In 1983, Conway and Gordon [J Graph Theory 7 (1983), 445± 453] showed that every (tame) spatial embedding of K 7 , the complete graph on 7 vertices, contains a knotted cycle. In this paper, we adapt the methods of Conway and Gordon to show that K 3,3,1,1 contains a knotted cycle in every spatial embedding. In the process, we establish that if a graph satis®es a certain linking condition for every spatial embedding, then the graph must have a knotted cycle in every spatial embedding.

For every natural number n, we exhibit a graph with the property that every embedding of it in M 3 contains a non-split n-component link. Furthermore, we prove that our graph is minor minimal in the sense that every minor of it has an embedding in M 3 that contains no non-split n-component link. 1.1991 Mathematics Subject Classification. 57M25, 57M15.

Of course the circle is the least-perimeter way to enclose a region of prescribed area in the plane. This paper proves that a certain standard "double bubble" is the least-perimeter way to enclose and separate two regions of prescribed areas. The solution for three regions remains conjectural.

In [Adams, 1994; The Knot Book], Colin Adams states as an open question whether removing a vertex and all edges incident to that vertex from an intrinsically knotted graph must yield an intrinsically linked graph. In this paper, we exhibit an intrinsically knotted graph for which there is a vertex that can be removed, and the resulting graph is not intrinsically linked. We further show that this graph is minor minimal with respect to being intrinsically knotted. ß

We examine graphs that contain a nontrivial link in every embedding into real projective space, using a weaker notion of unlink than was used in Flapan, et al [5]. We call such graphs intrinsically linked in RP 3 . We fully characterize such graphs with connectivity 0, 1 and 2. We also show that only one Petersen-family graph is intrinsically linked in RP 3 and prove that K 7 minus any two edges is also minorminimal intrinsically linked. In all, 597 graphs are shown to be minor-minimal intrinsically linked in RP 3 .