Tangled Platonic Polyhedra

Home , Divina proportione, Orbifold notation, Szilassi polyhedron

Toen Castle, V. Robins and S. T. Hyde

Last typeset on February 7, 2010

Abstract A method to ﬁnd all embeddings of a given graph on the genus-one two-dimensional torus is used to construct a variety of entangled Platonic polyhedra graphs. These graph embeddings contain knotted and/or linked subgraphs of arbitrary complexity, the simplest of which include the ‘trefoil’ knot and the ‘Hopf’ and ‘Whitehead’ links. Analyses of knotting and linking, point group symmetries and an energy function are presented for embeddings of the topological tetrahedron, cube and octahedron.

1 Contents

1 Introduction 3 1.1 The Big Picture (surfaces in space) ...... 7

2 Method - generalising Platonic Polyhedra 10 2.1 Universal cover ...... 11 2.2 2-cell embeddings ...... 12 2.3 Tilings of the torus dissolving to polyhedron nets ...... 15 2.4 Tangling ...... 17 2.5 Knots and links ...... 18 2.6 Energy ...... 22 2.7 Symmetry ...... 22

3 Tetrahedron 25 3.1 2-cell embeddings of the tetrahedron ...... 26 3.2 The tetrahedron in the universal covers: <8, 4> and <9, 3> .... 30 3.3 Comparison of tetrahedral isotopes ...... 31

4 Cube 34 4.1 2-cell embeddings of the cube ...... 34 4.2 The cube in its universal cover ...... 35 4.3 Symmetries of tangled cubes ...... 39 4.4 High symmetry cube isotopes ...... 40 4.5 Honeycomb symmetries ...... 42 4.6 Cube isotope knots, links and energies ...... 43

5 Octahedron 49 5.1 2-cell embeddings of the octahedron ...... 49 5.2 The toroidal octahedron in its universal cover ...... 49 5.3 High symmetry toroidal octahedra isotopes ...... 52 5.4 Properties of toroidal octahedra ...... 52

6 Further constructions 60 6.1 Non- 2-cell torus graph embeddings ...... 60 6.2 Duals ...... 61 6.3 Chirality ...... 64

7 Conclusion 64

2 1 Introduction

Polyhedra have long been of interest to mathematicians and philosophers, the fas- cination aided by the spatial, geometric, and combinatorial patterns commensurate with their simple structures. One notable early discussion of regular polyhedra was by Plato in his Timaeus [1]. He identified the ‘classical elements’ of fire, air, earth, water and quintessence or aether with the five regular polyhedra - tetrahedra, octrahedra, cubes, icosahedra and dodecahedra - as universal building blocks in his early ‘theory of everything’. Just as the understanding of the constitutive nature of elements has since grown, so too has the conception of polyhedra expanded. Polyhedra were initially envisaged as solid bodies with planar polygonal faces, such as used intrinsically by the ancient Egyptians in the design for their pyramids. The polyhedra now known as Platonic polyhedra due to the interest recorded in them by Plato, are defined by being regular, as each face is a congruent regular polygon, with the same number of faces meeting around each vertex. Later the concept had developed so that in the fifteenth century, Leonardo da Vinci famously illustrated Fra Luca Bartolomeo de Pacioli’s De divina proportione with an image of a rhombicuboctahedron that emphasised only the edges and vertices; omitting the faces and solid body (see Fig. 1). Keppler broadened the concept further by considering stellated polyhedra [2], before the idea was extended further still to infinite polyhedra [3] and myriad other still broader generalisations. Each generalisation allows interesting new structures to be created as the principles of the Platonic polyhedra are relaxed or extended. Within the mathematical literature, the definitions of polyhedra have expanded to mirror this growing variety of structures. We follow the conventions of tiling theory, as described by Grunbaum¨ [5], in identifying the face of a polyhedron with the cyclically ordered set of vertices and connecting edges that surrounds it. Using this definition allows us to maintain the intuitive link with regular polyhedra while creating topologically interesting finite structures, embedded in oriented manifolds of genus greater than zero. This definition is suitably broad to allow a freedom of structure that is not constrained to convexity, planar faces, or topologically spherical polyhedra. In turn, this freedom allows the presence of knots and links, as well as other entanglement modes [6] within the polyhedra. Interestingly, topologically toroidal polyhedra can be created within a definition more restrictive than ours, such as the dual pair of the Szilassi and Csasz´ ar´ polyhedra which have planar faces and straight edges [7, 8]. However, these requirements of straightness and linearity need to be dropped to allow a fuller variety of structures. Our method generates topologically distinct ways in which graphs (sets of vertices and edges with all edges joining pairs of vertices) can be embedded in space. Some of these ways are topologically trivial, whereas up to ambient isotopy

3 Figure 1: A ‘skeletal’ ployhedron is composed of just the edges and corners, and omits the faces and solid body. This illustration of a rhombicuboctahedron allows viewing of both the front and back elements[4].

4 (stretching and deformations without passing edges through each other or them- selves) others are inherently knotted or interlinked. While the method is applicable to any graph, in each case considered in this paper, the graph will be that of a tetrahedron, cube, or octahedron; but the embeddings of the constituent vertices and edges will vary between examples. If these graphs are considered in a chemical context – as atoms and bonds, or metal complexes and ligands – the results shed light on various topologically distinct conformational isomers that may be attain- able by the compound. As these isomers differ by their ambient isotopy, we follow the nomenclature of [9] in calling them isotopes. In literature regarding structural DNA manipulation [10] they are denoted topoisomers.

A C D D B E F A G H G H B

F E C

Figure 2: A topologically simple cube and a tangled version. Both are shown reticulating underlying surfaces.

There are different ways to quantify the ‘tangledness’ of a graph embedded in space. These methods are mostly imported from or inspired by knot theory: cycles within the graph embedding form loops which may be knotted or linked, and so can be described by knot and link tables, minimal crossing numbers, or knot polynomials etc. [11]. Indeed there has been some recent interest in the knots and links present in embedded graphs [12], however this work focuses on the knots and links unavoidably created in the embedding of sufﬁciently complex graphs, rather than the means of introducing controlled complexity into the embeddings of otherwise simple structures. An alternative to analysing the knots and links contained within an isotope is a generalisation of the knot polynomials called the Yamada polynomial [13] which provides an invariant (up to factors) of the entire embedded graph. This method

5 has been used to analyse the entanglement of the ‘theta graph’ – the graph with two vertices connected by three edges[14], however its utility for providing useful information about embeddings of larger graphs is unclear. Instead we prefer to use the hierarchy of graph embedding complexity provided by the minimum genus of the oriented surface necessary for a graph embedding as introduced in [9]. As the genus increases, so too does the complexity of knots [15], links and other entanglement modes [6] which manifest in the graph embedding. This hierarchy fits naturally with our method for generating these structures, which uses the 2-cell embeddings (see Section 2.2) of graphs to reticulate an oriented surface with the vertices and edges of the graph. A ‘reticulation’ of a surface indicates that the vertices and edges of a graph form a network on the surface, with edges only meeting or crossing at designated vertices. The use of 2-cell embeddings is a common technique within graph theory [16]. It is used in [17] and with a different emphasis in [18]. As we only consider small (comparatively combinatorically simple) graphs, we can omit many of the group theoretic aspects discussed in these works. As we are considering isotopes, within the equivalence class of ambient isotopy, it suffices to work with a representative in a canonical form. As this paper focuses on the genus-one surface, we consider only the ‘doughnut’ manner of torus embedding in space. This avoids issues arising from self-interesection of the surface, and facilitates the analysis using its universal cover: an abstraction which represents the surface, irrespective of its specific shape in space. The universal cover of the torus is the plane, with a two-periodic translational symmetry echoing the longitudinal and meridional lines of the surface structure of the torus. The particular geometric placement of the points within the universal cover is determined to be the average of the placements of the connected points (barycentric). This gives a unique and canonical representation for a given isotopy class of graph surface embeddings [19]. It is also well-formed, in terms of points and lines having distinct placements, for all graphs we consider here. In the representation in the universal cover, the graph becomes a translational unit cell of a periodic repeating pattern. In an alternative formulation, the surface can be cut open to form a surface patch reticulated with one unit of the repeating pattern in the universal cover. The manner that the patches fit together depends upon the periodicity of the universal cover. To find the actual placement of vertices, edges, and implicit faces in three-dimensional space, the surface is embedded in space in a suitable way, and then dissolved, leaving behind the points and lines, which can form a topologically tangled structure.

While the isotopes in this paper generalise Platonic polyhedra by virtue of shar- ing their graphs, some can share another feature of Platonic polyhedra: equally

6 sized faces. These isotopes have a high degree of symmetry within the universal cover, corresponding to the high point-group symmetry of traditional Platonic polyhedra. Less symmetric isotopes have varying face sizes, as do the Archimedean polyhedra such as the rhombicuboctahedron of Fig. 1. These polyhedra are de- ﬁned by being vertex-transitive i.e. the structure looks identical from any vertex. Other isotopes have less symmetry in the universal cover, and are merely generic genus-one versions of traditional polyhedra. For reasons of relevance to physical and chemical phenomena and simplicity of generation and analysis, in this paper we limit our analysis to oriented surfaces of genus one — the torus — and consider only the smaller Platonic polyhedra – tetrahedra, cubes and octahedra. We also preferentially focus on the isotopes with higher symmetry in the universal cover as we believe that of all the glut of potential structures produced by the method, they best represent the ‘Platonic ideal’.

1.1 The Big Picture (surfaces in space) It is common in the chemistry literature to represent certain structural elements within crystals as tetrahedra or other small solid shapes, connected to other atoms or solid shapes. In this sense the solid shapes represent an assembly of atoms, usually rigidly located with respect to each other. The structure of beta-cristobalite, described by an arrangement of tetrahedra in space, is an example of this (Figure 3). Some crystals can be represented by generalised space-ﬁlling polyhedron pack-

Figure 3: The chemical structure of beta-christobalite. Each tetrahedron represents a Silicon atom centred between four tetrahedrally arranged Oxygen atoms. Such structural motifs are found in many chemical structures. ings [20]. In these cases the chemical signiﬁcance of the polyhedral space partition

7 is that the symmetry-group of the tiling matches the symmetry-group of the crystal, and repeating motifs can be readily identified. An alternative viewpoint similar to this method maintains the idea of there being surfaces in space upon which the atoms and bonds are embedded (as points and lines), but no longer partitions space into modular building blocks of small regular shapes, but rather into infinite periodic components [21, 22, 23]. The periodicity of the crystalline structures is echoed in the periodicity of the surface that the atoms and bonds reticulate. This has the mathematically convenient consequence of allowing the information about the embedding of the points and lines in the infinite surface to be captured on a finite compactification of the surface. Many common crystal structures embed on the P, D and G [24, 22, 25] surfaces which can be compactly represented by the genus-3 torus. We combine elements of both the modular polyhedra and infinite surfaces methods to extend the complexity of the structures which can be described within these paradigms. For example, the structures in Fig. 2 suggest an immediate way in which to generalise structures such as the augmented structure of ReO3 shown in Fig. 4 or the beta-cristobalite shown in Fig. 3: each untangled cube or tetrahedral module can be replaced by a tangled version, with identical connectivity, as shown in [6].

Figure 4: The augmented structure of ReO3, denoted reo in [26, 27].

The method can also be used to extend the epinet project[24] to generate further structures within its rubric. The epinet project generates crystal structures from points and lines on the genus-3 torus. These objects are mapped into space by slic-

8 ing open reticulated tritori in a speciﬁc way, and packing the resulting specially- shaped surface patches together in space. Using this method, there is a large number of ways in which different points and lines can be mapped to various crystals. The variety of resulting crystals can be further extended by considering less regular cuttings of the tritorus. This produces twisted and tangled periodic crystals which generalise the ﬁnite tangled polyhedra presented here.

9 2 Method - generalising Platonic Polyhedra

The purpose of this method is to generate an embedding of a graph that by construction sits on a torus in three-dimensional space. The motivating idea for embedding the graph in a surface of non-zero genus is that increasing the genus allows additional structural complexity or ‘tangling’ to occur within the graph embedding. This tangling takes the form of knotted cycles and pairs of disjoint interpenetrating cycles as well as other tangling modes, which cannot exist in the standard genus- zero Platonic polyhedra. The method can also generate spatial embeddings of the graph that are ambient isotopic to the standard genus-zero embeddings as special simple cases. The first step is to choose the underlying graph topology; the examples in this chapter are the cube, tetrahedron or octahedron. The vertices of the graph are by definition located in a surface, as are the edges, so the edges that connect to a vertex must do so with a certain cyclic ordering around that vertex. If we presume that each face is disc shaped – rather than say, annular – then the information about the edge orderings around vertices is enough to determine how the faces fit together and thus the genus of the surface. Formally, this is called a 2-cell embedding, and is discussed further in Section 2.2. The process can be compared to stitching together a patchwork quilt, where the patches correspond to faces and the relative placement of the patches is determined by the ordering information around corners where three or more patches meet. With the genus of the surface determined, all that remains is to determine how the surface wears the ‘quilt’, and where the surface is in space. On the genus-one torus most of the potential choices make small or no difference to the final structure (up to ambient isotopy and chirality) so all that matters is how much, and in which direction, the ‘quilt’ is stretched onto the surface in space. Once this has been decided, the surface can be removed, leaving just the graph embedded in three-dimensional space, possibly tangled, and the knotting and linking of the cycles of the graph can be analysed to determine the extent and nature of the tangling. In this way, the surface can be seen as a ‘template’ for the placement of vertices and edges. As it is desirable to do this in a canonical way, we only consider surfaces which have no self-intersections, as self-intersections prohibit a canonical embedding into three-dimensional space. This rules out embedding the graph into non-oriented surfaces, which have no embedding into three-dimensional space that avoid self-intersections. A direct consequence of self-intersection of the template surface is that for any 2-cell embedding of adequate complexity there correspond multiple graph embeddings in three-dimensional space: an unnecessary extra layer of complexity, given the breadth of structures we can create using a single canonical surface embedding. In a paper which considers the creation and analysis of

10 2-cell embeddings, potentially for use in chemistry applications, [17], the authors also omit non-oriented surfaces. However they consider the lack of a non-self- intersecting embedding of such surface to directly annul the chemical relevance due to the presence of self-intersections of such surfaces. Having decided that the template surface must be of genus one and not contain self-intersections, the only candidates are tori which follow a loop in space as their core. If this loop is knotted it will automatically induce the same knot in a cycle which traverses that direction of the surface, an unnecessary extra mode of entanglement. Thus thoughout this paper we use the unknotted ‘doughnut’ manner of embedding the torus in space as it is simple and satisﬁes the other criteria.

2.1 Universal cover The space where the vertices, edges and faces are assembled together, before they are explicitly ‘laid out’ onto a speciﬁc surface is called the universal cover.

A covering space of a topological space M consists of a space X and a map Π: X M such that Π is continuous and given a suitably small open set U M, → ∈ the inverse is continuous on each preimage of U.

A universal cover is such a covering space in which X is simply connected. It is universal because any other space which covers M is itself covered by X.

Of the oriented surfaces, the universal cover of the sphere is itself, the cover of the torus is the two-dimensional Euclidean plane, and surfaces of genus-two or higher are covered by the two-dimensional Hyperbolic plane. In the case of the torus, multiple copies of the surface of the torus are laid out in a grid in the Euclidean plane. A path whose ends differ by (p, q) corresponds to a cycles that winds around the fundamental cycles of the torus (longitude and meridian) p, q { } times respectively. The longitudinal cycles are by convention those cycles which traverse the torus parallel to the equatorial plane, while the meridional cycles wrap around the torus in a perpendicular direction (encompassing a disc of the interior of the torus). The way in which a graph embeds in the universal cover is similar to its embedding in the surface, as translational lattice vectors correspond to the generators of the group of the torus. Thus each pattern of faces on the torus generates a doubly- periodic tiling pattern in the Euclidean plane. The area corresponding to a single cover of the torus is a unit cell of the periodic tiling, and for reasons of simplicity the unit cell is drawn as a parallelogram bounded by the lines of meridian and longitude.

11 A cycle within the graph that is null-homotopic on the surface – i.e. can con- tract down to a point – corresponds to a closed cycle in the universal cover. A cycle which wraps around the meridian and longitude p, q times respectively will cor- { } respond to a path which travels (p, q) in the universal cover, starting and ﬁnishing at equivalent vertices in different copies of the unit cell. For example, consider the difference between the path ABCDA and the path ABDCA on the torus in Fig. 5.

2.2 2-cell embeddings A 2-cell embedding is analogous to the example of a patchwork quilt. Faces correspond to patches of cloth, edges to the lines of sewing joining the patches, and vertices to where multiple patches or lines of sewing meet. Formally, a 2-cell embedding of a graph into a surface, is one in which every face is homeomorphic to an open disk, and so has only one boundary component. Vertices of the graph are associated with points in the surface, edges associate with arcs, and the faces are defined as the complement of the union of the points and arcs – the pieces of the surface left if the points and arcs are removed. We also apply the reasonable restriction of prohibiting intersections of arcs elsewhere from the locations of the appropriate arc endpoints. When a graph is embedded into a surface without edges crossing, there is an ordering of edges around each vertex, known as a rotation scheme, and the information encoded in this scheme is enough to recover the homeomorphic equivalence class of the embedding [16, 17]. This means that we can find all embeddings of a graph into a closed surface, up to homeomorphism, by considering the combinatoric ensemble of all rotation schemes commensurate with the 2-cell embedding of the graph. Fig. 5 shows an example of this process for the tetrahedron. As we limit our embeddings to oriented surfaces, the surfaces underlying each embedding can be identified up to homeomorphism by only one parameter, the genus. The genus is found using Euler’s formula:

V E + F = χ =2 2g, (1) − − where the letters refer respectively to the number of vertices, edges and faces, the Euler characteristic, and the genus, g. The number of faces in the embedding can be found by starting at an arbitrary edge, and turning (say) left at each vertex until the perimeter of the polygon has been traced, and repeating until all such faces have been found. The genus can then be calculated, as the number of vertices and edges is known from the graph. A 2-cell embedding is necessary for this process, as each face must have only one boundary component. Euler’s formula (Eq. 1) tells us the allowable genera of the 2-cell embeddings of the Platonic polyhedra graphs. The minimum genus is always zero (the sphere),

12 B B B B B B

C A C A C A C A C A C A C A B D D D D D D

B B B B B B D C A B C A C A C A C A C A C A D D D D D D Isotope A

D C B B B B B B A B A B C A C A C A C A C A C A D D D D D D D C D C A B C A B B B B B B C A C A C A C A C A C A A B D C D D D D D D A B A B D B C D B A B B B B B B D C D C C A C A C A C A C A C A C D A B D D D D D D a. D A B C b. c. A B A B B B B B C A C A C A C A C A C A D D D D D D D

D B B B B B B C A C A C A C A C A C A B D D D D D D B

C C d. e. Figure 5: The construction of a tangled tetrahedron. (a) A rotation scheme scheme is chosen, which can only fit together one way to form an oriented 2-cell complex. The vertices connect to the others in a manner determined by the source graph, in this case a tetrahedron: each vertex is connected by an edge to every other. (b) This rotation scheme forms an <8.4> tiling of the universal cover. A unit cell is chosen, bounded by two pairs of side vectors, such that there is only one translationally equivalent point in each cell and cell is of unit size. (c) The unit cell is ‘rectified’ to a square by a linear transformation. (d) The universal cover maps onto the torus by identifying opposite sides of the square, carrying the tiling with it into E3. In this example, the top and bottom of the unit cell are glued first, to form a tube, then the side vectors are glued, bending the tube into a torus shape. (e) The surface is removed, leaving the points and lines in space. The embedded graph has the same connectivity between vertices as its source graph, and the choice of rotation scheme and unit cell shape ensures that there is a single trefoil in the isotope, in cycle ABDCA.

13 and the minimum number of faces is two, to ensure an even value of χ, as necessary for oriented surfaces.

Polyhedron V E F g min ⇒ max Tetrahedron 4 6 2 1 Cube 8 12 2 2 Octahedron 6 12 2 3 Dodecahedron 20 30 2 5 Icosohedron 12 30 2 9 It is clear from this table that graphs with more edges (partially offset by more vertices) have 2-cell embeddings in surfaces of higher genus. Thus they can sustain more complex entanglement modes, as suggested in [9]. This paper concentrates on embeddings in the torus, however higher genus embeddings can model periodic structures: The genus-2 surface (bitorus) can be cut open into an ‘X’-shaped hollow • tube. Multiple copies of this hollow tube can be joined together, with the cuts of each tube aligning with that of its neighbour, and so respecting the local geometry of the bitorus, even across the cuts. These ‘X’-shaped surfaces have a channel inside of each tube, with the four tubes meeting at the centre of the ‘X’; in a graph of the channel structure of a surface formed by joined ‘X’s, all vertices will have four neighbours. Thus the bitorus is the quotient surface of many graphs embedded on a surface with a four-valent channel structure skeleton, including doubly periodic crystal structures. The genus-3 torus can be cut open into a surface homeomorphic to the Prim- • itive, Diamond or Gyroid minimal surface unit cell. Thus any graph embedding on the tritorus has a natural projection into a triply periodic inﬁnite structure. This is of particular relevance to generating and modeling triply periodic crystal structures [24]. Higher genus tori could be considered as the quotient surfaces of cubic re- • peating structures which require larger unit cells due to decreased symmetry, and perhaps increased structural complexity. The tilings generated by the 2-cell embedding process can have a variety of face sizes. The notation used in this chapter to describe these tilings is simply a list of the set of faces present, in the form for a tiling containing faces of size a, b, c and d. In the special cases of where distinct tilings have the same set of face sizes, the relevant tilings are given special names to avoid confusion. Regular tilings of p-gons with q of them meeting around each vertex have their conventional notation: p, q . { }

14 2.3 Tilings of the torus dissolving to polyhedron nets Finding the possible surface reticulations of the edge skeleton of a polyhedron onto the torus is a two stage process. Firstly we find its image in the universal cover: this information is provided by the 2-cell embedding rotation scheme. We then find all the ways to map this information onto the torus, by choosing differently shaped unit cells. There are two equivalent methods to map from the universal cover to the torus: in the first of them, the parallelogram unit cell has side vectors (q, r) and (s, t) in the universal cover, and is wrapped onto the surface in such a way that one of the side vectors of the parallelogram becomes a meridian loop of the torus, and the other becomes a longitudinal loop. Alternatively and equivalently, a unit cell parallelogram is formed by the canonical unit vectors of the crystal structure wrapped onto the torus in such a way that the two side vectors have homotopy type (u, v) and (w, x) with respect to the meridian and longitude lines. The variables mentioned in these two methods are related by: 1 qr uv− = , st wx ! " ! " In the chosen method, requirements on the length and orientation of the sides are inherent in the choice of the ‘parallelogram’ side vectors. The variables are restricted to integer values satisfying qr uv det = det = 1 st wx ± ! " ! " which can be more compactly written as qt rs = ux vw =1 (2) | − | | − | with the components of each vector being co-prime. Throughout the rest of this paper we use the first method outlined above, as shown in Figure 13b. The vector components are co-prime as the vector cannot pass through any translationally equivalent point in the universal cover more than once. If the vector is a scalar multiple k of a smaller integer-valued vector, say v = ku then in the process of tracing out the vector v in the universal cover, it is necessary to pass through k points translationally equivalent under u, which corresponds to an unwanted multiple covering of the torus. This final requirement means that by an application of Euclid’s algorithm known as Bezout’s´ identity1 that any choice of initial vector (q, r) (q and r co-prime) has

1Bezout’s´ Identity states that: If a and b are integers not both equal to 0, then there exist integers c and d such that GCD(a, b)=ac + bd, where GCD(a, b) is the greatest common divisor of a and b.

15 a matching vector (s, t) which satisfies the above constraints, as indeed does any second vector of the form (s, t)+k(q, r) for the same (s, t), with integer k. This means that any toroidal 2-cell embedding generates a 3-parameter family of structures: q, r and k can vary across any integer values. Allowing q, r, s, t and k to take high absolute values ‘stretches’ the graph on the torus, allowing circuits to become highly knotted or linked. The standard embedding map we use identifies two opposite sides of a parallelogram unit cell, effectively gluing them together into a cylinder. Then the remaining two sides are identified, gluing the ends of the cylinder together into a torus. A hexagonal unit cell could also be used, but is eschewed for simplicity of analysis and because it adds nothing new. The explicit mapping of the universal cover into E3 is to linearly transform the parallelogram unit cell to a unit square, then map: (x, y) ((R + r cos 2πy) cos 2πx, (R + r sin 2πy) cos 2πx, r sin 2πy) (3) → where r and R are the tube radius and distance from the tube centre to the torus centre, respectively. Equation 3 describes a mapping which is the composition of rolling the unit square into a cylinder of unit length and circumference, then bending the cylinder in the obvious way such that its two boundaries meet, forming a ‘doughnut’-shaped torus. Specifically, the equation maps the (1, 0) cycle to the length of the cylinder and the (0, 1) cycle to its circumference. Then the side of the cylinder maps to the longitudinal cycle of the torus embedding, and the perimeter of the cylinder maps to the meridian of the torus embedding. Several steps within this process could be altered to produce an equally valid map from the universal cover to a torus in E3, by varying the geometric offsets of the unit cell, or varying which vectors map to which cycles, including the direction of the cycles. However, these alterations produce isotopically equivalent structures, up to chirality, as shown in Fig. 6. This is least intuitively obvious in the case of exchanging which side vector of the unit cell maps to the meridanal and longitudinal cycles of the torus: If a small disk is removed from the torus and the torus is turned inside-out through this hole before the disk is replaced, then the resulting torus will have the latitudinal and meridional lines exchanged, see Fig. 8. If the puncture occurs away from any embedded lines, then the embedding of the vertices and lines in E3 is not affected by the removal of the disk, so during the inversion of the punctured torus the embedded lines are carried in a non-crossing manner. Thus the isotopy of the embedded graph is preserved, while the role of meridional and latitudinal lines on the surface are reversed. Different embeddings of isotopes can have different point group symmetries, as well as varying geometric properties, as is clear from Fig. 6. Depending on the

16 a. b.

c. d. Figure 6: Isotopically equivalent toroidal octahedral isotopes. (a) A highly symmetric isotope. (b) The same isotope offset by 1/2 of the meridian from the ref- erence image in (a). (c) The result if the side vectors of the original unit cell are glued together in the other order. (d) Once the surface is removed these structures are ambient isotopic. tiling, these differing geometric embeddings may both be of interest, as is the case for the ‘E type’ cube from [9], see Fig. 7.

2.4 Tangling The relaxation of the requirements for convexity and symmetric equivalence between vertices, edges and faces allows these generalised Platonic solids to have features not found in their simple cousins. One such feature is tangling, which in- cludes the presence of knotted and linked cycles, in the knot-theory sense. A cycle is a non-self-intersecting alternating sequences of vertices and edges. It is knotted if it cannot be deformed into the unit circle without passing edges or vertices through each other. Two disjoint cycles are linked if they each interpenetrate the

17 H D H B D F C G B E A F C A E G

Figure 7: Two embeddings of the D isotope, related to each other by eversion of the torus. other, such as two adjacent links in a chain – known as the ‘Hopf link’. Other more subtle forms of entanglement exist, such as ‘Brunnian rings’, which are a collection of three or more cycles inseparably bound together, despite no two of the cycles being linked; as well as ravels[6]. Brunnian links, such as the Borromean rings, cannot embed in a surface of genus one, and ravels are hypothesised to have no genus one embedding, so these entanglement modes are beyond the scope of this paper. The standard Platonic Polyhedra are untangled. This follows immediately from the fact that the standard Platonic polyhedra embed on the sphere: no loop on the surface of the sphere can have a knot in it without crossing itself; and no two loops can interlink without crossings. Thus knots and links are impossible in graphs embedded on the sphere, such as the standard embeddings of the Platonic polyhedra and other convex bodies.

2.5 Knots and links One way of partially identifying and understanding the structures of isotopes is to examine all the knots and links contained in the cycles therein. As the isotopes are created on the surface of a torus, the knots are restricted to being torus knots. As the name implies, torus knots are knots which can be embedded in the surface of a torus without crossings. They can be characterised by the number of times they traverse the torus in the meridional and longitudinal directions. A non-crossing closed path on a torus which traverses each direction p and q times is a (p, q) torus knot, if both p and q are at least two. In order for such a path to not cross itself, p

18 a b

c d

Figure 8: The eversion of a torus. (a) A small patch is removed from the torus, and (b) is stretched open enough to (c) allow the ‘core’ of the torus to pass through it. (d) The hole is then shrunk back down in size, and the patch is replaced. Meridional and longitudinal cycles are exchanged in this process. Images are screenshots from [28].

19 and q must be co-prime. If either p or q takes an absolute value of two or smaller, the cycle will be the unknot – ambient isotopic to the unit circle. A (p, q) torus knot is isotopic to a (q, p) one, following the same argument about torus inversion as in Section 2.3. The knot is not affected by the removal of the disk, so during the transformation it does not cross itself. However as the role of meridional and latitudinal lines are exchanged, a torus knot which previously wrapped around the torus p times in the meridional direction and q times in the latitudinal direction will wrap q times in the meridional direction and p times in the latitudinal direction after the inversion.

Figure 9: the trefoil is a (2, 3) torus knot.

Loops that interlink are known as links. If all the loops can simultaneously embed on a torus without crossings, then they are torus links. In order to not cross, each component must have the same (p, q) homotopy type, and for the components to interlink, both p and q must be non-zero. In the universal cover, an n-component (p, q) torus link consists of n parallel (p, q) vectors. It is denoted as an (np, nq) torus link, and is seen to be distinct from a torus knot by the common factor of n in each term. A Hopf link is composed of two (1, 1) loops, and is thus a (2, 2) torus link, while a Whitehead link is composed of two (1, 2) loops, and is thus a (2, 4) torus link. Similarly to torus knots, an (np, nq) torus link is isotopic to a (nq, np) one via the same torus eversion.

In an isotope, there will sometimes be a knot or link type which occurs in multiple places. In this case, when the knots and links in an isotope are present with a multiplicity m, they will be reported in the form m (p, q), or m (np, nq) ∗ ∗

20 indicating that there are m distinct (p, q) knots or (np, nq) links present.

Figure 10: Hopf (2,2) link and Whitehead (2,4) link, on and off the torus.

A (p, q) vector in the universal cover maps to a (p, q) torus knot, an observation which follows directly from the definitions. Thus analysis of the knots and links present can be done in the universal cover, where the path traced out by a closed circuit on the torus is easily measured. In order to find the knots present in an isotope, all distinct cycles must be considered, while to find links, all combinations of disjoint cycles must be analysed.

21 2.6 Energy For any given graph in the universal cover, there is a triply infinite number of embeddings into the torus, characterised by several variables, as discussed in section 2.3. In order to facilitate comparison between differing isotopes, we define a measure of energy upon the graph embeddings to provide an intuitively reasonable numeric value to discern the differing nature of ‘highly wound’ isotopes compared with their relaxed kindred. Being ‘highly wound’ requires that the edges are longer, to allow the cycles to have high homotopy types. Energy is thus (somewhat arbi- trarily) defined as an extension of the sum of the squares of the side lengths of the isotope: as the measure should be independent of the meridional and longitudinal rotation of the graph within the surface, it is calculated from the universal cover rather than from the specific embedding. However, a rectangle of unit area covers the torus equally well as a unit square, following the mapping x x (x, y) ((R + r cos 2πyα) cos 2π , (R + r sin 2πyα) cos 2π ,rsin 2πyα) → α α modified from equation 3 where α and 1/α are the side lengths of the unit-area rectangle. The energy is thus defined as the lowest sum-of-squares of the side lengths in the universal cover, for the most optimal value of α. A rectangular unit area has the advantage over a unit square of being a more naturally shaped unit cell of normalised long, thin parallelogram unit cells.

When applied to three-dimensional isotopes, this deﬁnition of energy best describes a toroidal graph embedding rather than the actual isotope with the tem- plating surface removed; distinct embedded graphs can be ambient isotopic. The deﬁnition of energy applying to embedded graphs thus has the natural extension to isotopes by considering the minimum energy amoung the ensemble of toroidally embedded graphs ambient isotopic to a given isotope.

2.7 Symmetry The graph embeddings have varying levels of symmetry in the universal cover and in their three-dimensional (E3) embedded structures. These symmetry consider- ations offer us another way to rank isotopes in some order of ‘simplicity’, complementary to the ranking provided by ‘energy’. The point group symmetries of toroidal graph embeddings depend upon both the possible symmetries of the torus embedding, the symmetries of the reticulation and the compatibility between these symmetries. The symmetry operations that map the torus to itself are mirror planes through the central axis, forming angles of π/n at the centre, for n = 1, 2, ... , a mirror { }

22 through the equatorial plane, rotational symmetries around the central axis, and rotation by π around an axis which lies in the equatorial plane and passes through the centre of the torus as described above. The presence of multiple equatorial 2- fold rotational axes is coupled with rotational symmetry around the central axis of the torus. In the universal cover, the symmetry is maximised by barycentric placement of vertices [19]. As the mapping from the plane to the torus is a conformal one, the point group symmetries of the embedding of the torus reticulation rely on the same symmetry elements being present in the universal cover. Thus, if there is a two-fold axis of rotation through the equatorial plane of the torus, then that axis of symmetry punctures the torus in four places. These four locations correspond to two-fold rotation points in the universal cover at half-unit offsets in directions parallel to the unit cell sides. Rotational symmetries around the axis of the torus correspond to translational symmetries in the universal cover parallel to the side of the unit cell which maps to the meridian of the torus.

Figure 11: An example ‘E-type’ toroidal cube isotope from [9] with four equatorial two-fold rotational axes. This isotope can be rotated by π around each line skewering the torus, mapping the structure back onto itself.

So what is the highest symmetry that an isotope can have in three dimensional

23 space, while embedded on a torus?2. The answer to this query depends upon the isotope in question, but for the small Platonic Polyhedra isotopes considered here, the most symmetric embedding is one with two-fold rotational axes through the equatorial plane, as in the case of toroidally embedded polyhedra, only the untangled isotopes can have mirror planes[29]. Such a torus being punctured by n skewers has Point Group symmetry 22n in Conway notation, for n>1, and in its universal cover has n 4-tuples of points, whose alignment along a common equator 1 in the embedding imply a separation from each other by a distance of 2n of the 1 longitudinal side vector and 2 of the meridional vector. The Conway notation is compared to other common Point Group symmetry notation in Table 1.

Table 1: The relationship between the level of structural symmetry, the symmetry type in 3D space (shown in both standard crystallographic Point Group notation and Schonﬂies¨ symbol), the surface orbifold type (using Conway orbifold notation), and the number of ‘skewers’ – 2-fold rotational lines in 3D space – for the cube. Order Point Group Schonﬂies¨ 2D Orbifold symmetry elements in E3 8 mmm D2h *222 3 mirrors 8 422 D4 2222 4 skewers 4 222 D2 2222 2 skewers 2 2 C2 2222 1 skewer 1 1 o none none

2The question of whether a structure can have a higher Point Group Symmetry once the surface is dissolved and the structure can move up to ambient isotopy is still an open one.

24 3 Tetrahedron

For our purposes, the tetrahedron is the graph of the corners and edges of the solid tetrahedron - its 1-skeleton. It is a small and simple structure, with a cor- respondingly small number of possible 2-cell embeddings. After reductions due to automorphisms, there are only three different 2-cell embeddings: the familiar 3, 3 sphere embedding (of four triangles) shown in Fig. 12, as well as two torus { } tilings each with two faces. One toroidal tiling has faces of size eight and four, and will be denoted <8, 4>, while the other has faces of size nine and three, denoted <9, 3>. Example untangled isotopes derived from these three 2-cell embeddings, and their universal covers, are shown in Figs. 12 and 13.

Figure 12: The tetrahedron as a sphere-embedded tiling. There are three triangular faces around each vertex, so the tiling is 3, 3 . { } Each of the toroidal tetrahedron tilings has an assortment of polygon sizes, so many of the surface ‘regularity’ properties of the Platonic tetrahedron such as high point group symmetry are lost in the toroidal generalisation. This is due to both the lower symmetry of the torus compared with the sphere, as well as the generally low level of two-dimensional symmetry in the universal cover; and the interaction of these two phenomena as discussed in Section 2.7 . The <8, 4> tiling can however be considered to be a generalised Archimedean tiling. Archimedean tilings have looser requirements than Platonic (or ‘regular’) tilings: they are tilings in which every vertex is equivalent but which contain a variety of face sizes. Familiar sphere-embedded examples of Archimedean tilings include truncated versions of the classical Platonic polyhedra such as the truncated

25 cube, which has two octagons and one triangle around each vertex. The rhombicuboctahedron of Fig. 1 is another example. This property of vertex congruence deﬁnes ‘uniform polyhedra’ which are not normally expected to be convex – including for example the stellated polyhedra – so this genus-one extension only varies from the previous deﬁnitions of polyhedra by allowing curved faces and edges in the three-dimensional embedding. The torus-embedding of the <9, 3> tiling has no symmetries greater than the translational symmetries of the torus, and so it is a more general case still. As each face is automatically a polygon – considered in the universal cover – the <9, 3> tiling generalises Johnson solids such as the square pyramid to the torus by allowing curved edges and faces. The universal cover of the barycentric <8, 4> tiling has 244 symmetry and is ∗ tile- and edge- 2-transitive and is vertex-1-transitive, while the barycentric <9, 3> tiling has 333 symmetry also has tile- and edge- 2-transitivity but is vertex-2- ∗ transitive. As discussed in Section 2.7, the presence of 2- and 4- four fold symmetry elements in the <8, 4> tiling at appropriate locations allows the cover to be mapped onto the torus comensurate with a two-fold rotational axis in the equatorial plane. The absence of any twofold rotational points or periodic behaviour within the unit cell in the universal cover of the <9, 3> tiling forbids any resultant isotope from possessing any rotational symmetry. The three-fold mirror points and the mirror planes are not preserved in the mapping to the torus.

3.1 2-cell embeddings of the tetrahedron As described in Section 2 and shown in Fig. 5, we omit the usually presumed relationship between the vertices and edges with the faces, as our method of generation ﬁnds new edges and corners to bound each face. To generate our novel tetrahedra, we presume only that the points and edges lie on a closed oriented surface, and that each face is homeomorphic to a disk.

The tetrahedron is equivalent to the complete graph K4 and is combinatorically simple: each vertex connects to the other three vertices. Thus there are two possible orientations of edges around each vertex, so the composition of these different possibilities allows 16 (= 24) different 2-cell embeddings. As there is no possi- bility of confusion - the graph has a maximum of one edge between each vertex pair - we label each edge around any given vertex by its other endpoint. Using the method described in Section 2.3, we ﬁnd the faces from the edge orderings around each vertex. The possible 2-cell embeddings are shown in Table 2.

There are either two or four faces in each of these 2-cell embeddings, so a

26 A C A C A C A C A C B B B B B B

D D D D D D A C A C A C A C A C B B B B B B

D D D D D D A C A C A C A C A C B B B B B B A D D D D D D B A C A C A C A C A C B B B B B B C D D D D D D A C A C A C A C A C B B B D a. B B B b. D D D D D D AD B AD B A D B A DB D C C C C A C A C A C A C A C

D D D D A B A B A B A B A B C C C C C A D D D D D D B A B A B A B A B C C C C

D D D D c.A B A B A B A B A d.B C Figure 13: Example universal covers and torus embeddings of the two different 2-cell embeddings, <8, 4> (a and b) and <9, 3> (c and d).

27 A C A C A C A C A A C C A C A C A C A C B B B B B B B B B B B B

D D D D D D D D D D D D A C A C A C A C A A C C A C A C A C A C B B B B B B B B B B B B

D D D D D D D D D D D D D D D D D C A C A C A C A C A A C A C A C A C A A C C A C A C A C A C B B B B B B a. B B B B B b.B B B B B B D D D D D D C D A C D A C D A C D A D C D A D D D D D A C AB C AB C BA C BA A C CB A C A C A C A C

D D D D D B C A C A C A C A C A B B B B B A

D D D D D C A C A C A C A C A B B B B B C

D D D D D C A C A C A C A C A D c. B B B B d. B Figure 14: This sequence of images illustrates how tangled toroidal isotopes are created by stretching the standard unit cell. The tetrahedral isotope of Fig. 13b corresponds to the universal cover in (a); it has canonical unit cell side vectors with the vector (0, 1) corresponding to the meridion and (1, 0) corresponding to the longitude. However, if the vector (1, 1) is instead chosen to wrap to the meridion, as in (b), a different unit cell is formed. (c) shows the parallelogram resulting from the choice of (1, 1) and (1, 0) as side vectors straightened back to a square, for ease of mapping to the torus; (d) is the resultant isotope. The torus embedding is different from that corresponding to the canonical side vectors (Fig. 13a), but both are still ambient isotopic to each other and to the sphere-embedded tetrahedron of Fig. 12, once the surface is removed. The sides of the unit cell need to be skewed further to allow a knot to form, giving the resultant isotope a different isotopy, such as Fig. 15a in which the side vectors are (1, 2) and (1, 1).

28 Table 2: The rotation schemes of all 2-cell embeddings of the tetrahedron into an oriented surface. The ﬁnal entry corresponds to Fig. 5. Orientation around vertex Cycles Face Sizes Genus ABCD BCD ACD ABD ABC ACDBCADBA, ABCDA 8,4 1 BCD ACD ABD BAC ABCDBACBDA, ADCA 9,3 1 BCD ACD ADB ABC ABCADBACDA, BDCB 9,3 1 BCD ACD ADB BAC ABCADCBDA, ACDBA 8,4 1 BCD ADC ABD ABC ABDCADBCDA, ACBA 9,3 1 BCD ADC ABD BAC ABDA, ACBA, ABDA, BCDB 3,3,3,3 0 BCD ADC ADB ABC ABDCBACDA, ADBCA 8,4 1 BCD ADC ADB BAC ACDBCADCBA, ABDA 9,3 1 BDC ACD ABD ABC ABCDACBDCA, ADBA 9, 3 1 BDC ACD ABD BAC ABCDBADCA, ACBDA 8, 4 1 BDC ACD ADB ABC ABCA, ACDA, ADBA, BDCB 3,3,3,3 0 BDC ACD ADB BAC ADCBDACDBA,ABCA 9,3 1 BDC ADC ABD ABC ADBCDACBA, ABDCA 8,4 1 BDC ADC ABD BAC ABDACBADCA, BCDB 9,3 1 BDC ADC ADB ABC ABDCBADBCA, ACDA 9,3 1 BDC ADC ADB BAC ABDACBDCA, ADCBA 8, 4 1

29 calculation of the Euler Characteristic gives

V E + F =4 6 + (2 or 4) = 0 or 2; − − so from Equation 1 the genus is respectively 1 or 0, and the surfaces are topologically tori or spheres. Note that the information in the lower half of Table 2 is redundant, insofar as it can be deduced from the information of the top half: In the table, the ordering is anti-symmetric about the half-way line, i.e. the orientations are reversed at each vertex, leading the faces to be traversed with opposite orientation. This can be (loosely) viewed as one half of the embeddings being embedded on the ‘outside’ of the surface, matching the other half generated on the ‘inside’. In each case the two embeddings create the same isotope.

3.2 The tetrahedron in the universal covers: <8, 4> and <9, 3> Considering only one surface orientation, there are eight combinatorially possible 2-cell embeddings of the tetrahedron. They have three distinct combinations of face sizes as shown in Figure 2: four triangles; octagon and square; nonagon and triangle.The standard tetrahedron is the tiling of the sphere by four triangles as shown in Fig. 12. The two remaining tilings are torus tilings and correspond to Schlaﬂi¨ symbols < 8, 4 > and < 9, 3 >. The universal covers for these 2-cell embeddings and example derived isotopes are shown in Fig. 13. An explanation for the number of each type of 2-cell embedding lies in their symmetries: there are three different Hamiltonian cycles of length four on the tetrahedron, each of which can correspond to the 4-cycle in the <8, 4> tiling. Similarly, there are four separate triangles which can correspond to the 3-cycle of the <9, 3> tiling; but only one distinct labelling of the spherical 3, 3 2-cell surface embed- { } ding. As previously described in Section 2.3, we can choose stretched unit cells, sub- ject to the restrictions given there, to generate knotted isotopes. The tetrahedron has no pairs of non-intersecting cycles, which prohibits the existence of links, however isotopes containing knots can be created. Some of these isotopes are shown in Figure 15. A table of the homotopy types of the cycles present in each 2-cell embedding is presented in Table 3. This information indicates the knotted cycles. Certain homotopy values shown in the table are constant between different cycles. This can be explained geometrically by observing that the common values correspond to the paths in the universal cover splitting and rejoining as they pass around a null-homotopic triangle or square. As evident from Fig. 13 each of the two hor- izontal and the two vertical paths in the <8, 4> tiling can be considered to be the composition of the other corresponding path with the null-homotopic 4-cycle, giv-

30 ing the same homotopy type for each. In the case of the <9, 3> cycle, each 4-cycle differs from a 3-cycle, by traversing two sides of a triangle rather than one.

A D

B B D

C C a. b. Figure 15: Two tetrahedral isotopes. (a) is from the <8, 4> cover with side vectors (1, 2) and (1, 1); (b) is from the <9, 3> cover with side vectors (2, 1) and (1, 1). The (a) isotope has a single (2,3) torus knot (trefoil) in cycle ABDCA, whereas the (b) isotope has two trefoils in cycles ACDA and ABDCA.

Table 3: Homotopy types of cycles on the tetrahedron, for unit cells taken with edges (q, r) and (s, t). The multiplicative factor of qt rs = 1 from Equation − ± 2 for all homotopies is not shown. There are no disjoint cycles in the tetrahedral graph, so there are no links. <8, 4> 2-cell embedding <9, 3> 2-cell embedding Cycles of length 3 Cycles of length 3 ABDA ABCA (0, 0) ( s, q) BDCB − ABDA (s, q) # − ACBA ACDA (s + t, q r) ( t, r) − − ACDA − BCDB (t, r) # − Cycles of length 4 Cycles of length 4 ABCDA (0, 0) ABCDA (s, q) − ABDCA ( s + t, q r) ABDCA ( t, r) − − − ACBDA ( s t, q + r) ACBDA (s + t, q r) − − − −

3.3 Comparison of tetrahedral isotopes The energy for each unit cell for a given 2-cell embedding can be readily calculated as described in Section 2.6. This information is shown in Table 4, together with the sphere-embedded tetrahedadron with normalised area, for purposes of comparison.

31 After embedding into three-dimensional space, many of these isotopes are ambient isotopic. This is especially pronounced for the untangled tetrahedron isotopes, due to the structural simplicity. The energy of an isotope is determined to be the lowest energy of any of the generating 2-cell embeddings and unit cell shapes, i.e. when the same isotope is formed from different embeddings, the energy is set to the lowest generating energy. This is shown in the table of tetrahedral isotope energies, Table 5, which by construction is a contracted version of Table 4.

Table 4: The 20 lowest energy toroidal tetrahedron unit cells together with the tetrahedral sphere tiling. Note that the untangled isotope can be generated in many ways. The knots are labeled by homotopy type, with the subscript indicating the length of each knotted cycle, and a multiplicative factor indicating the multiplicity of that knot. The energy of a symmetric sphere-embedded isotope such as the one shown in Fig. 12 is shown for comparison. The energy is normalised in this case by scaling the sphere to have unit surface area. Energy Tiling Example unit cell Knots 1.74 3, 3 Sphere embedding — { } 2.33 <9, 3> (1,0), (0,1) — { } 2.41 <8, 4> (1,0), (0,1) — { } 2.77 <8, 4> (1,0), (1,1) — { } 2.93 <9, 3> (1,0), (1,1) — { } 3.41 <8, 4> (1,0), (2,1) — { } 3.57 <9, 3> (1,0), (2,1) — { } 3.89 <8, 4> (1,1), (1,2) (3, 2) { } 4 3.99 <8, 4> (1,0), (3,1) — { } 4.14 <9, 3> (1,0), (3,1) — { } 4.51 <8, 4> (1,0), (4,1) — { } 4.55 <9, 3> (1,1), (1,2) (3, 2) , (3, 2) { } 3 4 4.64 <9, 3> (1,0), (4,1) — { } 4.79 <8, 4> (1,1), (2,3) (5, 2) { } 4 4.98 <8, 4> (1,0), (5,1) — { } 5.10 <9, 3> (1,0), (5,1) — { } 5.41 <8, 4> (1,0), (6,1) — { } 5.52 <9, 3> (1,0), (6,1) — { } 5.56 <8, 4> (1,1), (3,4) (7, 2) { } 4 5.75 <9, 3> (1,1), (2,3) (5, 2) , (5, 2) { } 3 4 5.94 <8, 4> (1,2), (1,3) 2 (3, 2) , (4, 3) { } ∗ 3 4 Considering the possible point group symmetry of these isotopes, the key is to

32 Table 5: Tetrahedral isotope energies: indexed by their lowest energy generating 2- cell embedding and unit cell shape in the universal cover. This table is a contracted version of Table 4, and uses the same labeling conventions. Energy Tiling Example unit cell Knots 1.74 3, 3 Sphere embedding — { } 2.33 <9, 3> (1,0), (0,1) — { } 3.89 <8, 4> (1,1), (1,2) (3, 2) { } 4 4.55 <9, 3> (1,1), (1,2) (3, 2) and (3, 2) { } 3 4 4.79 <8, 4> (1,1), (2,3) (5, 2) { } 4 5.56 <8, 4> (1,1), (3,4) (7, 2) { } 4 5.75 <9, 3> (1,1), (2,3) (5, 2) and (5, 2) { } 3 4 5.94 <8, 4> (1,2), (1,3) 2 (3, 2) and (4, 3) { } ∗ 3 4 observe that only the untangled structure can possess mirror planes, as shown in [29]. Neither of the universal cover tilings have translational symmetry inside the unit cell, which prohibits rotational symmetries around the central axis of the torus. Thus the maximum symmetry that an <8, 4>-generated isotope can have is a single equatorial twofold axis of rotation, i.e. a Point Group of ‘2’ (Schonﬂies¨ symbol C2, universal cover orbifold 2222). The <9, 3>-generated isotopes have no symmetry at all. Because of the tiling structure of <9, 3>, isotopes derived from this cover cannot have single knotted cycles. The triangle ABCA is null-homotopic, thus any cycle which passes through one of this triangle’s sides produces a loop embedded isotopically to the cycle which passes through the complementary two sides. As every cycle on the tetrahedron must pass through at least one edge of the triangle, there is a 1-1 pairing between 3- and 4- cycles as shown in Table 3, with the exception of the null-homotopic cycle ABCA itself.

33 4 Cube

The cube is a structure whose tangled toroidal isotopes can exhibit higher levels of symmetry than the tetrahedron. We now develop tangled versions of the familiar cube.

4.1 2-cell embeddings of the cube In studying tangled cubes, we do just as in the case of the tangled structures of tetrahedra: we consider the combination of corners and edges that comprise the solid cube. Just as before, to generate our novel cubes, we presume only that the points and edges lie on a closed oriented surface, and that each face is homeomorphic to a disk.

C D B A G H

F E

Figure 16: A labelled cube embedded on the sphere. This familiar 2-cell embedding of the cube graph onto the sphere, which forms a 4, 3 tiling, can be { } described by the ordering of edges around each vertex. For example, the edges around vertex A lead to vertices B, E and D, in a clockwise fashion.

We enumerate the 2-cell embeddings of the cube by labeling the vertices of the cube graph, then considering the orientation of edges around each vertex in the local surface patch. There are two possible distinct edge-orderings around each vertex as each vertex is 3-valent, and eight different vertices, resulting in 28 = 256 different 2-cell embeddings. When these embeddings are analysed, two of them have six faces, 54 have four faces and 200 have two faces. These correspond to embeddings on the sphere, torus and bitorus (genus-2 surface), respectively. As

34 is the case using our method, half of these embeddings are inversions of the other half - as if the tiling was reticulated on the inside of the surface, instead of the outside. For simplicity of analysis we omit the inverted versions by ﬁxing the edge orientation around a certain vertex. With this constraint, there is one sphere tiling, 27 torus tilings and 100 bi-torus tilings. The sphere tiling recovers the conventional cube (via homeomorphism), as shown in Figure 16. Of the 27 torus tilings, there are four different sets of face sizes, as shown in Table 6. The face sizes of a cover are not generally enough to identify a speciﬁc tiling in the universal cover, as one set of face sizes can correspond to multiple tilings.

Table 6: The 27 different genus-one 2-cell embeddings of the cube have four different sets of face sizes.

Number Face sizes 8 4, 4, 4, 12 12 4, 4, 6, 10 3 4, 4, 8, 8 4 6, 6, 6, 6

4.2 The cube in its universal cover There are ﬁve distinct tilings of the plane that can form a universal cover of the cube, one for each of the rows of Table 6 plus an additional tiling with faces of size six. Both these tiling patterns with three hexagons around each corner forms a ‘honeycomb’ 6, 3 pattern, each with a slightly different arrangement of vertices. { } Isotopes from these tilings are a close generalisation of the Platonic cube: All faces, edges and vertices are equivalent in the universal cover. The four different 2-cell embeddings with faces of size six lift to two translational subgroups of 236 in the universal cover. These translational subgroups ∗ are both homomorphic to the fundamental group of the torus, Π1. One 2-cell embedding lifts to a symmetry group which is a normal subgroup of 236, which we ∗ denote as the honeycomb universal cover due to its regular appearance. The other three tilings are symmetrically equivalent conjugate subgroups, which break some of the symmetry of the 236 lattice, and so we denote them brick wall universal ∗ covers to represent the manner in which the regular hexagon grid is distorted. The three remaining tiling types with mixed face sizes do not have the same topological regularity; this coincides with the tilings having lower symmetry i.e. fewer symmetric units per translational cell. It is not clear a priori that each set

35 of face sizes corresponds to only one tiling, so it is necessary to reconstruct each tiling from its 2-cell rotation scheme information. Comparison of these tilings then confirms in these cases that 2-cell embeddings with different rotation schemes but the same set of face sizes come from the same tiling. A more engaging explanation for why this phenomena occurs uses a counting argument following the 2-cell tiling reconstruction. Examining the tilings shown in Fig. 17 shows that for the < 4, 4, 4, 12 > tiling, there is one vertex which is surrounded by only cycles of size twelve. This vertex can correspond to each of the corners of a cube, so the exhaustive search of the rotation schemes finds eight such 2-cell embeddings. Similarly in the <4, 4, 6, 10> tiling, the edge between the two 4-cycles can correspond to each of the twelve edges of the cube, which is enough to uniquely determine the identity of the rest of the points. Finally, in the <4, 4, 8, 8> tiling, the two four-cycles must correspond to the three paired opposite faces of the cube. Images of each of these tilings in the universal cover are shown in Figure 17, while the corresponding cube isotopes with unit side vectors appear in Figure 18. Several tangled cube isotopes were presented in [9], from both the honeycomb and brick wall universal covers.They were previously named A, B, C, D and E, in order of increasing complexity of knots and links that they contain, as shown in Table 7. The criteria used for selecting these isotopes was that there must be a path between any pair of vertices within the unit cell that corresponded to a non-self- intersecting path in the finite graph. One obvious consequence of this requirement is that all vertices have to be within a distance of eight from each other, the length of a Hamiltonian cycle and the number of vertices being eight. In our method this restriction is equivalent to limiting the side vectors to (0, 1), (1, 0), (1, 1), (1, 2) and (2, 1), as well as reflections/rotations into other quadrants. With these criteria there are five different resulting isotopes, from six different combinations of gluing vector and universal cover structure.

Table 7: The side vectors and tiling type of the cube isotopes previously named in [9]. Despite originating from different universal covers, A and A" are isotopic.

Isotope name Universal Cover Side vectors A honeycomb (1,0),(0,1) A" brick wall (1,1),(0,1) B brick wall (1,2),(0,1) C brick wall (1,0),(1,1) D honeycomb (1,1),(1,0) E brick wall (1,2),(1,1)

36 B D B D B D C A C A C A B G B G B G B brickwallF C F honeycombC F C F <4, 4, 8, 8> G E G E G E F E F E F E F H F H F HE D E D E D E A H A H A H A G H G H G H B D B D B D D C D C D C D C C A C A C A B G B G B G B F C F C F C AF B A B A B A B F E F E F E G E G E G E E D E D E D E F H F H F AH H A H A H A G H G H G H D C D C D C D C B D B D B DB G B G B G B C A C A C A F C F C F C AF B A B A B A B F E F E F E G E G E G E E D E D E D E F H F H F AH H A H A H A G H G H G H D C D C D C D C B D<4, 4B, 6, 10D> C A C A C A A A A A B A B A B A B E F E F E F A B A B A B B B B F F F C G C G C G D C D C D C D H E D H E D H E H G H G H G A A A E F E F E F B B B A B A B A B F F F C G C G C G D C D C D C D H E D H E D H E A A A H G H G H G E F E F E F B B B F F F A B A B A B C G C G C G D H E D H E D H E D C D C D C H G H G H G Figure 17: Images of the ﬁve different 2-cell embeddings of the torus cube in the universal cover. The images are of representatives of the 2-cell embeddings shown in Figure 18 with shortest side vectors, rectiﬁed to the unit square for clarity in the mapping to the torus.

37 brickwall honeycomb <4, 4, 8, 8>

C A G D H G D B E F F H E C H G B D F A B E C A

<4, 4, 6, 10><4, 4, 4, 12>

H E A B E G D A F F D C G C B H

Figure 18: Cube isotopes with side vectors (1, 0) and (0, 1) from each of the ﬁve universal cover tilings. The associated universal cover tilings are shown in Figure 17. All these cube isotopes, except for the one from the honeycomb tiling are untangled, and so are ambient isotopic to the spherical cube.

38 4.3 Symmetries of tangled cubes The symmetries of an isotope embedded on a torus can vary according to a number of factors, as discussed in Sections 2.7 and 3.3. Recall that the search for high symmetry tangled toroidal isotopes requires two-fold rotational axes to puncture the torus in the equatorial plane. Thus a torus punctured by n such axes must have n 1 4-tuples of points, separated from each other by a distance of 2n of the longitudinal 1 side vector, and 2 of the meridional vector in its universal cover. One such example is shown in Fig. 19 which corresponds to the isotope shown in Fig. 11.

G E B D G E B D G E B

H C F A H C F A H C F

G E B D G E B D G E B

H C F A H C F A H C F

G E B D G E B D G E B

H C F A H C F A H C F

Figure 19: The universal cover of the ‘E-type’ isotope from [9], shown in Fig. 11. The points where the skewers puncture the torus in that ﬁgure correspond to two- fold points of rotation in the universal cover, marked in this diagram with dots.

With this requirement in mind, observe that the 2-fold rotation points in the 236 lattice occur at the 2 and 6 positions. For convenience of calculation, the ∗ ∗ ∗ lattice can be distorted so that the 2-fold rotation points have integer coordinates (x, y) with every second integer grid-point occupied: when x = y mod 2, as shown in Figure 20. Although this stretching ruptures much of the symmetry of 236 it importantly maintains the 2-fold rotation points. ∗ The possibilities for mirror planes in the three dimensional toroidal cube isotopes are restricted to the untangled cube, with a maximum of order eight symmetry found. In comparison, the most symmetric reticulation of the cube onto the sphere has order 48, reinforcing energy and conventional arguments to show that the natural embedding of the untangled cube in E3 is onto the sphere. The maximum symmetry embeddings of all other toroidally embedded cubes are of lower order, and solely contain equatorial two-fold axes of rotation, (as well as the accompa- nying rotation around the central axis) as shown in Table 8. Table 1 shows the

39 Figure 20: The stretched universal covers for the ‘brick wall’ (left) and ‘honeycomb’ (right) universal covers. Black dots show 2-fold rotation points, grid- lines have integer values, and red dots identify translationally equivalent elements. Hexagon corners and edges correspond to isotope corners and edges. relationship between different symmetry notations and concepts in the context of toroidally embedded polyhedra.

Table 8: The maximum point group symmetry of cube isotopes from each universal cover. Isotopes named in [9] are represented by their letter, while previously unnamed torus cubic isotopes are represented by a ‘!’. The untangled cube is ‘U’, and the isotope A" is isotopic to the isotope A. Blank ﬁelds denote the absence of any toroidal embedding with the given maximum symmetry. The level of point group symmetry in the 3-dimensional structure decreases as the symmetry and the number of inheritable symmetry elements in the universal cover decreases. Order Brick wall Honeycomb <4, 4, 8, 8> <4, 4, 6, 10> <4, 4, 4, 12> 8 U, B, E, ! 4 C, ! A, D, ! ! 2 A", ! ! ! 1 !

4.4 High symmetry cube isotopes Any graph that can embed in space to make a toroidal isotope can make an un- bounded number of toroidal isotopes through varying the vectors in the universal cover that map to the meridional and longitudinal cycles. These inﬁnite number of isotopes can be ranked by the energy function used in Section 2.6, however this single parameter can only represent certain features of the isotope. Another feature

40 of isotopes which can be used to distinguish certain special cases is the maximum embedded symmetry. This motivates the search for high symmetry toroidal cube isotopes. High symmetry tangled toroidal cubic isotopes must have the ‘brick wall’ tiling in their universal cover, as it is the only one to allow order 8 subgroups of the torus, as shown in Table 8. As discussed in section 4.3, isotopes with 224 Point Group symmetries must have four equatorial 2-fold axes of rotation, corresponding to two lines of eight 2-fold rotation points within a unit cell, the space between points in 1 a line being 8 of the longitudinal side vector, and the two lines being separated by 1 2 of the meridional vector. The 4-fold rotational axis along the central axis of the torus, which must accompany the 2-fold axes, manifests in the universal cover as translational symmetries along the meridional vector. Its existence is guaranteed by the presence of the equatorial rotational axes. The allowable gluing vectors which generate such high symmetry isotopes can now be determined. Considering the ‘brick wall’ tiling image from Figure 20, this task is equivalent to choosing two vectors between translationally equivalent points in the universal cover that intersect the appropriate number of 2-fold rotation points. Using lower case letters to refer to integers, we denote the ﬁrst gluing vector (q, r). This sets the vector between adjacent 2-fold rotation points to be (q/8, r/8). However 2-fold rotation points are only to be found at locations (a/8, b/4), with a = b mod 2. So then,

(q, r) = (a, 2b)= q = a and r =2b, (4) ⇒ with q and r being co-prime, and a = b mod 2 further implying that a and b are both odd and co-prime. Thus any first gluing vector of the form (a, 2b) with a, b odd, co-prime and of size at least 1, will ensure a toroidal embedding with eight evenly spaced longitudinal 2-fold rotation points. The second gluing vector must take values such that there is a second line of 2-fold rotational points like the first, but offset by half of the second gluing vector. By inspection or similar arithmetic, every vector (s, t) satisfies this requirement, so all that is required is that qt rs =1, following Equation 2 of Section 2.3. | − | Clearly the requirements upon the values that can be taken by the side vectors to create a highly symmetric cube isotope are still so loose as to allow an infinite number of isotopes with the same degree of symmetry. However, when the dual constraints of high symmetry and low energy are simultaneously applied, the elegant-looking isotopes from [9], named ‘B’ and ‘E’ result. These simple isotopes with high point group symmetry are generated from the ‘brick wall’ tiling in the universal cover by vector pairs ((1, 2), (0, 1)) and ((1, 2), (1, 1)). The E-type isotope is shown in Fig. 19, whilst isotope B and another more ‘highly wound’

41 isotope are shown in Figure 21. This extra example has side vectors ((3, 2), (4, 3)) and is of dubious scientiﬁc interest or aesthetic merit but is included to demonstrate the limits of the power of high symmetry to determine worthwhile structures.

D A H C F G G D F A E H B E a. b. C B Figure 21: Highly symmetric isotopes generated by the vectors (a) ((1, 2), (0, 1)) and (b) ((3, 2), (4, 3))from the ‘brick wall’ tiling in the universal cover. Isotope (a) is the ‘B-type’ isotope from [9]. Both isotopes have high point group symmetry: 224 in Conway orbifold notation.

Other simple torus embeddings from the ‘brick wall’ universal cover, whose gluing vectors do not satisfy the above requirements have lower Point Group symmetry: the ‘C’ isotope has order 4 and the‘ A"’ isotope has order 2 when derived from the ‘brick wall’ tiling. It has order 4 symmetry when derived from the ‘honeycomb’ tiling, as shown in Fig. 23. These two isotopes are shown in Figure 22.

These 224 point group symmetry isotopes have the maximum symmetry of any cube isotope. This is made clear by observing that in these isotopes, every 2-fold point of rotation in the universal cover corresponds to one of the equatorial rotation axes with the torus surface. Thus all rotational symmetries are exhausted, while mirror symmetries are prohibited. No other tiling pattern in the universal cover can provide as many 2-fold rotation points, with the exception of the ‘honeycomb’ hexagon packing, in which the points are misaligned.

4.5 Honeycomb symmetries Using an analysis identical to that in the previous section, it can be seen that every toroidal cube embedding from the ‘honeycomb’ universal cover has maximum symmetry 222. In fact, there are four lines of four 2-fold rotation points parallel to

42 A G H F D D E G C C B A B H F E

Isotope C Isotope A"

Figure 22: Isotope C and an alternate torus embedding of isotope A, denoted A", have point group symmetry 222 (D2) and 2, (C2) respectively, less than the isotopes in Figure 21. They come from the ‘brick wall’ universal cover with gluing vectors ((1, 0), (1, 1)) and ((1, 1), (0, 1)) respectively, and have low energies. Iso- tope A" is ambient isotopic to isotope A, which is derived from the ‘honeycomb’ tiling and has point group symmetry 222. both gluing vectors for any unit cell chosen, so any isotope can have only one pair on its equator, ‘wasting’ the other points in an embedding-symmetry sense. Two simple isotopes from the honeycomb universal cover, named A and D in [9] are shown in Fig. 23.

4.6 Cube isotope knots, links and energies The knots and links present in the cube isotopes derived from the two hexagonally tiled universal covers are presented in Table 11, while the isotopes deriving from the <4, 4, 6, 10> and <4, 4, 4, 12> universal covers are shown in Table 12 and the isotopes from the <4, 4, 8, 8> universal cover have the homotopy types of their cycles shown in Table 13. The named isotopes from [9] - those with gluing vectors not longer than a Hamiltonian circuit - are summarised in Table 9. The table shows that the honeycomb isotope with gluing vectors ((1, 0), (0, 1)) - isotope A - is isotopic to the ‘brick wall’ isotope with gluing vectors ((1, 1), (0, 1)). This isotopy follows from the knot and link information by having the only tangling present being a single Hopf link.

All low energy cube isotopes are shown in Table 10. The notable absence from this list is the <4, 4, 4, 12> tiling. Due to its tiling, it cannot contain links, and for small perturbations to its unit cell contains no knots, while for larger perturbations

43 A H D D C E F G B H G B F

C A E

Isotope A Isotope D

Figure 23: Isotopes A and D have point group 222. They come from the honeycomb universal cover with gluing vectors ((1, 0), (0, 1)) and ((1, 1), (1, 0)) respectively.

Table 9: Cube isotope information for the lowest energy isotopes, named in [9]. The ‘tiling’ column abbreviations are H for ‘honeycomb’ and B for ‘brick wall’. Multiplicative factors indicate the corresponding multiplicity of a certain knot or link type. Name Tiling Gluing vectors knots links A H (1,0) (0,1) - (2,2) A" B (1,1) (0,1) - (2,2) B B (1,2) (0,1) - (2,4) C B (1,0) (1,1) (3,2) 2*(2,2) D H (1,1) (1,0) 2*(2,3) (2,2),(2,4) E B (1,2) (1,1) 4*(2,3),(3,4) 2*(2,2),(2,4)

44 it contains at least six knots, see Fig. 12. Such an isotope has too much energy to appear on this list.

Table 10: The low energy cube isotopes: energies, universal cover tilings, unit cell side vectors, knots and links. The isotopes discovered in [9] are named. Iso- topes generated by small deformations of the unit cell may correspond to multiple embeddings and hence have multiple universal covers. In this case the tiling and unit cell of the lowest energy embedding is presented. The energy of the sphere- embedded octahedron of Fig. 16 – normalised to have the same surface area – is include for comparison. In the notation, multiples indicate the multiplicity of the knot within the isotope, while the subscript indicates the cycle length in which the knot occurs.

Energy Name Tiling Side vectors Links Knots 1.45 - Sphere - - - 1.33 A Honeycomb (1,0), (0,1) (2,2) - 1.76 C Brick wall (1,0), (1,1) (3,2) 2*(2,2) 2.20 B Brick wall (1,2), (0,1) - (2,4) 2.31 D Honeycomb (1,1), (1,0) 2*(2,3) (2,2),(2,4) 2.82 - <4, 4, 6, 10> (1,1), (1,2) (2,2) 4*(3,2) 2.91 - Brick wall (1,0),(2,1) 2*(4,2) (3,2), (5,2) 2.98 - <4, 4, 6, 10> (1,1), (2,1) (4,2) 4*(3,2) 3.14 - <4, 4, 6, 10> (3,1),(1,0) (6,2) - 3.16 - <4, 4, 8, 8> (0,1),(1,3) (2,2), (4,2) 4*(3,2) 3.53 E Brick wall (1,2), (1,1) 4*(2,3),(3,4) 2*(2,2),(2,4)

45 ‘Brick wall’ tangled cube cycles ‘Honeycomb’ tangled cube cycles Cycles of length 4 Cycles of length 4 AEFBA/CGHDC AEFBA/CGHDC ( s, q) ( s, q) − AEHDA/BCGFB − AEHDA/BCGFB ( t, r) # − ADCBA/EHGFE ( t, r) ABCDA/EFGHE ( s t, q + r) − − − Cycles of length 6 Cycles of length 6 ABCGFEA ABCGHEA ABFEHDA ABFGHDA (0, 0) (0, 0) ADCGHEA  ADCGFEA  BCDHGFB  BCDHEFB  ADCGFEA  ABCDHEA  ADHGFBA ABFGCDA ( t, r) ( s, q) AEHGCBA  − ADHGFEA  − BFEHDCB  BFEHGCB  ADCGFBA  ABCGHDA  AEHDCBA AEFBCDA ( s t, q + r) ( t, r) AEHGFBA  − − AEHGFBA  − CGFEHDC  CGFEHDC  ABCGHDA  ABCGFEA  AEFBCDA ABFEHDA (t s, q r) ( s t, q + r) AEFGHDA  − − BFGHDCB  − − BCGHEFB  AEHGCDA  Cycles of length 8 Cycles of length 8 ABCGFEHDA ABFGCDHEA (t s, q r) ADCGHEFBA ADHGCBFEA − − ( s, q) AEFGHDCBA  − ABCDHGFEA ' ( 2s t, 2q + r) AEHGFBCDA  ABFEHGCDA − − AEHDCGFBA ( 2s t, 2q + r) ABCGFEHDA '  − − ( s 2t, q +2r) AEFBCGHDA (t 2s, 2q r) AEHGFBCDA − − − − ' Table 11: Homotopy types of cycles on the cube, for unit cells taken with edges (q, r) and (s, t) for the conjugate (brick wall) and normal (honeycomb) subgroups of *236 symmetry in the universal cover. This information explicitly describes any knot or link which is present. The multiplicative factor of qt rs = 1 for all − ± homotopies is omitted for simplicity of presentation.

46 <4, 4, 4, 12> cube cycles Cycles of length 4 <4, 4, 6, 10> cube cycles BCGFB Cycles of length 4 CDHGC (0, 0) AEFBA/CGHDC ( s, q) EFGHE # − AEHDA/BCGFB ( t, r) ABCDA ( s, q) − − ABCDA/EFGHE ( s t, q + r) AEFBA (s + t, q r) − − − − Cycles of length 6 AEHDA (t, r) − ABCGHEA Cycles of length 6 ABFGHDA BCDHEFB (0, 0) ADCGFEA  BCDHGFB (0, 0) BCDHEFB  BCGHEFB  ABCDHEA CDHEFGC   ABFGCDA ABCGHDA ( s, q)  ADHGFEA  − ABFEHDA ( s, q) BFEHGCB  ABFGCDA  − ABCGHDA ABFGHDA   AEFBCDA AEFBCDA ( t, r)  AEHGFBA  − AEFGCDA (t, r) CGFEHDC  AEFGHDA  − ABCGFEA AEHGCDA   ABFEHDA AEFGCBA ( s t, q + r)  BFGHDCB  − − AEHDCBA (s + t, q r) AEHGCDA  AEHGCBA  − − Cycles of length 8 AEHGFBA   ABFGCDHEA Cycles of length 8 (t s, q r)  ADHGCBFEA − − ABFEHGCDA ( s, q) ABCDHGFEA ' ABCGFEHDA − ( 2s t, 2q + r) ABFEHGCDA − − ADHGCBFEA ' ( t, r) ABCGFEHDA ' ADCBFGHEA − ( s 2t, q +2r) AEHGFBCDA − − ABFGCDHEA ' ( s t, q + r) ' ABCDHGFEA − − ' Table 12: Homotopy types of cycles on the cube, for unit cells taken with edges (q, r) and (s, t) for the <4, 4, 4, 12> and <4, 4, 6, 10> universal covers. The covers are named after their largest face, as shown in Table 6 and Fig. 17. This information explicitly describes any knot or link which is present. The multiplicative factor of qt rs = 1 for all homotopies is omitted for simplicity of presentation. − ±

47 Type <4, 4, 8, 8> tangled cube Cycles of length 4 ABCDA (0, 0) EFGHE ABFEA / CGHDC ' ( s, q) − ADHEA / BFGCD (t, r) − Cycles of length 6 ABCGHDA ABFGHEA (t, r) ADCBFEA  − CGFEHDC  ABCDHEA  ADCGFBA ( s, q) ADHGFEA  − BCGHEFB  ABCGFEA  ABCGHEA (t s, q r) ADCGFEA  − − ADCGHEA  ADHEFBA  ADHGFBA (s + t, q r) BFEHDCA  − − BFGHDCA  Cycles of length 8 ABFGCDHEA (0, 0) ADHGCBFEA ABCGFEHDA ' (t, r) ADCBFGHEA − ABCDHGFEA ' (s, q) ADCGHEFBA − ' Table 13: Homotopy types of cycles on the cube for unit cells with edges (q, r) and (s, t) in the <4, 4, 8, 8> tiling of the universal cover. This information explicitly describes any knot or link which is present. The multiplicative factor of qt rs = − 1 for all homotopies is omitted for simplicity of presentation. ±

48 5 Octahedron

The next largest Platonic polyhedron after the cube is the octahedron. In terms of the combinatorics of the ways that its edges can connect the vertices, it is at the limit of what we can currently exhaustively enumerate and examine without further reﬁnements of the technique.

5.1 2-cell embeddings of the octahedron The octahedron has six vertices, each connected to four others. There are six cyclic ordering permutations of edges around these vertices when embedded into a surface, leading to 66 = 46656 different possible rotation schemes, or half that number if inversions are ignored. The distribution of the genus of the oriented surfaces 46656 under these 23328 = 2 different 2-cell embeddings is: Faces Genus Number of tilings Most symmetric tiling 8 0 1 3, 4 { } 6 1 262 4, 4 { } 4 2 7218 6, 4 { } 2 3 15846 12, 4 { } Clearly the combinatorics heavily favour tiling surfaces with negative Gaussian curvature. We omit these examples to instead build toroidally embedded polyhedral from the Euclidean tilings. Among the 262 combinatorial possibilities for tiling the Euclidean plane, there are 17 unique tiling patterns, the rest of the 262 being related to these by automorphism. The most symmetric cover is the 4, 4 { } cover, with six vertices and six faces in the unit cell. Isotopes derived from the regular sphere and torus tilings, 3, 4 and 4, 4 are shown in Fig. 24. { } { } 5.2 The toroidal octahedron in its universal cover There are 17 symmetrically distinct tilings of the toroidal octahedron in its universal cover, the Euclidean plane. They are shown in Fig. 25 together with the 2-fold rotation points which are important for determining the possible high symmetry toroidal embeddings. The homotopy types of the cycles in these covers are tabulated in Tables 14 and 15. For isotopes generated by non-canonical side vectors, say q, r , s, t , { } { } x x the new homotopy " is related to the tabulated homotopy by y y ! " " ! " 1 x x qr− " = . y y st ! " " ! "! "

49 A F D E B E C D B C F A

Figure 24: The octahedron’s regular tilings of the sphere and the torus. Each embedding has four faces around each vertex; in the sphere embedding the faces are of size three while in the torus the faces are of size four. The torus embedding is the lowest energy embedding of the 4, 4 tiling and contains a Hopf link in cycles { } ABEA and CDFC. There is another higher-energy isotope from the 4, 4 tiling { } produced using a stretched unit cell which was no knots or links, indicating that in this case the presence of the link allows a more ‘relaxed’ structure.

50 A A A A A A B B B B B B B B B B E B A E B A E D A E D A E D A E D A D D F F F C FF C FF C FF C F C F C F F A F A F A F A F A F A C C C C C C E C E C E C DE C DE C DE C E D A E D A E D A E A E A E A B B B B B B B D B D B D B D E B D A E B D A E D A EB D A EB D A EB D A B D B D B

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B D E B D E BD D E C 5.D E C D CE D D A CE DC6. D A CE CD D A CE C D A E CC D A E C D A C E E E E E E E D E E D E D E E D A D A E D A E D A E D A D A B B B B B B B BB B BB B BB B B B B B C B B C B C B C B C B C B F A F A F A F A F A F A F C A F C A F C A F E C A F E C A F E C A F B E E F B E E F B E E F B E F B E F B E C C C D C E D C E D C E D E D E D E F B D F B D F B D F B D F AB D F AB D A A A A A A A D A D A D A D FD A FD A F A F A F A F A C CC CC C CC C C C C C C C B D B D B D B FAD B FAD B FAD FA FA FA E F E F D E F D E FD F D E C FD F E C FD F C D FD A C DC FD A C CD FD A C C D A C D A C D A C E D E E DD E E DD E E DD E E D E E D E E CA E CA E CA E CA E CA E CA B B B B B B B B B B B B D D D D D B D B B F F F C B F C B F C B F C B C B C B C A C A C A E C A A E C A A E C A F AB E E F AB E E F AB E E F AB E F B E F B E F F F F C F C F C C C C F F F F F F D E D E D E D E D E D E E B D E B D F E B D F E B D F E AB D E AB D A F A AA F A A F A A F AA F AA F AA A A A D A D A D A D D D B C B CC B CC BC CC B BC C B BC C B C B C B C B B D B D B D B F FAD B FFAD B FFAD FDFA C FDFA C FDFA C D C D C D C E F E F D E F D E FD D E C FD E C FD C D C A CB D C A CB D C A CB C A B C A B C A B E D E E DD E E DD E E DD E E D E E D E E CA E CA E CA E CA E CA E CA F F F F F F A A B A B B B B B D D D B D B D B D B D BE D BE D E D E D E D E F F F C B F C B F C B F C B C B C B C A C A C A E C A A E C A A E C A F AB E E F AB E E F AB E E F AB E F B E F B E F F myF name is 20 101F!1 C F C F Cmy name is 17 1001C C C my name is 16 1001 F E F E F E F E F E F E D E D A E D A E D A E D A E D A E A B D 7.B D E B D DE B D DEA8.B D D AB D D A FD A 9.A FD A A F A AA F AA F A F A B C B CC B CC BC CC B BC C B CBC C B CC B CC B C B B D B D B D B F FD B FFD B FFD FDF C FDF C FDF C D C D C D C E F E F D E F D E FD D E C FD F E C FD F C D F C A CB D C A CB D C A CB C A B C A B C A B D E D EE D E D E D C E B D CE B C B E CA E CA E CA E CA E CA E D CA D F D F F F F F B B B B D BE D BE D E D E D E D E D D D D D D C B C B D EC B D EC B D EC B C B A A A E A A E A A E A F A E F A E F A E F A F F D E D E D E D E D E D E F E F E F E F E F EA F E A A A A A A A AA AA A A B B B B B A B A E A E A E A E A E E B C B C B C BC EC B BC EC B CBC EC B CC E B CC E B C E B D D D B F FD F B F FD F B F FD F BD F F F BD F F F BD F F F D D D F F F FD C FD F C FD F C D F C A CB D C A CB D C A CB C A B C A B C A B D E D E D E D E D A C E B D A CE B A C B A A A E CA E CA E CA E CA E CA EE D CA E D EF D F F F F F B B B B C D BE C D BE C D E C D E C D E C D E A D A D A D C A D C A D C A D C C B C C B D EC C B D EC B D EC B C B B B B B B B A A A E A A E A A E A F A E F A E F A E F A F F D F E D F E D F E D F E D F E D F E F E F E F E F E F EA F E A A A A A A A AA AA A A E B B E B B E B B E B BD E B ABD E B ABD E A D E A D E A D E A E E F B C F B C F B C F BC EC B F BC EC B F CBC EC B CC E B CC E B C E B C DD C DD C DD B CF FDD F B CF FDD F B CF FDD F BD F F F BD F F F BD F F F D D D F F F FD C FD F C FD F C D F C CB D CC CB D C CBB C BB C B C B E E E E D EE D EEE D E D E DA E D A E A A A A C C my name isC 29 101!1 C C my nameC is 34 101!1 F F my Fname is 3 1001 F F F C D E C D E C D E C D EE C D E C D E A D A D A D C A D C A D C A D C B C B C B B B B 10. A A A 11. B 12. B B B B B A A A A A A A A A A A A F F F F F F E E E E E D E D D A A A A A A E B E B E B E BD E ABD E ABD E A D E A D EE A D EE A EE E F B F B F B F BC E B F BC E B F BC E B C E B C E B C E B C D C D C D B CF D F B CF D F B CF D F BD F F BD F F BDD F F DD DD D F F C D F C D F C D F F F B F B C FB B C B C B C B C B C B D E D E D E D E D A E D A E A A A A C C C C C C F F A F A F A F F C C C C C C AA D AA D AA D C AA D A C AA D A C AA D A C D AE C D AE C D AE D E D E D E A A A B B B B B B A A A ED C A EDA C A EDA C A EDA C EDA C EDA C A F F F F F F F F F F F F E E E E E D E D D A A A A A A E C BB E C BB E C BB E C BBD EBC ABBD EBC AABBD B E A D B E A D B EE A D B EE A EE E F B F F B F F B F F BC EFB F BC EFB F BC EFB C E B C E B C E B C EDD C EDD C EDD B CFEDD F B CFEDD F B CFEDD F BD F F BD F F BDD F F DD DD D F F C D F C D F C D F F F B F B C FB B C B C B C B C B C B D E D E D E D E D A E D A E A A A A C C C C C C F F A F A F A F F C C C C C C AA D AA D AA D C AA D A C AA D A C AA D A C D AE C D AE C D AE D E D E D E A A A B B B B B B ED C ED C ED C ED C ED C ED C F F F F F F F F F F F F E E E E E E A A A A A A E C BB E C BB E C BB E C BBD EBC ABBD EBC ABBD B E A D B E A D B EE A D B EE A EE E F B F F B F myF nameB isF 6 1001F B EFB F B EFB F myB nameEFB is 5 101!1 E B E B E B C EDD CEEDD CEEDD B CEEDD F B C EDD F B C EDD F BD F BD F BDD F DD DD D C B A C B A C B A C B A C B A C B A 13. 14. F 15.F F F F F C C C C C C AA AA AA C AA A C AA A C AA A C D AE C D AE C D AE D E D E D E D D DD DD DDB DB B B B B E C EF C EEF C EEF C EEF C EF C F F F F F F F F F F E C BB E C BB E C BB E C BBD EBC ABBD EBC ABBD B E A D B E A D B E A D B E A E E F F F F F F F EF F EF F EF E E E C EDD CEEDD CEEDD B CEEDD F B C EDD F B C EDD F BD F BD F BD F D D D A A A A A A C C C C C C AA AA AA C AA A C AA A C AA A C A C A C A D D DD DD DDB DB B B B B E C EF C EEF C EEF C EEF C EF C F F F F F F F F F F E C BB E C BB E C BB E C BBD EBC ABBD EBC ABBD B E A D B E A D B E A D B E A E E F F F F F F F EF F EF F EF E E E C EDD CEEDD CEEDD B CEEDD F B C EDD F B C EDD F BD F BD F BD F D D D A A A A A A C C C C C C AA AA AA C AA A C AA A C AA A C A C A C A D D DD DD DDB DB B B B B E C EF C EEF C EEF C EEF C EF C F F F F F F F F F F E C BB E C 16. BB E C BB E C BBD EBC 17.BBD EBC BBD B D B D B D B F F F F F F F F F F F F51 C EDD C EDD C EDD C EDD C EDD C EDD

A A A A A A A A A A A A D D D D D D Figure 25: TheE 17C symmetricallyEF C EF C distinctEF C toroidalEF C octahedraEF C inF their universal C B C B C B C B BC B BC B B B B B F F F F F F ED cover,ED orderedED in reverseED orderED of the energyED of their canonical isotope. The dots

A mark 2-foldA rotationA pointsA A whichA A mapA theA tiling ontoA itself.A A D D D D D D E C EF C EF C EF C EF C EF C F C B C B C B C B BC B BC B B B B B F F F F F F ED ED ED ED ED ED Due to the special qualities of the octahedral 4, 4 tiling, the homotopy types { } of cycles in isotopes derived from q, r , s, t side vectors are explicitly tabulated { } { } in Table 17. Some simple tangled octahedra from the 4, 4 tiling, and their liftings to the { } universal cover are shown in Figures 28 and 26.

5.3 High symmetry toroidal octahedra isotopes Of the 262 tilings of the Euclidean plane generated by considering all rotation schemes of the 2-cell embedded octahedron, there are four that tile the plane with six 4-cycles per unit cell. They are all symmetrically related, so all isotopes generated by the regular 4, 4 tiling can be found by considering stretched unit cells, { } as previously demonstrated for the cube and tetrahedron. Just as in the case of the cube, there are no mirror planes possible in the toroidal isotopes so the key to generating high symmetry isotopes is to ﬁnd tiling patterns in the universal cover that have two lines of 2-fold rotation points within the unit cell, parallel to one side vector. From Fig. 25 it is readily apparent that the 4, 4 tiling is the only candidate { } matching this criteria. An example high-symmetry isotope can be generated by the gluing vector (1, 2) in Fig. 25a coupled with a second vector (0, 1).

These high symmetry isotopes have six lines of two-fold rotation through their equators coupled with a six-fold line of rotation passing through the centre of the torus. Isotopes formed in this way are known as ‘wreath octrahedra’ due to their shape. Several examples are shown in Figs. 28 and 27.

5.4 Properties of toroidal octahedra The lowest energy octahedron isotopes are shown in Table 16. As there are 17 different octahedral tilings, the isotopes formed by slight deformations of the unit cell between different tilings often coincide. Consequently, the source tiling and side vectors identiﬁed should be considered as an example universal cover source, rather than an exclusive one. In these cases the tiling and unit cell associated with the lowest energy isotope is given. The combination of knots and links present in an isotope is not necessarily enough to uniquely identify it, so all potentially duplicitous isotopes are manually checked. Toroidal octahedra derived from the regular 4, 4 tiling have a number of in- { } teresting proprerties. One such feature is that the ‘canonical’ isotope from this tiling, shown in Fig. 24 – with side vectors ((1,0),(0,1)) – contains a Hopf link, while the ‘stretched’ isotope ((1,-1),(0,1)) is untangled, see Fig. 26. The cycle ho-

52 Table 14: The homotopy type of cycles of length three and four from the seventeen canonical unit cells of tilings of Fig. 25. Table 15 shows the continuation of this table: homotopy types of the cycles of length ﬁve and six. The homotopy types for cycles in any one of these tilings with non-canonical side vectors is as described in the text of Section 5.2. The ordering of the tilings refers to the images shown in Fig. 25. 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 ABCA (-1, 1) (-1, 0) (-1, 0) (0, -1) (0, -1) (-1, 0) (-1, 0) (0, -1) (-1, 0) (0, 1) (0, -1) (0, 1) (0, -1) (0, -1) (0, -1) (-1, 1) (0, 1) ABEA (0, 1) (0, -1) (0, -1) (0, 0) (0, 0) (0, -1) (0, -1) (-1, 0) (0, -1) (0, 0) (-1, -1) (-1, 0) (-1, 0) (-1, 0) (-1, 0) (-1, 0) (-1, 1) ACDA (0, 0) (0, -1) (0, 0) (0, 0) (0, 0) (0, -1) (0, -1) (0, 0) (0, -1) (0, 0) (0, 0) (0, 0) (-1, 0) (0, 0) (-1, 1) (0, -1) (0, -1) ADEA (0, 0) (0, 0) (0, 0) (0, 0) (0, 1) (1, 0) (0, 0) (0, 0) (1, 0) (0, -1) (0, 0) (0, -1) (0, 0) (0, 0) (0, 0) (0, 0) (0, 1) BCFB (-1, 0) (-1, 0) (-1, 0) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (-1, 0) (0, 0) (1, 0) (0, 1) (-1, 0) (0, -1) (0, 0) (0, 1) (0, 1) BEFB (0, 0) (0, 0) (0, 0) (1, 0) (1, 0) (0, 0) (0, 0) (0, 0) (0, 0) (1, -1) (0, -1) (0, 0) (-1, 0) (0, -1) (-1, 1) (0, 0) (0, 1) CDFC (0, 0) (0, 0) (0, 1) (1, -1) (1, 0) (0, 0) (0, 0) (1, 0) (0, 0) (1, 0) (0, 0) (1, 0) (0, 0) (1, 0) (-1, 0) (-1, 0) (1, -1) 53 DEFD (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0) (-1, 0) (0, -1) (1, 0) (0, -1) (0, -1) (0, -1) (0, -1) (0, -1) (0, 1) (1, -1) (0, 1) ABCDA (-1, 1) (-1, -1) (-1, 0) (0, -1) (0, -1) (-1, -1) (-1, -1) (0, -1) (-1, -1) (0, 1) (0, -1) (0, 1) (-1, -1) (0, -1) (-1, 0) (-1, 0) (0, 0) ABEDA (0, 1) (0, -1) (0, -1) (0, 0) (0, -1) (-1, -1) (0, -1) (-1, 0) (-1, -1) (0, 1) (-1, -1) (-1, 1) (-1, 0) (-1, 0) (-1, 0) (-1, 0) (-1, 0) ABFCA (0, 1) (0, 0) (0, 0) (0, -1) (0, -1) (-1, 0) (-1, 0) (0, -1) (0, 0) (0, 1) (-1, -1) (0, 0) (1, -1) (0, 0) (0, -1) (-1, 0) (0, 0) ABFDA (0, 1) (0, -1) (0, -1) (-1, 0) (-1, -1) (-1, -1) (-1, -1) (-1, -1) (0, -1) (-1, 1) (-1, -1) (-1, 0) (0, -1) (-1, 0) (0, 0) (0, -1) (-1, 0) ABFEA (0, 1) (0, -1) (0, -1) (-1, 0) (-1, 0) (0, -1) (0, -1) (-1, 0) (0, -1) (-1, 1) (-1, 0) (-1, 0) (0, 0) (-1, 1) (0, -1) (-1, 0) (-1, 0) ACBEA (1, 0) (1, -1) (1, -1) (0, 1) (0, 1) (1, -1) (1, -1) (-1, 1) (1, -1) (0, -1) (-1, 0) (-1, -1) (-1, 1) (-1, 1) (-1, 1) (0, -1) (-1, 0) ACDEA (0, 0) (0, -1) (0, 0) (0, 0) (0, 1) (1, -1) (0, -1) (0, 0) (1, -1) (0, -1) (0, 0) (0, -1) (-1, 0) (0, 0) (-1, 1) (0, -1) (0, 0) ACFDA (0, 0) (0, -1) (0, -1) (-1, 1) (-1, 0) (0, -1) (0, -1) (-1, 0) (0, -1) (-1, 0) (0, 0) (-1, 0) (-1, 0) (-1, 0) (0, 1) (1, -1) (-1, 0) ACFEA (0, 0) (0, -1) (0, -1) (-1, 1) (-1, 1) (1, -1) (1, -1) (-1, 1) (0, -1) (-1, 0) (0, 1) (-1, 0) (-1, 1) (-1, 1) (0, 0) (0, 0) (-1, 0) ADFEA (0, 0) (0, 0) (0, 0) (0, 0) (0, 1) (1, 0) (1, 0) (0, 1) (0, 0) (0, 0) (0, 1) (0, 0) (0, 1) (0, 1) (0, -1) (-1, 1) (0, 0) BCDEB (-1, 0) (-1, 0) (-1, 1) (0, -1) (0, 0) (0, 0) (-1, 0) (1, -1) (0, 0) (0, 0) (1, 0) (1, 0) (0, -1) (1, -1) (0, 0) (0, 0) (1, 0) BCDFB (-1, 0) (-1, 0) (-1, 1) (1, -1) (1, 0) (0, 0) (0, 0) (1, 0) (-1, 0) (1, 0) (1, 0) (1, 1) (-1, 0) (1, -1) (-1, 0) (-1, 1) (1, 0) BCFEB (-1, 0) (-1, 0) (-1, 0) (-1, 0) (-1, 0) (0, 0) (0, 0) (0, 0) (-1, 0) (-1, 1) (1, 1) (0, 1) (0, 0) (0, 0) (1, -1) (0, 1) (0, 0) BEDFB (0, 0) (0, 0) (0, 0) (1, 0) (1, 0) (0, 0) (1, 0) (0, 1) (-1, 0) (1, 0) (0, 0) (0, 1) (-1, 1) (0, 0) (-1, 0) (-1, 1) (0, 0) CDEFC (0, 0) (0, 0) (0, 1) (1, -1) (1, 0) (0, 0) (-1, 0) (1, -1) (1, 0) (1, -1) (0, -1) (1, -1) (0, -1) (1, -1) (-1, 1) (0, -1) (1, 0) Table 15: The continuation of Table 14 showing cycles of length ﬁve and six. 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 ABCDEA (-1, 1) (-1, -1) (-1, 0) (0, -1) (0, 0) (0, -1) (-1, -1) (0, -1) (0, -1) (0, 0) (0, -1) (0, 0) (-1, -1) (0, -1) (-1, 0) (-1, 0) (0, 1) ABCFDA (-1, 1) (-1, -1) (-1, -1) (-1, 0) (-1, -1) (-1, -1) (-1, -1) (-1, -1) (-1, -1) (-1, 1) (0, -1) (-1, 1) (-1, -1) (-1, -1) (0, 0) (0, 0) (-1, 1) ABCFEA (-1, 1) (-1, -1) (-1, -1) (-1, 0) (-1, 0) (0, -1) (0, -1) (-1, 0) (-1, -1) (-1, 1) (0, 0) (-1, 1) (-1, 0) (-1, 0) (0, -1) (-1, 1) (-1, 1) ABEDCA (0, 1) (0, 0) (0, -1) (0, 0) (0, -1) (-1, 0) (0, 0) (-1, 0) (-1, 0) (0, 1) (-1, -1) (-1, 1) (0, 0) (-1, 0) (0, -1) (-1, 1) (-1, 1) ABEFCA (0, 1) (0, 0) (0, 0) (1, -1) (1, -1) (-1, 0) (-1, 0) (0, -1) (0, 0) (1, 0) (-1, -2) (0, 0) (0, -1) (0, -1) (-1, 0) (-1, 0) (0, 1) ABEFDA (0, 1) (0, -1) (0, -1) (0, 0) (0, -1) (-1, -1) (-1, -1) (-1, -1) (0, -1) (0, 0) (-1, -2) (-1, 0) (-1, -1) (-1, -1) (-1, 1) (0, -1) (-1, 1) ABFCDA (0, 1) (0, -1) (0, 0) (0, -1) (0, -1) (-1, -1) (-1, -1) (0, -1) (0, -1) (0, 1) (-1, -1) (0, 0) (0, -1) (0, 0) (-1, 0) (-1, -1) (0, -1) ABFDCA (0, 1) (0, 0) (0, -1) (-1, 0) (-1, -1) (-1, 0) (-1, 0) (-1, -1) (0, 0) (-1, 1) (-1, -1) (-1, 0) (1, -1) (-1, 0) (1, -1) (0, 0) (-1, 1) ABFDEA (0, 1) (0, -1) (0, -1) (-1, 0) (-1, 0) (0, -1) (-1, -1) (-1, -1) (1, -1) (-1, 0) (-1, -1) (-1, -1) (0, -1) (-1, 0) (0, 0) (0, -1) (-1, 1) ABFEDA (0, 1) (0, -1) (0, -1) (-1, 0) (-1, -1) (-1, -1) (0, -1) (-1, 0) (-1, -1) (-1, 2) (-1, 0) (-1, 1) (0, 0) (-1, 1) (0, -1) (-1, 0) (-1, -1) ACBEDA (1, 0) (1, -1) (1, -1) (0, 1) (0, 0) (0, -1) (1, -1) (-1, 1) (0, -1) (0, 0) (-1, 0) (-1, 0) (-1, 1) (-1, 1) (-1, 1) (0, -1) (-1, -1) ACBFDA (1, 0) (1, -1) (1, -1) (-1, 1) (-1, 0) (0, -1) (0, -1) (-1, 0) (1, -1) (-1, 0) (-1, 0) (-1, -1) (0, 0) (-1, 1) (0, 1) (1, -2) (-1, -1) ACBFEA (1, 0) (1, -1) (1, -1) (-1, 1) (-1, 1) (1, -1) (1, -1) (-1, 1) (1, -1) (-1, 0) (-1, 1) (-1, -1) (0, 1) (-1, 2) (0, 0) (0, -1) (-1, -1) ACDFEA (0, 0) (0, -1) (0, 0) (0, 0) (0, 1) (1, -1) (1, -1) (0, 1) (0, -1) (0, 0) (0, 1) (0, 0) (-1, 1) (0, 1) (-1, 0) (-1, 0) (0, -1) ACFBEA (0, 0) (0, -1) (0, -1) (0, 1) (0, 1) (1, -1) (1, -1) (-1, 1) (0, -1) (0, -1) (0, 0) (-1, 0) (-2, 1) (-1, 0) (-1, 1) (0, 0) (-1, 1) ACFDEA (0, 0) (0, -1) (0, -1) (-1, 1) (-1, 1) (1, -1) (0, -1) (-1, 0) (1, -1) (-1, -1) (0, 0) (-1, -1) (-1, 0) (-1, 0) (0, 1) (1, -1) (-1, 1) ACFEDA (0, 0) (0, -1) (0, -1) (-1, 1) (-1, 0) (0, -1) (1, -1) (-1, 1) (-1, -1) (-1, 1) (0, 1) (-1, 1) (-1, 1) (-1, 1) (0, 0) (0, 0) (-1, -1)

54 ADCBEA (1, 0) (1, 0) (1, -1) (0, 1) (0, 1) (1, 0) (1, 0) (-1, 1) (1, 0) (0, -1) (-1, 0) (-1, -1) (0, 1) (-1, 1) (0, 0) (0, 0) (-1, 1) ADCFEA (0, 0) (0, 0) (0, -1) (-1, 1) (-1, 1) (1, 0) (1, 0) (-1, 1) (0, 0) (-1, 0) (0, 1) (-1, 0) (0, 1) (-1, 1) (1, -1) (0, 1) (-1, 1) ADFBEA (0, 0) (0, 0) (0, 0) (1, 0) (1, 1) (1, 0) (1, 0) (0, 1) (0, 0) (1, -1) (0, 0) (0, 0) (-1, 1) (0, 0) (-1, 0) (-1, 1) (0, 1) BCDEFB (-1, 0) (-1, 0) (-1, 1) (1, -1) (1, 0) (0, 0) (-1, 0) (1, -1) (0, 0) (1, -1) (1, -1) (1, 0) (-1, -1) (1, -2) (-1, 1) (0, 0) (1, 1) BCDFEB (-1, 0) (-1, 0) (-1, 1) (0, -1) (0, 0) (0, 0) (0, 0) (1, 0) (-1, 0) (0, 1) (1, 1) (1, 1) (0, 0) (1, 0) (0, -1) (-1, 1) (1, -1) BCFDEB (-1, 0) (-1, 0) (-1, 0) (-1, 0) (-1, 0) (0, 0) (-1, 0) (0, -1) (0, 0) (-1, 0) (1, 0) (0, 0) (0, -1) (0, -1) (1, 0) (1, 0) (0, 1) BEDCFB (0, 0) (0, 0) (0, -1) (0, 1) (0, 0) (0, 0) (1, 0) (-1, 1) (-1, 0) (0, 0) (0, 0) (-1, 1) (-1, 1) (-1, 0) (0, 0) (0, 1) (-1, 1) ABCDFEA (-1, 1) (-1, -1) (-1, 0) (0, -1) (0, 0) (0, -1) (0, -1) (0, 0) (-1, -1) (0, 1) (0, 0) (0, 1) (-1, 0) (0, 0) (-1, -1) (-2, 1) (0, 0) ABCFDEA (-1, 1) (-1, -1) (-1, -1) (-1, 0) (-1, 0) (0, -1) (-1, -1) (-1, -1) (0, -1) (-1, 0) (0, -1) (-1, 0) (-1, -1) (-1, -1) (0, 0) (0, 0) (-1, 2) ABCFEDA (-1, 1) (-1, -1) (-1, -1) (-1, 0) (-1, -1) (-1, -1) (0, -1) (-1, 0) (-2, -1) (-1, 2) (0, 0) (-1, 2) (-1, 0) (-1, 0) (0, -1) (-1, 1) (-1, 0) ABEDFCA (0, 1) (0, 0) (0, 0) (1, -1) (1, -1) (-1, 0) (0, 0) (0, 0) (-1, 0) (1, 1) (-1, -1) (0, 1) (0, 0) (0, 0) (-1, -1) (-2, 1) (0, 0) ABEFCDA (0, 1) (0, -1) (0, 0) (1, -1) (1, -1) (-1, -1) (-1, -1) (0, -1) (0, -1) (1, 0) (-1, -2) (0, 0) (-1, -1) (0, -1) (-2, 1) (-1, -1) (0, 0) ABEFDCA (0, 1) (0, 0) (0, -1) (0, 0) (0, -1) (-1, 0) (-1, 0) (-1, -1) (0, 0) (0, 0) (-1, -2) (-1, 0) (0, -1) (-1, -1) (0, 0) (0, 0) (-1, 2) ABFCDEA (0, 1) (0, -1) (0, 0) (0, -1) (0, 0) (0, -1) (-1, -1) (0, -1) (1, -1) (0, 0) (-1, -1) (0, -1) (0, -1) (0, 0) (-1, 0) (-1, -1) (0, 0) ABFEDCA (0, 1) (0, 0) (0, -1) (-1, 0) (-1, -1) (-1, 0) (0, 0) (-1, 0) (-1, 0) (-1, 2) (-1, 0) (-1, 1) (1, 0) (-1, 1) (1, -2) (-1, 1) (-1, 0) ACBEFDA (1, 0) (1, -1) (1, -1) (0, 1) (0, 0) (0, -1) (0, -1) (-1, 0) (1, -1) (0, -1) (-1, -1) (-1, -1) (-1, 0) (-1, 0) (-1, 2) (1, -2) (-1, 0) ACBFDEA (1, 0) (1, -1) (1, -1) (-1, 1) (-1, 1) (1, -1) (0, -1) (-1, 0) (2, -1) (-1, -1) (-1, 0) (-1, -2) (0, 0) (-1, 1) (0, 1) (1, -2) (-1, 0) ACBFEDA (1, 0) (1, -1) (1, -1) (-1, 1) (-1, 0) (0, -1) (1, -1) (-1, 1) (0, -1) (-1, 1) (-1, 1) (-1, 0) (0, 1) (-1, 2) (0, 0) (0, -1) (-1, -2) ACDFBEA (0, 0) (0, -1) (0, 0) (1, 0) (1, 1) (1, -1) (1, -1) (0, 1) (0, -1) (1, -1) (0, 0) (0, 0) (-2, 1) (0, 0) (-2, 1) (-1, 0) (0, 0) ACFBEDA (0, 0) (0, -1) (0, -1) (0, 1) (0, 0) (0, -1) (1, -1) (-1, 1) (-1, -1) (0, 0) (0, 0) (-1, 1) (-2, 1) (-1, 0) (-1, 1) (0, 0) (-1, 0) ADCBFEA (1, 0) (1, 0) (1, -1) (-1, 1) (-1, 1) (1, 0) (1, 0) (-1, 1) (1, 0) (-1, 0) (-1, 1) (-1, -1) (1, 1) (-1, 2) (1, -1) (0, 0) (-1, 0) ADCFBEA (0, 0) (0, 0) (0, -1) (0, 1) (0, 1) (1, 0) (1, 0) (-1, 1) (0, 0) (0, -1) (0, 0) (-1, 0) (-1, 1) (-1, 0) (0, 0) (0, 1) (-1, 2) ADFCBEA (1, 0) (1, 0) (1, 0) (1, 0) (1, 1) (1, 0) (1, 0) (0, 1) (1, 0) (1, -1) (-1, 0) (0, -1) (0, 1) (0, 1) (-1, 0) (-1, 0) (0, 0) Table 16: The knots and links and tiling and unit cell information regarding the lowest-energy isotopes. In the case where there are multiple tilings that generate the same isotope with the same energy, they are bracketed. The energy of the sphere-embedded octahedron of Fig. 24 – normalised to have the same surface area – is include for comparison. In the notation, multiples indicate the multiplicity of the knot within the isotope, while the subscript indicates the cycle length in which the knot occurs.

Energy Tiling Side vectors Links Knots 2.36 Sphere - - - 1.90 15 (5) (1,-1) (1,0) - (2,2) 2.65 9 (0,1) (1,-1) 2*(2,2) (3,2)6 2.83 1 (1,-1) (1,0) 3*(2,2) (3,2)6 3.07 1 (0, 1), (1, 1) (4,2) - 3.12 5 (1, 1),( 2, 1) (2, 2) (3, 2)5, 2*(3, 2)6 3.29 15 (1, 2), (1, 1) (2, 2) 2*(2, 3)5, 2*(2, 3)6 3.75 15 (2,1), (1, 1) (2, 4) 2*(2, 3)5, 2*(2, 3)6 3.89 1 (1, 0), (1, 1) 3*(2, 2), (4,2) 3*(3, 2)6 3.93 4 (0, 1), (1, 2) (2, 2), (4, 2) 2*(3, 2)5, 2*(3, 2)6 3.94 8 (1, 1), (2, 1) (4, 2) (3, 2)5, 2*(3, 2)6

55 motopy types for this tiling, as necessary to deduce the knottedness and linkedness of the isotope, are shown in Table 17. The equivalent information about the homotopy types of cycles present in toroidal octahedra derived from all 17 tilings is presented in the appendix.

F B

C D C F E E D A A a. b. Figure 26: Simple toroidal isotopes from the 4, 4 tiling, side vectors (a) { } ((1,0),(0,1)) and (b) ((1,-1),(0,1)). The isotope in (a) has a Hopf link, while the isotope in (b) is untangled, as is clear from its already near-planar embedding.

56 B B B B B B

F A F A F A F A F A F A

E C E C E C E C E C E C

B D B D B D B D B D B D

F AA FF my nameA is 120F 1001A F A F A F A

E CC EE CC E C E C E C E C

B D B D BB D B D B D B D

F AA F A F A F A F A F A

E CC E C E C E C E C E C

B D B D BB D B D B D B D

F AA FF A F A F A F A F A

E CC EE CC E C E C E C E C

B D B D BB D B D B D B D

F AA FF A F A F A F A F A

E C E C E C E C E C E C Figure 27: Some unit cells and side vectors of the octahedron in its 4, 4 tiling of { } B D B itsD universalB cover.D TheB green,D yellow,B blueD and redB linesD mark the vectors (1,0), F A (0,1),F (-1,1),A (1,2) respectively.F A TheF green/yellowA F and blue/yellowA F unitA cells correspond to the isotopes in Fig. 26a and b respectively. Features present in these E C isotopesE areC deducibleE fromC theirE unit cells:C cyclesE ABEAC andE CFDCC are disjoint and have homotopy type ( 1, 1) with respect to the green/yellow unit cell and so B D B D B D B − D B D B D form a Hopf link. The same cycles are parallel to the ( 1, 1) vector which corre- − F A spondsF to theA meridionalF loopA in Fig.F 26b;A so inF that embeddingA F theseA cycles are co-planar and parallel to the equatorial plane of the torus. The red line is the vec- E C E C E C E C E C E C tor (1, 2) which aligns with the two parallel lines of dots that mark 2-fold rotation D pointsD of the tiling.D Thus isotopesD which haveD this vectorD map to their meridian can possess high point group symmetry, 226 in Conway notation. The ‘6-fold’ part of this symmetry follows from the six 2-fold rotation points which align along the (2, 1) vector which induces a 6-fold rotation round the axis of the torus. Such high symmetry isotopes are called ‘wreath’ octahedra due to their appearance, as shown in Fig. 28c and d.

57 F C

B D F D E A E C A B a. b. F F B E B E

D C C D A A c. d.

Figure 28: Toroidal octahedra, ordered with increasing structural complexity. All these isotopes are from the 4, 4 tiling, with varying side vectors: ((1,0),(0,1)), { } ((1,0),(-1,1)), ((1,2),(0,1)),((1,2),(-1,-1)). Isotopes (c) and (d) are called ‘wreath’ octahedra, due to their characteristic highly symmetric shape.

58 Cycles of length 4 ABCDA Cycles of length 3 ABFCA ABCA/DEFD  ACDEA ADCA/BEFB ( s, q)  (0, 0)  − ADFEA  ADEA/BCFB   BCFEB  ABEA/CFDC (t s, q r) − − BEDFB Cycles of length 5  ADEBA  ABCDEA  ADFBA  ABEFCA   ADFCA  ADCFBA    ( s, q) AEBCA  ADFBEA  −   AEFBA  ( t, r) AEFDCA  − AEFCA  BCFDEB   BCDEB ABCFDA    BCDFB  ABCFEA   CDEFC  ABEDCA    Cycles of length 6 ABEFDA    ABCDFEA ABFDCA   ABEDFCA ABFDEA    (t s, q r) ABEFCDA ACFBEA  − −  (0, 0)  ABFCDEA  ACFDEA   ACDFBEA  ADCBEA   ADFCBEA  ADCFEA    ACDEFBA  BEDFCB    ADEFCBA BEFDCB    ADEBFCA ADEBCA   ( t, r)  ADFEBCA  − ADEFBA    AEDFBCA  ADEFCA  ( s t, q + r) AEFBCDA  ADFBCA  − −   ABCFDEA  AEFBCA   ABEFDCA  (t 2s, 2q r) BCDEFB   − −  ADCFBEA    ADEFBCA ( 2s t, 2q + r)  − − Table 17: Homotopy types of cycles in the toroidal octahedron from the 4, 4 { } tiling, for unit cells with edges (q, r) and (s, t). The multiplicative factor of qt − rs = 1 for all homotopies is omitted for simplicity of presentation. The paired ± 3-cycles are the only disjoint cycles present, and so are the only potential source of interpenetrating links in the structure.

59 6 Further constructions

6.1 Non- 2-cell torus graph embeddings The 2-cell graph embeddings presented so far are not the only way to embed a graph onto a torus. Alternative embeddings relax the requirement that every ‘face’ of the graph embedding has a single connected boundary component. These embeddings may correspond to the standard sphere embedding, with a ‘bridge’ (or ‘tunnel’) joining two faces, raising the genus of the surface so that it becomes a topological torus. The bridge must not itself be knotted, to allow a smooth deformation of the surface to the standard embedding of the torus. In the universal cover these embeddings correspond to periodic ‘strips’ of the graph embedding stacked next to each other in the plane, separated from each other by an empty ribbon of space: see for example Fig. 29a. Thus while the cover is still two-periodic the graphs can only be periodically traversed along one of those dimensions. Just as the unit cell of the 2-cell embeddings in the universal cover can be wrapped onto the torus in inﬁnitely many ways, so too can these other embeddings. The embedded structure that results once the surface is removed depends on two factors: the deformation of the unit cell, and the relationship between the faces joined by the bridge. If both ends of bridge are within the same face, the graph embedding is ambient isotopic to the spherical embedding. If the bridge joins adjacent faces, no new links are brought into the structure, but every cycle which passes through the edge that separates the newly-joined faces will contain a knot corresponding to the chosen deformation of the unit cell. In effect, by judicious choice of the unit cell a (p, q) torus knot is created for any co-prime p and q. Finally, if the faces joined by the bridge are non-adjacent on the sphere embedding, then the torus embedding will allow both new knots and links, again determined by the unit cell deformation. By appropriate choice of unit cell side vectors, a (p, q) torus knot will be in every cycle that circumnavigates the sphere passing once between the newly-joined faces. Every set of k 2 disjoint cycles which circumnavigate the sphere in such ≥ a manner are part of a k(p, q) torus link. One such example is shown in Fig. 29b. Embedded graphs of this sort have degenerate embeddings in the universal cover if barycentric placement is used to determine vertex location: all vertices and edges converge to lie along a single line. This makes the ”energy” of these structures problematic to reconcile with the 2-cell embedded structures’ energies, so comparison is best done by comparing the knots and links present within the isotope. As this paper focuses on isotopes generated by 2-cell embeddings, these kind of isotopes are generally ignored except for simple isotopes such as that presented in Fig. 29 which are presented for comparison. Note that just as some isotopes can be generated by multiple 2-cell embeddings in the universal cover, some can also be

60 generated by both 2-cell and non- 2-cell embeddings: the twisted cube in Fig. 29 is ambient isotopic to the ‘B-type’ shown in Fig. 21 whilst if the example of Fig. 29 had a single twist, rather than a double – i.e. a (1, 1) alignment rather than a (1, 2) – then it would be equivalent to the ‘A-type’ isotope of Fig. 23.

a. b.

Figure 29: A cube embedded onto a torus in a non- 2-cell manner. The faces with a single boundary component are coloured blue in a) the universal cover and b) the embedding in three dimensions, while the face with two boundary components is coloured green for emphasis. In this example, two loops form a Whitehead link, as per Fig. 10.

6.2 Duals An interesting side note to the isotopes constructed in this paper is the structure of the embedded graphs’ duals. The dual of an embedded graph is constructed by mapping the faces of the graph to the vertices of the dual graph, and connecting them if the two faces of the original graph are connected. In this way faces map to nodes, nodes map to faces, and edges of the original graph cross the edges of the dual graph. Several examples are shown in Fig. 30.

The dual relationship of sphere-embedded Platonic polyhedra is well known: the cube and octahedron have a dual relationship, as do the icosahedron and dodecahedron, while the tetrahedron is self-dual. In tiling terms, the dual of a p, q { } tiling is a q, p tiling. { } Structures reticulated onto the torus have different duals to those reticulated

61 onto the sphere: the dual of either 6, 3 cube tiling – 4 faces – is the 3, 6 torus { } { } tiling – with 4 vertices. The high-symmetry octahedron embedding, 4, 4 , is self- { } dual, whereas the duals of non-regular tilings have mixed vertex types due to the varying face sizes of the original tilings. Both these regular tiling examples are displayed in Fig. 30, as the dual of the <4, 4, 8, 8> tiling, which is irregular.

There is a remarkable connection between the tangledness of a 2-cell graph embedding and its dual: for every knot or link in an isotope, there is the corresponding same knot or link in the dual of its graph embedding. This phenomenon is apparent in the universal cover: for every (non-null-homotopic) path corresponding to a closed cycle in the isotope there is a sequence of adjacent faces immediately be- side it, and bounded by a neighbouring ”parallel” translationally equivalent path. If there is more than one parallel path per unit cell - as is the case for links - then there is a corresponding number of ribbons of adjacent faces bounded on each side by these parallel paths. Each ribbon of faces has the same displacement in the universal cover, and hence the same homotopy type on the torus, as its bounding paths; and there are the same number of ribbons as paths. Thus any knot or link present in the closed paths is also present in the ribbons, and thus by deﬁnition in the dual of the graph embedding. The length of such knotted or linked cycles is generally not consistent between the original embedded graph and its dual.

A search for a toroidally embedded isotope which contains a unique set of knots and links (counting multiplicities) is thus limited to the 4, 4 octahedral { } tiling, as any other isotope shares the same knot and links with its dual. However any graph embedding which contains the 4, 4 octahedral surface tiling as a mi- { } nor within it will contain the same knots and links as that graph embedding. The minor of a graph is any graph that results from a source graph by removing edges and vertices, including merging two edges across a vertex which connects to only those two edges, or shrinking edges down to merge the vertices at each end. The concept of a graph minor extends in the natural way to graph embeddings and periodic tilings. Graph minors and some of their applications are discussed in detail in Chapter ??.

It is not clear that there must be an extra knot or link in every toroidal isotope that is larger than the octahedron and contains the octahedron. So in the search for an isotope with a unique set of knots and links, the further limitations on this larger graph required to provide a positive result dilute the interest of this phenomenon beyond that required to deserve further investigation.

62 a. b.

c. d.

e. f.

Figure 30: Cubes and an octahedron, superimposed on their duals, shown on the torus and in their universal cover. (a) and (c) show different torus embeddings of the untangled cube isotope, with (b) and (d) their universal covers. Although the dual of a surface graph embedding is unique, different embeddings of an isotope vary, and hence the (embedded) graphs dual to each cube embedding are different. (e) and (f) show an embedding and the universal cover of the octahedral 4, 4 { } tiling. This tiling is self-dual – the structure and its dual are identical – and so is the genus-one analogue of the tetrahedron63 embedded on the sphere. 6.3 Chirality All isotopes shown in this chapter, except for those containing no knots or links, are chiral. This can be confirmed by examination, and motivates the result presented in [29] and Chapter three: that any polyhedral toroidally embedded isotope that contains a knot or link is chiral. That result still leaves the question of which specific chirality any given isotope may take. The chirality is determined by many factors during its generation, each alternate choice of which is capable of reflecting the resultant isotope between its left- and right- handed conformation. These factors include:

Interpreting the handedness of the rotation scheme, leading to a choice be- • tween two mirrored tilings in the universal cover.

Mapping the universal cover to the torus. Several choices are contained • within equation 3:

– The sign of the side vectors. – The universal cover is mapped to the ‘outside’ of the torus rather than its ‘inside’. – When the unit square is rolled into a cylinder, this deformation can occur ‘into’ or ‘out from’ the page.

and so some attention must be paid to these factors in generating an isotope with a given chirality. Note that the ordering choice of which pair of side vectors to identify ﬁrst does not alter the chirality. This due to the process of torus eversion shown in Fig. 8 which exchanges the meridian and longitudinal cycles without al- tering direction or chirality.

In practice it is simplest to make a convenient choice for the factors above, as has been done in this paper, and then vary the signs of the side vectors to generate isotopes with alternate chirality.

7 Conclusion

Graphs can embed into three dimensional space in different ways. Some graphs, such as polyhedral graphs have a simple embedding which contains no tangling. On the other hand, there is no upper bound on the potential complexity of the embedding: both cycles and individual edges can be knotted up to arbitrary complexity, while these knots and cycles can also interlink, or be entangled in other ways,

64 forming a giant entangled mess. Torus embeddings allow us to walk a fine line between these two extremes, allowing a controlled level of structural complexity. This chapter has introduced a technique to generate 2-cell toroidally embedded versions of arbitrary graphs, with a simple method to vary the knots and links present within the isotopes of these embedded graphs. If no such toroidal embedding exists for a given graph, that will be readily apparent from the 2-cell information. This chapter focuses on the isotopes generated by three specific Platonic polyhedra: the tetrahedron, cube and octahedron. The knots and links present within all toroidal 2-cell embeddings have been comprehensively enumerated and symmetry and ‘energy’ arguments have been used to identify interesting isotopes from the panoply of tangled structures generable using this method. The Euclidean plane – the universal cover of the torus – is a convenient locale to study the effect of varying the shape of the unit cell upon the resultant surface embeddings. The results provided in this chapter inform an understanding of a more difficult setting: varying the unit cell shape of Hyperbolic tilings such as those used in the [24] project. Hyperbolic tilings can be used to make 3-periodic surface embeddings, which then produce repeating crystals once the surface is removed, in direct analogy with the method shown in this paper. Varying the shape of the unit cells in this context will gernerate ‘twisted’ and otherwise deformed 3-periodic patterns with novel structural complexity that can be altered at the users whim. This work has direct application to chemistry in the form of ‘DNA origami’ and the design of carcerands – ‘host’ molecules which encage their ‘guest’ species. Conventionally such carcerands are topologically spherical but should they be tem- plated upon a topologically toroidal substrate, such as a molecule which contains a single large-enough cycle, they could provide a new class of topologically distinct carcerands. In ‘DNA origami’, too, this work is relevant. Complex microstructures are often built with ‘sticky ends’ connected to DNA junctions. A ‘sticky end’ occurs when one strand of the DNA double helix is severed, leaving a strand of A,C,G and T bases which will stick only to the complementary base sequence – the sequence that was removed. In this technique, DNA junctions have multiple strands of DNA emerging from a junction, each of which is tipped with a specially tailored ‘sticky end’ which should ensure that it connects only to the complementary sticky end, which is also present. This process is a direct chemical analogy with the technique presented in this paper; our technique predicts that such reactions designed to produce spherically-embedded structures will also produce tangled toroidal isotopes (or ‘topoisomers’ as they are referred to in that field).

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