Related papers: From tangle fractions to DNA
We construct a new family of skein exact triangles for link Floer homology. The skein triples are described by a triple of rational tangles $(R_0,R_1,R_{2n+1})$, where $R_0$ is the trivial tangle and $R_k$ is obtained from it by applying…
Knots and links which are closed 3-braids are a very special class. Like 2-bridge knots and links, they are simple enough to admit a complete classification. At the same time they are rich enough to serve as a source of examples on which,…
In this work we introduce a convolution operation over the tangent bundle of Riemann manifolds in terms of exponentials of the Connection Laplacian operator. We define tangent bundle filters and tangent bundle neural networks (TNNs) based…
A novel basis of discrete analytic polynomials on a rhombic lattice is introduced and the associated convolution product is studied. A class of discrete analytic functions that are rational with respect to this product is also described.
We revisit finite racks and quandles using a perspective based on permutations which can aid in the understanding of the structure. As a consequence we recover old results and prove new ones. We also present and analyze several examples.
Doubly periodic tangles (DP tangles) are configurations of curves embedded in the thickened plane, invariant under translations in two transversal directions. In this paper we extend the classical theory of DP tangles by introducing the…
In this paper we summarise the work discussed in Ref. [1] and [2] (q-alg/9505003), in which we introduced a method helpful in solving the problem of knot classification. We also present results obtained since then.
This paper introduces the use of cable dynamics models as a means to explore the mechanics of DNA on long-length scales. It is on these length scales that DNA forms twisted and curved three-dimensional shapes known as supercoils and loops.…
We present a new and very concrete connection between cluster algebras and knot theory. This connection is being made via continued fractions and snake graphs. It is known that the class of 2-bridge knots and links is parametrized by…
We present an elementary introduction to one of the most important today knot theory approaches, which gives rise to a representation for a class of knot polynomials in terms of quantum groups. Historically, the approach was at the same…
The study of a certain class of matrix integrals can be motivated by their interpretation as counting objects of knot theory such as alternating prime links, tangles or knots. The simplest such model is studied in detail and allows to…
In this paper, we describe a reproduction of the Relational Graph Convolutional Network (RGCN). Using our reproduction, we explain the intuition behind the model. Our reproduction results empirically validate the correctness of our…
In this study we determine the structure of reversible DNA codes obtained from skew cyclic codes. We show that the generators of such DNA codes enjoy some special properties. We study the structural properties of such family of codes and we…
Rationals are known to form interesting and computationally rich structures, such as Farey sequences and infinite trees. Little attention is being paid to more general, systematic exposition of the basic properties of fractions as a set.…
The thesis deals with applications of fractional calculus to fractals. It introduces the notion of local fractional derivative (LFD). Fractal and multifractal functions have been studied in the thesis using LFD. New kind of equations are…
This article presents a reformulation of the Theory of Functional Connections: a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The reformulation presented in this paper exploits…
Construction of (colored) knot polynomials for double-fat graphs is further generalized to the case when "fingers" and "propagators" are substituting R-matrices in arbitrary closed braids with m-strands. Original version of arXiv:1504.00371…
This paper explores the problem of unknotting closed braids and classical knots in mathematical knot theory. We apply evolutionary computation methods to learn sequences of moves that simplify knot diagrams, and show that this can be…
Unlike polynomials, rational functions can represent functions having poles or branch cuts with root-exponential convergence and no Runge phenomenon. Recent developments of the AAA and greedy Thiele algorithms have sparked renewed interest…
Most genes are part of larger families of evolutionary related genes. The history of gene families typically involves duplications and losses of genes as well as horizontal transfers into other organisms. The reconstruction of detailed gene…