arXiv subjects

Hakan Akgün

Publications and source records attributed to Hakan Akgün.

4 recordsLinked to original sources

arXiv2026

KnottedGraph: Scalable knotted-graph topology for scientific and mathematical discovery

Scientific data span heterogeneous structures, including coordinates, networks, surfaces, volumes and fields, yet their topology can be quantified within a common framework through graph connectivity, cycle structure, genus and spatial embedding. Graph- and homology-based summaries do not determine spatial embedding, while standard knot and link polynomials require extensions to accommodate branching graphs. Here, we introduce KnottedGraph, a computational framework that converts such scientific representations to knotted graphs that retain graph connectivity and spatial embedding together. It constructs projected diagrams and PD codes, enabling various topological analyses, including Yamada-polynomial evaluation for topological classification. For scalable exact evaluation, it combines partial resolutions that leave the same unresolved connections and optimizes their processing order; the resulting algorithm is verified against published topological invariants of knotted graphs with up to 500 crossings. This scalability enables us to introduce an LLM-assisted mathematical-discovery methodology, in which computational topological data generated across knotted-graph families are used to identify candidate closed-form formulas. With this approach, we identify analytical Yamada-polynomials for generic graph motif families exhibiting Abelian and non-Abelian word sequences. Together, these scalable capabilities make knotted-graph topology computationally accessible across scientific domains, enabling large-scale classification and introducing a route from topological data to LLM-assisted AI4Math discovery.

cs.MS↗

arXiv2026

Topological classification through knotted graphs: Fermi surface dispersions and Lifshitz transitions

Knot theory has provided a rich topological taxonomy for band structures, but its reach is fundamentally limited: knot invariants classify only 1D nodal lines at gap closure, and cannot encode the full dispersion or rich Fermi surface structure of realistic materials. Here we show that knotted graphs (knots that admit graph-like intersections in 3D space) - which have so far been elusive in condensed matter literature - provide a unified topological language for classifying the entire band dispersion, and even the eigenstate topology in some contexts. We propose a new framework beyond the existing Yamada polynomials that can topologically characterize the intricacies of realistic Fermi surfaces completely, crucially including how their multiple disconnected pieces are nested. This yields the Yamada set, a boundary-resolved extension which organizes the full topological evolution across energy into a Yamada sequence: a compact dispersion-level fingerprint directly tied to experimental signatures of Lifshitz transitions. Our framework is demonstrated with DFT-based band structures of real materials. Beyond dispersion-level classifications, this framework can be extended to non-Hermitian exceptional surfaces, where Berry-curvature flux further equips the knotted-graph skeleton with a directed Abelian edge flow that also captures the eigenstate topology.

cond-mat.mes-hall↗

arXiv2026

HSG-12M: A Large-Scale Benchmark of Spatial Multigraphs from the Energy Spectra of Non-Hermitian Crystals

AI is transforming scientific research by revealing new ways to understand complex physical systems, but its impact remains constrained by the lack of large, high-quality domain-specific datasets. A rich, largely untapped resource lies in non-Hermitian quantum physics, where the energy spectra of crystals form intricate geometries on the complex plane -- termed as Hamiltonian spectral graphs. Despite their significance as fingerprints for electronic behavior, their systematic study has been intractable due to the reliance on manual extraction. To unlock this potential, we introduce Poly2Graph: a high-performance, open-source pipeline that automates the mapping of 1-D crystal Hamiltonians to spectral graphs. Using this tool, we present HSG-12M: a dataset containing 11.6 million static and 5.1 million dynamic Hamiltonian spectral graphs across 1401 characteristic-polynomial classes, distilled from 177 TB of spectral potential data. Crucially, HSG-12M is the first large-scale dataset of spatial multigraphs -- graphs embedded in a metric space where multiple geometrically distinct trajectories between two nodes are retained as separate edges. This simultaneously addresses a critical gap, as existing graph benchmarks overwhelmingly assume simple, non-spatial edges, discarding vital geometric information. Benchmarks with popular GNNs expose new challenges in learning spatial multi-edges at scale. Beyond its practical utility, we show that spectral graphs serve as universal topological fingerprints of polynomials, vectors, and matrices, forging a new algebra-to-graph link. HSG-12M lays the groundwork for data-driven scientific discovery in condensed matter physics, new opportunities in geometry-aware graph learning and beyond.

cs.LG↗

arXiv2025

Protected chaos in a topological lattice

The erratic nature of chaotic behavior is thought to erode the stability of periodic behavior, including topological oscillations. However, we discover that in the presence of chaos, non-trivial topology not only endures but also provides robust protection to chaotic dynamics within a topological lattice hosting non-linear oscillators. Despite the difficulty in defining topological invariants in non-linear settings, non-trivial topological robustness still persists in the parametric state of chaotic boundary oscillations. We demonstrate this interplay between chaos and topology by incorporating chaotic Chua's circuits into a topological Su-Schrieffer-Heeger (SSH) circuit. By extrapolating from the linear limit to deep into the non-linear regime, we find that distinctive correlations in the bulk and edge scroll dynamics effectively capture the topological origin of the protected chaos. Our findings suggest that topologically protected chaos can be robustly achieved across a broad spectrum of periodically-driven systems, thereby offering new avenues for the design of resilient and adaptable non-linear networks.

cond-mat.mes-hall↗