Abstract
We announce results about flat (linkless) embeddings of graphs in 3space. A piecewiselinear embedding of a graph in 3space is called flat if every circuit of the graph bounds a disk disjoint from the rest of the graph. We have shown: (i) An embedding is flat if and only if the fundamental group of the complement in 3space of the embedding of every subgraph is free. (ii) If two flat embeddings of the same graph are not ambient isotopic, then they differ on a subdivision of K_{5} or K_{3},_{3}. (iii) Any flat embedding of a graph can be transformed to any other flat embedding of the same graph by “3switches”, an analog of 2switches from the theory of planar embeddings. In particular, any two flat embeddings of a 4connected graph are either ambient isotopic, or one is ambient isotopic to a mirror image of the other. (iv) A graph has a flat embedding if and only if it has no minor isomorphic to one of seven specified graphs. These are the graphs that can be obtained from K_{6} by means of Y∆ and ∆Yexchanges.
Original language  English (US) 

Pages (fromto)  8489 
Number of pages  6 
Journal  Bulletin of the American Mathematical Society 
Volume  28 
Issue number  1 
DOIs 

State  Published  Jan 1993 
Externally published  Yes 
All Science Journal Classification (ASJC) codes
 General Mathematics
 Applied Mathematics