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.
By Tuomas Kelom\"aki, Abel Lacabanne, Daniel Tubbenhauer, Pedro Vaz, Victor L. Zhang
The paper introduces a new mathematical framework for polynomial group convolutional neural networks (PGCNNs) using graded group algebras. It presents two natural parametrizations of the architecture—based on Hadamard and Kronecker products—that are related by a linear map. The authors compute the dimension of the resulting neuromanifold, show it depends only on the number of layers and group size, and describe the general fiber of the Kronecker parametrization, conjecturing a similar description for the Hadamard case, supported by explicit computations for small groups and shallow networks.
By Yacoub Hendi, Daniel Persson, Magdalena Larfors
arXiv:2609.40290v1 Announce Type: cross
Abstract: In algorithmic statistics a string x is explained by a finite set containing it, and Kolmogorov's structure function records the smallest such model...
By Romie Banerjee
arXiv:2607. 12026v1 Announce Type: cross Abstract: Finite groups are rigid algebraic objects, whose Cayley graphs expose a rich network geometry through which group-theoretic structure can be measured, compared, and learned.
By Rashid Barket, Enrico Grimaldi, Yacoub Hendi, Edward Hirst, Adam Onus, Harmeet Singh
arXiv:2606. 26660v1 Announce Type: cross Abstract: Triangulations, i.
By Charles Arnal, Jacky H. T. Yip, Fran\c{c}ois Charton, Gary Shiu
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...
By Sel\c{c}uk Kayacan
arXiv:2609. 28500v1 Announce Type: cross Abstract: We study exact representability by bias-free shallow polynomial neural networks using algebraic geometry.
By Sepehr Akbari, Shahrzad Jamshidi
arXiv:2606. 26212v1 Announce Type: new Abstract: A Graph Neural Network (GNN) framework for predicting the solvability of finite groups from their Cayley graph representations was introduced in [1].
By Tal Weissblat
arXiv:2606. 07619v1 Announce Type: new Abstract: We present a Graph Neural Network (GNN) framework for the classification of finite groups according to their solvability.
By Tal Weissblat
arXiv:2609.07031v1 Announce Type: new
Abstract: Learning with group invariances is central to many scientific and geometric learning problems, yet its computational foundations remain poorly understo...
By Ashkan Soleymani, Behrooz Tahmasebi, Patrick Jaillet, Stefanie Jegelka
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.
By Chris Kapulkin, Nathan Kershaw
arXiv:2608. 08154v1 Announce Type: cross Abstract: The Zarankiewicz number Z(m,n,s,t) is the maximum number of edges in a bipartite graph with parts of orders m and n containing no copy of Ks,t.
By Koyar Afrasyab