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- Ian Agol, David Gabai, Iain R. Aitchison, Hyam Rubinstein
- 2002

Given a hyperbolic 3{manifold M containing an embedded closed geodesic, we estimate the volume of a complete hyperbolic metric on the complement of the geodesic in terms of the geometry of M . As a corollary, we show that the smallest volume orientable hyperbolic 3{manifold has volume > :32. AMS Classi cation numbers Primary: 57M50 Secondary: 53C15, 53C22

- CUBED MANIFOLDS, Iain R. Aitchison, Sho Matsumoto, J. Hyam Rubinstein
- 2006

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… (More)

- Iain R. Aitchison, J. Hyam Rubinstein
- Experimental Mathematics
- 1997

- Iain R. Aitchison, David Gabai
- 2004

In this paper, 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 in [?], 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… (More)

- Iain R. Aitchison, Daniel S. Silver, Iain R. Aitchison, DANIEL S. SILTER
- 2009

We prove that for any free group automorphism +* having a specified form there exists an invertible ribbon disc pair (B4,D2) such that the closure of B4 nbd(D2) fibres over the circle with fibre a handlebody and monodromy equal to X*. We apply this to obtain results about ribbon 1and 2-knots. 0. Introduction and preliminaries. This paper is motivated by the… (More)

We associate to each finite presentation of a group G a compact CWcomplex that is a 3-manifold in the complement of a point, and whose fundamental group is isomorphic to G. We use this complex to define a notion of genus for G and give examples, and also define a notion of ‘closed group’. A group has genus 0 if and only if it is the fundamental group of a… (More)

- Haken 3–manifolds, Iain R. Aitchison, J. Hyam Rubinstein
- 1999

A combinatorial condition is obtained for when immersed or embedded incompressible surfaces in compact 3–manifolds with tori boundary components remain incompressible after Dehn surgery. A combinatorial characterisation of hierarchies is described. A new proof is given of the topological rigidity theorem of Hass and Scott for 3–manifolds containing immersed… (More)

It is known that about 70% of surgeries on the figure 8 knot give manifolds which contain immersed incompressible surfaces. We improve this to about 80% by giving a very simple proof that all even surgeries give manifolds containing such a surface. Moreover, we give a quick proof that every (6k, t) surgery is virtually Haken, thereby partially dealing with… (More)

We give a short, simple and conceptual proof, based on spin structures, of sphere eversion: an embedded 2-sphere in R can be turned inside out by regular homotopy. Ingredients of this eversion are seamlessly connected. We also give the mathematical origins of the proof: the Hopf fibration, and the topological structure of real-projective 3-space.

- Daniel S. Silver, Iain R. Aitchi, Iain R. Aitchison, Daniel S. Silver
- 2010

Construction by isotopy is a technique introduced by Iain R. Aitchison for obtaining doubly slice fibered knots in any dimension. We show that if k is any doubly slice fibered (n 2)-knot, n > 5 , such that nx (Sn k) = Z , then k is constructible by isotopy. We also prove that the m-twist-spin of any doubly slice knot is constructible by isotopy.… (More)