Twisted torus knots with essential tori in their complements (Q2848024)

Let \(\Sigma\) be a genus one Heegaard surface of the \(3\)--sphere \(S^3\), which bounds two solid tori \(V_1\) and \(V_2\). Let \(T(p, q)\) \((p > q > 0)\) be a \((p, q)\)--torus knot which lies on \(\Sigma\). Choose an unknotted circle \(C\) so that it intersects \(V_i\) in a trivial arc and a disk \(D\) bounded by \(C\) intersects \(T(p, q)\) in \(r\) \((2 \leq r \leq p+q)\) points in the same direction. Adding \(s\) full twists along \(D\), we obtain a twisted torus knot \(T(p, q, r, s)\). Twisted torus knots were introduced by \textit{J. C. Dean} [Hyperbolic knots with small Seifert-fibered Dehn surgeries. Ph.D. thesis, University of Texas at Austin (1996), Algebr. Geom. Topol. 3, 435--472 (2003; Zbl 1021.57002)]) in the study of Seifert fibered surgeries on knots.NEWLINENEWLINEIt is interesting to classify non-hyperbolic twisted torus knots. If \(q\) divides \(r\), there are infinitely many twisted torus knots which are cables of torus knots [\textit{K. Morimoto} and \textit{Y. Yamada}, Kobe J. Math. 26, No. 1--2, 29--34 (2009; Zbl 1190.57005)], [\textit{S. Lee}, J. Knot Theory Ramifications 21, No. 1, Article ID 1250005, 4 p. (2012; Zbl 1234.57008)]. Even when \(q\) does not divide \(r\), \textit{K. Morimoto} [J. Knot Theory Ramifications 22, No. 9, Article ID 1350049, 12 p. (2013; Zbl 1278.57014), Tokyo J. Math. 35, No. 2, 499--503 (2012; Zbl 1261.57009)] gives an infinite family of non-cabled satellite twisted torus knots. For instance, \(T(9, 5, 7, -1)\) is a connected sum of two torus knots \(T(2,3)\) and \(T(2, -5)\).NEWLINENEWLINEIn the paper under review, the author proves that if \(T(p, q, r, s)\) is a satellite knot, then \(q\) divides \(r\) or \(|s| \leq 2\). To prove the result, the author applies a skillful combinatorial technique developed in study of Dehn surgery. At the end of the paper, he asks if there is a satellite twisted torus knot \(T(p, q, r, s)\) such that \(q\) does not divide \(r\) and \(|s| = 2\).