Topological realizations of ortho-projection graphs
type: Item
instance of: scholarly article
title: Topological realizations of ortho-projection graphs
zbMATH Open document ID: 1179.57005
author: L. Traldi
author: Louis Zulli
published in: Topology and its Applications
publication date: 2009-09-28T00:00:00Z
review text: The present paper takes into account ortho-projection graphs, i.e. undirected, looped graphs \(G\) (without multiple loops and without multiple edges) satisfying the following adjacency condition: arbitrary (not necessarily distinct) vertices are neighbors in \(G\) if and only if they have an odd number of common neighbors. These graphs, which arise arise naturally in knot theory and in low-dimensional topology, are closely related to circle graphs (see for example [\textit{A. Bouchet}, J. Comb. Theory, Ser. B 60, No.~1, 107--144 (1994; Zbl 0793.05116) and \textit{H. de Fraysseix}, Eur. J. Comb. 5, 223--238 (1984; Zbl 0551.05056)]); in particular, in [\textit{B. Shtylla, L. Traldi} and \textit{L. Zulli}, Discrete Math. 309, No. 6, 1769--1773 (2009; Zbl 1205.05006)], interesting relations are proved to exist between ortho-projection graphs and the problem of realizing circle graphs by generic closed curves. Here, the authors discuss the production of ortho-projection graphs from alternating knot diagrams, and introduce a more general construction of such graphs from splittings of closed, non-orientable surfaces. As their main result, they prove that this new topological construction generates all ortho-projection graphs. Moreover (despite the computer assisted analysis proving that each ortho-projection graph with eight or fewer vertices is the looped interlacement graph of an alternating classical knot diagram [see \textit{B. Shtylla, L. Traldi} and \textit{L. Zulli}, J. Knot Theory Ramifications 15, No. 1, 81--100 (2006; Zbl 1093.57008)]), they present a minimal example of an ortho-projection graph that does not arise from a knot diagram, and provide a surface-splitting that realizes this graph.
reviewed by: Maria Rita Casali
Mathematics Subject Classification ID: 57M15
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 05C62
zbMATH DE Number: 5608087
zbMATH Keywords: non-orientable surface splitting
zbMATH Keywords: orthogonal projection
zbMATH Keywords: circle graph
zbMATH Keywords: signed chord diagram
zbMATH Keywords: trip matrix
zbMATH Keywords: alternating knot
MaRDI profile type: Publication
full work available at URL: j.topol.2009.07.019
OpenAlex ID: W1983578315
cites work: Circle graph obstructions
cites work: A characterization of circle graphs
cites work: Q4172067
cites work: On the realization of double occurrence words
cites work: AN EXTENSION OF THE JONES POLYNOMIAL OF CLASSICAL KNOTS
cites work: Q4004234
cites work: A BRACKET POLYNOMIAL FOR GRAPHS, I
cites work: A matrix for computing the Jones polynomial of a knot
cites work: A MATRIX INVARIANT OF CURVES IN S2
DOI: 10.1016/J.TOPOL.2009.07.019
description: scientific article; zbMATH DE number 5608087