A sharp-move is known as an unknotting operation for knots. A self sharp-move is a sharp-move on a spatial graph where all strings in the move belong to the same spatial edge. We say that two spatial embeddings of a graph are sharp edge-homotopic if they are transformed into each other by self sharp-moves and ambient isotopies. We investigate how is the sharp edge-homotopy strong and classify all spatial theta curves completely up to sharp edge-homotopy. Moreover we mention a relationship between...
We study new equivalence relations in spatial graph theory. We consider natural generalizations of delta link-homotopy on links, which is an equivalence relation generated by delta moves on the same component and ambient isotopies. They are stronger than edge-homotopy and vertex-homotopy on spatial graphs which are natural generalizations of link-homotopy on links. Relationship to existing familiar equivalence relations on spatial graphs are stated, and several invariants are defined by using the...
An ordered and oriented 2-component link L in the 3-sphere is said to be achiral if it is ambient isotopic to its mirror image ignoring the orientation and ordering of the components. Kirk-Livingston showed that if L is achiral then the linking number of L is not congruent to 2 modulo 4. In this paper we study orientation-preserving or reversing symmetries of 2-component links, spatial complete graphs on 5 vertices and spatial complete bipartite graphs on 3 + 3 vertices in detail, and determine...
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