scholarly journals The Hopf Monoid of Hypergraphs and its Sub-Monoids: Basic Invariant and Reciprocity Theorem

10.37236/8740 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Jean-Christophe Aval ◽  
Théo Karaboghossian ◽  
Adrian Tanasa
Keyword(s):  
Simplicial Complexes ◽  
Reciprocity Theorem ◽  
Combinatorial Interpretation ◽  
Polynomial Invariant ◽  
Basic Invariant ◽  
Polynomial Invariants ◽  
Combinatorial Objects ◽  
Monoid Structure ◽  
Building Sets ◽  
Hopf Monoid

In arXiv:1709.07504 Aguiar and Ardila give a Hopf monoid structure on hypergraphs as well as a general construction of polynomial invariants on Hopf monoids. Using these results, we define in this paper a new polynomial invariant on hypergraphs. We give a combinatorial interpretation of this invariant on negative integers which leads to a reciprocity theorem on hypergraphs. Finally, we use this invariant to recover well-known invariants on other combinatorial objects (graphs, simplicial complexes, building sets, etc) as well as the associated reciprocity theorems.

GENERALIZED INDEX POLYNOMIALS FOR VIRTUAL LINKS

2012 ◽  
Vol 21 (14) ◽  
pp. 1250128
Author(s):  
KYEONGHUI LEE ◽  
YOUNG HO IM
Keyword(s):  
Knot Theory ◽  
Recursive Method ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knots ◽  
Virtual Links

We construct some polynomial invariants for virtual links by the recursive method, which are different from the index polynomial invariant defined in [Y. H. Im, K. Lee and S. Y. Lee, Index polynomial invariant of virtual links, J. Knot Theory Ramifications19(5) (2010) 709–725]. We show that these polynomials can distinguish whether virtual knots can be invertible or not although the index polynomial cannot distinguish the invertibility of virtual knots.

A general time-dependent invariant for and integrability of the quadratic system

1994 ◽  
Vol 444 (1922) ◽  
pp. 643-650 ◽  
Keyword(s):  
Direct Method ◽  
First Integral ◽  
Quadratic System ◽  
Time Dependent ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Linear Polynomial ◽  
A Direct Method ◽  
Special Case ◽  
General Time

We derive a general time-dependent invariant (first integral) for the quadratic system (QS) that requires only one condition on the coefficients of the QS. The general invariant could yield asymptotic behaviour of phase-space trajectories. With more conditions imposed on the coefficients, the general invariant reduces to polynomial form and is equivalent to polynomial invariants found using a direct method. For the special case of a linear polynomial invariant where one of the variables is analytically invertible, the solution of the QS is reduced to a quadrature.

KAUFFMAN-TYPE POLYNOMIAL INVARIANTS FOR DOUBLY PERIODIC STRUCTURES

2007 ◽  
Vol 16 (06) ◽  
pp. 779-788 ◽  
Author(s):  
SERGEI A. GRISHANOV ◽  
VADIM R. MESHKOV ◽  
ALEXANDER V. OMEL'CHENKO
Keyword(s):  
Periodic Structures ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Doubly Periodic Structures

A two-variable polynomial invariant of non-oriented doubly periodic structures is proposed. A possible application of this polynomial for the classification of textile structures is suggested.

INDEX POLYNOMIAL INVARIANTS OF TWISTED LINKS

2013 ◽  
Vol 22 (04) ◽  
pp. 1340005 ◽  
Author(s):  
NAOKO KAMADA
Keyword(s):  
Virtual Link ◽  
Alternative Definition ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Intersection Index ◽  
Definition Of

Bourgoin defined the notion of a twisted link. In a sense, it is a non-orientable version of a virtual link. Im, Lee and Lee defined a polynomial invariant of a virtual link by using the virtual intersection index. In this paper, we give an alternative definition of index polynomial by using indices of real crossings and extend it to a twisted links.

scholarly journals A POLYNOMIAL INVARIANT OF VIRTUAL LINKS

2013 ◽  
Vol 22 (12) ◽  
pp. 1341002 ◽  
Author(s):  
ZHIYUN CHENG ◽  
HONGZHU GAO
Keyword(s):  
Knot Theory ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knots ◽  
The Third ◽  
Linking Number ◽  
Virtual Links ◽  
Definition Of ◽  
Knots And Links

In this paper, we define some polynomial invariants for virtual knots and links. In the first part we use Manturov's parity axioms [Parity in knot theory, Sb. Math.201 (2010) 693–733] to obtain a new polynomial invariant of virtual knots. This invariant can be regarded as a generalization of the odd writhe polynomial defined by the first author in [A polynomial invariant of virtual knots, preprint (2012), arXiv:math.GT/1202.3850v1]. The relation between this new polynomial invariant and the affine index polynomial [An affine index polynomial invariant of virtual knots, J. Knot Theory Ramification22 (2013) 1340007; A linking number definition of the affine index polynomial and applications, preprint (2012), arXiv:1211.1747v1] is discussed. In the second part we introduce a polynomial invariant for long flat virtual knots. In the third part we define a polynomial invariant for 2-component virtual links. This polynomial invariant can be regarded as a generalization of the linking number.

scholarly journals Formal Group Laws and Chromatic Symmetric Functions of Hypergraphs

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Jair Taylor
Keyword(s):  
Power Series ◽  
Symmetric Functions ◽  
Formal Group ◽  
Combinatorial Interpretation ◽  
Recursive Structure ◽  
Combinatorial Objects ◽  
Formal Group Law ◽  
International Audience ◽  
Formal Group Laws ◽  
Group Law

International audience If $f(x)$ is an invertible power series we may form the symmetric function $f(f^{-1}(x_1)+f^{-1}(x_2)+...)$ which is called a formal group law. We give a number of examples of power series $f(x)$ that are ordinary generating functions for combinatorial objects with a recursive structure, each of which is associated with a certain hypergraph. In each case, we show that the corresponding formal group law is the sum of the chromatic symmetric functions of these hypergraphs by finding a combinatorial interpretation for $f^{-1}(x)$. We conjecture that the chromatic symmetric functions arising in this way are Schur-positive. Si $f(x)$ est une série entière inversible, nous pouvons former la fonction symétrique $f(f^{-1}(x_1)+f^{-1}(x_2)+...)$ que nous appelons une loi de groupe formel. Nous donnons plusieurs exemples de séries entières $f(x)$ qui sont séries génératrices ordinaires pour des objets combinatoires avec une structure récursive, chacune desquelles est associée à un certain hypergraphe. Dans chaque cas, nous donnons une interprétation combinatoire à $f^{-1}(x)$, ce qui nous permet de montrer que la loi de groupe formel correspondante est la somme des fonctions symétriques chromatiques de ces hypergraphes. Nous conjecturons que les fonctions symétriques chromatiques apparaissant de cette manière sont Schur-positives.

scholarly journals Quasisymmetric functions from combinatorial Hopf monoids and Ehrhart Theory

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Jacob White
Keyword(s):  
Commutative Algebra ◽  
Hopf Algebras ◽  
Simplicial Complexes ◽  
Hilbert Functions ◽  
Quasisymmetric Functions ◽  
Ehrhart Theory ◽  
International Audience ◽  
Hopf Monoid

International audience We investigate quasisymmetric functions coming from combinatorial Hopf monoids. We show that these invariants arise naturally in Ehrhart theory, and that some of their specializations are Hilbert functions for relative simplicial complexes. This class of complexes, called forbidden composition complexes, also forms a Hopf monoid, thus demonstrating a link between Hopf algebras, Ehrhart theory, and commutative algebra. We also study various specializations of quasisymmetric functions.

scholarly journals PROPERTIES OF GROUPS G OF DOUBLE ERRORS AND ITS INVARIANTS IN BCH CODES

2018 ◽  
pp. 40-46
Author(s):  
V. A. Lipnitskij ◽  
A. V. Serada
Keyword(s):  
Irreducible Polynomial ◽  
Galois Field ◽  
Bch Codes ◽  
Bch Code ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
The Core ◽  
Complex Process ◽  
Scope Of Application ◽  
Complete Set

The goal of the work is the further extending the scope of application of code automorthism in methods and algorithms of error correction by these codes. The effectiveness of such approach was demonstrated by norm of syndrome theory that was developed by Belarusian school of noiseless coding at the turn of the XX and XXI century. The group Г of the cyclical shift of vector component lies at the core of the theory. Under its action The error vectors are divided into disjoint Г-orbits with definite spectrum of syndromes. This allowed to introduce norms of syndrome of a family of BCH codes that are invariant over action of group Г. Norms of syndrome are unique characteristic of error orbit Г of any decoding set, hence it is the basis of permutation norm methods of error decoding. Looking over the Г-orbits of errors not the errors these methods are faster than classic syndrome methods of error decoding, are avoided from the complex process of solving the algebraic equation in Galois field, are simply implemented.A detailed theory for automorphism group G of BCH codes obtained by adding cyclotomic substitution to the group Г develops in the article. The authors held a detailed study of structure of G-orbit of errors as union of orbits Г of error vectors; one-to-one mapping of this structure on the norm structure of group Г. These norms being interconnected by Frobenius automorphism in the Galois field – field of BCH code constitute the complete set of roots of the only irreducible polynomial. It is a polynomial invariant of its orbit G. The main focus of the work is on the description of properties and specific features of groups G of double errors and its polynomial invariants.

A PARITY AND A MULTI-VARIABLE POLYNOMIAL INVARIANT FOR VIRTUAL LINKS

2013 ◽  
Vol 22 (13) ◽  
pp. 1350073 ◽  
Author(s):  
YOUNG HO IM ◽  
KYOUNG IL PARK
Keyword(s):  
Knot Theory ◽  
Virtual Link ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Link Diagrams ◽  
Virtual Links ◽  
Knot Diagrams

We introduce a parity of classical crossings of virtual link diagrams which extends the Gaussian parity of virtual knot diagrams and the odd writhe of virtual links that extends that of virtual knots introduced by Kauffman [A self-linking invariants of virtual knots, Fund. Math.184 (2004) 135–158]. Also, we introduce a multi-variable polynomial invariant for virtual links by using the parity of classical crossings, which refines the index polynomial introduced in [Index polynomial invariants of virtual links, J. Knot Theory Ramifications19(5) (2010) 709–725]. As consequences, we give some properties of our invariant, and raise some examples.

scholarly journals A Reciprocity Theorem for Domino Tilings

10.37236/1562 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
James Propp
Keyword(s):  
Recurrence Relation ◽  
Reciprocity Theorem ◽  
Algebraic Methods ◽  
Linear Recurrence ◽  
Combinatorial Interpretation ◽  
Domino Tilings ◽  
Linear Recurrence Relation

Let $T(m,n)$ denote the number of ways to tile an $m$-by-$n$ rectangle with dominos. For any fixed $m$, the numbers $T(m,n)$ satisfy a linear recurrence relation, and so may be extrapolated to negative values of $n$; these extrapolated values satisfy the relation $$T(m,-2-n)=\epsilon_{m,n}T(m,n),$$ where $\epsilon_{m,n}=-1$ if $m \equiv 2$ (mod 4) and $n$ is odd and where $\epsilon_{m,n}=+1$ otherwise. This is equivalent to a fact demonstrated by Stanley using algebraic methods. Here I give a proof that provides, among other things, a uniform combinatorial interpretation of $T(m,n)$ that applies regardless of the sign of $n$.

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