The Yamada polynomial of lens spatial graphs

2015 ◽  
Vol 08 (02) ◽  
pp. 1550029 ◽  
Author(s):  
Nafaa Chbili
Keyword(s):  
Braid Group ◽  
Lens Space ◽  
Spatial Graph ◽  
Spatial Graphs ◽  
Combinatorial Representation

Let [Formula: see text] be a spatial graph in the 3-sphere which covers a graph in the lens space. We introduce a combinatorial representation of [Formula: see text] which reflects the symmetry of the spatial graph. This representation involves a ribbon n-graph and the generator of the center of the n-braid group. We apply this result to study the Yamada polynomial of a class of lens spatial graphs.

Robot navigation algorithms using learned spatial graphs

Robotica ◽  
1986 ◽  
Vol 4 (2) ◽  
pp. 93-100 ◽  
Author(s):  
S. S. Iyengar ◽  
C. C. Jorgensen ◽  
S. V. N. Rao ◽  
C. R. Weisbin
Keyword(s):  
Robot Navigation ◽  
Sensor Data ◽  
Two Dimensional ◽  
Continuous Transition ◽  
Spatial Graph ◽  
Spatial Graphs ◽  
Navigation Algorithms ◽  
Sensor Information ◽  
Navigation Planning ◽  
Available Information

SUMMARYFinding optimal paths for robot navigation in a known terrain has been studied for some time but, in many important situations, a robot would be required to navigate in completely new or partially explored terrain. We propose a method of robot navigation which requires no pre-learned model, makes maximal use of available information, records and synthesizes information from multiple journeys, and contains concepts of learning that allow for continuous transition from local to global path optimality. The model of the terrain consists of a spatial graph and a Voronoi diagram. Using acquired sensor data, polygonal boundaries containing perceived obstacles shrink to approximate the actual obstacles surfaces, free space for transit is correspondingly enlarged, and additional nodes and edges are recorded based on path intersections and stop points. Navigation planning is gradually accelerated with experience since improved global map information minimizes the need for further sensor data acquisition. Our method currently assumes obstacle locations are unchanging, navigation can be successfully conducted using two-dimensional projections, and sensor information is precise.

Every finite group action on a compact 3-manifold preserves infinitely many hyperbolic spatial graphs

2014 ◽  
Vol 23 (06) ◽  
pp. 1450034 ◽  
Author(s):  
Toru Ikeda
Keyword(s):  
Finite Group ◽  
Group Action ◽  
Group Actions ◽  
Infinite Family ◽  
Finite Group Action ◽  
Finite Group Actions ◽  
Spatial Graph ◽  
Cell Decompositions ◽  
Spatial Graphs ◽  
Ideal Triangulations

We consider symmetries of spatial graphs in compact 3-manifolds described by smooth finite group actions. This paper provides a method for constructing an infinite family of hyperbolic spatial graphs with given symmetry by connecting spatial graph versions of hyperbolic tangles in 3-cells of polyhedral cell decompositions induced from triangulations of the 3-manifolds. This method is applicable also to the case of ideal triangulations.

scholarly journals Colorings, determinants and Alexander polynomials for spatial graphs

2016 ◽  
Vol 25 (04) ◽  
pp. 1650019
Author(s):  
Blake Mellor ◽  
Terry Kong ◽  
Alec Lewald ◽  
Vadim Pigrish
Keyword(s):  
Fundamental Group ◽  
Alexander Polynomial ◽  
Metacyclic Group ◽  
Alexander Polynomials ◽  
Spatial Graph ◽  
Spatial Graphs ◽  
Integer Weight

A balanced spatial graph has an integer weight on each edge, so that the directed sum of the weights at each vertex is zero. We describe the Alexander module and polynomial for balanced spatial graphs (originally due to S. Kinoshita, Alexander polynomials as isotopy invariants I, Osaka Math. J. 10 (1958) 263–271.), and examine their behavior under some common operations on the graph. We use the Alexander module to define the determinant and [Formula: see text]-colorings of a balanced spatial graph, and provide examples. We show that the determinant of a spatial graph determines for which [Formula: see text] the graph is [Formula: see text]-colorable, and that a [Formula: see text]-coloring of a graph corresponds to a representation of the fundamental group of its complement into a metacyclic group [Formula: see text]. We finish by proving some properties of the Alexander polynomial.

scholarly journals A study of projections of 2-bouquet graphs

2017 ◽  
Vol 26 (05) ◽  
pp. 1750025
Author(s):  
Elaina Aceves
Keyword(s):  
Spatial Graph ◽  
Spatial Graphs

We extend the concepts of trivializing and knotting numbers for knots to spatial graphs and 2-bouquet graphs, in particular. Furthermore, we calculate the trivializing and knotting numbers for projections and pseudodiagrams of 2-bouquet spatial graphs based on the number of precrossings and the placement of the precrossings in the pseudodiagram of the spatial graph.

scholarly journals Stick number of spatial graphs

2017 ◽  
Vol 26 (14) ◽  
pp. 1750100 ◽  
Author(s):  
Minjung Lee ◽  
Sungjong No ◽  
Seungsang Oh
Keyword(s):  
Research Program ◽  
Upper Bound ◽  
Upper Bounds ◽  
Crossing Number ◽  
Index Formula ◽  
Spatial Graph ◽  
Spatial Graphs ◽  
Number Formula ◽  
Arc Index

For a nontrivial knot [Formula: see text], Negami found an upper bound on the stick number [Formula: see text] in terms of its crossing number [Formula: see text] which is [Formula: see text]. Later, Huh and Oh utilized the arc index [Formula: see text] to present a more precise upper bound [Formula: see text]. Furthermore, Kim, No and Oh found an upper bound on the equilateral stick number [Formula: see text] as follows; [Formula: see text]. As a sequel to this research program, we similarly define the stick number [Formula: see text] and the equilateral stick number [Formula: see text] of a spatial graph [Formula: see text], and present their upper bounds as follows; [Formula: see text] [Formula: see text] where [Formula: see text] and [Formula: see text] are the number of edges and vertices of [Formula: see text], respectively, [Formula: see text] is the number of bouquet cut-components, and [Formula: see text] is the number of non-splittable components.

scholarly journals On chirality of toroidal embeddings of polyhedral graphs

2017 ◽  
Vol 26 (08) ◽  
pp. 1750050
Author(s):  
Senja Barthel
Keyword(s):  
Euler Characteristic ◽  
Planar Graphs ◽  
Graph Embedding ◽  
Alternative Proof ◽  
Torus Knots ◽  
Spatial Graph ◽  
Spatial Graphs ◽  
Knots And Links

We investigate properties of spatial graphs on the standard torus. It is known that nontrivial embeddings of planar graphs in the torus contain a nontrivial knot or a nonsplit link due to [2, 3]. Building on this and using the chirality of torus knots and links [9, 10], we prove that the nontrivial embeddings of simple 3-connected planar graphs in the standard torus are chiral. For the case that the spatial graph contains a nontrivial knot, the statement was shown by Castle et al. [5]. We give an alternative proof using minors instead of the Euler characteristic. To prove the case in which the graph embedding contains a nonsplit link, we show the chirality of Hopf ladders with at least three rungs, thus generalizing a theorem of Simon [12].

scholarly journals COLORING INVARIANTS OF SPATIAL GRAPHS

2010 ◽  
Vol 19 (06) ◽  
pp. 829-841 ◽  
Author(s):  
MACIEJ NIEBRZYDOWSKI
Keyword(s):  
Spatial Graph ◽  
Spatial Graphs

We define the fundamental quandle of a spatial graph and several invariants derived from it. In the category of graph tangles, we define an invariant based on the walks in the graph and cocycles from nonabelian quandle cohomology.

ON NORMALIZATIONS OF A REGULAR ISOTOPY INVARIANT FOR SPATIAL GRAPHS

2011 ◽  
Vol 22 (11) ◽  
pp. 1545-1559
Author(s):  
ATSUSHI ISHII
Keyword(s):  
Finite Type ◽  
Finite Type Invariants ◽  
Spatial Graph ◽  
Trivalent Graphs ◽  
Spatial Graphs ◽  
Graph Homology ◽  
Almost All

We give a framework to normalize a regular isotopy invariant of a spatial graph, and introduce many normalizations satisfying the same relation under a local move. We normalize the Yamada polynomial for spatial embeddings of almost all trivalent graphs without a bridge, and see the benefit to utilize our normalizations from the viewpoint of skein relations, the finite type invariants, and evaluations of the Yamada polynomial. We show that the collection of the differences between two of our normalizations is a complete spatial-graph-homology invariant.

SPATIAL GRAPH EXTERIORS REALIZING GIVEN JSJ DECOMPOSITION GRAPHS

2013 ◽  
Vol 22 (05) ◽  
pp. 1350020
Author(s):  
TORU IKEDA
Keyword(s):  
Connected Graph ◽  
Finite Graph ◽  
Jsj Decomposition ◽  
Spatial Graph ◽  
Spatial Graphs ◽  
Incompressible Boundary ◽  
Jsj Decompositions

If the exterior E(G) of a spatial graph G in a closed orientable 3-manifold is an irreducible 3-manifold with incompressible boundary, there is a unique finite graph Γ associated to the JSJ decomposition of E(G). This paper provides a method for constructing spatial graphs such that a given finite connected graph is associated to the JSJ decompositions of their exteriors.

GRAPH SKEIN MODULES AND SYMMETRIES OF SPATIAL GRAPHS

2012 ◽  
Vol 21 (09) ◽  
pp. 1250090 ◽  
Author(s):  
NAFAA CHBILI
Keyword(s):  
Necessary Conditions ◽  
Skein Modules ◽  
Spatial Graph ◽  
Spatial Graphs

In this paper, we compute the graph skein algebra of the punctured disk with two holes. Then, we apply the graph skein techniques developed here to establish necessary conditions for a spatial graph to have a symmetry of order p, where p is a prime. The obstruction criteria introduced here extend some results obtained earlier for symmetric spatial graphs.

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