Pleaese contact me for further information:

Email address: iain@ms.unimelb.edu.au

Selected publications, experience, supervision etc: to be updated as desired

Holistic sphere eversion: the `heart of the matter'

Animations related to Diophantine approximation by Gaussians, and the Diamond lattice

Artin Conjectures: Links with geometry, and an Elliptic Curve

pdf file of talk recently given at Monash University, Victorian Algebra Conference, October 2014

A Butterfly Effect: Crystallographic Phyllotaxy. Describing srs and the gyroid surface

pdf file of talk recently given at Okinawan Institute for technology, June 2014. Contains embedded movies (requires Adobe Reader/Acrobat to be viewed). The structure of the srs/Sunada K4/Laves network, a construction of the piece-wise linear minimal gyroid surface, deformations to the diamond lattice, and relationships of channel structures in symmetric networks to Diophantine approximation of complex numbers by Gaussian rationals. A slightly different version of the talk given at Palmerston North (194 megabytes)

THE ARAKARUM: From Plimpton 322 and Pythagorus, to Elliptic Curve Cryptography

pdf file of talk recently given at University of Melbourne, Victorian Algebra Conference, November 2013

A Butterfly Effect: Crystallographic Phyllotaxy

pdf file of talk recently given at Victoria University of Wellington, and at Massey University, Palmerston North. Contains embedded movies (requires Adobe Reader/Acrobat to be viewed). The structure of the srs/Sunada K4/Laves network, a construction of the piece-wise linear minimal gyroid surface, deformations to the diamond lattice, and relationships of channel structures in symmetric networks to Diophantine approximation of complex numbers by Gaussian rationals. (202 megabytes)

Geometry and Markoff's spectrum for Q(i), I

Trans. Amer. Math. Soc. 365 (2013), 6065-6102 MSC (2010): Primary 57M50, 20H10, 53C22, 11J06

On the genus of infinite groups,

Math. Res. Lett. 19 (2012), 601-612.

Geometry of Numbers: Structure of Diamond

pdf file of 2012 talk given at Okinawan Institute of Science and Technology, 35meg

The Star of Ishtar: From Plimpton 322 and Pythagorus, to Elliptic Curve Cryptography

pdf file of talk recently given at Victoria University of Wellington, January 2012

The (AB)C and A(BC) of Poincare's homology sphere

preprint 2010

Cusped hyperbolic 3-manifolds: canonically CAT(0) with CAT(0) spines,

preprint

All finitely presentable groups from link complements and Kleinian groups,

preprint 2010

The Holiverse: holistic eversion of the 2-sphere,

preprint (submitted version 2010)

Suspensions of alternating knots,

to appear, under revision, J. Knot Theory and its Ramifications

The geometry of oriented cubes,

invited for submission.

Localising Dehn's lemma and the loop theorem in 3-manifolds,

Math. Proc. Cambrdge Phil. Soc. 137 no. 6 (2004), 281 -- 292. MR2092060 (2005f:57022).

On Archimedean link complements,

J. Knot Theory and its Ramifications, Special Issue: Knots 2000 Korea (Volume 3), 11 no. 6 (2002), 833 -- 868. MR1936238 (2004b:57008).

Combinatorial Dehn surgery on cubed and Haken 3-manifolds,

Geometry and Topology 2 (1999), 1 -- 21. MR1734399 (2000j:57033).

Polyhedral metrics and 3-manifolds which are virtual bundles,

Bull. London Math. Soc. 31 (1999), 90 -- 96. MR1651060 (99h:57024).

Dehn surgery on the figure 8 knot: immersed surfaces,

Proc. Amer. Math. Soc. 127 (1999), no. 8, 2437 -- 2442. MR1485454 (99j:57012).

Surfaces in the figure-8 knot complement,

J. Knot Theory and its Ramifications 7 (1998), 1005 -- 1025. MR1671559 (2000k:57008).

Geodesic surfaces in knot complements,

J. Experimental Math. 6 (1997) 137--150. MR1474574 (98h:57011).

Immersed surfaces in cubed manifolds,

Asian J. Math. 1 (1997), 85 -- 95. MR1480991 (98k:57018).

Cabling the Vassiliev invariants,

J. Knot Theory and its Ramifications 6 (1997), 327 -- 358. MR1457192 (98k:57011a).

Surfaces in 3-manifolds: group actions on surface bundles,

Kodai Math. J. 17 (1994), 549 -- 559. MR1296926 (96a:57037).

Incompressible surfaces and the topology of 3-manifolds,

special issue of the Journal of the Australian Math. Soc., 55 (1993), 1-22. MR1231691 (94h:57024).

Cusp structures of alternating links,

Inventiones Math. 109 (1992), 473-494. MR1176199 (93h:57007).

Canonical surgery on alternating link diagrams,

in: Knots 90, (Osaka, 1990), ed. A. Kawauchi, (de Gruyter, Berlin-New York, 1992), pp. 543--558. MR1177446 (93h:57006).

Combinatorial cubings, cusps, and the dodecahedral knots,

in: Low Dimensional Topology, Ohio 1990, ed. L. Siebenmann and W. Neumann (de Gruyter, Berlin-New York, 1992), pp. 17-26. MR1184399 (93i:57016).

An introduction to polyhedral metrics of non-positive curvature on 3-manifolds,

in: Geometry of Low-Dimensional Manifolds: 2, ed. S.K. Donaldson and C.B. Thomas, Cambridge Univ. Press, Cambridge (1990) pp. 127 - 162. MR1171913 (93e:57018).

Jones polynomials and 3-manifolds,

in: Miniconference on Geometry and Physics, ed. M.N. Barber and M.K. Murray, Centre for Mathematical Analysis, Canberra (1989): pp. 18 - 49. MR 91a:57002; MR 90m:57010; Zbl:678.0018. MR1027860 (91a:57002).

Heaven and Hell,

in: Proceedings of the Sixth International Colloquium on Differential Geometry, ed. L.A. Cordero, Universidade de Santiago de Compostela (1989) pp. 5 -- 24 . MR 91e:57024. MR1040833 (91e:57024).

On certain ribbon disc pairs,

Trans. American Math. Soc. 306 (1988), 529 -- 551. MR 89f:57004.

Isotoping and twisting knots and links,

PhD Thesis Berkeley

Fibred knots and involutions on homotopy spheres,

Contemp. Math. 34 (1985), 1 - 75. MR 86h:570014.

MSc Thesis Melbourne.

Ross Street,

Higher categories, strings, cubes and simplex equations

Appl. Categ. Structures 3 (1995), no. 1, 29–77.

Robion C. Kirby,

The Topology of 4-Manifolds,

Lecture Notes in Mathematics 1374.

Preface and introduction.

`Dehn's lemma and handle decompositions of some $4$-manifolds'.

Pacific J. Math. 86 (1980), no. 2, 565--569.

MR0590570 (81m:57011): ` ... Next a quick proof of an unpublished result of I. Aitchison is given: Any smooth $S^1\times S^2$ in $S^4$ bounds a topological $B^2\times S^2$. ... '

On Aitchison's construction by isotopy.

Trans. Amer. Math. Soc. 305 (1988), no. 2, 641--652.

MR0924773 (89d:57030): `The author studies a construction of I. Aitchison ("construction by isotopy") which produces fibered "$n$-knots" ($n$-dimensional spheres imbedded in $(n+2)$-space) which are "double slice" (i.e. can be represented as an intersection of a hyperplane in $(n+3)$-space with a trivial $(n+1)$-knot). The motivating conjecture (of Aitchison) is that every double-slice fibered knot can be produced by this construction. ...'

On Aitchison's construction by isotopy. II

Trans. Amer. Math. Soc. 317 (1990), no. 2, 813--823.

MR0987168 (90g:57021): `In his Ph.D. dissertation I. Aitchison introduced a method for constructing double slice fibered knots called ``construction by isotopy'' (CBI) and conjectured that every such knot could be constructed this way. The present paper makes several contributions. ... '

Holistic sphere eversion: the `heart of the matter'

Animations showing part of an eversion of the 2-sphere in Euclidean 3-space. George Francis has dubbed this the `Holiverse', since it provides the first conceptually holistic version of sphere eversion since Smale's original proof 55 years ago.

Animations related to Diophantine approximation by Gaussians, and the Diamond lattice

The discrete parts of the Markoff and Lagrange spectra coincide. In the real case, this spectrum corresponds to real numbers poorly approximable by rationals via their continued-fraction expansion. This can be understood in terms of geodesics on a hyperbolic once-punctured torus, and via paths throughout the planar honeycomb lattice. These animations illustrate the analogue for approximating complex numbers via Gaussian rationals, related to geodesics in the hyperbolic Borromean ring complement, and to paths through the diamond lattice

A Butterfly Effect: Crystallographic Phyllotaxy

pdf file of talk recently given at Victoria University of Wellington, and at Massey University, Palmerston North. Contains embedded movies (requires Adobe Reader/Acrobat to be viewed).

This combines several pieces of work: ** The structure of the srs/Sunada K4/Laves network; ** Two constructions of the piece-wise linear minimal gyroid surface via 96 copies of a Euclidean isosceles triangle; ** Deformations between the K4 and the diamond lattices; ** Relationships of channel structures in symmetric networks to Diophantine approximation of complex numbers by Gaussian rationals. (202 megabytes)

On the genus of infinite groups,

Math. Res. Lett. 19 (2012), 601-612.

Abstract: `We associate to each finite presentation of a group G a compact CW- complex 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 compact orientable 3-manifold.' Quinn showed that every finitely-presentable group arises as the fundamental group of a (singular) non-orientable 3-manifold-with-boundary: each singular point has a neighborhood which is a cone on a projective plane, and boundary is allowed. This paper proves an orientable analogue, where singular points can be cones on either a torus, or on surfaces of arbitrary genus, and defines a concept of genus for finitely presentable groups such that a group has genus 0 if and only if it is a 3-manifold fundamental group.

In unpublished work, this can be improved to realization by singular 3-manifolds with empty boundary, and leads to an invariant of 3-manifolds detecting failure of embedding in Euclidean 4-space.'

Geometry of Numbers: Structure of Diamond

pdf file of 2012 talk recently given at Okinawan Institute of Science and Technology, 35meg, on work with Abe described above.

The Star of Ishtar: From Plimpton 322 and Pythagorus, to Elliptic Curve Cryptography

pdf file of talk recently given at Victoria University of Wellington, January 2012.

This collects together a number of concepts of historical significance, and has its origins in: ** The reason that SL(2,Z) actions on the hyperbolic modular diagram should arise naturally in a proof (such as Wiles') of Fermat's Last Theorem. ** How complex numbers and Gaussian integers naturally lead to a parametrization of all Pythagorean triples. ** How Thales Theorem and rational points on a circle essentially embodies the Cayley transform from the upper-half-space to the unit-disc models for hyperbolic geometry. ** Klein's construction of continued fractions in terms of Euclidean-planar straight lines and Z+Z lattice points can be naturally interpreted in terms of this construction ** That the content of Old Babylonian Plimpton 332 can be naturally understood in terms of triangle and trapezoid subdivision for inheritance problems, and that the Old Babylonian predilection for inversion suggests that the difference of two squares is a more appropriate underpinning of Plimpton 322. ** The Old Babylonian substitution x -> (1-x)/(1+x) manifests in Pythagorean triples, and in coordinatizing the Cayley transform as the rotation of an octahedron. ** This transformation yields a natural substitution for the integral definition of trigonometric functions. ** Iteration yields Fagnano's substitution yielding the elliptic integral describing the rectification of the lemniscate, recognized by Euler. ** Substitution applied to Hartshorne's `non-Euclidean Pythagorean equation', applicable to rational solutions for the multiplicative distance function as suggested by Lenstra in the context of Euler's Diophantine cuboid problem, and which gives rational points on K3 surfaces, yields exactly the Edwards form for elliptic curves, recently introduced, and which gives much simpler form for the group structure composition equations.

Geometry and Markoff's spectrum for Q(i), I

to appear, 2013, Transactions American Mathematical Society.

Abstract. We develop a study of the relationship between geometry of geodesics and Markoff's spectrum for Q(i). There exists a particular immersed totally geodesic twice punctured torus in the Borromean rings complement, which is a double cover of the once punctured torus having Fricke coordinates ??(2 sqrt(2), 2 sqrt(2), 4). The set of the simple closed geodesics on this once punctured torus is decomposed into two subsets. The discrete part of Markoff's spectrum for Q(i) (except for one element) is given by the maximal Euclidean height of the lifts of the simple closed geodesics composing one of the subsets.

The (AB)C and A(BC) of Poincare's homology sphere

preprint 2010

Abstract: There is considerable recent interest in such as (a) how a linear or cyclic string of symbols, for example in DNA protein encoding, might give rise to 3-dimensional structure, (b) whether the space we live in is Poincare's dodecahedral space, which can be obtained from plumbing on the E8 graph, in turn related to icosians, (c) how the ubiquitous E8 arises in unexpected places, and (d) the origin of structure of the sporadic simple groups, such as the Mathieu groups M12, M24, and the Monster. We introduce the concept of a transcription complex: given a symbolic word of length n, and an involution-permutation in Sn, we give three canonical constructions, yielding some familiar symmetric 2- and 3-dimensional cell complexes. We illustrate using the two transpositions abc -> acb and abc -> bac in the simplest symmetric group S3, yielding Poincare's homology sphere: the general construction, and other (equally!) interesting examples will be described in more detail in forthcoming papers.

Cusped hyperbolic 3-manifolds: canonically CAT(0) with CAT(0) spines,

preprint

We prove that every non-compact hyperbolic 3-manifold M of finite volume admits a natural complete piecewise-Euclidean singular metric of non-positive curvature, with a natural deformation retraction onto a CAT(0) 2-complex K(M, H), corresponding to each choice H = {Hi} of disjoint horotori Hi, one for each cusp.

All finitely presentable groups from link complements and Kleinian groups,

preprint 2010

Abstract. Klein defined geometry in terms of invariance under group actions; here we give a discrete converse of this, interpreting all finitely presentable groups in terms of the hyperbolic geometry of 3-manifolds (whose fundamental groups are, appropriately, Kleinian groups). For G' a Kleinian group of isometries of hyperbolic 3-space H3, with MG' = H3/G' a non-compact N-cusped orientable 3-manifold of finite volume, let PG' be its dense set of parabolic fixed points. Let M'G' := H3+PG' /G' be the 3-complex obtained by compactifying each cusp of MG' with an additional point. This is the 3-dimensional analogue of the standard compactifcation of cusps of hyperbolic Riemann surfaces. We prove that every finitely presentable group G is of the form G = \pi_1(M'G' ), in infinitely many ways: thus every finitely presentable group arises as the fundamental group of an orientable 3-complex M' -- denoted as a `link-singular' 3-manifold -- obtained from a hyperbolic link complement by coning each boundary torus of its exterior to a distinct point. We define the closed-link-genus, clg(G), of any finitely presentable group G, which completely characterizes fundamental groups of closed orientable 3-manifolds: clg(G) = 0 if and only if G is the fundamental group of a closed orientable 3-manifold. Moreover clg(G) gives an upper bound for the concept genus(G) of genus defined earlier by Aitchison and Reeves, and in turn is bounded by the minimal number of relations among all finite presentations of G. Our results place some aspects of the study of finitely presentable groups more centrally within both classical and modern 3-manifold topology: accordingly, proofs given are expressed in these terms, although some can be seen naturally in 4-manifold topology.

The Holiverse: holistic eversion of the 2-sphere,

preprint (submitted version 2010)

A succinct and conceptual proof of eversion of the 2-sphere in Euclidean 3-space. George Francis has dubbed this the `Holiverse', since it provides the first conceptually holistic version of sphere eversion since Smale's original proof 55 years ago.

Suspensions of alternating knots,

to appear, under revision, J. Knot Theory and its Ramifications

The decomposition of prime, non-splittable alternating link complements into cubes, yielding a CAT(0) metric, generalizes to analogous alternating virtual links. Abstract: We consider complements of links in the suspension of an orientable surface S of genus g > 0, and show that their fundamental groups naturally admit presentations which are analogues of Dehn presentations of classical link complements. For certain links, we show that they admit canonical cubings of non-positive curvature, and that certain groups admitting Dehn-like presentations contain closed surface subgroups.

The geometry of oriented cubes,

invited for submission.

We obtain a geometric interpretation of Street's simplicial orientals, by describing an oriental structure on the n-cube. This corresponds to describing nested sequences of embedded disks in the boundary of the n-cube, obtained consecutively as ordered deformations keeping the boundaries fixed. Canonical forms for the order are obtained, leading to definitions of cocycle conditions for application to non-abelian cohomology in a category. Street has defined an n-category structure on the k-simplex, leading to his notion of orientals. This was motivated by an approach to non-abelian cohomology, arising from quantum field theory, but also seems to have intriguing connections with homotopy coherence, loop space desuspension and the cobar construction, as well as the realisation of homotopy types. Intimately related to Street's structure on simplices is an analogous one on cubes, from which the simplicial results can be derived. This paper takes up Roberts' challenge that "no amount of staring at the low dimensional cocycle conditions would reveal the pattern for higher dimensions". The result is surprisingly simple, natural and beautiful. We define and describe this finer structure of oriented cubes, from a geometric point of view. This makes obvious the relevance to homotopy theory. This paper is self-contained, presupposing minimal familiarity with either category theory or homotopy theory. Categorical aspects and applications will be discussed in a subsequent paper.

Localising Dehn's lemma and the loop theorem in 3-manifolds,

Math. Proc. Cambrdge Phil. Soc. 137 no. 6 (2004), 281 -- 292.

MR2092060 (2005f:57022): `A new proof of the Dehn lemma and the loop theorem is given. The classical proof of Papakyriakopoulos uses towers of coverings, whereas the basic tools of the present proof are normal surfaces and the concepts of simple hierarchies and of boundary patterns of hierarchies: ``It seems reasonable to call the procedure `localizing' since we show that cutting up a singular compressing disk into pieces, using the hierarchy, means that we only have to solve Dehn's lemma and the loop theorem locally, i.e. in regions which are (possibly punctured) handlebodies with simple closed curves as boundary pattern. In this case, the procedure is to cut the handlebody further up into a (possibly punctured) 3-cell and the disk into subdisks.'' '

On Archimedean link complements,

J. Knot Theory and its Ramifications, Special Issue: Knots 2000 Korea (Volume 3), 11 no. 6 (2002), 833 -- 868.

MR1936238 (2004b:57008): `The authors continue the work started in [I. R. Aitchison, E. Lumsden and J. H. Rubinstein, Invent. Math. 109 (1992), no. 3, 473--494; MR1176199 (93h:57007)] on so-called balanced alternating links. The complements of such links admit decompositions into a pair of ideal hyperbolic polyhedra. In the paper under review, the authors define a slightly narrower class of links (which they call ``completely realizable'') and classify them completely.'

Combinatorial Dehn surgery on cubed and Haken 3-manifolds,

Geometry and Topology 2 (1999), 1 -- 21.

MR1734399 (2000j:57033): `Summary: 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 characterization of hierarchies is described. A new proof is given of the topological rigidity theorem of Hass and Scott for 3-manifolds containing immersed incompressible surfaces, as found in cubings of non-positive curvature.'

Polyhedral metrics and 3-manifolds which are virtual bundles,

Bull. London Math. Soc. 31 (1999), 90 -- 96.

MR1651060 (99h:57024): `The authors generalise an old construction due to Thurston which yields closed orientable 3-manifolds which are finitely covered by bundles over the circle. They give several simple constructions of large classes of examples of such manifolds.'

Dehn surgery on the figure 8 knot: immersed surfaces,

Proc. Amer. Math. Soc. 127 (1999), no. 8, 2437 -- 2442.

MR1485454 (99j:57012): `Let $M_{(p,q)}$ be the manifold obtained by $(p, q)$-Dehn surgery on the figure 8 knot, i.e., the manifold obtained by gluing a solid torus to the exterior $M_8$ so that the meridian of the solid torus represents $p\mu + q\lambda$ on the boundary of $M_8$. W. P. Thurston [``Geometry and topology of $3$-manifolds'', lecture notes, Princeton Univ., Princeton, NJ, 1977/78] showed that all but finitely many surgeries on the figure 8 knot yield non-Haken hyperbolic $3$-manifolds; in particular, these manifolds contain no closed incompressible surfaces. On the other hand, it is conjectured that every closed hyperbolic $3$-manifold contains an immersed incompressible surface. Several authors gave some congruence equations for $(p, q)$ such that $M_{(p, q)}$ has a finite cover with positive first Betti number; hence it contains an immersed incompressible surface, and these results show that about $70\%$ of surgeries on the figure 8 knot yield manifolds containing immersed incompressible surfaces. In the paper under review, the authors prove that for every $k$, $M_{(2k, q)}$ contains an immersed incompressible surface. This result, together with previous results, shows that essentially $80\%$ of surgeries on the figure 8 knot yield manifolds containing immersed incompressible surfaces. To prove the result the authors consider the $5$-fold irregular dihedral cover $M^*_8$ of $M_8$, which is the exterior of the link $8^3_4$ [G. Burde, Canad. J. Math. 23 (1971), 84--89; MR0281189 (43 #6908)]. It is known [I. R. Aitchison, E. Lumsden and J. H. Rubinstein, Invent. Math. 109 (1992), no. 3, 473--494; MR1176199 (93h:57007)] that $M^*_8$ contains an immersed incompressible surface $S$ which is still incompressible after any $(p, q)$-surgery except for at most $12$. The authors then prove that $M_{(p,q)}$ contains an immersed incompressible surface of genus greater than $1$ as the image of $S$ by the covering projection $M^*_{(p,q)} \to M_{(p,q)}$ when $p$ is even, except for $(0, 1), (\pm 2, 1)$ and $(\pm 4, 1)$ surgeries. (It is well known that $M_{(0,1)}, M_{(\pm 4,1)}$ contain embedded incompressible tori and $M_{(\pm 2,1)}$ contains an immersed incompressible torus.) In the case where $p = 6k$, $q \ne 0$, the authors show the stronger result that $M_{(p, q)}$ is finitely covered by a Haken manifold.'

Surfaces in the figure-8 knot complement,

J. Knot Theory and its Ramifications 7 (1998), 1005 -- 1025.

MR1671559 (2000k:57008): `Summary: ``We examine various closed normal surfaces immersed in the triangulated figure-8 knot complement. We give some conditions for surfaces to be regular (without branch points) and a sufficient condition for compressibility. We end with a criterion for incompressibility of normal surfaces in cubed 3-manifolds.'' '

Geodesic surfaces in knot complements,

J. Experimental Math. 6 (1997) 137--150.

MR1474574 (98h:57011): `There is a good deal of interest in understanding the closed $\pi_1$-injective surfaces which are immersed or embedded in hyperbolic 3-manifolds. In particular, the simplest case is that of a surface which is totally geodesic in the hyperbolic metric. There are nonexistence results due to Menasco and Reid in the embedded case and arguments coming from work of Maclachlan and Reid give positive and negative results for immersed surfaces in the arithmetic case. However, it follows from a theorem of Reid that the figure-8 knot is the only arithmetic knot, so different techniques are required in general. This paper exhibits two hyperbolic knots and a nonanalytic proof that the complement of each of these knots contains a closed immersed totally geodesic surface. The authors go on to examine various aspects of the geometry and algebraic properties of the knot complements.'

Immersed surfaces in cubed manifolds,

Asian J. Math. 1 (1997), 85 -- 95.

MR1480991 (98k:57018): `The main thrust of this paper is to prove that certain 3-manifolds are virtually Haken. The main tool is the theory of cubed manifolds. A cubed $n$-manifold $M^n$ is a manifold that admits a decomposition into $n$-cubes together with the Euclidean length-space metric. Every closed PL $n$-manifold $M^n$ admits a decomposition into $n$-cubes by further subdivision of each simplex. A cubed manifold is said to be non-positively curved in the sense of Gromov if the link of each cell has no closed geodesic in the induced spherical metric of length less than $2\pi$. Every cubed manifold $M$ contains a canonical immersed hypersurface $S$ which is obtained as the union of all the hypercubes within each cube of $M$ that are parallel to and equidistant from all pairs of parallel opposite faces. The main result of the paper is that if $M$ is an orientable, non-positively curved, cubed $n$-manifold such that all the codimension two faces of the cubing have even degree, then there is a finite covering space of $M$ to which the canonical hypersurface $S$ of $M$ lifts to an embedding. This result is applied to prove that an orientable, non-positively curved, cubed $3$-manifold such that all the edges of the cubing have even degree is virtually Haken. In the proof of the main result, the group $O_n$ of orientation-preserving symmetries of the $n$-cube is not described correctly. The group $O_n$ is an extension of an elementary 2-group of rank $n-1$ by the symmetric group $S_n$ on $n$ elements which splits only in odd dimensions. The 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.'

Cabling the Vassiliev invariants,

J. Knot Theory and its Ramifications 6 (1997), 327 -- 358.

MR1457192 (98k:57011a): `In [P. M. Melvin and H. R. Morton, Comm. Math. Phys. 169 (1995), no. 3, 501--520; MR1328734 (96g:57012)] Melvin and I put forward a conjecture about a formula for retrieving the Alexander polynomial of a knot from the collection of all its ${\rm SU}(2)_q$ invariants, which we termed its ``coloured Jones function''. This Melvin-Morton conjecture was subsequently proved and generalised by D. Bar-Natan and S. Garoufalidis [Invent. Math. 125 (1996), no. 1, 103--133; MR1389962 (97i:57004)]. Our original thoughts had been prompted by the simple behaviour of the Alexander polynomial of the $n$-fold cable of a knot, and the knowledge of how Vassiliev invariants of type $m$ would behave, at the level of weight systems, under such cabling. Our attempts to establish a proof were held up on two counts, one in handling the transition from framed to unframed invariants, and the other in envisaging a passage from a weight system argument to an exact knot invariant relation. Bar-Natan and Garoufalidis dealt successfully with both these problems, but, rather to my surprise, their argument at the weight system level did not exploit the cabling route in identifying the Alexander polynomial. The first of the papers reviewed here examines the action of cabling operations on weight systems, and identifies the eigenvalues and eigenvectors. The authors show that $n$-fold cabling, operating on chord diagrams of degree $m$, has eigenvalues $n^k$ with $k\leq m$, and they identify the eigenspace where $k=m$, or rather the dual weight system, with the ubiquitous immanent weight system arising from the Alexander polynomial. In the second paper Kricker uses weight system information to look at a quantum invariant based on a representation $\lambda$ of a classical Lie algebra or superalgebra and to consider its dependence on the number of strands when it is applied to a cable with $n$ strands about a knot $K$. Having found the terms of highest degree in the parameter $n$ for each coefficient of the power series invariant, he shows how to identify these with the ratio of products of Alexander polynomials of $K$ evaluated at a range of different powers of the variable $t$, depending on the data for $\lambda$. The proof depends on the detailed analysis of the highest degree terms in the cabling action on the weight systems, and uses the work of the first paper to relate this to the Alexander polynomials. The results certainly underline the expectation of a widespread occurrence of the Alexander polynomial among families of invariants. One minor criticism is that Fox's basic cabling formula for the Alexander polynomial, which shows very clearly the involvement of the highest degree eigenvalues in the related Vassiliev invariants, is not given any initial prominence. At its simplest, an $f$-fold cable of a knot $K$ has Alexander polynomial $\Delta_K(t^n)$, so that the coefficient of $h^m$ in $\Delta_K(e^h)$ is multiplied by $n^m$ in the cable, showing that this coefficient gives a Vassiliev invariant of degree $m$ in the eigenspace with highest possible eigenvalue under cabling.'

Surfaces in 3-manifolds: group actions on surface bundles,

Kodai Math. J. 17 (1994), 549 -- 559.

MR1296926 (96a:57037): `A compact but very informative survey of the present-day state and key problems of 3-dimensional topology reported at the Workshop on Geometry and Topology (Hanoi, 1993) is presented. Special attention is paid to surfaces in 3-manifolds, Haken, atoroidal and hyperbolic 3-manifolds. A wide bibliography is added. The author also presents and sketches proofs of two new results on finite group actions on surface bundles over the circle: (1) Given a finite group $G$, there exist finitely many hyperbolic 3-manifolds $M$, each fibering over the circle, on which $G$ acts freely by isometries. (2) ``Most'' genus 2 hyperbolic bundles have a unique fibration of genus 2. The quotient of any such bundle under a regular covering projection is Haken. '

Incompressible surfaces and the topology of 3-manifolds,

special issue of the Journal of the Australian Math. Soc., 55 (1993), 1-22.

Again, a survey paper, with more statements of claimed results than proofs:

MR1231691 (94h:57024): `This is a survey article, which describes some of the ways in which incompressible surfaces are used in the study of 3-manifolds. The emphasis is on the case when these surfaces are immersed. A number of related conjectures have been made along the lines of the following: ``If $M$ is a closed orientable irreducible 3-manifold with infinite fundamental group, then $M$ has an immersed closed orientable incompressible surface.'' One long-range goal is to shed enough light on the nature of non-Haken 3-manifolds (which have no embedded incompressible surfaces) to permit their topological classification and/or establish that they all admit Riemannian metrics of constant sectional curvature $-1$, extending the ``classical'' results for Haken 3-manifolds. The authors have studied (singular) polyhedral metrics on 3-manifolds (e.g., the induced metric on a space obtained from Euclidean polyhedra by identifying faces isometrically), with the additional condition that the singularities correspond to concentrated nonpositive curvature (e.g., dihedral angles around each edge sum to at least $2\pi$). If all the polyhedra are cubes, it is easy to find immersed incompressible surfaces, by joining together square cross-sections of the cubes. Furthermore, the lifts of these immersions to the universal cover of the 3-manifold have the same nice intersection properties (4-plane, 1-line) as least-area surfaces. It then follows [J. Hass and P. Scott, Topology 31 (1992), no. 3, 493--517; MR1174254 (94g:57021)] that such a manifold is determined by its fundamental group. The authors establish the existence of a cubing of nonpositive curvature for all complements of (nonsplittable, prime) alternating links. By an unpublished theorem of Thurston, an immersed incompressible surface in such a link complement will remain incompressible after all but a finite number of Dehn surgeries (if there are no accidental parabolics). Since there are many known links where virtually all results of Dehn surgery are non-Haken manifolds, we end up with infinitely many non-Haken manifolds for which the conjecture mentioned in the first paragraph holds. The related results of many other mathematicians are mentioned in the course of developing this basic story. In summary, polyhedral metrics of nonpositive curvature, and especially cubings, provide a fertile meeting ground on which combinatorial and geometric techniques can combine in the pursuit of topological goals.'

Cusp structures of alternating links,

Inventiones Math. 109 (1992), 473-494.

MR1176199 (93h:57007). `The complement of an alternating link in $S^3$ can be decomposed as the union of two ideal polyhedra; the identification of the polyhedral faces is completely determined by the regular projection of the knot. This observation was first made by Thurston in his 1978 lecture notes, in which it was used to describe the complement of the figure-8 knot and to analyze the geometric properties of that space, as well as the spaces obtained by surgery on the figure-8. This decomposition has since been exploited by a number of authors, most notably in work of W. Menasco [Topology 23 (1984), no. 1, 37--44; MR0721450 (86b:57004)], where it was proved that prime alternating knots contain no nonperipheral incompressible tori. (Hence, with the exception of $(2,n)$ torus knots, such knots are hyperbolic.) The paper under review examines a special class of alternating knots and links, those that have reduced projections that are nicely balanced. A knot projection is called balanced if each vertex of the projection is the vertex of exactly one bigon in the corresponding polygonal decomposition of $S^2$, and it is called nicely balanced if there are no triangular regions in that decomposition. (The authors include the description of a number of constructions of such links.) For such links the authors are able to extract a number of geometric properties of the link complement, and of its peripheral tori, or cusps. One observation in the paper is that the complements of such links contain immersed, $\pi_1$-injective surfaces satisfying the 4-plane, 1-line condition of J. Hass and P. Scott [Topology 31 (1992), no. 3, 493--517]. A deeper observation, based on a careful analysis of the cusp structure, states that for most surgeries on these links, the surface remains $\pi_1$-injective. It follows from the results of Hass and Scott that the resulting closed 3-manifolds are determined by their fundamental groups.'

Canonical surgery on alternating link diagrams,

in: Knots 90, (Osaka, 1990), ed. A. Kawauchi, (de Gruyter, Berlin-New York, 1992), pp. 543--558.

MR1177446 (93h:57006): `The authors explain how to associate a closed orientable $3$-manifold $M_G$ to any planar graph $G$ in a canonical way. The manifold is built from two identical polyhedra by gluing pairs of faces together. Given an irreducible alternating knot diagram of a nonsplit link $L$ there is a canonical chessboard shading of the plane in which the unbounded region is coloured white. This determines two dual graphs $G$ and $G^*$, which have their vertices in the black and white regions, respectively. $L$ can be reconstructed from either graph. The shading is used to specify two combinatorially equivalent polyhedra which are glued together to form a noncompact orientable $3$-manifold canonically homeomorphic to $S^3-L$. Performing surgery on $L$ gives the closed manifold $M_G$. The framing curve for this surgery is determined by the diagram. The solution of some of the Tait conjectures by way of the Jones polynomial means that the crossing number and writhe of an irreducible alternating diagram of a knot are invariant. The authors apply this result to show that when $L$ is a knot the surgery curve, and thus $M_G$, depends only on $L$ and not on the diagram chosen.'

Combinatorial cubings, cusps, and the dodecahedral knots,

in: Low Dimensional Topology, Ohio 1990, ed. L. Siebenmann and W. Neumann (de Gruyter, Berlin-New York, 1992), pp. 17-26.

This paper introduces the dodecahedral knots, which prove to have certain exceptional properties relating number theory and hyperbolic geometry, in the context of conjectures of Neumann and Reid.

MR1184399 (93i:57016): `Regular tessellations of 3-dimensional space-forms are studied, yielding constant curvature finite-volume 3-manifolds. New examples are introduced via dodecahedral knots for the tessellation $\{5,3,6\}$. These examples arise as part of a general construction which connects them with previously known examples, using techniques developed for 4-valent graphs in the authors' joint work with E. Lumsden [Invent. Math. 109 (1992), no. 3, 473--494; MR1176199 (93h:57007)]. Results are also given on $\pi_1$-injective surfaces in the complements of the links being studied, which remain $\pi_1$-injective after ``most'' Dehn surgeries. '

An introduction to polyhedral metrics of non-positive curvature on 3-manifolds,

in: Geometry of Low-Dimensional Manifolds: 2, ed. S.K. Donaldson and C.B. Thomas, Cambridge Univ. Press, Cambridge (1990) pp. 127 - 162.

An up-to-date more elegant version of this paper will be forthcoming shortly, including some unpublished results from that time (with proofs also from that period).

MR1171913 (93e:57018): `This survey paper introduces singular Euclidean metrics for 3-manifolds based on Euclidean polyhedra and cubes, and gives a number of unproved results concerning applications of incompressible surfaces in cubed 3-manifolds of non-positive curvature (in the sense of Gromov). Proofs of a number of these results are given in a number of papers written prior to 1992, listed here: publication of some other proofs followed subsequent to 1997 (and some errors remain uncorrected). From Math Reviews: `In this paper, the authors give a very thorough introduction to their ideas on polyhedral metrics of nonpositive curvature on 3-manifolds. Their aim is to give an overview of what is known while sketching their arguments. In this they have succeeded extremely well. The details should appear in later papers. The idea of a polyhedral metric on a manifold is not new. One decomposes a manifold $M$ into polyhedra meeting in faces and gives each polyhedron a Euclidean metric chosen so that metrics agree on common faces. This results in a metric on $M$, obtained from shortest paths joining points, which is Euclidean except possibly on a codimension two subset. If the singularities are all of the correct type one says that one has a polyhedral metric of nonpositive curvature. ``Correct type'' means basically that if one considers the universal cover of $M$ equipped with the naturally induced polyhedral metric, then geodesic rays leaving a point keep diverging as one moves away from the point, so that the picture is similar to that in a non-positively curved Riemannian manifold. The authors consider mainly polyhedral metrics of nonpositive curvature in which all the polyhedra are Euclidean cubes. They call such a metric a cubing. They construct infinite families of closed 3-manifolds with cubings. They also prove a rigidity theorem for such manifolds. If $M$ is a cubed 3-manifold and $N$ is an irreducible 3-manifold which is homotopy equivalent to $M$ then $M$ is homeomorphic to $N$. It is not necessary to assume that $N$ is also cubed. The key ingredient in this rigidity result is a beautiful construction of compact surfaces in a cubed manifold which, although singular, are not very singular. This means that they can apply earlier work of the reviewer and Hass. They consider analogues of cubings for nonclosed 3-manifolds and obtain some new kinds of results. Let $M$ be a knot complement which admits a cubing in this new sense. (They show that many knot and link complements admit the required kind of cubing.) They show that for all except finitely many Dehn fillings of $M$, the closed manifolds $M_{(p,q)}$ obtained by Dehn filling all satisfy the same rigidity result as closed cubed manifolds, though they do not assert that they actually admit cubings. Again they use singular surfaces. This time they show that there are certain singular incompressible surfaces in $M$ which, for all except finitely many Dehn fillings, remain incompressible in $M_{(p,q)}$ and also satisfy the conditions required to apply the rigidity result of the reviewer and Hass.'

Jones polynomials and 3-manifolds,

in: Miniconference on Geometry and Physics, ed. M.N. Barber and M.K. Murray, Centre for Mathematical Analysis, Canberra (1989): pp. 18 - 49. MR 91a:57002; MR 90m:57010; Zbl:678.0018.

MR1027860 (91a:57002): `This paper gives an exposition of the statistical mechanics approach to the Jones polynomial of oriented links [V. F. R. Jones, Ann. of Math. (2) 126 (1987), no. 2, 335--388; MR0908150 (89c:46092)], along with some more recent developments. The author gives a comparison of the Jones polynomial with the classical Alexander polynomial, highlighting the apparent lack of connection of the Jones polynomial with 3-manifold topology. He also outlines some of the standard constructions of 3-manifolds, emphasizing those related to braids and to fibered links, which might yield the hoped-for relations with the new polynomials. '

Heaven and Hell,

in: Proceedings of the Sixth International Colloquium on Differential Geometry, ed. L.A. Cordero, Universidade de Santiago de Compostela (1989) pp. 5 -- 24 .

MR1040833 (91e:57024): `This paper is, at least partly, an attempt to relate some mathematics to the world of art and even to some mystical vision of reality. The ``Heaven and Hell'' of the title is the famous work of M. C. Escher, known also as ``Circle Limit IV''. The authors associate with it a tessellation of the hyperbolic plane by hexagons with colored edges, and consider the group $\Gamma^3$ of symmetries of this tessellation. They define a generic (abstract) polyhedron to be a 3-ball with a 3-valent graph embedded in its boundary. Regions of the boundary complementary to this graph are called faces. Then the authors prove that to every decomposition of a 3-manifold into generic polyhedra is associated a unique conjugacy class of torsion-free, orientation-preserving subgroups of $\Gamma^3$. Several related topics are also discussed, including pseudo-Anosov maps, decompositions of 3-manifolds into Platonic solids, singular geometry of 3-manifolds, and others. For the ``mystic'' reference, see the ``Final remark'' section (numbered 13!).'

On certain ribbon disc pairs,

Trans. American Math. Soc. 306 (1988), 529 -- 551. MR 89f:57004.

Abstract: We prove that for any free group automorphism X* 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 1- and 2-knots. The examples enabling the proof of the main claim of the paper were constructed using the techniques of my PhD thesis.

Isotoping and twisting knots and links,

PhD Thesis Berkeley.

A construction is given for infinitely many distinct fibered ribbon knots in the 3-sphere of genus 2 which are simultaneously slices of an unknotted 2-sphere in the 4-sphere, and slices of a 0-spun figure-8 knot, which have monodromies extending over handle bodies, but which cannot be distinguished by any known Abelian (Alexander/Seifert) invariants at the time. They can be distinguished by invariants of the hyperbolic structure of their complements, namely the stretching factors of their pseudo-Anosov monodromies. The construction generalizes to prove that all simple odd-dimensional knots can be constructed by isotopy, that is, their monodromies arise by an isotopy of a handle-body within a sphere of the same dimension. This latter result was subsequently published by Silver with the erroneous claim that the proof given in this thesis was incorrect. This work was submitted as a substitute for the originally agreed-upon thesis providing a geometric construction for showing that the spin-cobordism group in dimension 4 is the integers. (My advisor Kirby had incorrectly advised earlier that agreeing to incorporate this work as part of a project reformulating the foundations of smooth 4-dimensional manifolds would not compromise my being allowed to submit the work for my thesis).

Fibred knots and involutions on homotopy spheres,

Contemp. Math. 34 (1985), 1 - 75. MR 86h:570014.

MSc Thesis Melbourne. This generalizes Akbulut and Kirby's construction of a handle-decomposition for a single Cappell-Shaneson homotopy 4-sphere, to give handle decompositions for infinitely many CS-4-spheres, and shows that infinitely many CS-4-spheres are diffeomorphic to the standard sphere. It is also shown that Akbulut and Kirby's original claim is not correct: they erroneously identify the double cover of an exotic projective 4-space as being the standard 4-sphere, when in fact the double cover is the result of a Gluck construction on a knotted 2-sphere in the standard sphere, and hence remained (in 1982) as a potential counter-example to the smooth 4-dimensional Poincare Conjecture.

Ross Street,

Higher categories, strings, cubes and simplex equations

Appl. Categ. Structures 3 (1995), no. 1, 29–77.

Robion C. Kirby,

The Topology of 4-Manifolds,

Lecture Notes in Mathematics 1374.

Preface and introduction.

`Dehn's lemma and handle decompositions of some $4$-manifolds'.

Pacific J. Math. 86 (1980), no. 2, 565--569.

MR0590570 (81m:57011): ` ... Next a quick proof of an unpublished result of I. Aitchison is given: Any smooth $S^1\times S^2$ in $S^4$ bounds a topological $B^2\times S^2$. ... '

On Aitchison's construction by isotopy.

Trans. Amer. Math. Soc. 305 (1988), no. 2, 641--652.

MR0924773 (89d:57030): `The author studies a construction of \n I. Aitchison\en ("construction by isotopy") which produces fibered "$n$-knots" ($n$-dimensional spheres imbedded in $(n+2)$-space) which are "double slice" (i.e. can be represented as an intersection of a hyperplane in $(n+3)$-space with a trivial $(n+1)$-knot). The motivating conjecture (of Aitchison) is that every double-slice fibered knot can be produced by this construction. The main results of this paper concern the case of "simple" knots (roughly speaking, knots whose complements are as nearly homotopically equivalent to the circle as possible without being trivial). For $n$ large enough, these knots can be classified by homological/homotopy invariants. When $n$ is odd, the crucial invariant is the Seifert matrix, and it is known how to characterize double-slice and fibered knots via the Seifert matrix. The author shows here how to realize every permissible matrix by "construction by isotopy", thus verifying Aitchison's conjecture for these knots. When $n$ is even and the knot is fibered and "$\bold Z$-torsion free", the classifying invariant is the "$LP$-matrix". The author shows how to characterize double-slice in terms of the $LP$-matrix and then realizes any such $LP$-matrix by a "construction by isotopy". This then verifies Aitchison's conjecture for this class of knots also.'

On Aitchison's construction by isotopy. II

Trans. Amer. Math. Soc. 317 (1990), no. 2, 813--823. MR0987168 (90g:57021): `In his Ph.D. dissertation \n I. Aitchison\en introduced a method for constructing double slice fibered knots called ``construction by isotopy'' (CBI) and conjectured that every such knot could be constructed this way. The present paper makes several contributions. One result is that CBI produces double slice knots with fibered slicing disks and that, for higher-dimensional knots, all such knots are CBI. As a consequence any higher-dimensional double slice fibered knot with group $\bold Z$ is CBI. A special class of double slice knots are the doubled disk knots. It is shown that the double of any fibered disk knot is CBI, but it is also shown that there exist non-doubled disk knots which are CBI, as a consequence of the general result that any twist-spin of a doubled disk knot is CBI. Finally the conjecture of Aitchison is disproved by noting a particular geometric property of CBI knots and invoking a theorem of Cappell and Ruberman to find a (higher-dimensional) fibered double disk knot which is not CBI.'