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I am seeking literature on 3D orthogonal drawings of knots, especially minimum bend drawings. An orthogonal drawing employs segments parallel to the axes of a Cartesian coordinate system. A bend is a vertex at which two segments meet orthogonally. A drawing insists on simplicity in the sense that nonadjacent segments are disjoint, and adjacent segments meet only at their shared endpoint.

One can imagine first drawing a 2D projection with a minimal number of crossings and then removing the crossings. For the trefoil below, naive crossing-removal increments the $8$ bends in the 2D drawing to $8 + 3 \cdot 4 = 20$ bends, but the trefoil can be drawn with $12$ bends:
           TrefoilMinBends
I would be especially interested in algorithmic methods to derive the right 3D drawing above from the left 2D projection. Thanks for ideas and pointers!

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up vote 7 down vote accepted

Chad Giusti calls these "Plumbers' Knots":

http://front.math.ucdavis.edu/0811.2215

http://front.math.ucdavis.edu/1107.4717

In the first paper Giusti gives the space of plumbers knots a natural stratification which is a CW-decomposition. This allows him to enumerate components of the space (knot types) in a fairly algebraic manner.

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@Ryan: Great references! I would never have thought of searching using the word "plumber"! –  Joseph O'Rourke May 16 '12 at 19:55
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