Amit A Chatwani.

Big Up to Professors John Sullivan and George Francis, Elizabeth Denne, and Yana Malyshev
NSF VIGRE Supported REU at University of Illinois

NOTE: If the sphere in the left panel does not appear in full, adjust the scroll panels to the left!

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The Crossing Map captures the change in the number of crossings of a knot depending on the viewers perspective. The knot is shown and the red target on the sphere corresponds to the viewpoint.

Curves
There are several curves which may be toggled on and off.
  • Tantrix/Antipodal Tantrix Curves: the number of crossings changes by 1.
  • 2-Curves: the number of crossings changes by 2.
  • Trisecant Curves: gives those directions from which three collinear points of the knot (a trisecant) may be seen. This is still being constructed (the case shown is the case where a trisecant is formed only when two edges of the knot are adjacent and the third pierces the plane spanned by these edges.
  • Vertex-Eye View Curves: those directions where the knot is seen by a chosen vertex.
Loading
Knots may be loaded in two ways, from the pull down menu, or by entering a number pair from standard knot tables. By default, the knot will have a minimal stick representation, a smooth representation may be selected.
Viewing Modes
Rotate: Rotate the knot and watch the red target move on the sphere. Alternatively, move the sphere.
Linked: The sphere and knot's movements are linked based on the mouse movement (The target stays fixed)
Orbit: The viewer orbits around the sphere (most useful for projected curves).
Other Features
Red Dot Trail: Leaves a path on the sphere giving the change in viewpoint.

Change Knot Vertices: You may move a selected knot vertex, and all the curves are automatically recomputed.

Pick Points for VEV: This displays the Vertex-Eye View curves only from the vertices that are selected.
Knot coordinates provided by Robert G. Scharein from KnotPlot.com.



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