Algebraic & Geometric Topology

Partial duals of plane graphs, separability and the graphs of knots

Iain Moffatt

Full-text: Open access

Abstract

There is a well-known way to describe a link diagram as a (signed) plane graph, called its Tait graph. This concept was recently extended, providing a way to associate a set of embedded graphs (or ribbon graphs) to a link diagram. While every plane graph arises as a Tait graph of a unique link diagram, not every embedded graph represents a link diagram. Furthermore, although a Tait graph describes a unique link diagram, the same embedded graph can represent many different link diagrams. One is then led to ask which embedded graphs represent link diagrams, and how link diagrams presented by the same embedded graphs are related to one another. Here we answer these questions by characterizing the class of embedded graphs that represent link diagrams, and then using this characterization to find a move that relates all of the link diagrams that are presented by the same set of embedded graphs.

Article information

Source
Algebr. Geom. Topol., Volume 12, Number 2 (2012), 1099-1136.

Dates
Received: 10 January 2012
Revised: 23 February 2012
Accepted: 25 February 2012
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715382

Digital Object Identifier
doi:10.2140/agt.2012.12.1099

Mathematical Reviews number (MathSciNet)
MR2928906

Zentralblatt MATH identifier
1245.05030

Subjects
Primary: 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25] 57M15: Relations with graph theory [See also 05Cxx]
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 05C75: Structural characterization of families of graphs

Keywords
$1$–sum checkerboard graph dual embedded graph knots and links Partial duality plane graph ribbon graph separability Tait graph Turaev surface

Citation

Moffatt, Iain. Partial duals of plane graphs, separability and the graphs of knots. Algebr. Geom. Topol. 12 (2012), no. 2, 1099--1136. doi:10.2140/agt.2012.12.1099. https://projecteuclid.org/euclid.agt/1513715382


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