Tossup

Thurston proved that a knot must have this property if it is not a torus knot or a satellite knot. A space with this property has as its orientation-preserving isometries the group of real Möbius transformations with determinant one. Spaces with this property have curves called horocycles, can be modeled (10[1])in two dimensions by Poincaré’s half-plane and disk, and have constant negative curvature. This property is the first word in a class of functions derived from the pair “e-to-the-x plus-or-minus e-to-the-negative-x all over (10[1])two.” Lobachevsky names a non-Euclidean geometry with this (10[1])property, which breaks the parallel postulate and has (10[1])triangles whose angles sum to less than 180 degrees. For 10 points, what property derives from the name of a conic section exemplified (10[1])by the function one-over-x? ■END■ (0[2])

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