Knot homotopy in subspaces of the 3-sphere (Q265530)

A knot, \(K\), in a compact connected proper \(3\) dimensional submanifold, \(M\), of the 3-sphere, \(S^3\) is called \textit transient \textrm if \(K\) is homotopic in \(M\) to a knot \(K'\) that is trivial in \(S^3\). Otherwise the knot is called \textit persistent \textrm. The authors show that the complement of such a submanifold, \(M\), is a disjoint union of handlebodies if and only if every knot in \(M\) is transient. They also examine some persistent knots.NEWLINENEWLINE\quad The authors define the \textit transient number \textrm of a knot, \(K\), to be the minimal number of disjoint arcs needed to attach to \(K\) so that \(K\) is transient in a small neighborhood of the union of \(K\) and the arcs. They show the transient number of a knot is less or equal to both the unknotting number and the tunnel number. Lest it be suspected that the transient number is always equal to one of those, a knot is exhibited with transient number \(1\), unknotting number \(2\) and tunnel number \(2\).NEWLINENEWLINE\quad Methods of combinatorial group theory as well as low dimensional geometric topology are utilized in the proofs.NEWLINENEWLINE\quad Consideration is also given to compact \(3\)-manifolds other than \(S^3\).