Category: Topology Geometric

The Animated Gif that Woo Shik made which wa…

The Animated Gif that Woo Shik made which was based upon an example of mine and Atsushi Ishii. Think of the endpoints of the arcs as being fixed in the boundary of a thin half-space ({(x,y,z): 0 \ge x, and y is in a small interval} The z-axis is vertical. The knotted trivalent graph that you see is embedded in this half-space with its y-coordinate indicated by over/under crossing information. The half space rotates in 4-space around a central plane which has coordinates x=0, w=0 —- this is the plane that contains the endpoints. As it rotates the figure moves. While the gif is infinite, it is meant to indicate two full twists. The colors can be used to track when the twists are complete. The example has non-trivial quandle cocycle invariant.

Gordian (Un)knots

Gordian (Un)knots

The unknot arises in the mathematical theory of knots. Intuitively, the unknot is a closed loop of rope without a knot in it. A knot theorist would describe the unknot as an image of any embedding that can be deformed, i.e. ambient-isotoped, to the standard unknot, i.e. the embedding of the circle as a geometrically round circle. The unknot is also called the trivial knot. An unknot is the identity elementwith respect to the knot sum operation.

GIF.

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