that are not flat: for instance, if G is a graph formed by two disjoint cycles, and it is embedded to form the Whitehead link, then the embedding is linkless but not flat. A graph is said to be intrinsically linked if, no matter how it is embedded, the embedding is always linked. Although linkless and flat embeddings are not the same, the graphs that have linkless embeddings are the same as the graphs that have flat embeddings. Examples and counterexamples As showed, each of the seven graphs of the Petersen family is intrinsically linked: no matter how each of these graphs is embedded in space, they have two cycles that are linked to each other. These graphs include the complete graph K6, the Petersen graph, the graph formed by removing an edge from the complete bipartite graph K4,4, and the complete tripartite graph K3,3,1. Every planar graph has a flat and linkless embedding: simply embed the graph into a plane and embed the plane into space. If a graph is planar, this is the only way to embed it flatly and linklessly into space: every flat embedding can be continuously deformed to lie on a flat plane. And conversely, every nonplanar linkless graph has multiple linkless embeddings. An apex graph, formed by adding a single vertex to a planar graph, also has a flat and linkless embedding: embed the planar part of the graph on a plane, place the apex above the plane, and draw the edges from the apex to its neighbors as line segments. Any closed curve within the plane bounds a disk below the plane that does not pass through any other graph feature, and any closed curve through the apex bounds a disk above the plane that does not pass through any other graph feature. If a graph has a linkless or flat embedding, then modifying the graph by subdividing or unsubdividing its edges, adding or removing multiple edges between the same pair of points, and performing Y-Δ transforms that replace a degree-three vertex by a triangle connecting its three neighbors or the reverse all preserve flatness and linklessness. In particular, in a cubic planar graph (one in which all vertices have exactly three neighbors, such as the cube) it is possible to make duplicates of any independent set of vertices by performing Y-Δ transforms, adding multiple copies of the resulting triangle edges, and then