Edge-Homotopy,SHARP Edge-Homotopy And DELTA Vertex-Homotopy On Spatial Graphs

A C_k-move is a concept of local move on oriented links, it was first introduced by K.Habiro, then extended to spatial graphs by K.Taniyama and A.Yasuhara. Some properties of -move are of significant importance to study the classification of spatial graph. In this paper, we first study the relationships between adjacent delta-move and quasi-adjacent delta-move of spatial graph, and get: For any two spatial embeddings f and g of a spatial graph G, if they can be transformed into each other by adjacent delta-moves, then they can be transformed into each other by quasi-adjacent delta-moves and ambient isotopies. As a main application of this result, we have the following theorem: For spatial graph C₃,â–³-homotopy,delta vertex-homotopy and edge-homotopy are mutually equivalent. Furthermore, we give a list of edge-homotopy classes of C₃. Here, our main work is to overcome the difficulty of requesting each vertex of the spatial graph has degree 3 and generalize some corresponding results.Chen Juanjuan gave a sufficient condition for vertex homotopy implying delta vertex-homotopy. As another application of the result, we give a weaker condition for vertex homotopy implying delta vertex- homotopy.Furthermore, R.Nikkuni gave another equivalence relation of spatial graph in 2005—sharp edge-homotopy, and pointed out that it shares a property in common with delta vertex-homotopy—weaker than delta edge-homotopy and stronger than edge-homotopy. In this paper, we have this result: For a bouquent Bm , sharp edge-homotopy and delta vertex-homotopy are equivalent.