Celebratio Mathematica — Rubinstein

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E. Kang and J. H. Rubinstein: “Ideal triangulations of 3-manifolds, I. Spun normal surface theory,” pp. 235–265 in Proceedings of the Casson Fest: Arkansas and Texas 2003 (Fayetteville, AR, 10–12 April 2003 and Austin, TX, 19–21 May 2003). Edited by C. Gordon and Y. Rieck. Geometry & Topology Monographs 7. Mathematical Sciences Publishers (Berkeley, CA), 2004. Part II was published in Algebr. Geom. Topol. 5 (2005). MR 2172486 Zbl 1085.57016 ArXiv 0410541 incollection

In this paper, we will compute the dimension of the space of spun and ordinary normal surfaces in an ideal triangulation of the interior of a compact 3-manifold with incompressible tori or Klein bottle components. Spun normal surfaces have been described in unpublished work of Thurston. We also define a boundary map from spun normal surface theory to the homology classes of boundary loops of the 3-manifold and prove the boundary map has image of finite index. Spun normal surfaces give a natural way of representing properly embedded and immersed essential surfaces in a 3-manifold with tori and Klein bottle boundary [Kang 1999, 2005]. It has been conjectured that every slope in a simple knot complement can be represented by an immersed essential surface [Baker 1996; Baker and Cooper 2000]. We finish by studying the boundary map for the figure-8 knot space and for the Gieseking manifold, using their natural simplest ideal triangulations. Some potential applications of the boundary map to the study of boundary slopes of immersed essential surfaces are discussed.

Andrew J. Casson	Related
Cameron McAllan Gordon	Related	Volume
Yo'av Rieck	Related
Ensil Kang	Related

E. Kang and J. H. Rubinstein: “Ideal triangulations of 3-manifolds, II: Taut and angle structures,” Algebr. Geom. Topol. 5 (2005), pp. 1505–1533. Part I was published in Proceedings of the Casson Fest (2004). MR 2186107 Zbl 1096.57018 ArXiv 0502437 article

This is the second in a series of papers in which we investigate ideal triangulations of the interiors of compact 3-manifolds with tori or Klein bottle boundaries. Such triangulations have been used with great effect, following the pioneering work of Thurston. Ideal triangulations are the basis of the computer program SNAPPEA of Weeks, and the program SNAP of Coulson, Goodman, Hodgson and Neumann. Casson has also written a program to find hyperbolic structures on such 3-manifolds, by solving Thurston’s hyperbolic gluing equations for ideal triangulations. In this second paper, we study the question of when a taut ideal triangulation of an irreducible atoroidal 3-manifold admits a family of angle structures. We find a combinatorial obstruction, which gives a necessary and sufficient condition for the existence of angle structures for taut triangulations. The hope is that this result can be further developed to give a proof of the existence of ideal triangulations admitting (complete) hyperbolic metrics. Our main result answers a question of Lackenby. We give simple examples of taut ideal triangulations which do not admit an angle structure. Also we show that for ‘layered’ ideal triangulations of once-punctured torus bundles over the circle, that if the manodromy is pseudo Anosov, then the triangulation admits angle structures if and only if there are no edges of degree 2. Layered triangulations are generalizations of Thurston’s famous triangulation of the Figure-8 knot space. Note that existence of an angle structure easily implies that the 3-manifold has a \( \operatorname{CAT}(0) \) or relatively word hyperbolic fundamental group.