On the transient number of a knot (Q6644086)

A knot \(K\) in the \(3\)-sphere \(S^3\) is said to be transient in a submanifold \(M\) of \(S^3\) if \(K\) can be homotoped within \(M\) to the trivial knot in \(S^3\). The transient number of \(K\), denoted \(tr(K)\), is the minimal number of simple arcs that have to be attached to \(K\), in order for \(K\) to be homotoped to a trivial knot in a regular neighborhood of the union of \(K\) and the arcs. The transient number is related to other knot invariants, namely \(tr(K)\leq u(K)\), where \(u(K)\) is the unknotting number, and \(tr(K) \leq t (K)\), where \(t (K)\) is the tunnel number.\N\NIn this paper, the authors prove that the rank of the first homology group of a cyclic branched cover of a knot gives lower bounds for the transient number. In particular, if \(tr(K) = 1\), then the first homology group of the double branched cover of K is cyclic. Using this, they give examples of knots with arbitrary large transient number and explore the transient number of knots in the tables. They also consider the transient number of a connected sum of knots, prove some facts and propose some problems.