Annular Dehn Surgery on Knots in a Solid Torus

摘要

Let K be a knot in a solid torus V, and N (K) a regular neighbourbood of K in V. K is called a hyperbolic knot if M=V-intN(K) is hyperbolic. By Thurston's Geometrization Theorem, K is hyperbolic if and only if M is irreducible, α-irreducible, Atoroidal, anannular. A slope on T=αN(K) is a T-isotopy class of essential, unoriented, simple closed Curves on T, and the distance between two slopes r_1 and r_2, denoted by Δ(r_1,R_2), is the Minimal geometric intersection number among all the curves representing the slopes.