Editing Linkless embedding (section)

=== Knotless graphs ===
[[File:Blue_Trefoil_Knot.png|thumb|A closed curve forming a [[trefoil]], the simplest nontrivial knot.]]

Related to the concept of linkless embedding is the concept of knotless embedding, an embedding of a graph in such a way that none of its simple cycles form a nontrivial [[knot (mathematics)|knot]]. The graphs that do not have knotless embeddings (that is, they are ''intrinsically knotted'') include ''K''<sub>7</sub> and ''K''<sub>3,3,1,1</sub>.<ref>{{harvtxt|Conway|Gordon|1983}}; {{harvtxt|Foisy|2002}}.</ref> However, there also exist minimal forbidden minors for knotless embedding that are not formed (as these two graphs are) by adding one vertex to an intrinsically linked graph, but the list of these is unknown.<ref>{{harvtxt|Foisy|2003}}.</ref>

One may also define graph families by the presence or absence of more complex knots and links in their embeddings,<ref>{{harvtxt|Nešetřil|Thomas|1985}}; {{harvtxt|Fleming|Diesl|2005}}.</ref> or by linkless embedding in [[3-manifold|three-dimensional manifolds]] other than Euclidean space.<ref>{{harvtxt|Flapan|Howards|Lawrence|Mellor|2006}}</ref> {{harvtxt|Flapan|Naimi|Pommersheim|2001}} define a graph embedding to be triple linked if there are three cycles no one of which can be separated from the other two; they show that ''K''<sub>9</sub> is not intrinsically triple linked, but ''K''<sub>10</sub> is.<ref>For additional examples of intrinsically triple linked graphs, see {{harvtxt|Bowlin|Foisy|2004}}.</ref> More generally, one can define an ''n''-linked embedding for any ''n'' to be an embedding that contains an ''n''-component link that cannot be separated by a topological sphere into two separated parts; minor-minimal graphs that are intrinsically ''n''-linked are known for all ''n''.<ref>{{harvtxt|Flapan|Pommersheim|Foisy|Naimi|2001}}</ref>