arXiv:2603.07955v4 Announce Type: replace-cross Abstract: We develop a reinforcement learning pipeline for simplifying knot diagrams. A trained agent learns move proposals and a value heuristic for n...
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 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.
AutoGraphForge is a computational pipeline designed to automate the discovery, refutation, formalization, and proving of graph-theoretic conjectures. It generates conjectures using a Graffiti3 generator, filters out known results with a novelty filter, tests candidates against a large dataset of graphs, and refines surviving conjectures through counterexample search. The pipeline then translates each conjecture into Lean 4, verifies proofs with neural provers, and integrates the results into a formal library.
arXiv:2609. 29881v1 Announce Type: cross Abstract: Choosing Shellsort gaps is a well-known open problem.
The paper reports a new record for the snake-in-the-box problem, presenting a length‑191 snake in the 9‑dimensional hypercube, surpassing the previous 190‑length record that had stood for 14 years. It also establishes new lower bounds for dimensions 10 through 13. The authors introduce a novel search technique called snakepits, which expands the search space by considering collections of disjoint snakes, and propose a new benchmark, Snakepit‑in‑the‑Box, that evaluates maximal edge counts across multiple components. Additionally, they present Beam Anchor, a learned construction algorithm that generates 100 inequivalent length‑190 snakes in dimension 9.
arXiv:2606. 18001v1 Announce Type: new Abstract: Knowledge graph (KG) foundation models (KGFMs) are zero-shot generalizers: trained once, they can predict links on unseen graphs without retraining.
The paper introduces the Free Inference dimension (dFI) as a new combinatorial measure of environmental complexity for value‑mixture agents in finite meta‑reinforcement learning. It shows that dFI is smaller than the VC‑dimension and relates to the Natarajan dimension, providing PAC‑style generalization bounds. The authors also define a complementary PMS identification dimension and demonstrate that a hybrid strategy—averaging until the first collision and then selecting—achieves optimal performance, supported by grid‑world experiments.
arXiv:2509. 06108v2 Announce Type: replace-cross Abstract: Graph drawing concerns the algorithmic visualization of graphs.
arXiv:2602. 16807v2 Announce Type: replace Abstract: A collection of hyperplanes $\mathcal{H}$ slices all edges of the $n$-dimensional hypercube $Q_n$ with vertex set $\{-1,1\}^n$ if, for every edge $e$ in the hypercube, there exists a hyperplane in $\mathcal{H}$ intersecting $e$ in its interior.
We study certain extremal problems in combinatorial geometry that ask about configurations of points in an $n \times n$ grid that satisfy strict, global geometric constraints. Classical exact solvers suffer from combinatorial explosion for these types of problems, and standard reinforcement learning and transformer-based models struggle with the sparse reward "validity cliff" and quadratic token-consumption limits.
The paper proposes using concepts from low‑dimensional topology—specifically Morse theory and cobordism—to enhance graph diffusion models, introducing the MG‑Diff pipeline. It provides theoretical guarantees that the Morse‑theoretic guidance remains stable under small perturbations when a positive decision‑gap exists. The authors demonstrate the approach on spatio‑temporal graph forecasting and graph regeneration, suggesting broader potential for topology in machine learning.