AnsweredbyAgol: 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.
The figure eight knot (complement) is the starting point for much of hyperbolic geometry. Althoughotherhyperbolicmanifoldswerediscoveredbeforeit,thefigureeightknotcomplementhasoneofthesimplesthyperbolicstructurestoanalyze.ThurstonfirstprovedhishyperbolicDehnsurgerytheoremforthefigureeightknotcomplement-afterunderstandingtheproofinthiscase,thegeneralcaseisnotmuchhardertounderstand.Itisthesimplestknotforwhichevery3-manifoldisabranchedcoveroverit.Itwasoneofthefirst(non-torus)knotsforwhichtheknot-complementproblemwasproven.Ithasthemostnumberofnon-hyperbolicDehn-fillingsoveranyone-cuspedhyperbolic3-manifold.Itisthesmallestvolumeorientablehyperbolicmanifoldwithonecusp.Itwasthefirstknotproventhatallnon-trivialDehnfillingshaveafinite-sheetedcoverwithpositivefirstbettinumber.Itwasthefirstknotforwhichthevolumeconjecturehasbeenverified.