Nontrivial elements in a knot group which are trivialized by Dehn fillings

Abstract

Let K be a nontrivial knot in the 3-sphere with the exterior E(K), and u in G(K), the fundamental group of E(K), a slope element represented by an essential simple closed curve on the boundary of E(K). Since the normal closure of u in G(K) coincides with that of the inverse of u, and u and its inverse u correspond to a slope r, a rational number or 1/0, we write << r >> = << u >>. The normal closure << u >> describes elements which are trivialized by r-Dehn filling of E(K). In this article, we prove that << r_1 >> =<< r_2 >> if and only if r_1 = r_2, and for a given finite family of slopes S = {r_1, ..., r_n}, the intersection of << r_1 >> , << r_2>>, ..., and << r_n >> contains infinitely many elements except when K is a (p, q)-torus knot and pq belongs to S. We also investigate inclusion relation among normal closures of slope elements.