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Linearly embedded graphs in 3-space with homotopically free exteriors

Title
Linearly embedded graphs in 3-space with homotopically free exteriors
Author
허영식
Keywords
linear embedding; complete graph; fundamental group; free
Issue Date
2015-04
Publisher
GEOMETRY & TOPOLOGY PUBLICATIONS
Citation
ALGEBRAIC AND GEOMETRIC TOPOLOGY, v. 15, NO 2, Page. 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) (R-3 - 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.
URI
http://msp.org/agt/2015/15-2/p18.xhtmlhttp://hdl.handle.net/20.500.11754/23777
ISSN
1472-2739
DOI
10.2140/agt.2015.15.1161
Appears in Collections:
COLLEGE OF NATURAL SCIENCES[S](자연과학대학) > MATHEMATICS(수학과) > Articles
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