Generalized Equidistant Chebyshev Polynomials and Alexander Knot Invariants

A. M. Pavlyuk

doi:10.15407/ujpe63.6.488

Generalized Equidistant Chebyshev Polynomials and Alexander Knot Invariants

Ukrainian Journal of Physics ◽

10.15407/ujpe63.6.488 ◽

2018 ◽

Vol 63 (6) ◽

pp. 488

Author(s):

A. M. Pavlyuk

Keyword(s):

Chebyshev Polynomials ◽

Polynomial Invariants ◽

Knot Invariants ◽

Positive Integers

We introduce the generalized equidistant Chebyshev polynomials T(k,h) of kind k of hyperkind h, where k, h are positive integers. They are obtained by a generalization of standard and monic Chebyshev polynomials of the first and second kinds. This generalization is fulfilled in two directions. The horizontal generalization is made by introducing hyperkind ℎ and expanding it to infinity. The vertical generalization proposes expanding kind k to infinity with the help of the method of equidistant coefficients. Some connections of these polynomials with the Alexander knot and link polynomial invariants are investigated.

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Recurrent Generalization of F-Polynomials for Virtual Knots and Links

Symmetry ◽

10.3390/sym14010015 ◽

2021 ◽

Vol 14 (1) ◽

pp. 15

Author(s):

Amrendra Gill ◽

Maxim Ivanov ◽

Madeti Prabhakar ◽

Andrei Vesnin

Keyword(s):

Local Symmetry ◽

Weight Functions ◽

Virtual Link ◽

Link Diagram ◽

Polynomial Invariants ◽

Knot Invariants ◽

Virtual Knot ◽

Virtual Knots ◽

Virtual Links ◽

Knots And Links

F-polynomials for virtual knots were defined by Kaur, Prabhakar and Vesnin in 2018 using flat virtual knot invariants. These polynomials naturally generalize Kauffman’s affine index polynomial and use smoothing in the classical crossing of a virtual knot diagram. In this paper, we introduce weight functions for ordered orientable virtual and flat virtual links. A flat virtual link is an equivalence class of virtual links with respect to a local symmetry changing a type of classical crossing in a diagram. By considering three types of smoothing in classical crossings of a virtual link diagram and suitable weight functions, there is provided a recurrent construction for new invariants. It is demonstrated by explicit examples that newly defined polynomial invariants are stronger than F-polynomials.

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Two-variable polynomial invariants of virtual knots arising from flat virtual knot invariants

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216518420154 ◽

2018 ◽

Vol 27 (13) ◽

pp. 1842015 ◽

Cited By ~ 4

Author(s):

Kirandeep Kaur ◽

Madeti Prabhakar ◽

Andrei Vesnin

Keyword(s):

Sufficient Conditions ◽

Polynomial Invariants ◽

Knot Invariants ◽

Virtual Knot ◽

Virtual Knots ◽

Knot Invariant ◽

Index Value

We introduce two sequences of two-variable polynomials [Formula: see text] and [Formula: see text], expressed in terms of index value of a crossing and [Formula: see text]-dwrithe value of a virtual knot [Formula: see text], where [Formula: see text] and [Formula: see text] are variables. Basing on the fact that [Formula: see text]-dwrithe is a flat virtual knot invariant, we prove that [Formula: see text] and [Formula: see text] are virtual knot invariants containing Kauffman affine index polynomial as a particular case. Using [Formula: see text] we give sufficient conditions when virtual knot does not admit cosmetic crossing change.

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Polynomial Invariant of Molecular Circuit Topology

Symmetry ◽

10.3390/sym13091751 ◽

2021 ◽

Vol 13 (9) ◽

pp. 1751

Author(s):

Alireza Mashaghi ◽

Roland van der Veen

Keyword(s):

Knot Theory ◽

Linear Chain ◽

Polynomial Invariant ◽

Circuit Topology ◽

Polynomial Invariants ◽

Knot Invariants ◽

Topological Framework ◽

Critical Chain ◽

Chain Entanglements ◽

Different Parts

The topological framework of circuit topology has recently been introduced to complement knot theory and to help in understanding the physics of molecular folding. Naturally evolved linear molecular chains, such as proteins and nucleic acids, often fold into 3D conformations with critical chain entanglements and local or global structural symmetries stabilised by formation contacts between different parts of the chain. Circuit topology captures the arrangements of intra-chain contacts within a given folded linear chain and allows for the classification and comparison of chains. Contacts keep chain segments in physical proximity and can be either mechanically hard attachments or soft entanglements that constrain a physical chain. Contrary to knot theory, which offers many established knot invariants, circuit invariants are just being developed. Here, we present polynomial invariants that are both efficient and sufficiently powerful to deal with any combination of soft and hard contacts. A computer implementation and table of chains with up to three contacts is also provided.

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The fourth Skein Module and the Montesinos-Nakanishi Conjecture for 3-algebraic links

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216501001281 ◽

2001 ◽

Vol 10 (07) ◽

pp. 959-982 ◽

Cited By ~ 10

Author(s):

Józef H. Przytycki ◽

Tatsuya Tsukamoto

Keyword(s):

Necessary Conditions ◽

Braid Groups ◽

Linear Combinations ◽

Polynomial Invariants ◽

Knot Invariants ◽

Skein Module ◽

Linearly Independent ◽

Kauffman Polynomial ◽

Better Than ◽

Skein Relation

We study the fourth skein module of 3-manifolds, based on the skein relation b0L0 + b1L1 + b2L2 + b3L3 = 0 and a framing relation L(1) = aL (a, b0, b3 invertible). We give necessary conditions for trivial links to be linearly independent in the module. We show how elements of the skein module behave under the n-move and we compute the values for (2, n)-torus links and twist knots as elements of the skein module. Using mutants and rotors, we find different links which represent the same element in the skein module. We also show that algebraic links (in the sense of Conway) and closed 3-braids are linear combinations of trivial links. We introduce the concept of n-algebraic tangles (and links) and analyze the skein module for 3-algebraic links. As a by product we prove the Montesinos-Nakanishi 3-moves conjecture for 3-algebraic links (including 3-bridge links). For links in S3, the structure of our skein module suggests the existence of three new polynomial invariants of unoriented framed (or unframed) links. One of them would generalize the Kauffman polynomial of links and another one could be used to analyze amphicheirality of links (and may work better than the Kauffman polynomial). In the conclusion, we speculate that our new knot invariants are related to a deformation of the symplectic quotient of braid groups.

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Language, procedures, and the non-perceptual origin of natural number concepts

10.31234/osf.io/3q4mf ◽

2016 ◽

Author(s):

David Barner

Keyword(s):

General Framework ◽

Ad Hoc ◽

Building Blocks ◽

Number Words ◽

Linguistic Markers ◽

Logical Foundations ◽

Large Numbers ◽

Perceptual Representations ◽

Positive Integers ◽

Perceptual Systems

Perceptual representations – e.g., of objects or approximate magnitudes –are often invoked as building blocks that children combine with linguisticsymbols when they acquire the positive integers. Systems of numericalperception are either assumed to contain the logical foundations ofarithmetic innately, or to supply the basis for their induction. Here Ipropose an alternative to this general framework, and argue that theintegers are not learned from perceptual systems, but instead arise toexplain perception as part of language acquisition. Drawing oncross-linguistic data and developmental data, I show that small numbers(1-4) and large numbers (~5+) arise both historically and in individualchildren via entirely distinct mechanisms, constituting independentlearning problems, neither of which begins with perceptual building blocks.Specifically, I propose that children begin by learning small numbers(i.e., *one, two, three*) using the same logical resources that supportother linguistic markers of number (e.g., singular, plural). Several yearslater, children discover the logic of counting by inferring the logicalrelations between larger number words from their roles in blind countingprocedures, and only incidentally associate number words with perception ofapproximate magnitudes, in an *ad hoc* and highly malleable fashion.Counting provides a form of explanation for perception but is not causallyderived from perceptual systems.

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Short Generating Functions for some Semigroup Algebras

The Electronic Journal of Combinatorics ◽

10.37236/1729 ◽

2003 ◽

Vol 10 (1) ◽

Cited By ~ 23

Author(s):

Graham Denham

Keyword(s):

Rational Function ◽

Generating Functions ◽

Hilbert Series ◽

Simple Expression ◽

Arbitrary Field ◽

Graded Algebra ◽

Semigroup Algebras ◽

Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

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On a Two-Sided Turán Problem

The Electronic Journal of Combinatorics ◽

10.37236/1735 ◽

2003 ◽

Vol 10 (1) ◽

Cited By ~ 1

Author(s):

Dhruv Mubayi ◽

Yi Zhao

Keyword(s):

Minimum Size ◽

Turan Problem ◽

Turan Problems ◽

Turán Problem ◽

Positive Integers

Given positive integers $n,k,t$, with $2 \le k\le n$, and $t < 2^k$, let $m(n,k,t)$ be the minimum size of a family ${\cal F}$ of nonempty subsets of $[n]$ such that every $k$-set in $[n]$ contains at least $t$ sets from ${\cal F}$, and every $(k-1)$-set in $[n]$ contains at most $t-1$ sets from ${\cal F}$. Sloan et al. determined $m(n, 3, 2)$ and Füredi et al. studied $m(n, 4, t)$ for $t=2, 3$. We consider $m(n, 3, t)$ and $m(n, 4, t)$ for all the remaining values of $t$ and obtain their exact values except for $k=4$ and $t= 6, 7, 11, 12$. For example, we prove that $ m(n, 4, 5) = {n \choose 2}-17$ for $n\ge 160$. The values of $m(n, 4, t)$ for $t=7,11,12$ are determined in terms of well-known (and open) Turán problems for graphs and hypergraphs. We also obtain bounds of $m(n, 4, 6)$ that differ by absolute constants.

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A Note on Differential Identities in Prime and Semiprime Rings

Contemporary Mathematics ◽

10.37256/cm.000127.77-83 ◽

2020 ◽

pp. 77-83

Author(s):

Mohammad Shadab Khan ◽

Mohd Arif Raza ◽

Nadeemur Rehman

Keyword(s):

Prime Ring ◽

Semiprime Ring ◽

Nonzero Ideal ◽

Differential Identities ◽

Standard Identity ◽

Prime And Semiprime Rings ◽

Semiprime Rings ◽

Positive Integers

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d ( r ○ s)(r ○ s) + ( r ○ s) d ( r ○ s)n - d ( r ○ s))m for all r, s ϵ I, then R is commutative. (ii) If (d ( r ○ s)( r ○ s) + ( r ○ s) d ( r ○ s)n - d (r ○ s))m ϵ Z(R) for all r, s ϵ I, then R satisfies s4, the standard identity in four variables. Moreover, we also examine the case when R is a semiprime ring.

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