For every knot K in the 3-sphere, there exists a one-vertex triangulation of S^3 with an edge that forms K, with an explicit construction from any diagram of K.
Lutz, Knots in collapsible and non-collapsible balls , Electron
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Triangulations of the 3-sphere with knotted edge
For every knot K in the 3-sphere, there exists a one-vertex triangulation of S^3 with an edge that forms K, with an explicit construction from any diagram of K.