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REGULAR PROJECTIONS OF GRAPHS WITH AT MOST THREE DOUBLE POINTS

Department of Mathematics, School of Natural Sciences, Hanyang University, Seoul 133-791, Korea

and

Department of Mathematical Sciences, School of Arts and Sciences, Tokyo Woman's Christian University, 2-6-1 Zempukuji, Suginami-ku, Tokyo 167-8585, Japan

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https://doi.org/10.1142/S0218216510008261Cited by:0 (Source: Crossref)

Abstract

A generic immersion of a planar graph into the 2-space is said to be knotted if there does not exist a trivial embedding of the graph into the 3-space obtained by lifting the immersion with respect to the natural projection from the 3-space to the 2-space. In this paper, we show that if a generic immersion of a planar graph is knotted then the number of double points of the immersion is more than or equal to three. To prove this, we also show that an embedding of a graph obtained from a generic immersion of the graph (does not need to be planar) with at most three double points is totally free if it contains neither a Hopf link nor a trefoil knot.

Keywords:

AMSC: 57M15, 57M25

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