Hyperbolic geometry of multiply twisted knots (Q614509)

In the paper under review, the geometric properties of knots and links with diagrams with a high amount of twisting of multiple strands are investigated. This extends the author's work [Algebr. Geom. Topol. 7, 93--108 (2007; Zbl 1135.57005)] where only twists on two strands were considered. For a diagram of a knot or link, one can choose a maximal twist region selection, which is not necessarily unique. The generalized augmented link is obtained by inserting a simple closed loop, called a crossing circle, encircling the strands at each twist region. Clearly, there is a relation between the complements of the original knot or link and the augmented link given by Dehn surgery. The first result shows that if a knot or link \(K\) in the \(3\)-sphere has a diagram with a maximal twist region selection where each twist region contains at least six half-twists, and if the generalized augmented link is hyperbolic, then \(K\) is also hyperbolic. After estimating the lengths of slopes of Dehn surgery, which recovers the complement of \(K\) from that of the augmented link, the proof appeals to the famous six-theorem by \textit{I. Agol} [Geom. Topol. 4, 431--449 (2000; Zbl 0959.57009)] and \textit{M. Lackenby} [Invent. Math. 140, No.~2, 243--282 (2000; Zbl 0947.57016)]. It is interesting that a reflection on the complement of the augmented link, whose fixed point set is a surface, is used effectively to estimate the slope lengths. As a generalization of \textit{D. Futer, E. Kalfagianni} and \textit{J. S. Purcell} [J. Differ. Geom. 78, No.~3, 429--464 (2008; Zbl 1144.57014)], the volume of the generalized augmented link is estimated from below, when it is hyperbolic. From this, the author shows that if a knot or link \(K\) in the \(3\)-sphere has a diagram \(D\) with a maximal twist region selection where each twist region contains at least seven half-twists, and if the augmented link is hyperbolic, then the volume of the complement of \(K\) is at least \(0.64756(\mathrm{tw}(D)-1)\), where \(\mathrm{tw}(D)\) denotes the number of generalized twist regions. Finally, each crossing circle is shown to be isotopic to a geodesic in the hyperbolic structure of \(K\) under a certain condition on the number of half-twists.