Suppose we call a knot an equivalence class of embeddings of S1 --> R3 under ambient isotopy, a knot representative a particular such embedding, and a knot diagram the "2 1/2 dimensional" shadow of such a knot representative on S2 from a particular vantage point P, i.e. the light source for the shadow is at P, and the "2 1/2" means that the over and under information is shown.


My question involves how the set of knot diagrams of a particular fixed knot representative vary as P, the viewpoint varies. We know about the Reidemeister rules, which generate a group of transformations of knot diagrams in a way such that the underlying knot remains invariant under changes of both ambient isotopy and projection. Intuitively there is also a smaller group of pure "projective" transformations of knot diagrams under which any fixed knot representative remains invariant. Assuming that, what are the relations between these two groups of transformations of knot diagrams?



Can you characterize the group of transformations of knot diagrams which preserve the knot embedding?
I remember attending a talk by Barbara Jablonska at Knots in Washington (2009) in which she studied a knot in a geometric, rigid fashion. As the direction of projection varies over $S^2$, she obtained interesting surfaces by looking at the locus of a particular crossing (if memory serves). Here's an abstract of the talk, but I cannot find anything published.


<http://atlas-conferences.com/c/a/x/q/18.htm>


