Twisting of composite torus knots (Q2396614)

For the unknot \(U\), take a disk \(D\) intersecting \(U\) transversely in its interior. By performing a twisting along \(D\), the unknot \(U\) often changes into a non-trivial knot. The resulting knot is said to be a twisted knot. Of course, many knots are twisted, but there exist non-twisted knots. For example, the granny knot, the torus knot \(T(5,8)\) and the torus knots \(T(p,p+7)\) \((p\geq 7)\) are known to be non-twisted. On the other hand, some composite knots are twisted, but all known examples of twisted composite knots have two prime factors. Hence it might not be too much to expect that any composite knot with more than two prime factors is non-twisted. The paper under review gives a new supporting evidence of this conjecture. More precisely, the connected sum of three torus knots \(T(2,p)\sharp T(2,q)\sharp T(2,r)\) is non-twisted for any \(p,q,r\geq 3\). The argument involves \(4\)-dimensional techniques.