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Identifiable projections of spatial graphs. (English)
J. Knot Theory Ramifications 13, No. 8, 991-998 (2004).
In general, a given regular projection $φ: G\rightarrow S^2$ of a finite graph $G$ lifts to different embeddings $f : G\rightarrow S^3$. The regular projection $φ$ is called an {\it identifiable} projection if any two embeddings of $G$ into $S^3$ having projection $φ$ are ambient isotopic. It is known that if $G$ is homeomorphic to a circle then the identifiable projections of $G$ are embeddings $φ: G\rightarrow S^2$ up to Type $I$ Reidemeister moves. The authors show that if a finite graph $G$ has an identifiable projection then $G$ is planar. Furthermore they show that the identifiable projections for a certain class of planar graphs are embeddings up to Type $I$ Reidemeister moves.
Wolfgang Heil (Tallahassee)