https://arxiv.org/pdf/1702.04862.pdf

# Fig8_slopes

Steve Trettel

Medium | Mathematica Notebook. |
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The Art | A slope (that is, a curve going p times around the longitude and q times around the meridian) on the boundary of a regular neighborhood of the figure-8 knot in the 3 sphere, viewed under sterographic projection from a point on the surface of the knot (making the visible region homeomorphic to the knot complement) |

The Math | The figure 8 knot is one of the simplest knots, able to be drawn as a planar diagram with only four crossings. Its complement in the 3-sphere admits a hyperbolic structure, and doing hyperbolic dehn surgery on the torus cross section of the cusp produces infinitely many examples of closed hyperbolic 3-manifolds. Such a surgery is determined by choosing a slope (a closed curve ) on the boundary, such that when a solid torus is glued in, the disk cross section of the solid torus trivializes this curve. |

Categories | Mathematical Illustration, Geometric Topology, Knot Theory, Stereographic Projection |