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2-IRREDUCIBILITY OF SPATIAL GRAPHS

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China

Department of Mathematics, School of Education, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo, 169-8050, Japan

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GENGYU ZHANG

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China

Present address: Department of Mathematics, Tokyo Institute of Technology, Tokyo, 152-8551, Japan.

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https://doi.org/10.1142/S0218216506004336Cited by:0 (Source: Crossref)

Abstract

An embedded graph G in the 3-sphere S³ is called 2-irreducible if there are no separating spheres, cutting spheres, singular separating spheres, singular cutting spheres or 2-cutting spheres of G. Let D be a 2-disk in S³ that is very good for G. Let G′ be an embedded graph in S³ obtained from G by contracting D to a point. We show that if G′ is 2-irreducible then G is 2-irreducible. By this criterion certain graphs are easily shown to be 2-irreducible. As an application we show a pair of embedded graphs in the 3-sphere which is distinguished by 2-irreducibility.

Keywords:

AMSC: Primary 57M25, Secondary 57M15, Secondary 05C10

References

J. Milnor, Ann. Math. 59, 177 (1954). Crossref, Web of Science, Google Scholar
T. Motohashi, Topol. Appl. 93, 161 (1999). Crossref, Web of Science, Google Scholar
T. Motohashi, Topol. Appl. 108, 267 (2000). Crossref, Web of Science, Google Scholar
M. Scharlemann, Some pictorial remarks on Suzuki's Brunnian graph, Topology '90, Ohio State University Mathematical Research Institute Publications 1 (de Gruyter, Berlin, 1992) pp. 351–354. Google Scholar
T. Soma, Osaka J. Math. 35, 235 (1998). Web of Science, Google Scholar
S. Suzuki, Topology and Computer Science, ed. S. Suzuki (Kinokuniya, Tokyo, 1987) pp. 259–276. Google Scholar
K. Taniyama, J. Knot Theory Ramifications 11, 121 (2002). Link, Web of Science, Google Scholar