## Cubic Orbifold Atlas - Notes on Figures and Main Table

Underlying Topological Space

Before interpreting the orbifold drawings, one should first note the underlying topological space listed at the top of each atlas page. The 3-sphere S3 may be considered regular 3-space plus a point at infinity. Thus in #196 we have a tetrahedral singular set with one W2 invariant limiting lattice complex at the center of the tetrahedron and a second W2 on the point at infinity. Thus there are two geodesic paths joining the two J2 invariant lattice complexes as indicated by the dotted line in the left-hand drawings for #196.

A 3-disk D3, has a mirror boundary made up of one to three Wyckoff mirror regions delineated by thicker lines. There are two Wyckoff mirror regions in #221, each of which has two subregions to forming the four faces of the tetrahedron. Note that mirror regions are usually depicted as spherical (e.g., #223) rather than flat as in #221.

Projective planes RP2 and RP3 cannot be drawn directly in regular 3-space. The dashed line around a cone base signifies the 2-dimensional antipodal convention (e.g., #201), and the opposing arrows on a sphere signify the 3-dimensional antipodal convention (e.g., #197). Antipodal means that upon exiting the figure surface at a given angle of incidence, you go half way around the (cone) axis or (sphere) center, then reenter the surface at the same angle of incidence.

Characteristic (resp. Non-Characteristic) Orbits = Lattice (resp. Limiting Lattice) Complexes

A set of points generated from a point Xo by the space group G is called a crystallographic orbit Og(Xo). Let E be the symmetry (eigensymmetry) of this set of points. If E=G, Og(Xo) is a characteristic crystallographic orbit. However if E is a supergroup of G, Og(Xo) is a non-characteristic crystallographic orbit with respect to G. The table lists the standard (characteristic) Wyckoff sets (top rows without cover column entries), followed by the non-characteristic ones. The characteristic and non-characteristic sites characterize symmetry and pseudo-symmetry respectively. Characteristic and non-characteristic orbits are also called lattice complexes and limiting lattice complexes, respectively.

Each invariant (fixed point) lattice complex is a single point within the orbifold labeled with an invariant lattice complex symbol in the left-hand drawing. However, the present drawings sometimes do not show all the invariant lattice complex points listed in the table. This is partially to prevent clutter but more often it means we we are not sure where some of them should be positioned at present.

The orbifold tables are based on the International Tables for Crystallography, Vol. a; the book by P. Engel, T. Matsumoto, G. Steinmann and H. Wondratshek, "The Non-Characteristic Orbits of the Space Groups", Zeit. fur Krist. (1984), Supplement Issue No. 1; and the lattice complexes references given in our chapter on that subject.

The left singular set figure of the 3-orbifold drawing may display a pseudo-symmetry among equal, but distinct, Wyckoff sites. Wyckoff sites are labeled by small letters in the right-hand drawing. The pseudo symmetry is related to the group normalizer discussed under Wyckoff Sets below, but we do not have an analytical correlation at present. Empirically, it appears as a local geometric point symmetry (222, 2mm, 2/m, 2 or m) within the left drawing and is listed in the top line of each page to aid visual recognition of the FPS. The corresponding parent supergroups containing a true symmetry subcomponent (i.e., m or 2) of the FPS are listed in the FPS and 2[4]-Cover columns.

Each row in the table represents a Wyckoff set which may contain two or more equivalent but distinct Wyckoff sites. The first number in the Mult column gives the multiplicity of the Wyckoff site and the second the number of Wyckoff sites in the Wyckoff set.

The invariant lattice complex sites in the cubic space groups, listed in increasing Wyckoff-site unit-cell multiplicity [n], are: [1] P; [2] I; [3] J; [4] F, +Y, -Y; [6] J*, W; [8] D, +Y*, -Y*; [12] S, +V, -V, W*; [16] T, Y**; and [24] S*, V*. A subscript 2 on a symbol denotes a 2 x 2 x 2 block of unit cells, which increases the above multiplicity by a factor of 8.

A pair of invariant lattice complex points (X and Y) of multiplicity m and n, respectively, joined by an axis of multiplicity p has the univariant lattice complex symbol (p/m)X[-](p/n)Y. For example, the invariant lattice complex F's and W's of #208 (Wyckoff site multiplicity 4 and 6, respectively) are joined by 2-fold axes (l1, l2, k1, k2) of multiplicity 12, producing the symbol F3[-]W2. The numbers 2 and 3 in this univariant lattice complex symbol denote coordination multiplicity around the invariant lattice complex sites F and W, respectively.

One or more limiting lattice complex points may be on a geodesic line between two invariant lattice complex (or limiting lattice complex) points. Limiting lattice complexes always have the multiplicity of the site on which they lie. For example, there is an univariant lattice complex P4[F]P4 in #195 with F on a 3-axis of multiplicity 4; thus F has multiplicity 4. If only one limiting lattice complex and no other punctuation symbols is listed in brackets between two invariant lattice complexes, it is positioned midway way between them along the geodesic line.

There is also a F3[-]W2 univariant limiting lattice complex in #195 with F and W on axes of multiplicity 4 and 6, respectively, joined by an "axis" of site symmetry 1 with multiplicity 12, which makes it homeomorphic to the univariant (non-limiting) lattice complex example from #208 discussed above. This relation is recorded in Col. 6 of #195. Also note from the left orbifold figure of #195 that the two F3[-]W2 univariant limiting sites are on a common geodesic (straight line in this case) and form a 2-fold axis of normalizer pseudo-symmetry (i.e., the Figure Pseudo-Symmetry in Col. 1).

The singular set of the orbifold forms a graph, and all components of the graph (i.e., nodes and links) are spherical orbifolds. A link between nodes is a subgroup of both nodes. Cubic orbifold #201 has 332, 30 and 222 nodes joined by 2- and 3-fold links and a 2-fold loop (via the antipodal relationship). The group graph symbols for the univariant lattice complexes are 33<2>22, 32<3>0 and 2<2>& with <2>, <3> the links, <2>&, the loop, and 33, 22, 32 unmatched half links. If the link is a 1-fold "axis" there are more unmatched half links such as 332 and 222 in 332<1>222 of #201.

Pseudo-symmetry in the Group Graph column is denoted by a * following the symbol such as 2* and m* for pseudo 2-fold axes and pseudo mirrors, respectively. For example in the above example we have 2*=332<1>222. This * notation is consistent with the 2-color cubic orbifold notation in Appendices A and B where symmetry and antisymmetry (i.e., pseudo symmetry) are discussed.

All space groups G, other than the last two cubics (#229 and #230) have a normalizer N(G) which is a proper supergroup of G. For the cubic space groups, all normalizers (n.b., affine = Euclidean here) are other cubic space groups. A Wyckoff set is the set of Wyckoff positions that are interchanged by the normalizer group N(G).

Each row in the table denotes a Wyckoff set that will contain one or more equal but distinct Wyckoff sites. The Wyckoff site notation of International Tables For Crystallography, Volume A, Space-Group Tables, T. Hahn, Ed. (1992) is used. It is often necessary to divide a Wyckoff site into sub regions denoted by numbers following the Wyckoff letter. For example, the 3-axis 'e' in #196 is partitioned into 4 components, e1, e2, e3 and e4. The required partitioning is much more apparent with orbifold drawings than it is with space group symmetry drawings.

In the Wyckoff Set column, the Wyckoff point, line, plane, and 3-space sites are ordered hierarchically with points as subcomponents of lines, lines and points as subcomponents of planes, etc. There are also footnotes attached to the first half of the cubic space groups to help clarify the correlation between space-group geometry and orbifold singular-set topology.

For the non-characteristic Wyckoff sets, the second (or fourth order if necessary) supergroup that has the corresponding characteristic orbit is given for the univariant and divariant limiting lattice complexes.

A special cubic group/subgroup table, with pointer just above the "clickable" 6 by 6 main entry table for the cubic space group orbifolds, is also clickable. That table provides a mechanism for surfing the orbifold tables by second order subgroups or supergroups (i.e., those which have solid connecting arcs). Fourth (dashed connecting arc)order group/subgroup relationships are also shown but not clickable. See our chapter on cubic groups/subgroups for a detailed description of the table. Only the cubic entries in the group/subgroup table are clickable.

## Notes on Heegaard Splitting Table

Crystal Structures are represented as Morse function critical nets which are the topological critical points and interconnecting separatrix lines of overlapping Gaussian atomic thermal-motion density functions within the crystal. An orbifold contains one wrapped-up (orbifolded) asymmetric unit (fundamental domain) of the crystal structure which must be topologically compatible with the singular set of the orbifold. Minimum gradient separatrix lines go downhill between adjacent peak-pass, pass-pale, and pale-pit pairs of critical points and maximum gradient ones connect adjacent peak-pit pairs.

A smooth surface intersecting all pass-pale lines without intersecting peak-pass and pale-pit lines or critical points is a Heegaard surface. This surface characterizes the crystal structure and is ideally a constant density (level) surface. Enumerating the set of irreducible Heegaard surfaces is a classical approach for characterizing the topology of a manifold (or orbifold). This corresponds to finding a basis set of different real or conceptual simple critical nets for each space group expressed in terms of Heegaard surfaces, which is the goal in mind, with possible application is crystal structure prediction.

Heegaard splitting along the Heegaard surface divides an orientable Euclidean 3-orbifold into two orientable handlebody 3-orbifolds. For non-orientable (i.e., centrosymmetric space group) Euclidean 3-orbifolds the two parts are called compression bodies, or (if there are no boundary components present) non-orientable handlebodies. For each case, one component contains the peak and pass critical points and the other contains the pales and pits.

Critical net drawings for most of the cubic structures in the atlas are shown in the lattice complex chapter. The Heegaard surface in such drawings is a plane midway up the drawing and parallel to the bottom edge.

The three example splittings for #196 ,i.e., (a) ZnS, (b) NaCl, and (c) FCC, yield the orientable handlebody 3-orbifold pairs with group graphs (a) two 32<3>32, (b) two 33<2>33, and (c) 32<3>2<3>32 + 332<1>22. However, this handlebody group-graph information is not listed explicitly in the current orbifold atlas. Instead we provide information on the type of invariant lattice complexes sites occupied by each critical point. For instance in the above example we have: peak / pass /pale / pit = (a) FF / T / T / FF, (b) FF / J2 / J2 / FF, and (c) FFF / TT / J2 / F. Information on the Heegaard surface 2-orbifold nomenclature and its approximate positioning within the 3-orbifold also is given.

Structure types include generic types such as body and face centered cubic (BCC, FCC), simple cubic (SCub), diamond (Diam), as well as specific chemical formuli such as NaCl, CsCl, NaTl, ZnS, etc. Certain hypothetical structure types use the peak site lattice complex symbol such as Y** in #230. The dual (interchanging peak/pass/pale/pit with pit/pale/pass/peak) of Y** is S, and the dual of FCC is Li2O.

Here a multiplicity of n means that the relevant Wyckoff set multiplicities of the orbifold allow the complete critical net of the structure to be imbedded in n equivalent but different orientations, thus there are n equivalent Heegaard surfaces. An s following the multiplicity signifies symmetric splitting. For example, the high Wyckoff set multiplicities for orbifold #196 allows multiplicity two for the symmetric ZnS splitting, multiplicity one for the symmetric NaCl splitting, and multiplicity four for the asymmetric FCC splitting.

The critical points are tabulated in the sequence peaks / passes / pales / pits. In the #196 example ZnS and NaCl have their peak + pit critical points on invariant lattice complex point of type F, but the two structures have their passes + pales on lattice complex points of types T and J2, respectively. The FCC derivative in this example (#196) has 1 peak, 1 pass, 2 pales and 3 pits.

A Heegaard surface in a Euclidean 3-orbifold is occasionally an Euclidean 2-orbifold but more often is a hyperbolic 2-orbifold. Nomenclature for the hyperbolic 2-orbifolds is similar to that of the spherical (point group) and Euclidean (plane group) 2-orbifolds. The nomenclature is determined by noting the boundary mirrors, n'-axes in boundary mirrors, n-axes, and all pass-pale separatrix lines which intersect the Heegaard surface. We record the first three as 2-orbifold boundaries, dihedral corners, and cone points, respectively. We then calculate the Euler characteristic X from those elements and distinguish the hyperbolic and Euclidean cases by noting if X is < 0 or = 0, respectively.

For example, the Heegaard surface entry H3232{11} for ZnS in #196 denotes a hyperbolic (H) surface with four cone points (3232). The {11} indicates that two (nonsingular) pass-pale lines pass through the Heegaard surface. This information can often be obtained directly from the corresponding critical net drawing in the chapter on critical nets. This nomenclature scheme is under review and may be replaced by something that reflects the two-sided character of Heegaard surfaces.

The Wyckoff elements cut by the Heegaard surface are listed in this column. Sometimes a element may be cut twice and this is indicated by dividing the element into subcomponent, as indicated in the trailing number, and listing all relevant subcomponents. For example, the FCC Heegaard splitting in #196 has four different Wyckoff cut sets, because the multiplicity is four, but only one of the four is listed (e1 e2 g f1 f2). The Wyckoff 3-axis e is already partitioned into four parts (e1 e2 e3 e4) in the figure because a 3-axis automatically continues through a 332 node as discussed in our web tutorial; but f is shown as a single link in the figure. Thus we must conceptually partition f into two equal parts, f1 and f2, then cut each sub partition. Thus the Heegaard surface is a triangular tunnel going along the base (e1 e2 f) of the tetrahedra entering at the f e3 e4 face and chopping off the e1 e2 g vertex.

The Heegaard surfaces for the ZnS and NaCl examples in #196 are simple normal surface quadralateral planes which cut through two pairs of opposite edges in the tetrahedra. The FCC example can be made into a normal surface problem by subdividing the tetrahedron into sub-tetrahedra, but that is a piece-wise linear combinatorial topology approach and we prefer to use smooth manifold methods when posssible.