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

TL;DR

This paper characterizes when the normal closures of slope elements in a knot group are equal or intersect infinitely, revealing unique properties of nontrivial knots and their Dehn fillings.

Contribution

It proves the uniqueness of slope elements' normal closures and describes their intersection properties, especially highlighting the special case of (p, q)-torus knots.

Findings

Normal closures of different slopes are equal only if the slopes are identical.

The intersection of normal closures of multiple slopes is infinite unless the knot is a (p,q)-torus knot with pq in the set.

Unique characterization of slope elements' normal closures in knot groups.

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.