Vol. 252, No. 2, 2011

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On intrinsically knotted or completely 3-linked graphs

Ryo Hanaki, Ryo Nikkuni, Kouki Taniyama and Akiko Yamazaki

Vol. 252 (2011), No. 2, 407–425
Abstract

We say that a graph is intrinsically knotted or completely 3-linked if every embedding of the graph into the 3-sphere contains a nontrivial knot or a 3-component link each of whose 2-component sublinks is nonsplittable. We show that a graph obtained from the complete graph on seven vertices by a finite sequence of Y-exchanges and Y-exchanges is a minor-minimal intrinsically knotted or completely 3-linked graph.

Keywords
spatial graph, intrinsic knottedness, Y-exchange, Y-exchange
Mathematical Subject Classification 2000
Primary: 57M15
Secondary: 57M25
Milestones
Received: 10 August 2010
Revised: 20 January 2011
Accepted: 20 January 2011
Published: 29 October 2011
Authors
Ryo Hanaki
Department of Mathematics
Nara University of Education
Takabatake
Nara 630-8305
Japan
Ryo Nikkuni
Department of Mathematics, School of Arts and Sciences
Tokyo Woman’s Christian University
2-6-1 Zempukuji, Suginami-ku
Tokyo 167-8585
Japan
Kouki Taniyama
Department of Mathematics, School of Education
Waseda University
Nishi-Waseda 1-6-1, Shinjuku-ku
Tokyo 169-8050
Japan
Akiko Yamazaki
Division of Mathematics, Graduate School of Science
Tokyo Woman’s Christian University
2-6-1 Zempukuji, Suginami-ku
Tokyo 167-8585
Japan