scholarly journals Ribbon graphs and bialgebra of Lagrangian subspaces

2016 ◽  
Vol 25 (12) ◽  
pp. 1642006 ◽  
Author(s):  
Victor Kleptsyn ◽  
Evgeny Smirnov
Keyword(s):  
Vector Space ◽  
Symplectic Form ◽  
Chord Diagram ◽  
Ribbon Graph ◽  
Dimensional Vector Space ◽  
Ribbon Graphs ◽  
Chord Diagrams ◽  
Intersection Matrix ◽  
Lagrangian Subspace ◽  
Lagrangian Subspaces

To each ribbon graph we assign a so-called [Formula: see text]-space, which is a Lagrangian subspace in an even-dimensional vector space with the standard symplectic form. This invariant generalizes the notion of the intersection matrix of a chord diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual) and Vassiliev moves on ribbon graphs are reinterpreted nicely in the language of [Formula: see text]-spaces, becoming changes of bases in this vector space. Finally, we define a bialgebra structure on the span of [Formula: see text]-spaces, which is analogous to the 4-bialgebra structure on chord diagrams.

The genus of graphs associated with vector spaces

2019 ◽  
Vol 19 (05) ◽  
pp. 2050086 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
K. Prabha Ananthi
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Undirected Graph ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Vertex Connectivity ◽  
Dimensional Vector Space ◽  
Nonzero Component ◽  
Vertex Set ◽  
Component Graph

Let [Formula: see text] be a k-dimensional vector space over a finite field [Formula: see text] with a basis [Formula: see text]. The nonzero component graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as nonzero vectors of [Formula: see text] such that there is an edge between two distinct vertices [Formula: see text] if and only if there exists at least one [Formula: see text] along which both [Formula: see text] and [Formula: see text] have nonzero scalars. In this paper, we find the vertex connectivity and girth of [Formula: see text]. We also characterize all vector spaces [Formula: see text] for which [Formula: see text] has genus either 0 or 1 or 2.

scholarly journals THE INVARIANTS FIELD OF SOME FINITE PROJECTIVE LINEAR GROUP ACTIONS

2011 ◽  
Vol 85 (1) ◽  
pp. 19-25
Author(s):  
YIN CHEN
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Vector Space ◽  
Orthogonal Group ◽  
Homogeneous Polynomial ◽  
Classical Group ◽  
Projective Group ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Projective Linear Group

AbstractLet Fq be a finite field with q elements, V an n-dimensional vector space over Fq and 𝒱 the projective space associated to V. Let G≤GLn(Fq) be a classical group and PG be the corresponding projective group. In this note we prove that if Fq (V )G is purely transcendental over Fq with homogeneous polynomial generators, then Fq (𝒱)PG is also purely transcendental over Fq. We compute explicitly the generators of Fq (𝒱)PG when G is the symplectic, unitary or orthogonal group.

scholarly journals Genus Ranges of Chord Diagrams

2015 ◽  
Vol 24 (04) ◽  
pp. 1550022 ◽  
Author(s):  
Jonathan Burns ◽  
Nataša Jonoska ◽  
Masahico Saito
Keyword(s):  
Outer Boundary ◽  
Orientable Surface ◽  
Fixed Number ◽  
Chord Diagram ◽  
Boundary Circle ◽  
Line Segments ◽  
Chord Diagrams ◽  
Computer Calculations

A chord diagram consists of a circle, called the backbone, with line segments, called chords, whose endpoints are attached to distinct points on the circle. The genus of a chord diagram is the genus of the orientable surface obtained by thickening the backbone to an annulus and attaching bands to the inner boundary circle at the ends of each chord. Variations of this construction are considered here, where bands are possibly attached to the outer boundary circle of the annulus. The genus range of a chord diagram is the genus values over all such variations of surfaces thus obtained from a given chord diagram. Genus ranges of chord diagrams for a fixed number of chords are studied. Integer intervals that can be, and those that cannot be, realized as genus ranges are investigated. Computer calculations are presented, and play a key role in discovering and proving the properties of genus ranges.

scholarly journals Identities Generalizing the Theorems of Pappus and Desargues

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1382
Author(s):  
Roger D. Maddux
Keyword(s):  
Projective Plane ◽  
Vector Space ◽  
Invariant Theory ◽  
Lattice Theory ◽  
Three Dimensional ◽  
Dimensional Vector ◽  
Dimensional Vector Space

The Theorems of Pappus and Desargues (for the projective plane over a field) are generalized here by two identities involving determinants and cross products. These identities are proved to hold in the three-dimensional vector space over a field. They are closely related to the Arguesian identity in lattice theory and to Cayley-Grassmann identities in invariant theory.

scholarly journals Sets of idempotents that generate the semigroup of singular endomorphisms of a finite-dimensional vector space

1982 ◽  
Vol 25 (2) ◽  
pp. 133-139 ◽  
Author(s):  
R. J. H. Dawlings
Keyword(s):  
Vector Space ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Mathematical System

IfMis a mathematical system and EndMis the set of singular endomorphisms ofM, then EndMforms a semigroup under composition of mappings. A number of papers have been written to determine the subsemigroupSMof EndMgenerated by the idempotentsEMof EndMfor different systemsM. The first of these was by J. M. Howie [4]; here the case ofMbeing an unstructured setXwas considered. Howie showed that ifXis finite, then EndX=Sx.

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