Torus knots that cannot be untied by twisting (Q5956917)

Let \(K\) be an oriented knot in the \(3\)-sphere \(S^3.\) Let \(D^2\) be a disk imbedded in \(S^3\) intersecting \(K\) in its interior. Let \(\omega = |lk(L, K)|,\) where \(L\) denotes the boundary of \(D^2,\) and \(n\) an integer. A \(-{1 \over n}\) Dehn surgery along \(L,\) equivalently \(n\) full twists along \(D^2,\) changes \(K\) into a new knot \(K'\) in \(S^3.\) We say that \(K'\) is obtained from \(K\) by (\(n, \omega\))-twisting or simply twisting. Let \(T\) denote the set of knots that are obtained from a trivial knot by a single (\(n, \omega\))-twisting along a disk \(D^2\) for some integer \(n.\) \textit{Y. Ohyama} showed that any knot can be obtained from a knot in \(T\) by a single twisting [Rev. Mat. Univ. Complutense Madr. 7, No. 2, 289-305 (1994; Zbl 0861.57015)]. The paper under review is concerned with torus knots. A torus knot \(T(p, q)\) \((0 < p < q)\) is exceptional if \(q \equiv \pm 1 \pmod p,\) and non-exceptional if it is not exceptional. Any exceptional torus knot belongs to \(T.\) In this paper, the authors give some conditions on \(n, \omega, p, q\) and Tristram's \(d\)-signature for a torus knot \(T(p, q)\) that is obtained from a trivial knot by a single (\(n, \omega\))-twisting. By using this result, they give some necessary conditions on \(p\) and \(q\) for a non-exceptional torus knot \(T(p, q)\) to belong to \(T.\) As a consequence they show that there are infinitely many non-exceptional torus knots that are not contained in \(T.\)