# Knot diagrams and knot projection diagrams

One of Mathematica's lovely curated databases is KnotData, as you can see by running:

KnotData[{"PretzelKnot", {5, 2, 3}}]


1. Why do some knots have KnotDiagrams (KnotData["Trefoil", "KnotDiagram"]) but others not (KnotData[{"PretzelKnot", {5, 3, 2}}, "KnotDiagram"])?
2. Is there an elegant way to get a knot projection (where crossing data is absent) for direct computation, without having to create a three-dimensional representation and then project it down to two dimensions and extract crossing numbers and such?

(I'm running 10.4.0.0 on Mac OS 10.10.3, though this shouldn't matter because the data is on the Wolfram Cloud.)

Here's a somewhat kludgy way to answer this question.

Here's a knot:

KnotData[{3, 1}]


Here's its three-dimensional space curve:

f = KnotData[{3, 1}, "SpaceCurve"]


Here is the projection of the space curve onto the $z = 0$ plane:

ParametricPlot3D[{f[t][[1]], f[t][[2]], 0},
{t, 0, 2 Pi},
Axes -> False]


or

ParametricPlot[{f[t][[1]], f[t][[2]]},
{t, 0, 2 Pi},
Axes -> False]


While this gives a reasonable figure it does not easily provide the number of intersections and their topological relations, which is important for a number of problems in knot theory.