Intrinsically linked graphs and even linking number

Fleming, Thomas; Diesl, Alexander

doi:10.2140/agt.2005.5.1419

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Intrinsically linked graphs and even linking number

Thomas Fleming and Alexander Diesl

Algebraic & Geometric Topology 5 (2005) 1419–1432

DOI: 10.2140/agt.2005.5.1419

arXiv: math.GT/0511133

Abstract

We study intrinsically linked graphs where we require that every embedding of the graph contains not just a non-split link, but a link that satisfies some additional property. Examples of properties we address in this paper are: a two component link with $l k (A, L) = k 2^{r}, k \neq 0$ , a non-split $n$ -component link where all linking numbers are even, or an $n$ -component link with components $L, A_{i}$ where $l k (L, A_{i}) = 3 k, k \neq 0$ . Links with other properties are considered as well.

For a given property, we prove that every embedding of a certain complete graph contains a link with that property. The size of the complete graph is determined by the property in question.

Keywords

intrinsically linked graph, spatial graph, graph embedding, linking number

Mathematical Subject Classification 2000

Primary: 57M15

Secondary: 57M25, 05C10

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Publication

Received: 22 April 2004

Revised: 13 September 2005

Accepted: 20 September 2005

Published: 15 October 2005

Authors

Thomas Fleming
University of California San Diego Department of Mathematics 9500 Gilman Drive La Jolla CA 92093-0112 USA

Alexander Diesl
University of California Berkeley Department of Mathematics 970 Evans Hall Berkeley CA 94720-3840 USA

Volume 5, issue 4 (2005)