74.完全グラフの準線形埋め込みについて

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Let K_n be a complete graph with n-vertices v_1, v_2,…, v_n and e_1, e_2,…, e_m, m=n (n-1)/2, be its edges. We introduce the barycentric coordinate for each e_k (k=1, 2,…, m), that is, for each point x of e_k we correspond (1-t)v_i+tv_i (0≦t≦1). Let p: D^2×R^1→D^2⊂R^2 be the projection and φ: K_n→D^2×R^1 be an embedding such that the restriction (pоφ)|e_i: e_i→D^2⊂R^2 is linear with respect to the barycentric coordinate of e_i and the coordinate of R^2. We call such φ a semi-linear embedding of K_n. In this paper we study the ambient isotopic classification of semi-linear embeddings of K_n. It is the same classification of the following. Let K_n be a linear realization of K_n on the unit disk D^2 with the vertices v_i contained in ∂D^2. We make the over and the under crossing for each intersection of edges. Then we get a "semi-linear" realization in the 3-space (or 3-sphere). And classify such "semi-linear" realizations by the ambient isotopy of the 3-space.
1987-03-15