arXiv research

Study on the topology of ordered disc configurations, revealing nontrivial homotopy classes.

Study shows challenges in reinforcement learning math problems, proposing enhancements and a hardness measure.

Topological quantum computers use hyperbolic knots for computations.

New proof shows a link problem is hard without complex links.

Researchers prove quantum invariants remain hard even when restricted.

The Hard Lefschetz Theorem extends to certain Kähler Lie Algebroids with ellipticity.

We prove that the problem of deciding whether a 2- or 3-dimensional simplicial complex embeds into $\mathbb{R}^3$ is NP-hard. Our construction also shows that deciding whether a 3-manifold with boundary tori admits an $\mathbb{S}^{3}$ filling is NP-hard. The former stands in contrast with the lower dimensional cases wh…

Algorithm calculates quantum invariants of 3-manifolds with polynomial time complexity.

We investigate the complexity of finding an embedded non-orientable surface of Euler genus $g$ in a triangulated $3$-manifold. This problem occurs both as a natural question in low-dimensional topology, and as a first non-trivial instance of embeddability of complexes into $3$-manifolds. We prove that the problem is NP…

Quantum algorithm approximates Khovanov homology ranks.

We prove that deciding if a diagram of the unknot can be untangled using at most $k$ Riedemeister moves (where $k$ is part of the input) is NP-hard. We also prove that several natural questions regarding links in the $3$-sphere are NP-hard, including detecting whether a link contains a trivial sublink with $n$ componen…

Polynomial invariants classify molecular chains based on their contact arrangements.

Study monopole h-invariants from a topological viewpoint.

The paper explores conditions for topological rigidity in quotients of the Davis complex.

Constructing compact non-Kähler manifolds with and without the Hard Lefschetz Condition

The study of higher-order homology embeddings for manifold topology.

Study vineyards linking TDA and knot theory, showing rich topological features.

(1) We show that if a presentation of the trivial group is "hard to trivialize", in the sense that lots of Tietze moves are necessary to transform it into the trivial presentation, then the associated presentation complex (which is a contractible 2-dimensional cell complex) is "hard to embed in $\mathbb{R}^3$", in the …

Paper calculates topological complexity of robot movement in narrow aisles.

Computer experiments reveal complex knots that don't simplify.

Complex manifolds with compatible metric have a naturally defined subspace of harmonic differential forms that satisfy Serre, Hodge, and conjugation duality, as well as hard Lefschetz duality. This last property follows from a representation of $sl(2,\mathbb{C})$, generalizing the well known structure on the harmonic f…

FibeRed reduces complex data dimensions while preserving topology.

The well-known Kähler identities naturally extend to the non-integrable setting. This paper deduces several geometric and topological consequences of these extended identities for compact almost Kähler manifolds. Among these are identities of various Laplacians, generalized Hodge and Serre dualities, a generalized hard…

This paper proposes a new method for learning covers of geometric datasets to improve topological inference and visualization.

In this paper, we use an aerial base station (aerial-BS) to enhance fairness in a dynamic environment with user mobility. The problem of optimally placing the aerial-BS is a non-deterministic polynomial-time hard (NP-hard) problem. Moreover, the network topology is subject to continuous changes due to the user mobility…

We investigate the computational complexity of some problems in three-dimensional topology and geometry. We show that the problem of determining a bound on the genus of a knot in a 3-manifold, is NP-complete. Using similar ideas, we show that deciding whether a curve in a metrized PL 3-manifold bounds a surface of area…

In the recent years money laundering schemes have grown in complexity and speed of realization, affecting financial institutions and millions of customers globally. Strengthened privacy policies, along with in-country regulations, make it hard for banks to inner- and cross-share, and report suspicious activities for th…

The increasing penetration of distributed energy resources poses numerous reliability issues to the urban distribution grid. The topology estimation is a critical step to ensure the robustness of distribution grid operation. However, the bus connectivity and grid topology estimation are usually hard in distribution gri…

In this empirical paper, we investigate how learning agents can be arranged in more efficient communication topologies for improved learning. This is an important problem because a common technique to improve speed and robustness of learning in deep reinforcement learning and many other machine learning algorithms is t…

Paper tackles optimal network compression for financial systems.

Finding optimal correction of errors in generic stabilizer codes is a computationally hard problem, even for simple noise models. While this task can be simplified for codes with some structure, such as topological stabilizer codes, developing good and efficient decoders still remains a challenge. In our work, we syste…

Researchers use discrete Morse theory to improve the topology of matching complexes of complete graphs.

This paper poses some basic questions about instances (hard to find) of a special problem in 3-manifold topology. "Important though the general concepts and propositions may be with the modern industrious passion for axiomatizing and generalizing has presented us...nevertheless I am convinced that the special problems …

Study shortest non-separating curves on non-orientable surfaces, proving NP-hardness and tractability.

Paper proves hardness of learning various complex models under local pseudorandom generators.

This work connects hardness of approximation and learning.

Study on hard Legendrian unknots using normal rulings.

Constructs real algebraic functions with specified preimages.

Moving between 3-manifold triangulations is NP-hard

New method finds large counterexamples by selectively exploring triangulations.

Hard instances, which require a long time for a specific algorithm to solve, help (1) analyze the algorithm for accelerating it and (2) build a good benchmark for evaluating the performance of algorithms. There exist several efforts for automatic generation of hard instances. For example, evolutionary algorithms have b…

Tackles the computational hardness of HPC detection, conjecturing equivalence to PC detection.

The main goal of this work is to present a detailed study of the foundations of Complex Geometry, highlighting its geometrical, topological and analytical aspects. Beginning with a preliminary material, such as the basic results on holomorphic functions in one or more variables and the definition and first examples of …

Three hard diagrams of the unknot require extra crossings to simplify.

Hardness proven for learning neural networks with polynomial size and Gaussian inputs.

In this article, we introduce a fixed parameter tractable algorithm for computing the Turaev-Viro invariants TV(4,q), using the dimension of the first homology group of the manifold as parameter. This is, to our knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parame…

Study categorizes knots and links as rigid or shaky based on Reidemeister moves.

This paper gives infinitely many examples of unknot diagrams that are hard, in the sense that the diagrams need to be made more complicated by Reidemeister moves before they can be simplified. In order to construct these diagrams, we prove theorems characterizing when the numerator of the sum of two rational tangles is…