#### Volume 15, issue 2 (2015)

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Linearly embedded graphs in $3$–space with homotopically free exteriors

### Youngsik Huh and Jung Hoon Lee

Algebraic & Geometric Topology 15 (2015) 1161–1173
##### Abstract

An embedding of a graph into ${ℝ}^{3}$ is said to be linear if any edge of the graph is sent to a line segment. And we say that an embedding $f$ of a graph $G$ into ${ℝ}^{3}$ is free if ${\pi }_{1}\left({ℝ}^{3}-f\left(G\right)\right)$ is a free group. It is known that the linear embedding of any complete graph is always free.

In this paper we investigate the freeness of linear embeddings by considering the number of vertices. It is shown that the linear embedding of any simple connected graph with at most 6 vertices whose minimal valency is at least 3 is always free. On the contrary, when the number of vertices is much larger than the minimal valency or connectivity, the freeness may not be an intrinsic property of the graph. In fact we show that for any $n\ge 1$ there are infinitely many connected graphs with minimal valency $n$ which have nonfree linear embeddings and furthermore that there are infinitely many $n$–connected graphs which have nonfree linear embeddings.

##### Keywords
linear embedding, complete graph, fundamental group, free
##### Mathematical Subject Classification 2010
Primary: 57M25
Secondary: 57M15, 05C10
##### Publication
Received: 17 June 2014
Revised: 31 August 2014
Accepted: 17 September 2014
Published: 22 April 2015
##### Authors
 Youngsik Huh Department of Mathematics College of Natural Sciences Hanyang University Seoul 133-791 South Korea Jung Hoon Lee Department of Mathematics and Institute of Pure and Applied Mathematics Chonbuk National University Jeonju 561-756 South Korea