We say that a graph is intrinsically knotted or completely 3-linked if every embedding of the graph into the 3-sphere contains a nontrivial knot or a 3-component link each of whose 2-component sublinks is nonsplittable. We show that a graph obtained from the complete graph on seven vertices by a finite sequence of Y-exchanges and Y -exchanges is a minor-minimal intrinsically knotted or completely 3-linked graph.

In 1993 K. Habiro defined C k -move of oriented links and around 1994 he proved that two oriented knots are transformed into each other by C k -moves if and only if they have the same Vassiliev invariants of order ≤ k−1. In this paper we define Vassiliev invariant of type (k 1 , ..., k l ), and show that, for k = k 1 + · · · + k l , two oriented knots are transformed into each other by C k -moves if and only if they have the same Vassiliev invariants of type (k 1 , ..., k l ). We introduce a concept 'band description of knots' and give a diagram-oriented proof of this theorem. When k 1 = · · · = k l = 1, the Vassiliev invariant of type (k 1 , ..., k l ) coincides with the Vassiliev invariant of order ≤ l − 1 in the usual sense. As a special case, we have Habiro's theorem stated above.

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We study the relations of knots and links contained in aspatial graph. This is an survey article on the results about knots and links contained in aspatial graph. We do not intend to cover all results in this topic. We only treat some of them here.