All knot groups are metric (Q791928)

Let K be a knot manifold and \(\pi_ 1(K)\) the associated knot group. \(\{gN| g\in \pi_ 1(K)\) and N is a normal subgroup of \(\pi_ 1(K)\) with finite inde\(x\}\) forms a basis for a topology \(\tau\) on \(\pi_ 1(K)\). \((\pi_ 1(K),\tau)\) is a topological group homeomorphic to the rationals. When M is a regular covering space of K we say that \(g_ 1\) and \(g_ 2\) lift alike if and only if either \(g_ 1\) and \(g_ 2\) lift to loops in M or both \(g_ 1\) and \(g_ 2\) lift to paths in M. If N is any infinite normal subgroup then there exists a nonconstant sequence \(\{g_ i\}\) in gN with the property that for any finite sheeted regular covering space M of K, g and all but finitely many elements of \(\{g_ i\}\) lift alike. g and all but finitely many elements of \(\{g_ i\}\) lift alike if and only if for any finite sheeted regular covering space M of K all but finitely many elements of \(\{g_ ig^{-1}\}\) lift to loops. Obviously these results can be generalized.