Geodesic
Linearly embedded graphs in 3–space with homotopically free exteriors
Huh, Youngsik1 ; Lee, Jung Hoon2
1Department of Mathematics, College of Natural Sciences, Hanyang University, Seoul 133-791, South Korea
2Department of Mathematics and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 561-756, South Korea
Algebraic and Geometric Topology, Tome 15 (2015) no. 2, pp. 1161-1173
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An embedding of a graph into ℝ3 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 ℝ3 is free if π1(ℝ3 − f(G)) 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 ≥ 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.

DOI : 10.2140/agt.2015.15.1161
Classification : 57M25, 57M15, 05C10
Keywords: linear embedding, complete graph, fundamental group, free

Huh, Youngsik  1   ; Lee, Jung Hoon  2

1 Department of Mathematics, College of Natural Sciences, Hanyang University, Seoul 133-791, South Korea
2 Department of Mathematics and Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 561-756, South Korea 
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Huh, Youngsik; Lee, Jung Hoon. Linearly embedded graphs in 3–space with homotopically free exteriors. Algebraic and Geometric Topology, Tome 15 (2015) no. 2, pp. 1161-1173. doi: 10.2140/agt.2015.15.1161

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