Editing Knot theory (section)

===Hyperbolic invariants===
[[William Thurston]] proved many knots are [[hyperbolic knot]]s, meaning that the [[knot complement]] (i.e., the set of points of 3-space not on the knot) admits a geometric structure, in particular that of [[hyperbolic geometry]]. The hyperbolic structure depends only on the knot so any quantity computed from the hyperbolic structure is then a knot invariant {{Harv|Adams|2004}}.

{{multiple image
 | align = right
 | total_width = 320
 | image1 = BorromeanRings.svg
 | width1 = 626 | height1 = 600
 | caption1 = The [[Borromean rings]] are a link with the property that removing one ring unlinks the others.
 | image2 = SnapPea-horocusp_view.png
 | width2 = 560 | height2 = 416
 | caption2 = [[SnapPea]]'s cusp view: the [[Borromean rings]] complement from the perspective of an inhabitant living near the red component.
}}

Geometry lets us visualize what the inside of a knot or link complement looks like by imagining light rays as traveling along the [[geodesic]]s of the geometry. An example is provided by the picture of the complement of the [[Borromean rings]]. The inhabitant of this link complement is viewing the space from near the red component. The balls in the picture are views of [[horoball]] neighborhoods of the link. By thickening the link in a standard way, the horoball neighborhoods of the link components are obtained. Even though the boundary of a neighborhood is a torus, when viewed from inside the link complement, it looks like a sphere. Each link component shows up as infinitely many spheres (of one color)  as there are infinitely many light rays from the observer to the link component. The fundamental parallelogram (which is indicated in the picture), tiles both vertically and horizontally and shows how to extend the pattern of spheres infinitely.

This pattern, the horoball pattern, is itself a useful invariant. Other hyperbolic invariants include the shape of the fundamental parallelogram, length of shortest geodesic, and volume. Modern knot and link tabulation efforts have utilized these invariants effectively. Fast computers and clever methods of obtaining these invariants make calculating these invariants, in practice, a simple task {{Harv|Adams|Hildebrand|Weeks|1991}}.