Ryo Nikkuni

We say that a set of pairs of disjoint cycles [Formula: see text] of a graph [Formula: see text] is linked if for any spatial embedding [Formula: see text] of [Formula: see text] there exists an element [Formula: see text] of [Formula: see text] such that the [Formula: see text]-component link [Formula: see text] is nonsplittable, and also say minimally linked if none of its proper subsets are linked. In this paper, (1) we show that the set of all pairs of disjoint cycles of [Formula: see text] is minimally linked if and only if [Formula: see text] is essentially same as a graph in the Petersen family, and (2) for any two integers [Formula: see text], we exhibit a minimally linked set of Hamiltonian [Formula: see text]-pairs of cycles of the complete graph [Formula: see text] with at most 18 elements.

In this paper our main object is the graph embedded into the 3-space, called the spatial graph. We study a clasp-pass equivalence on spatial graphs, which is an equivalence relation generated by clasp-pass moves and ambient isotopies. Our approach is an analogy of the delta equivalence classification on spatial graphs, where a delta equivalence is an equivalence relation generated by delta moves and ambient isotopies and implied by a clasp-pass equivalence. Consequently, clasp-pass classifications on spatial embeddings of several non-planar graphs and a specified planar graph are given. This is a preliminary report on our recent work and details will appear elsewhere.