Intrinsically Linked And Intrinsically Knotted Graphs

The spatial graph theory is the natural generalization of the knotted topology theory. It is the active branch of the topology. Intrinsically linked and intrinsically knotted graphs are very important graphs in spatial graphs. This paper combine the two properties of intrinsic linking and intrinsic knotting to produce these two kinds of graphs: 1) Intrinsically linked graphs with knotted components, that is, every spatial embedding of this graph contains a non-split link, where at least one of the components of this link is a nontrivial knot. 2) Intrinsically knotted and 3-linked graphs, that is, every spatial embedding of this graph contains nontrivial knot and a non-split 3-component link.Furthermore, E. Flapan, H. Howards, D. Lawrence, B. Mellor showed that a graph is intrinsically linked in an arbitrary 3-manifold if and only if it is intrinsically linked in R 3. On the base, Firstly, this paper defined the disjoint linked graphs with respect to the unknotted embedding in 3-manifold . Secondly, we showed that =1 for every spatial embedding of the graph . Finally, we showed that that is disjoint linked graphs in w( K 3,1,1,1,1) K3,1,1,1,1H 12R 3 is a disjoint linked graphs with respect to the unknotted embedding in 3-manifold .