Twisted torus knots with graph manifold Dehn surgeries (Q2787663)

Let \(H\) be a genus two handlebody and a twisted torus knot \(K(p,q,r,m,n)\) where \(0< q < p\) and \(m,n >0\), \(0 < r \leq p + q\), lying in \(\partial H\). Let \(H' = \overline {S^3 - H}\) and \(\Sigma = \partial H = \partial H'\). Then \((H, H' ;\Sigma)\) forms a genus two Heegaard splitting of \(S^3\). The definition and properties of twisted torus knots were introduced by \textit{J. C. Dean} [Algebr. Geom. Topol. 3, 435--472 (2003; Zbl 1021.57002)]. A simple closed curve \(k\) in the boundary of a genus two handlebody \(H\) is said to be Seifert if \(H[k]\), i.e., the \(3\)-manifold obtained by adding a \(2\)-handle to \(H\) along \(k\), is a Seifert-fibered space and not a solid torus. Suppose \(K\) is a knot in \(S^3\) which lies in \(\Sigma\). The knot \(K\) in \(\Sigma\) is called Seifert/Seifert if it is Seifert with respect to both \(H\) and \(H'\). Dean also provided three criteria for twisted torus knots to be Seifert. The criteria give three types of Seifert curves called: hyper Seifert-fibered; middle Seifert-fibered; end Seifert-fibered. Then there are several kinds of twisted torus knots which are Seifert/Seifert: a twisted torus knot is called hyper/middle if it is hyper Seifert-fibered in one handlebody and middle Seifert-fibered in the other handlebody; the kinds middle/middle and hyper/hyper are defined in a similar manner. In two previous papers [J. Korean Math. Soc. 51, No. 6, 1221--1250 (2014; Zbl 1302.57024)] and [Bull. Korean Math. Soc. 52, No. 1, 313--321 (2015; Zbl 1309.57007)], the author classified all hyper/hyper and all hyper/middle twisted torus knots respectively. Let \(S(a_1,\ldots , a_n)\) be the Seifert-fibered space over a surface \(S\) with \(n\) exceptional fibers of indexes \(a_1,\ldots , a_n\). In the paper under review, it is proved that there are only two possibilities for \(H[k]\) for a Seifert curve \(k\): either \(D^2(a,b)\) or an orientable Seifert-fibered space over the Möbius band with at most one exceptional fiber. By a result in [Dean, loc. cit.] it follows that if \(K\) is Seifert/Seifert, then the \(\gamma\)-Dehn surgery \(K(\gamma)\) is either \(S^2(a,b,c,d)\), \({\mathbb{R}}P^2(a,b,c)\), \(K^2(a,b)\), or a graph manifold, where \(K^2\) is a Klein bottle (but this case can be ruled out for homological reasons). The author finds all middle/middle twisted torus knots and shows the main theorem of the paper, which says that all the middle/middle twisted torus knots possibly except a few admit a Dehn surgery producing non-Seifert-fibered graph manifolds.