Exceptional Dehn Surgeries on Some Infinite Series of Hyperbolic Knots and Links

We study closed connected orientable 3-manifolds obtained by Dehn surgery along the oriented components of a link, introduced and considered by Motegi and Song (2005) and Ichihara et al. (2008). For such manifolds, we find a finite balanced group presentation of the fundamental group and describe exceptional surgeries. This allows us to construct an infinite family of tunnel number one strongly invertible hyperbolic knots with three parameters, which admit toroidal surgeries and Seifert fibered surgeries. Among the obtained results, we mention that for every integer \(n >5\) $n>5$ there are infinitely many hyperbolic knots in the 3–sphere, whose \((n-2)\) $(n-2)$ and \((n+1)\) $(n+1)$ -surgeries are toroidal, and \((n-1)\) $(n-1)$ and n-surgeries are Seifert fibered.