A circular embedding of a graph in Euclidean 3-space |
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Authors: | Kumi Kobata Toshifumi Tanaka |
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Affiliation: | aInterdisciplinary Graduate School of Science and Engineering, Kinki University, Kowakae 3-4-1, Higashiosaka-shi 577-8502, Osaka, Japan;bOsaka City University, Advanced Mathematical Institute, Sugimoto 3-3-138, Sumiyoshi-ku 558-8585, Osaka, Japan |
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Abstract: | A spatial embedding of a graph G is an embedding of G into the 3-dimensional Euclidean space . J.H. Conway and C.McA. Gordon proved that every spatial embedding of the complete graph on 7 vertices contains a nontrivial knot. A linear spatial embedding of a graph is an embedding which maps each edge to a single straight line segment. In this paper, we construct a linear spatial embedding of the complete graph on 2n−1 (or 2n) vertices which contains the torus knot T(2n−5,2) (n4). A circular spatial embedding of a graph is an embedding which maps each edge to a round arc. We define the circular number of a knot as the minimal number of round arcs in among such embeddings of the knot. We show that a knot has circular number 3 if and only if the knot is a trefoil knot, and the figure-eight knot has circular number 4. |
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Keywords: | Knots Spatial graph Complete graph Circular number |
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