The Journal of Experimental Mathematics is a Diamond Open Access journal, with no charges to authors or readers. JXM publishes research articles in all branches of mathematics that feature experimental results with important implications. There are no strict page limits. A published article may be accompanied when appropriate by links to other media, such as repositories for code or data, notes or videos on background material, and videos of related lectures.
Repulsion of Zeros Close to s = 1/2 for Modular Form L-Functions
In this paper, we present results of several experiments in which we model the repulsion of low-lying zeros of L-functions using random matrix theory. Previous work has typically focused on the twists of L-functions associated to elliptic curves and on families that can be modeled by unitary and orthogonal matrices. We consider families of L-functions of modular forms of weight greater than 2 and we consider families that can be modeled by symplectic matrices. Additionally, we explore a model for low-lying zeros of twists that incorporates a discretization arising from the Kohnen-Zagier theorem. Overall, our numeric evidence supports the expectation that the repulsion of zeros decreases as the conductor of the twist increases. Surprisingly, though, it appears that using the discretization that arises from the Kohnen-Zagier theorem does not model the data better than if the discretization is not used for forms of weight 4 or above.
Copyright (c) 2026 Nicol´as Coloma, Maria Espericueta Sandoval, Erika Lopez, Francisco Ponce, Gustavo Rama, Nathan C. Ryan, and Alejandro Vargas-Altamirano https://creativecommons.org/licenses/by-nc/4.0
Big Data Approach to Kazhdan-Lusztig Polynomials
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.
Copyright (c) 2026 Abel Lacabanne, Daniel Tubbenhauer, and Pedro Vaz https://creativecommons.org/licenses/by-nc/4.0
A few years ago Morier-Genoud and Ovsienko introduced an interesting quantization of the real numbers as certain power series in a quantization parameter q. It is known now that the golden ratio has minimal radius among all these series. We study the rational numbers having maximal radius of convergence equal to 1, which we call Kronecker fractions. We prove that the corresponding continued fraction expansions must be palindromic and describe all Kronecker fractions with prime denominators. We found several infinite families of Kronecker fractions and all Kronecker fractions with denominator less than 10000. We also comment on the irrational case and on the relation with braids, rational knots and links.
Copyright (c) 2026 Sam J. Evans, Alexander P. Veselov, and Brian Winn https://creativecommons.org/licenses/by-nc/4.0
Uniquely Optimal Codes of Low Complexity are Symmetric
We formulate explicit predictions concerning the symmetry of optimal codes in compact metric spaces. This motivates the study of optimal codes in various spaces where these predictions can be tested.
Copyright (c) 2026 Emily J. King, Dustin G. Mixon, Hans Parshall, and Chris Wells https://creativecommons.org/licenses/by-nc/4.0
Computation of P-Functions on Plane Algebraic Curves
Numerical tools for computation of P-functions, also known as Kleinian, or multiply periodic, are proposed. In this connection, computation of periods of both the first and second kinds is reconsidered. An analytical approach to constructing Riemann surfaces of plane algebraic curves of low gonalities is used. The approach is based on explicit radical solutions to quadratic, cubic, and quartic equations, which serve for hyperelliptic, trigonal, and tetragonal curves, respectively. The proposed analytical models of Riemann surfaces give full control over computation of the Abel image of any point or divisor. Therefore, computation of P-functions at Abel images of given divisors can be done directly. An alternative computation with the help of the Jacobi inversion problem is used for verification. Hyperelliptic and trigonal curves are considered in detail, and illustrated by examples. A method of finding the unique characteristic corresponding to the vector of Riemann constants is suggested for non-hyperelliptic and hyperelliptic curves.
Copyright (c) 2026 Julia Bernatska https://creativecommons.org/licenses/by-nc/4.0
Classification of Cellular Fake Surfaces
Generic polyhedra are interesting mathematical objects to study in their own right. In this paper, we initiate a systematic study of two-dimensional generic polyhedra with an eye towards applications to low-dimensional topology, especially the Andrews-Curtis and Zeeman conjectures. After recalling the basic notions of generic polyhedra and fake surfaces, we derive some interesting properties of fake surfaces. Our main result is a complete classification of acyclic cellular fake surfaces up to complexity 4 and a classification of acyclic cellular fake surfaces without small disks of complexity 5. From this classification, we prove the contractibility conjecture for acyclic cellular fake surfaces of complexity 4, and the embedded disk conjecture up to complexity 5. We provide evidence for the conjectures that the probability of being a spine among fake surfaces is 0 and that every contractible fake surface has an embedded disk.
Copyright (c) 2026 Lucas Fagan, Yang Qiu, and Zhenghan Wang https://creativecommons.org/licenses/by-nc/4.0
Concatenative Nonmonotonicity and Optimal Links in HP Protein Folding Models
The hydrophobic-polar (HP) model represents proteins as binary strings embedded in lattices, with fold quality measured by an energy score. We prove that the optimal fold energy is not monotonic under concatenation for several standard lattices, including the 2D and 3D rectangular, hexagonal, and triangular lattices. In other words, concatenating two polymers can produce a fold with strictly worse optimal energy than one of the polymers alone. For closed chains, we show that under the levels-of-hydrophobicity model of Agarwala et al. (1997), proper links can arise as uniquely optimal folds, revealing an unexpected connection between HP models and knot/link theory.
Copyright (c) 2026 Bjørn Kjos-Hanssen https://creativecommons.org/licenses/by-nc/4.0
Prime Numbers and Dynamics of the Polynomial x² - 1
Let n ∈ Z>=2. By P(n) we denote the set of all prime divisors of the integers in the sequence n, n2 - 1, (n2 - 1)2 - 1,... . We ask whether the set P(n) determines n uniquely under the assumption that n ≠ m2 - 1 for m ∈ Z>=2. This problem originates in the structure theory of infinite-dimensional Lie algebras. We show that the sets P(n) generate infinitely many equivalence classes of positive integers under the equivalence relation n1 ∼ n2 ⇔ P(n1) = P(n2). We also prove that the sets P(n) separate all positive integers up to 229, and we provide some heuristics on why the answer to our question should be positive.
Copyright (c) 2026 Ivan Penkov and Michael Stoll https://creativecommons.org/licenses/by-nc/4.0
Continued Fractions and the Four Color Theorem: Erratum
This note corrects a minor mistake in my paper "Continued Fractions and the Four Color Theorem" (J. Exp. Math. vol. 1, 2025).
Copyright (c) 2026 Richard Evan Schwartz https://creativecommons.org/licenses/by-nc/4.0