Spines and surgery descriptions of graph manifolds (Q6053443)

Graph manifolds are compact orientable 3-manifolds obtained by gluing several copies of \(D^2\times S^1\) and \(N^2\times S^1\) together by homeomorphisms of some components of their boundaries (\(D^2\) is the \(2\)-disc and \(N^2\) denotes the 2-disc with two holes). In this paper, geometric presentations of fundamental groups of many orientable graph manifolds with three or four indices (invariants) are studied. A group presentation (with two generators and two relations) is geometric if it corresponds to a spine of the 3-manifold or, equivalently, arises from a genus 2 Heegaard diagram of the 3-manifold. The paper considers graph manifolds obtained by exceptional Dehn surgeries along hyperbolic knots and links, including many Takahashi manifolds and Teragaito manifolds. The homeomorphism types of them are decided by the fundamental groups. Takahashi manifolds are those obtained from the three component magic link, cf. [\textit{M.-o Takahashi}, Tsukuba J. Math. 13, No. 1, 175--189 (1989; Zbl 0677.57002)]. \textit{M. Teragaito} [J. Knot Theory Ramifications 17, No. 9, 1051--1061 (2008; Zbl 1297.57024)] found a family of hyperbolic knots admitting three toroidal Dehn surgeries. Those resulting manifolds are called Teragaito manifolds.