arXiv research

Enhanced Alexander module detects linking numbers in links.

Classifies modules of surface-knots in terms of their properties.

Corrects a paper on Alexander modules and answers a related question.

Constructs the medial quandle from link's peripheral structure.

Correspondence found between Askey-Wilson polynomials and genus-two handlebody skein module.

Novel method computes Kauffman bracket skein modules of small 3-manifolds.

Joyce observed that the Alexander invariant and the medial quandle of a classical knot are equivalent to each other, as invariants. In the present paper, we discuss the rather complicated extension of Joyce's observation to several different medial quandles and reduced (one-variable) Alexander modules associated with c…

Study on skein modules and character varieties of Seifert manifolds.

Skein modules over 3-manifolds are shown to form line bundles.

Holomorphic structures on quantum flag manifolds uniquely defined.

Extending ideas of Hedden-Ni, we show that the module structure on Khovanov homology detects split links. We also prove an analogue for untwisted Heegaard Floer homology of the branched double cover. Technical results proved along the way include two interpretations of the module structure on untwisted Heegaard Floer h…

Study meromorphic open-string vertex algebras and modules over Riemannian manifolds.

Proves effective Chen ranks conjecture for Koszul modules.

We calculate the Kauffman bracket skein module (KBSM) of the complement of all two-bridge links. For a two-bridge link, we show that the KBSM of its complement is free over the ring $\BC[t^{\pm 1}]$ and when reducing $t=-1$, it is isomorphic to the ring of regular functions on the character variety of the link group.

The paper computes groups and modules for wheel graphs using Fibonacci and Chebyshev polynomials.

By the work of Harer, the reduced homology of the complex of curves is a fundamental cohomological object associated to all torsion free finite index subgroups of the mapping class group. We call this homology group the Steinberg module of the mapping class group. It was previously known that the curve complex has the …

Let $(M,g)$ be a pseudo-Riemannian manifold of signature $(p,q)$. We construct mutually quasi-inverse equivalences between the groupoid of bundles of weakly-faithful complex Clifford modules on $(M,g)$ and the groupoid of reduced complex Lipschitz structures on $(M,g)$. As an application, we show that $(M,g)$ admits a …

Develops a new deep learning framework for privacy-preserving text representations.

Proves module structure on odd Khovanov homology and applies to ribbon 2-knots.

KM method reduces ConvNet parameters to 9% higher accuracy with minimal additional memory.

Spatiotemporal sequence prediction is an important problem in deep learning. We study next-frame(s) video prediction using a deep-learning-based predictive coding framework that uses convolutional, long short-term memory (convLSTM) modules. We introduce a novel reduced-gate convolutional LSTM(rgcLSTM) architecture that…

Phase Modulation on the Hypersphere (PMH) is a power efficient modulation scheme for the \textit{load-modulated} multiple-input multiple-output (MIMO) transmitters with central power amplifiers (CPA). However, it is difficult to obtain the precise channel state information (CSI), and the traditional optimal maximum lik…

Spectrum selectively trains LLMs based on SNR to save resources.

Study skein modules via gauge theory, finding non-TQFT dimensions.

We generalize our previous work on categorification of Kauffman bracket skein module of surfaces, by extending our homology to tangles in cylinders over surfaces, F x [0,1]. Our homology of 0-tangles and 1-tangles in D^3 coincides (up to normalization) with Khovanov link homology and the reduced Khovanov link homology.…

The reduced Burau representation $V_n$ of the braid group $B_n$ is obtained from the action of $B_n$ on the homology of an infinite cyclic cover of the $n$-punctured disc. In this note, we calculate $H_*(B_n;V_n)$ as a module over the Laurent polynomial ring $\mathbb{Q}[t,t^{-1}]$.

Aramayona and Leininger have provided a "finite rigid subset" $\mathfrak{X}(Σ)$ of the curve complex $\mathscr{C}(Σ)$ of a surface $Σ= Σ^n_g$, characterized by the fact that any simplicial injection $\mathfrak{X}(Σ) \to \mathscr{C}(Σ)$ is induced by a unique element of the mapping class group $\mathrm{Mod}(Σ)$. In this…

In this work, we investigate the feasibility and effectiveness of employing deep learning algorithms for automatic recognition of the modulation type of received wireless communication signals from subsampled data. Recent work considered a GNU radio-based data set that mimics the imperfections in a real wireless channe…

In arXiv:math/0508510, Rasmussen observed that the Khovanov-Rozansky homology of a link is a finitely generated module over the polynomial ring generated by the components of this link. In the current paper, we study the module structure of the middle HOMFLYPT homology, especially the Betti numbers of this module. For …

Ends and cohomology theory for noncompact spaces.

Proposes ContSup to boost local learning by supplying context between isolated modules.

We give new obstructions to the module structures arising in Heegaard Floer homology. As a corollary, we characterize the possible modules arising as the Heegaard Floer homology of an integer homology sphere with one-dimensional reduced Floer homology. Up to absolute grading shifts, there are only two. We use this coro…

The paper shows the computation of the noncommutative generalization of the A-polynomial of the trefoil knot. The classical A-polynomial was introduced by Cooper, Culler, Gillet, Long and Shalen, and was generalized to the context of Kauffman bracket skein modules by the author in joint work with Frohman and Lofaro. A …

Study connects knot contact homology to Chern-Simons theory's large N limit.

Lifelong learning is a very important step toward realizing robust autonomous artificial agents. Neural networks are the main engine of deep learning, which is the current state-of-the-art technique in formulating adaptive artificial intelligent systems. However, neural networks suffer from catastrophic forgetting when…

In financial field, a robust software system is of vital importance to ensure the smooth operation of financial transactions. However, many financial corporations still depend on operators to identify and eliminate the system failures when financial software systems break down. This traditional operation method is time…

COIN++ compresses multiple data types efficiently.

Abstract: Homotopy Poisson algebra models for reduced spaces derived from Poisson structures.

TEAFormers preserve multi-dimensional time series structures for better forecasting.

Torsion objects of von Neumann categories describe the phenomen "spectrum near zero" discovered by S. Novikov and M. Shubin. In this paper we classify Hermitian forms on torsion objects of a finite von Neumann category. We prove that any such form can be represented as a discriminant form of a degenerate Hermitian form…

In this paper, we suggest a construction of determinant lines of finitely generated Hilbertian modules over finite von Neumann algebras. Nonzero elements of the determinant lines can be viewed as volume forms on the Hilbertian modules. Using this, we study both $L^2$ combinatorial and $L^2$ analytic torsion invariants …

Sliced skein algebras and geometric Kauffman bracket study algebraic structures and their properties.

A training-free message passing module improves hypergraph neural networks.

In this short article we give a geometric meaning of the divisibility of $KO$-theoretical Euler classes for given two spin modules. We are motivated by Furuta's 10/8-inequality for a closed spin $4$-manifold. The role of the reducibles is clarified in the monopole equations of Seiberg-Witten theory, as done by Donaldso…

The paper classifies symbols of differential operators on vector bundles.

We discuss ways that the ring of coefficients for a TQFT can be reduced if one restricts somewhat the allowed cobordisms. When we apply these methods to a TQFT associated to SO(3) at an odd prime p, we obtain a functor from a somewhat restricted cobordism category to the category of free finitely generated modules over…

The artificial neural network shows powerful ability of inference, but it is still criticized for lack of interpretability and prerequisite needs of big dataset. This paper proposes the Rule-embedded Neural Network (ReNN) to overcome the shortages. ReNN first makes local-based inferences to detect local patterns, and t…

Adversaries with multiple antennas can fool deep learning modulators more effectively.