Revisions to Fundamental Examples

Copied image to imgur.com, as it was not being displayed because of the new https rule. Added link to original image source.

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edited Dec 10, 2017 at 23:42

jeq

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Answered by Agol: The figure eight knot (complement) is the starting point for much of hyperbolic geometry. Although other hyperbolic manifolds were discovered before it, the figure eight knot complement has one of the simplest hyperbolic structures to analyze. Thurston first proved his hyperbolic Dehn surgery theorem for the figure eight knot complement - after understanding the proof in this case, the general case is not much harder to understand. It is the simplest knot for which every 3-manifold is a branched cover over it. It was one of the first (non-torus) knots for which the knot-complement problem was proven. It has the most number of non-hyperbolic Dehn-fillings over any one-cusped hyperbolic 3-manifold. It is the smallest volume orientable hyperbolic manifold with one cusp. It was the first knot proven that all non-trivial Dehn fillings have a finite-sheeted cover with positive first betti number. It was the first knot for which the volume conjecture has been verified.

(see also this Wikipedia article.)

alt text http://mathworld.wolfram.com/images/eps-gif/FigureEightKnot_700.gif^(source)

Answered by Agol: The figure eight knot (complement) is the starting point for much of hyperbolic geometry. Although other hyperbolic manifolds were discovered before it, the figure eight knot complement has one of the simplest hyperbolic structures to analyze. Thurston first proved his hyperbolic Dehn surgery theorem for the figure eight knot complement - after understanding the proof in this case, the general case is not much harder to understand. It is the simplest knot for which every 3-manifold is a branched cover over it. It was one of the first (non-torus) knots for which the knot-complement problem was proven. It has the most number of non-hyperbolic Dehn-fillings over any one-cusped hyperbolic 3-manifold. It is the smallest volume orientable hyperbolic manifold with one cusp. It was the first knot proven that all non-trivial Dehn fillings have a finite-sheeted cover with positive first betti number. It was the first knot for which the volume conjecture has been verified.

(see also this Wikipedia article.)

alt text http://mathworld.wolfram.com/images/eps-gif/FigureEightKnot_700.gif

Answered by Agol: The figure eight knot (complement) is the starting point for much of hyperbolic geometry. Although other hyperbolic manifolds were discovered before it, the figure eight knot complement has one of the simplest hyperbolic structures to analyze. Thurston first proved his hyperbolic Dehn surgery theorem for the figure eight knot complement - after understanding the proof in this case, the general case is not much harder to understand. It is the simplest knot for which every 3-manifold is a branched cover over it. It was one of the first (non-torus) knots for which the knot-complement problem was proven. It has the most number of non-hyperbolic Dehn-fillings over any one-cusped hyperbolic 3-manifold. It is the smallest volume orientable hyperbolic manifold with one cusp. It was the first knot proven that all non-trivial Dehn fillings have a finite-sheeted cover with positive first betti number. It was the first knot for which the volume conjecture has been verified.

(see also this Wikipedia article.)

alt text ^(source)

replaced http://mathoverflow.net/ with https://mathoverflow.net/

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edited Apr 13, 2017 at 12:58

Community Bot

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Answered by AgolAgol: The figure eight knot (complement) is the starting point for much of hyperbolic geometry. Although other hyperbolic manifolds were discovered before it, the figure eight knot complement has one of the simplest hyperbolic structures to analyze. Thurston first proved his hyperbolic Dehn surgery theorem for the figure eight knot complement - after understanding the proof in this case, the general case is not much harder to understand. It is the simplest knot for which every 3-manifold is a branched cover over it. It was one of the first (non-torus) knots for which the knot-complement problem was proven. It has the most number of non-hyperbolic Dehn-fillings over any one-cusped hyperbolic 3-manifold. It is the smallest volume orientable hyperbolic manifold with one cusp. It was the first knot proven that all non-trivial Dehn fillings have a finite-sheeted cover with positive first betti number. It was the first knot for which the volume conjecture has been verified.

(see also this Wikipedia article.)

alt text http://mathworld.wolfram.com/images/eps-gif/FigureEightKnot_700.gif

Answered by Agol: The figure eight knot (complement) is the starting point for much of hyperbolic geometry. Although other hyperbolic manifolds were discovered before it, the figure eight knot complement has one of the simplest hyperbolic structures to analyze. Thurston first proved his hyperbolic Dehn surgery theorem for the figure eight knot complement - after understanding the proof in this case, the general case is not much harder to understand. It is the simplest knot for which every 3-manifold is a branched cover over it. It was one of the first (non-torus) knots for which the knot-complement problem was proven. It has the most number of non-hyperbolic Dehn-fillings over any one-cusped hyperbolic 3-manifold. It is the smallest volume orientable hyperbolic manifold with one cusp. It was the first knot proven that all non-trivial Dehn fillings have a finite-sheeted cover with positive first betti number. It was the first knot for which the volume conjecture has been verified.

(see also this Wikipedia article.)

alt text http://mathworld.wolfram.com/images/eps-gif/FigureEightKnot_700.gif

Answered by Agol: The figure eight knot (complement) is the starting point for much of hyperbolic geometry. Although other hyperbolic manifolds were discovered before it, the figure eight knot complement has one of the simplest hyperbolic structures to analyze. Thurston first proved his hyperbolic Dehn surgery theorem for the figure eight knot complement - after understanding the proof in this case, the general case is not much harder to understand. It is the simplest knot for which every 3-manifold is a branched cover over it. It was one of the first (non-torus) knots for which the knot-complement problem was proven. It has the most number of non-hyperbolic Dehn-fillings over any one-cusped hyperbolic 3-manifold. It is the smallest volume orientable hyperbolic manifold with one cusp. It was the first knot proven that all non-trivial Dehn fillings have a finite-sheeted cover with positive first betti number. It was the first knot for which the volume conjecture has been verified.

(see also this Wikipedia article.)

alt text http://mathworld.wolfram.com/images/eps-gif/FigureEightKnot_700.gif

added 256 characters in body

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edited Jun 22, 2010 at 6:22

Gil Kalai

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Answered by Agol: The figure eight knotfigure eight knot (complement) is the starting point for much of hyperbolic geometry. Although other hyperbolic manifolds were discovered before it, the figure eight knot complement has one of the simplest hyperbolic structures to analyze. Thurston first proved his hyperbolic Dehn surgery theorem for the figure eight knot complement - after understanding the proof in this case, the general case is not much harder to understand. It is the simplest knot for which every 3-manifold is a branched cover over it. It was one of the first (non-torus) knots for which the knot-complement problem was proven. It has the most number of non-hyperbolic Dehn-fillings over any one-cusped hyperbolic 3-manifold. It is the smallest volume orientable hyperbolic manifold with one cusp. It was the first knot proven that all non-trivial Dehn fillings have a finite-sheeted cover with positive first betti number. It was the first knot for which the volume conjecture has been verified.

(see also this Wikipedia article.)

alt text http://mathworld.wolfram.com/images/eps-gif/FigureEightKnot_700.gif

The figure eight knot (complement) is the starting point for much of hyperbolic geometry. Although other hyperbolic manifolds were discovered before it, the figure eight knot complement has one of the simplest hyperbolic structures to analyze. Thurston first proved his hyperbolic Dehn surgery theorem for the figure eight knot complement - after understanding the proof in this case, the general case is not much harder to understand. It is the simplest knot for which every 3-manifold is a branched cover over it. It was one of the first (non-torus) knots for which the knot-complement problem was proven. It has the most number of non-hyperbolic Dehn-fillings over any one-cusped hyperbolic 3-manifold. It is the smallest volume orientable hyperbolic manifold with one cusp. It was the first knot proven that all non-trivial Dehn fillings have a finite-sheeted cover with positive first betti number. It was the first knot for which the volume conjecture has been verified.

alt text http://mathworld.wolfram.com/images/eps-gif/FigureEightKnot_700.gif

Answered by Agol: The figure eight knot (complement) is the starting point for much of hyperbolic geometry. Although other hyperbolic manifolds were discovered before it, the figure eight knot complement has one of the simplest hyperbolic structures to analyze. Thurston first proved his hyperbolic Dehn surgery theorem for the figure eight knot complement - after understanding the proof in this case, the general case is not much harder to understand. It is the simplest knot for which every 3-manifold is a branched cover over it. It was one of the first (non-torus) knots for which the knot-complement problem was proven. It has the most number of non-hyperbolic Dehn-fillings over any one-cusped hyperbolic 3-manifold. It is the smallest volume orientable hyperbolic manifold with one cusp. It was the first knot proven that all non-trivial Dehn fillings have a finite-sheeted cover with positive first betti number. It was the first knot for which the volume conjecture has been verified.

(see also this Wikipedia article.)

alt text http://mathworld.wolfram.com/images/eps-gif/FigureEightKnot_700.gif