The paper proves that for many standard link invariants, the proportion of distinct invariant values and the detection probability among prime alternating links with at most n crossings decay exponentially in n, with an explicit universal rate. It shows that almost every such link lies in an invariant fiber whose size is itself exponential in n, a phenomenon that applies broadly, including to the Jones and HOMFLYPT polynomials and integral Khovanov homology. The companion website provides detailed data, including complete distributions of fiber sizes, separate alternating and non-alternating data, and topological data analysis.
arXiv:2610.10206v1 Announce Type: cross Abstract: Links are disjoint unions of circles smoothly embedded in $S^3$. We use reinforcement learning and Bayesian optimisation to obtain new upper bounds o...
arXiv:2412. 01283v3 Announce Type: replace-cross Abstract: We investigate the structure of Kazhdan-Lusztig polynomials of the symmetric group by leveraging computational approaches from big data, including exploratory and topological data analysis, applied to the polynomials for symmetric groups of up to 11 strands.
arXiv:2506. 15020v2 Announce Type: replace-cross Abstract: We propose persistent discrete homology as a tool for topological data analysis and discuss its advantages over the existing methods.
arXiv:2608. 18570v1 Announce Type: cross Abstract: Machine learning and data analysis techniques have recently emerged as powerful tools for identifying patterns and formulating conjectures in mathematical research, most notably in the field of low-dimensional topology.
arXiv:2512.23348v3 Announce Type: replace-cross Abstract: We introduce a data-analysis framework based on filtrations of finite topological spaces. Starting from a finite metric data set, we construc...
arXiv:2607. 20657v1 Announce Type: cross Abstract: Unique and rapid classification of knots and links is an open mathematical problem that is relevant to a range of (bio)physical systems, including polymer melts, DNA, and proteins.
arXiv:2512.13825v2 Announce Type: replace-cross Abstract: In unsupervised learning, the training data for deep learning does not come with any labels, thus forcing the algorithm to discover hidden pa...
arXiv:2605. 26234v2 Announce Type: replace-cross Abstract: A recent conjecture by Joel Fine posits a relationship between the coefficients of the HOMFLY polynomial of a knot $K$ in the 3-sphere $S^3$, and the signed count of minimal surfaces in hyperbolic 4-space $\mathrm{H}^4$ meeting the sphere at infinity at $K$, with prescribed genus and self-intersection number.
The paper presents numerical evidence that topological freezing in SU(3) Yang‑Mills theory can be reduced using a flow‑based method. By combining open boundary conditions with Stochastic Normalizing Flows, the authors analyze how computational costs scale at lattice spacings as fine as 0.03 fm. They find that both the integrated autocorrelation time of the squared topological charge and the statistical uncertainty on the topological susceptibility stay under control across the explored parameter range with current computational resources.
The article introduces a new non‑archimedean metric, the cophenetic metric, defined on persistent homology classes of all degrees. It demonstrates that zeroth persistent homology combined with this metric and various hierarchical clustering algorithms yields statistically verifiable, commensurate topological information on multiple datasets. The resulting clusters, evaluated by silhouette score and Rand index, perform well, and the metric enables visualization of inter‑relations among persistent homology classes across all degrees via rooted trees.
The paper introduces a topological method for detecting Hopf bifurcations directly from scalar time series. It combines delay-coordinate reconstruction with persistent homology, using the maximum persistence of one‑dimensional homology classes as a scalar descriptor of cyclic structure. Finite‑resolution persistence bounds are derived for the supercritical Hopf case, and a derivative‑based estimator localizes the critical parameter; the method is tested on the Hopf normal form, Lorenz system, and a reduced Belousov–Zhabotinsky model.
arXiv:2606. 06117v1 Announce Type: cross Abstract: We introduce pVR, a topological machine learning framework for alignment-free genomic sequence classification that combines $p$-adic numbers with topological data analysis.