Search

Gonz\'alez-Meneses, Manch\'on, and Silvero showed that the (hypothetical) extreme Khovanov homology of a link diagram is isomorphic to the reduced (co)homology of the independence simplicial complex of its Lando graph. Przytycki and Silvero conjectured that the extreme Khovanov homology of any link is torsion-free. In this paper, we investigate explicit geometric realizations of the real-extreme Khovanov homology of pretzel links. This partially solves the conjecture., Comment: 16 pages, 28 figures

The metapopulation network model is effectively used to study the spatial spread of epidemics with individuals mobility. Considering the time-varying nature of individual activity and the preferences for attractive destinations in population mobility, this paper develops a time-varying network model in which activity of a population is correlated with its attractiveness. Based on the model, the spreading processes of the SIR disease on different correlated networks are studied, and global migration thresholds are derived. It is observed that increasing the correlation between activity and attractiveness results in a reduced outbreak threshold but suppresses the disease outbreak size and introduces greater heterogeneity in the spatial distribution of infected individuals. We also investigate the impact of non-pharmacological interventions (self-isolation and self-protection) on the spread of epidemics in different correlation networks. The results show that the simultaneous implementation of these measures is more effective in negatively correlated networks than in positively correlated or non-correlated networks, and the prevalence is reduced significantly. In addition, both self-isolation and self-protection strategies increase the migration threshold of the spreading and thus slow the spread of the epidemic. However, the effectiveness of each strategy in reducing the density of infected populations varies depending on different correlated networks. Self-protection is more effective in positively correlated networks, whereas self-isolation is more effective in negatively correlated networks. These findings contribute to a better understanding of epidemic spreading in large-scale time-varying metapopulation networks and provide insights for epidemic prevention and control.

Appropriate weight initialization settings, along with the ReLU activation function, have become cornerstones of modern deep learning, enabling the training and deployment of highly effective and efficient neural network models across diverse areas of artificial intelligence. The problem of \textquotedblleft dying ReLU," where ReLU neurons become inactive and yield zero output, presents a significant challenge in the training of deep neural networks with ReLU activation function. Theoretical research and various methods have been introduced to address the problem. However, even with these methods and research, training remains challenging for extremely deep and narrow feedforward networks with ReLU activation function. In this paper, we propose a novel weight initialization method to address this issue. We establish several properties of our initial weight matrix and demonstrate how these properties enable the effective propagation of signal vectors. Through a series of experiments and comparisons with existing methods, we demonstrate the effectiveness of the novel initialization method., Comment: 13 pages

One of the most interesting tools that have recently entered the data science toolbox is topological data analysis (TDA). With the explosion of available data sizes and dimensions, identifying and extracting the underlying structure of a given dataset is a fundamental challenge in data science, and TDA provides a methodology for analyzing the shape of a dataset using tools and prospects from algebraic topology. However, the computational complexity makes it quickly infeasible to process large datasets, especially those with high dimensions. Here, we introduce a preprocessing strategy called the Characteristic Lattice Algorithm (CLA), which allows users to reduce the size of a given dataset as desired while maintaining geometric and topological features in order to make the computation of TDA feasible or to shorten its computation time. In addition, we derive a stability theorem and an upper bound of the barcode errors for CLA based on the bottleneck distance., Comment: 13 pages, 10 figures, 2 tables

We determine the Betti numbers for the (degenerate and normalized) set-theoretic Yang-Baxter (co)homology groups of cyclic biquandles and estimate their torsion subgroups. This partially settles the conjecture presented by Przytycki, Vojtechovsky, and Yang. We also obtain cocycles which are representatives of the elements of a basis for the free part of the cohomology group of a cyclic biquandle., Comment: 11 pages, 3 figures

Biracks and biquandles, which are useful for studying the knot theory, are special families of solutions of the set-theoretic Yang-Baxter equation. A homology theory for the set-theoretic Yang-Baxter equation was developed by Carter, Elhamdadi, and Saito in order to construct knot invariants. In this paper, we construct a normalized (co)homology theory of a set-theoretic solution of the Yang-Baxter equation. We obtain some concrete examples of non-trivial $n$-cocycles for Alexander biquandles. For a biquandle $X,$ its geometric realization $BX$ is discussed, which has the potential to build invariants of links and knotted surfaces. In particular, we demonstrate that the second homotopy group of $BX$ is finitely generated if the biquandle $X$ is finite., Comment: 16 pages, 9 figures

Racks and quandles are prominent set-theoretical solutions of the Yang-Baxter equation. We enumerate racks and quandles of orders $n\le 13$ up to isomorphism, improving upon the previously known results for $n\le 8$ and $n\le 9$, respectively. The enumeration is based on the classification of subgroups of small symmetric groups up to conjugation, on a representation of racks and quandles in symmetric groups due to Joyce and Blackburn, and on a number of theoretical and computational observations concerning the representation. We explicitly find representatives of isomorphism types of racks of order $\le 11$ and quandles of order $\le 12$. For the remaining orders we merely count the isomorphism types, relying in part on the enumeration of $2$-reductive racks and $2$-reductive quandles due to Jedli\v{c}ka, Pilitowska, Stanovsk\'y and Zamojska-Dzienio.

W. Rump showed that there exists a one-to-one correspondence between involutive right non-degenerate solutions of the Yang-Baxter equation and Rump right quasigroups. J. S. Carter, M. Elhamdadi, and M. Saito, meanwhile, introduced a homology theory of set-theoretic solutions of the Yang-Baxter equation in order to define cocycle invariants of classical knots. In this paper, we introduce the normalized homology theory of an involutive right non-degenerate solution of the Yang-Baxter equation and prove that the set-theoretic Yang-Baxter homology of certain solutions can be split into the normalized and degenerated parts., Comment: 14 pages, 6 figures

In this paper, we propose a generalized notion of a distance function, called a $g$-metric. The $g$-metric with degree $n$ is a distance of $n+1$ points, generalizing the ordinary distance between two points and $G$-metric between three points. Indeed, it is shown that the $g$-metric with degree 1 (resp. degree 2) is equivalent to the ordinary metric (resp. the $G$-metric). Fundamental properties and several examples for the $g$-metric are also given. Moreover, topological properties on the $g$-metric space including the convergence of sequences and the continuity of mappings on the $g$-metric space are studied. Finally, we generalize some well-known fixed point theorems including Banach contraction mapping principle and \'Ciri\'c fixed point theorem in the $g$-metric space., Comment: 41 pages, 2 figures

A qualgebra $G$ is a set having two binary operations that satisfy compatibility conditions which are modeled upon a group under conjugation and multiplication. We develop a homology theory for qualgebras and describe a classifying space for it. This space is constructed from $G$-colored prisms (products of simplices) and simultaneously generalizes (and includes) simplicial classifying spaces for groups and cubical classifying spaces for quandles. Degenerate cells of several types are added to the regular prismatic cells; by duality, these correspond to "non-rigid" Reidemeister moves and their higher dimensional analogues. Coupled with $G$-coloring techniques, our homology theory yields invariants of knotted trivalent graphs in $\mathbb{R}^3$ and knotted foams in $\mathbb{R}^4$. We re-interpret these invariants as homotopy classes of maps from $S^2$ or $S^3$ to the classifying space of $G$., Comment: 28 pages, 24 figures

In this paper we give a sufficient and necessary condition for two rooted trees with the same plucking polynomial. Furthermore, we give a criteria for a sequence of non-negative integers to be realized as a rooted tree., Comment: 13 pages, 12 figures

In the Khovanov homology of links, presence of $\mathbb{Z}_2$-torsion is a very common phenomenon. Finite number of examples of knots with $\mathbb{Z}_n$-torsion for $n>2$ were also known, none for $n>8$. In this paper, we prove that there are infinite families of links whose Khovanov homology contains $\mathbb{Z}_n$-torsion for $2 < n < 9$ and $\mathbb{Z}_{2^s}$-torsion for $s < 24$. We also introduce $4$-braid links with $\mathbb{Z}_3$-torsion which are counterexamples to the PS braid conjecture. We also provide an infinite family of knots with $\mathbb{Z}_5$-torsion in reduced Khovanov homology and $\mathbb{Z}_3$-torsion in odd Khovanov homology., Comment: 33 pages, 8 figures, and 33 tables

In 1993, Fenn, Rourke and Sanderson introduced rack spaces and rack homotopy invariants, and modifications to quandle spaces and quandle homotopy invariants were introduced by Nosaka in 2011. In this paper, we define the Cayley-type graph and the extended quandle space of a quandle in analogy to rack and quandle spaces. Moreover, we construct the shadow homotopy invariant of a classical link and prove that the shadow homotopy invariant is equal to the quandle homotopy invariant multiplied by the order of a quandle., Comment: 12 pages, 9 figures

We classify rooted trees which have strictly unimodal q-polynomials (plucking polynomial). We also give criteria for a trapezoidal shape of a plucking polynomial. We generalize results of Pak and Panova on strict unimodality of q-binomial coefficients. We discuss which polynomials can be realized as plucking polynomials and whether or not different rooted trees can have the same plucking polynomial., Comment: 26 pages, 14 figures

It is a classical result in reduced homology of finite groups that the order of a group annihilates its homology. Similarly, we have proved that the torsion subgroup of rack and quandle homology of a finite quasigroup quandle is annihilated by its order. However, it does not hold for connected quandles in general. In this paper, we define an $m$-almost quasigroup ($m$-AQ) quandle which is a generalization of a quasigroup quandle and study annihilation of torsion in its rack and quandle homology groups., Comment: 13 pages, 2 figures

In 1965, E. C. Zeeman proved that the (+/-)-twist spin of any knotted sphere in (n-1)-space is unknotted in the n-sphere. In 1991, Y. Marumoto and Y. Nakanishi gave an alternate proof of Zeeman's theorem by using the moving picture method. In this paper, we define a knotted 2-dimensional foam which is a generalization of a knotted sphere and prove that a (+/-)-twist spin of a knotted trivalent graph may be knotted. We then construct some families of knotted graphs for which the (+/-)-twist spins are always unknotted., Comment: 12 pages, 12 figures, some color

We prove that if Q is a finite quasigroup quandle, then |Q| annihilates the torsion of its homology. It is a classical result in reduced homology of finite groups that the order of a group annihilates its homology. From the very beginning of the rack homology (between 1990 and 1995) the analogous result was suspected. The first general results in this direction were obtained independently about 2001 by R.A.Litherland and S.Nelson, and P.Etingof and M.Grana. In Litherland-Nelson paper it is proven that if (Q;*) is a finite homogeneous rack (this includes quasigroup racks) then the torsion of homology is annihilated by |Q|^n. In Etingof-Grana paper it is proven that if (X;A) is a finite rack and N=|G^0_Q| is the order of a group of inner automorphisms of Q, then only primes which can appear in the torsion of homology are those dividing N (the case of connected Alexander quandles was proven before by T.Mochizuki). The result of Litherland-Nelson is generalized by Niebrzydowski and Przytycki and in particular, they prove that the torsion part of the homology of the dihedral quandle R_3 is annihilated by 3. In Niebrzydowski-Przytycki paper it is conjectured that for a finite quasigroup quandle, torsion of its homology is annihilated by the order of the quandle. The conjecture is proved by T.Nosaka for finite Alexander quasigroup quandles. In this paper we prove the conjecture in full generality. For this version, we rewrote the Section 3 totally and introduced the concept of the precubic homotopy. In Section 2, the main addition is Corollary 2.2 which summarizes identities observed in the proof of the main theorem as we use it later in Section 3., Comment: 13 pages, 1 figure; accepted for publication in Journal of Pure and Applied Algebra