arXiv subjects

Wonwoo Kang

Publications and source records attributed to Wonwoo Kang.

6 recordsLinked to original sources

arXiv2026

Higher $q$-Continued Fractions and Dimers on Band Graphs

In this paper, we explore the theory of higher dimers on band graphs. First, we provide a combinatorial interpretation for the trace of the $q$-deformed higher continued fraction matrices, by showing that with respect to a $q$-weighting on edges, the trace gives the dimer partition function on the set of good higher dimers, which generalizes the notion of good perfect matchings. We also show that the set of good higher dimer covers form a distributive lattice with respect to face flips on square faces. Finally, we attempt to generalize the symmetry result on circular fence posets to the case of good higher dimers, by showing that the dimer partition on a certain family of band graphs are palindromic, in particular, through an approach fitting in the context of dimer theory.

math.CO↗

arXiv2026

Stable Vectorization of Persistent Laplacians via Spectral Descriptors

Persistence images vectorize persistence diagrams into stable, finite-dimensional features. Inspired by this idea, we developed a vectorization framework for the spectral information encoded by the persistent Laplacian (PL). Given a scalar signature of a persistent Laplacian, we form a Persistent Laplacian Diagram (PLD) and smooth it into a Persistent Laplacian Image (PLI). We prove a stability theorem for PLIs with respect to the Wasserstein distance between PLDs under an admissibility condition on the signature. Through experiments on MNIST and QM7, we show that PLI with suitable signatures, especially the trace, provides an effective way to extract predictive topological and geometric information from PL, outperforming existing PL-based representations in these settings.

math.AT↗

arXiv2026

Cluster Expansions from Punctured Orbifolds

We provide multiple combinatorial expansion formulas - in terms of snake graphs, labelled posets, matrices, and $T$-walks - for elements in generalized cluster algebras associated to arcs on punctured orbifolds and illustrate their equivalence. This work generalizes and unifies existing work on combinatorial expansion formulas from surfaces and unpunctured orbifolds.

math.CO↗

arXiv2026

Unimodality and Cluster Algebras from Surfaces

We prove that the rank polynomial of the lattice of order ideals of a loop fence poset is unimodal. This poset arises as the poset of join-irreducibles in the lattice of good matchings of loop graphs associated with notched arcs. Equivalently, such polynomials can be obtained by evaluating all coefficient variables in an F-polynomial at a single variable q. We also conclude that the rank polynomial of any tagged arc, whether plain or notched, is not only unimodal but also satisfies a symmetry condition known as almost interlacing. Furthermore, when the lamination consists of a single curve, the cluster expansion-evaluated by setting all cluster variables to 1 and all coefficient variables to q-is also unimodal. We conjecture that polynomials in this case are log-concave.

math.CO↗

arXiv2024

Skein relations for punctured surfaces

We investigate skein relations in cluster algebras from punctured surfaces, extending the work of Çanakçi-Schiffler and Musiker-Williams on unpunctured surfaces. Using a combinatorial expansion formula by O{ğ}uz-Yıldırım and Pilaud-Reading-Schroll, we provide explicit formulas for these relations. This work demonstrates that the punctured analogues of the bangle and bracelet functions form spanning sets for cluster algebras associated with a punctured surfaces. For surfaces with boundary and closed surfaces of genus 0, we further show that the bangles and bracelets form bases.

math.CO↗

arXiv2024

Accelerated Evaluation of Ollivier-Ricci Curvature Lower Bounds: Bridging Theory and Computation

Curvature serves as a potent and descriptive invariant, with its efficacy validated both theoretically and practically within graph theory. We employ a definition of generalized Ricci curvature proposed by Ollivier, which Lin and Yau later adapted to graph theory, known as Ollivier-Ricci curvature (ORC). ORC measures curvature using the Wasserstein distance, thereby integrating geometric concepts with probability theory and optimal transport. Jost and Liu previously discussed the lower bound of ORC by showing the upper bound of the Wasserstein distance. We extend the applicability of these bounds to discrete spaces with metrics on integers, specifically hypergraphs. Compared to prior work on ORC in hypergraphs by Coupette, Dalleiger, and Rieck, which faced computational challenges, our method introduces a simplified approach with linear computational complexity, making it particularly suitable for analyzing large-scale networks. Through extensive simulations and application to synthetic and real-world datasets, we demonstrate the significant improvements our method offers in evaluating ORC.

stat.ML↗