Regular projections of graphs with at most three double points

Youngsik Huh; Ryo Nikkuni

doi:10.1142/s0218216510008261

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arXiv · 0808.4027

Regular projections of graphs with at most three double points

Youngsik Huh·Ryo Nikkuni

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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.

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Youngsik Huh, Ryo Nikkuni. 2008-08-29. Regular projections of graphs with at most three double points. https://doi.org/10.1142/s0218216510008261

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