arXiv subjects
Publications and source records attributed to Angus Gruen.
The snake-in-the-box problem, introduced by Kautz in 1958, asks for the longest induced (chordless) path, called a snake, in the hypercube graph $Q_n$. The maximum length $a(n)$ is known in each dimension $n \leq 8$. We give snakes that are longer than the previous best-known in every dimension from $9$ to $13$, improving the lower bound on $a(n)$. All record-length paths are provided in a computer-verifiable dataset.
Mathematical search problems present a unique challenge for Reinforcement Learning (RL) due to vast search spaces and sparse rewards. In previous works, the Andrews-Curtis (AC) conjecture was established as an illustrative example of such problems. In this work, we identify a critical structural barrier in the AC landscape: a "Two-Hump" distribution, where problem instances are either trivially solvable or effectively impossible, with a scarcity of intermediate "hard-but-solvable" instances required for effective learning. We tackle this challenge through two primary avenues: novel data generation techniques to populate the difficulty gap, and significant algorithmic enhancements including the introduction of supermoves and Transformer-based architectures. We demonstrate substantial performance improvements over previous baselines, and release new comprehensive benchmark datasets including AC-19 (125,192 AC-trivial presentations of varying difficulty with length at most 19) and AC-1M (1,136,154 hard AC-trivial presentations of length at most 30), the first large-scale, publicly available datasets of this kind.
We construct the invariant $F_K^{\mathfrak{sl}_3}\in\mathbb{Z}[q,q^{-1}][[x,y]]$ for any positive braid knot $K$, whose existence was conjectured by Park, building on earlier work of Gukov--Manolescu. The main step in our work extends a result by the first author on the invariant associated to symmetric representations of $\mathfrak{sl}_3$ to all irreducible representations. We conclude with a conjectural framework for constructing $F_K^{\mathfrak{sl}_N}$ for arbitrary $N$.
Using a long-standing conjecture from combinatorial group theory, we explore, from multiple perspectives, the challenges of finding rare instances carrying disproportionately high rewards. Based on lessons learned in the context defined by the Andrews-Curtis conjecture, we propose algorithmic enhancements and a topological hardness measure with implications for a broad class of search problems. As part of our study, we also address several open mathematical questions. Notably, we demonstrate the length reducibility of all but two presentations in the Akbulut-Kirby series (1981), and resolve various potential counterexamples in the Miller-Schupp series (1991), including three infinite subfamilies.
Using resurgent analysis we offer a novel mathematical perspective on a curious bijection (duality) that has many potential applications ranging from the theory of vertex algebras to the physics of SCFTs in various dimensions, to q-series invariants in low-dimensional topology that arise, e.g. in Vafa-Witten theory and in non-perturbative completion of complex Chern-Simons theory. In particular, we introduce explicit numerical algorithms that efficiently implement this bijection. This bijection is founded on preservation of relations, a fundamental property of resurgent functions. Using resurgent analysis we find new structures and patterns in complex Chern-Simons theory on closed hyperbolic 3-manifolds obtained by surgeries on hyperbolic twist knots. The Borel plane exhibits several intriguing hints of a new form of integrability. An important role in this analysis is played by the twisted Alexander polynomials and the adjoint Reidemeister torsion, which help us determine the Stokes data. The method of singularity elimination enables extraction of geometric data even for very distant Borel singularities, leading to detailed non-perturbative information from perturbative data. We also introduce a new double-scaling limit to probe 0-surgeries from the limiting $r\to\infty$ behavior of 1/r surgeries, and apply it to the family of hyperbolic twist knots.
For every knot $K$ and lie algebra $\mathfrak{g}$, there is a Gukov-Manolescu series denoted $F^{\mathfrak{g}}_K$ which serves as an analytic continuation of the quantum knot invariants associated to finite dimensional irreducible representations of $\mathfrak{g}$. There has been a great deal of work done on computing this invariant for $\mathfrak{g} = \mathfrak{sl}_2$ but comparatively less work has studied other lie algebras. In this paper we extend the large colour $R$ matrix from $\mathfrak{sl}_2$ to symmetrically coloured $\mathfrak{sl}_N$. This gives a definition for $F^{\mathfrak{sl}_N, sym}_K$ for positive braid knots and allows for predictions of $F^{\mathfrak{sl}_N, sym}_K$ for a much larger class of knots and links. It also provides further evidence towards a conjectural HOMFLY-PT analouge of $F_K$.
We generalize the $F_K$ invariant, i.e. $\widehat{Z}$ for the complement of a knot $K$ in the 3-sphere, the knots-quivers correspondence, and $A$-polynomials of knots, and find several interconnections between them. We associate an $F_K$ invariant to any branch of the $A$-polynomial of $K$ and we work out explicit expressions for several simple knots. We show that these $F_K$ invariants can be written in the form of a quiver generating series, in analogy with the knots-quivers correspondence. We discuss various methods to obtain such quiver representations, among others using $R$-matrices. We generalize the quantum $a$-deformed $A$-polynomial to an ideal that contains the recursion relation in the group rank, i.e. in the parameter $a$, and describe its classical limit in terms of the Coulomb branch of a 3d-5d theory. We also provide $t$-deformed versions. Furthermore, we study how the quiver formulation for closed 3-manifolds obtained by surgery leads to the superpotential of 3d $\mathcal{N}=2$ theory $T[M_3]$ and to the data of the associated modular tensor category $\text{MTC} [M_3]$.
Reducing a 6d fivebrane theory on a 3-manifold $Y$ gives a $q$-series 3-manifold invariant $\widehat{Z}(Y)$. We analyse the large-$N$ behaviour of $F_K=\widehat{Z}(M_K)$, where $M_K$ is the complement of a knot $K$ in the 3-sphere, and explore the relationship between an $a$-deformed ($a=q^N$) version of $F_{K}$ and HOMFLY-PT polynomials. On the one hand, in combination with counts of holomorphic annuli on knot complements, this gives an enumerative interpretation of $F_K$ in terms of counts of open holomorphic curves. On the other, it leads to closed form expressions for $a$-deformed $F_K$ for $(2,2p+1)$-torus knots. They suggest a further $t$-deformation based on superpolynomials, which can be used to obtain a $t$-deformation of ADO polynomials, expected to be related to categorification. Moreover, studying how $F_K$ transforms under natural geometric operations on $K$ indicates relations to quantum modularity in a new setting.
A formula for the modular data of $\mathcal{Z}(Vec^ωG)$ was given by Coste, Gannon and Ruelle in arXiv:hep-th/0001158, but without an explicit proof for arbitrary 3-cocycles. This paper presents a derivation using the representation category of the quasi Hopf algebra $D^ωG$. Further, we have written code to compute this modular data for many pairs of small finite groups and 3-cocycles. This code is optimised using Galois symmetries of the S and T matrices. We have posted a database of modular data for the Drinfeld center of every Morita equivalence class of pointed fusion categories of dimension less than 64.