Celebratio Mathematica

I. R. Aitchison and J. H. Rubinstein: “Fibered knots and involutions on homotopy spheres,” pp. 1–74 in Four-manifold theory (Durham, NH, 4–10 July 1982). Edited by C. Gordon and R. Kirby. Contemporary Mathematics 35. American Mathematical Society (Providence, RI), 1984. MR 780575 Zbl 0567.57015 incollection

Iain Roderick Aitchison	Related
Robion C. Kirby	Related	Volume
Cameron McAllan Gordon	Related	Volume

I. R. Aitchison and J. H. Rubinstein: “Heaven & hell,” pp. 5–24 in Proceedings of the sixth international colloquium on differential geometry (Santiago de Compostela, Spain, 19–23 September 1988). Edited by L. A. Cordero. Cursos e Congresos da Universidade de Santiago de Compostela 61. Universidad Santiago de Compostela, 1989. MR 1040833 Zbl 0737.57004 incollection

Iain Roderick Aitchison	Related
Luis A. Cordero	Related

I. R. Aitchison and J. H. Rubinstein: “An introduction to polyhedral metrics of nonpositive curvature on 3-manifolds,” pp. 127–161 in Geometry of low-dimensional manifolds (Durham, UK, July 1989), vol. 2: Symplectic manifolds and Jones–Witten theory. Edited by S. K. Donaldson and C. B. Thomas. London Mathematical Society Lecture Note Series 151. Cambridge University Press, 1990. MR 1171913 Zbl 0735.57005 incollection

Iain Roderick Aitchison	Related
Charles B. Thomas	Related
S. K. Donaldson	Related

I. R. Aitchison and J. H. Rubinstein: “Combinatorial cubings, cusps, and the dodecahedral knots,” pp. 17–26 in Topology ’90 (Columbus, OH, February–June 1990). Edited by B. Apanasov, W. D. Neumann, A. W. Reid, and L. Siebenmann. Ohio State University Mathematical Research Institute Publications 1. de Gruyter (Berlin), 1992. MR 1184399 Zbl 0773.57010 incollection

There are finitely many tessellations of 3-dimensional space-forms by regular Platonic solids. Explicit examples of constant curvature finite-volume 3-manifolds arising from these are well-known for all possibilities, except for the tessellation \( \{5,3,6\} \). We introduce the dodecahedral knots \( D_f \) and \( D_s \) in \( \mathbb{S}^3 \) to fill this gap. Techniques used illustrate the results on cusp structures and \( \pi_1 \)-injective surfaces of alternating link complements obtained by Aitchison, Lumsden and Rubinstein [1991].

The Borromean rings and figure-eight knot arise from the tessellation of hyperbolic 3-space by regular ideal octahedra and tetrahedra respectively. We produce exactly four new links in \( \mathbb{S}^3 \), corresponding to the tessellations \( \{4,3,6\} \) and \( \{5,3,6\} \) of \( \mathbb{H}^3 \), and united by a canonical construction from the Platonic solids.

The dodecahedral knot \( D_f \) is the third in an infinite sequence of fibred, alternating knots, the first member of which being the figure-eight. The complements of these new links contain \( \pi_1 \)-injective surfaces, which remain \( \pi_1 \)-injective after ‘most’ Dehn surgeries. The closed 3-manifolds obtained by such surgeries are determined by their fundamental groups, but are not known to be virtually Haken.

Laurent C. Siebenmann	Related
Alan William Reid	Related
Walter D. Neumann	Related	Volume
Boris Nikolaevich Apanasov	Related
Iain Roderick Aitchison	Related

I. R. Aitchison, E. Lumsden, and J. H. Rubinstein: “Cusp structures of alternating links,” Invent. Math. 109 : 1 (1992), pp. 473–494. MR 1176199 Zbl 0810.57010 article

An alternating link \( \mathcal{L}_{\Gamma} \) is canonically associated with every finite, connected, planar graph \( \Gamma \). The natural ideal polyhedral decomposition of the complement of \( \mathcal{L}_{\Gamma} \) is investigated. Natural singular geometric structures exist on \( \mathbb{S}^3 - \mathcal{L}_{\Gamma} \) , with respect to which the geometry of the cusp has a shape reflecting the combinatorics of the underlying link projection. For the class of ‘balanced graphs’, this induces a flat structure on peripheral tori modelled on the tessellation of the plane by equilateral triangles. Examples of links containing immersed, closed \( \pi_1 \)-injective surfaces in their complements are given. These surfaces persist after ‘most’ surgeries on the link, the resulting closed 3-manifolds consequently being determined by their fundamental groups.

E. Lumsden	Related
Iain Roderick Aitchison	Related

I. R. Aitchison and J. H. Rubinstein: “Canonical surgery on alternating link diagrams,” pp. 543–558 in Knots 90 (Osaka, Japan 15–19 August 1990). Edited by A. Kawauchi. de Gruyter (Berlin), 1992. MR 1177446 Zbl 0765.57005 incollection

Akio Kawauchi	Related
Iain Roderick Aitchison	Related

I. R. Aitchison and J. H. Rubinstein: “Incompressible surfaces and the topology of 3-dimensional manifolds,” J. Aust. Math. Soc., Ser. A 55 : 1 (1993), pp. 1–22. MR 1231691 Zbl 0813.57017 article

I. R. Aitchison and J. H. Rubinstein: “Geodesic surfaces in knot complements,” Exp. Math. 6 : 2 (1997), pp. 137–150. MR 1474574 Zbl 0891.57017 article

I. R. Aitchison, S. Matsumoto, and J. H. Rubinstein: “Immersed surfaces in cubed manifolds,” Asian J. Math. 1 : 1 (1997), pp. 85–95. MR 1480991 Zbl 0935.57033 article

We investigate the separability of the canonical surface immersed in cubed manifolds of non-positive curvature, partially answering the hyperbolicity question of cubed 3-manifolds. We show that the canonical surface is separable if we assume that the degrees of all edges are even. Further, the general argument also gives the same result for higher-dimensional manifolds admitting a cubing of non-positive curvature. This main result is extended to some other structures, including flying-saucer and polyhedral decompositions as well as surgeries on certain link complements; these constructions provide examples of non-Haken manifolds that are virtually Haken.

Iain Roderick Aitchison	Related
Saburo Matsumoto	Related

I. R. Aitchison, S. Matsumoto, and J. H. Rubinstein: “Surfaces in the figure-8 knot complement,” J. Knot Theory Ramifications 7 : 8 (1998), pp. 1005–1025. MR 1671559 Zbl 0924.57018 article

Iain Roderick Aitchison	Related
Saburo Matsumoto	Related

I. R. Aitchison and J. H. Rubinstein: “Combinatorial Dehn surgery on cubed and Haken 3-manifolds,” pp. 1–21 in Proceedings of the KirbyFest (Berkeley, CA, 22–26 June 1998). Edited by J. Hass and M. Scharlemann. Geometry & Topology Monographs 2. International Press (Cambridge, MA), 1999. Papers dedicated to Rob Kirby on the occasion of his 60th birthday. MR 1734399 Zbl 0948.57016 ArXiv 9911072 incollection

Robion C. Kirby	Related	Volume
Joel Hass	Related	Volume
Iain Roderick Aitchison	Related
Martin Scharlemann	Related	Volume

I. R. Aitchison, S. Matsumoto, and J. H. Rubinstein: “Dehn surgery on the figure 8 knot: Immersed surfaces,” Proc. Am. Math. Soc. 127 : 8 (1999), pp. 2437–2442. MR 1485454 Zbl 0926.57006 article

Iain Roderick Aitchison	Related
Saburo Matsumoto	Related

I. R. Aitchison and J. H. Rubinstein: “Polyhedral metrics and 3-manifolds which are virtual bundles,” Bull. London Math. Soc. 31 : 1 (1999), pp. 90–96. MR 1651060 Zbl 0930.57015 article

Throughout this paper, all 3-manifolds are closed and orientable. Our aim is to give some new examples of 3-manifolds which are virtual bundles, that is, have finite sheeted covers which are surface bundles over the circle. Thurston has raised the question as to whether all irreducible atoroidal 3-manifolds might have this property. The geometrisation conjecture would imply that all such 3-manifolds have hyperbolic metrics, and the virtual bundle question is then equivalent to showing the existence of geometrically infinite incompressible surfaces (compare [Thurston 1978]).

Not much progress has been made on this problem. Recently, Reid [1995] has given an explicit example using an arithmetic hyperbolic 3-manifold, and Cooper, Long and Reid [1994] have given methods of constructing immersed incompressible surface in surfaces bundles over the circle, which are sometimes geometrically infinite. In [Rubinstein and Wang 1998], examples are given of horizontal immersed incompressible surfaces in graph manifolds which do not lift to embeddings in any finite sheeted covering spaces. Similar examples are given in toroidal manifolds with cubings of non-positive curvatures in [Reid 1995]. So the assumption that the manifold has the atoroidal property is essential in Thurston’s question. Our aim is to give several simple constructions of large classes of examples using different polyhedral metrics of non-positive curvature on 3-manifolds (compare [Aitchinson and Rubinstein 1990]).

We follow closely the construction of Thurston (compare [Sullivan 1981]), where he observes that the right-angled regular hyperbolic dodecahedron can be viewed as a cube with arcs drawn on each face through midpoints of a pair of opposite sides so that no two such arcs share a common vertex. Then the foliation of the cube by planes orthogonal to a diagonal gives an induced foliation of any 3-manifold arising from a torsion-free finite index subgroup of the symmetry group of the tessellation of hyperbolic 3-space by the right-angled dodecahedron. Clearly, any leaf of this foliation is then an immersed geometrically infinite surface. One explicit example is then the 4-fold cyclic branched cover of the Borromean rings, since the Borromean rings has a flat 2-fold branched cover which is obtained by gluing two cubes together. This example is formed by a subgroup of index 4 in the above symmetry group.

Our examples come directly from this observation of Thurston, using different ways of dividing a 3-manifold into simple polyhedra so that there is an overall metric of non-positive curvature. We use some well-known tessellations of Euclidean 3-dimensional space, described in [Coxeter 1937], by truncated octahedra and tetrahedra, plus flying saucers (compare [Aitchinson and Rubinstein 1990]), as well as the easy case of cubes.

In the final section we show that our classes of cubed examples and flying saucers admit geometric decompositions in the sense of Thurston, since they are virtual bundles.

I. R. Aitchison and J. H. Rubinstein: “Localising Dehn’s lemma and the loop theorem in 3-manifolds,” Math. Proc. Camb. Philos. Soc. 137 : 2 (2004), pp. 281–292. MR 2092060 Zbl 1067.57009 article

We give a new proof of Dehn’s lemma and the loop theorem. This is a fundamental tool in the topology of 3-manifolds. Dehn’s lemma was originally formulated by Dehn, where an incorrect proof was given. A proof was finally given by Papakyriakopolous in his famous 1957 paper where the fundamental idea of towers of coverings was introduced. This was later extended to the loop theorem, and the version used most frequently was given by Stallings.

We have shown that hierarchies for Haken 3-manifolds could be understood by a ‘local version’ of Dehn’s lemma and the loop theorem. Developing the idea further enables us here to give a new proof of the classical theorems of Papakyriakopoulos, which do not use towers of coverings. A similar result was obtained by Johannson, with the added assumption that the 3-manifold in question was Haken. Our approach means that no extra hypotheses are necessary. Our method uses the concept of boundary patterns of hierarchies, as developed by Johannson. Marc Lackenby has independently produced a very similar proof in his lecture notes.

Note that the more difficult sphere theorem can then be deduced using Dehn’s lemma and the loop theorem, plus the PL theory of minimal surfaces. Other applications, like the important result that a covering of an irreducible 3-manifold is irreducible, then follow also by PL minimal surface theory.

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