by high dimensionality and the ``varying length'' problem across different filtration scales. We propose a compact spectral representation that distills the persistent Laplacian into three mathematically grounded invariants: Betti numbers, the spectral gap, and analytic torsion. Across benchmark datasets including MNIST, QM-3D, and SKEMPI WT, we demonstrate that this reduced feature space captures the essential predictive signal of the full spectrum, and in some cases outperforms it, while significantly reducing computational overhead and preventing the noise introduced by higher-frequency eigenvalues. Our results suggest that these invariants provide a principled, fixed-length interface between spectral geometry and topological learning.

cs.LG↗

arXiv2026

Action of the Witt algebra on categorified quantum groups

We construct an action of the positive Witt algebra on the categorified quantum group associated to a simply-laced Lie algebra. In the type A case, we show that this action induces an action of the positive Witt algebra on $\mathfrak{gl}_n$-foams, recovering the action of Qi, Robert, Sussan, and Wagner. We also show that this construction is compatible with the trace decategorification, inducing the action of the positive Witt algebra on the current algebra.

math.QA↗