Generalized bipyramids and hyperbolic volumes of alternating k-uniform tiling links

# Generalized bipyramids and hyperbolic volumes of alternating $k$-uniform tiling links

## Abstract.

We present explicit geometric decompositions of the hyperbolic complements of alternating -uniform tiling links, which are alternating links whose projection graphs are -uniform tilings of , , or . A consequence of this decomposition is that the volumes of spherical alternating -uniform tiling links are precisely twice the maximal volumes of the ideal Archimedean solids of the same combinatorial description, and the hyperbolic structures for the hyperbolic alternating tiling links come from the equilateral realization of the -uniform tiling on . In the case of hyperbolic tiling links, we are led to consider links embedded in thickened surfaces with genus and totally geodesic boundaries. We generalize the bipyramid construction of Adams to truncated bipyramids and use them to prove that the set of possible volume densities for all hyperbolic links in , ranging over all , is a dense subset of the interval , where is the volume of the ideal regular octahedron.

Acknowledgments: Thanks to the referee for extremely helpful comments and to Abhijit Champanerkar, Ilya Kofman and Jessica Purcell for their useful input. Thanks to the NSF Grant DMS1659037, the Williams College SMALL REU program, and NSF graduate fellowship DGE1122492 for supporting our research.

## 1. Introduction

For many links embedded in a 3-manifold , the complement admits a unique hyperbolic structure. In such cases we say that the link is hyperbolic, and denote the hyperbolic volume of the complement by , leaving the identity of implicit. Computer programs like SnapPy allow for easy numerical computation of these volumes, but simple theoretical computations are rare. In [CKP15a], Champanerkar, Kofman, and Purcell present an explicit geometric decomposition of the complement of the infinite square weave, the infinite alternating link whose projection graph is the square lattice (see Figure 1), into regular ideal octahedra, one for each square face of the projection. The volume of the infinite link complement is infinite, but it is natural to study instead the volume density, defined as where is the crossing number. For infinite links with symmetry like , we can make this well defined by taking the volume density of a single fundamental domain. Since faces of are in bijective correspondence with crossings, the decomposition of [CKP15a] shows the volume density to be , the volume of the regular ideal octahedron and the largest possible volume density for any link in .

In this paper we generalize their example and give explicit geometric decompositions and volume density computations for infinite alternating -uniform tiling links, alternating links whose projection graphs are edge-to edge tilings by regular polygons of the sphere (embedded in ), the Euclidean plane (in ), or the hyperbolic plane (in ) such that their symmetry groups have compact fundamental domain. (The terminology -uniform refers to the fact there can be transitivity classes of vertices, for any finite ). In the cases of Euclidean tilings and hyperbolic tilings, we avoid working directly with infinite links by taking a quotient of the infinite link complement by a surface subgroup of the symmetry group. In the case of a Euclidean tiling, we quotient by a subgroup and obtain a finite link complement in a thickened torus . Independently, in [CP17], the authors also determine the hyperbolic structures for these alternating tiling links.

In the case of a hyperbolic tiling, we consider thickened higher genus surfaces, denoted where is the genus. These surfaces are permitted to be non-orientable, in which case the genus is defined as where is the Euler characteristic. This definition differs from the standard “non-orientable genus” by a factor of 2, with the benefit that all surfaces of a given genus have equal Euler characteristic, though in exchange surfaces may have half-integer genus. When the specific genus is unimportant, we use to denote any thickened surface of genus . The crossing number of a link in is defined as the minimal number of crossings across all possible projections onto . It follows from [AFLT02] and[Fle03] that for an alternating link in , is realized only for a reduced alternating projection of onto . We assume throughout that the two boundary components of in the hyperbolic structure on are totally geodesic, and hence the metric is uniquely determined (see Theorem 2.3).

In Theorem 4.2, we prove that the hyperbolic structure on a alternating -uniform tiling link is realized when the bipyramids corresponding to the faces of the tiling have dihedral angles of the edges ending at the top and at the bottom apexes equal to the angles of the tiling when it is realized as an equilateral tiling.

This analysis of alternating -uniform tiling links yields independent results about all link complements in and that are hyperbolic. We show that volume densities for links in are bounded above by . This should be compared with the upper bound on volume density of for links in the -sphere, which follows from the octahedral decomposition of D. Thurston as noted below. For fixed genus , we show the volume density of links in is bounded by a number . The bounds correspondingly approach as increases.

We show that alternating tiling links derived from the 4-regular hyperbolic -gon tilings achieve the maximal volume density for each , and use these to prove the set of all possible volume densities of links in is dense in the interval . As a corollary to the proof, we have that volume densities for links in the thickened torus are dense in the interval . This extends previous results that volume densities are dense in for knots in from [Bur15] and [ACJ17].

In [Thu99], D. Thurston described a decomposition of the complement of a link in into octahedra, using one at each crossing, as in Figure 2(a). These octahedra have two ideal vertices, located on the cusps at that crossing, and four finite vertices which are identified in pairs to two finite points. The two points are thought of as being far above and below the projection sphere for the link, and we denote them and (for up and down). Any such octahedron has volume less than , so this gives an upper bound on the volume of the link, . There exist links such that asymptotically approaches . (cf. [CKP15a], [CKP15b]).

In [Ada17], the octahedral decomposition is rearranged into face-centered bipyramids. Thurston’s octahedra are cut open along the core vertical line connecting the ideal vertices, yielding four tetrahedra as in Figure 2(b). The tetrahedra each have two ideal and two finite vertices, identified with and . The edge connecting the finite vertices passes through the center of one of the four faces adjacent to the crossing for that octahedron, as shown in Figure 2. This edge is shared by one tetrahedron from each crossing bordering that face, which glue together to form a bipyramid. The apexes of the bipyramids are the finite vertices and , while the vertices around the central polygon are ideal. Thus we can think of the link complement as decomposing into bipyramids, one per face of the link projection, each such -bipyramid corresponding to a face of edges. Any such -bipyramid has volume less than the maximal volume ideal -bipyramid, shown in [Ada17] to be regular, with volume bounded above by and asymptotically approaching . These bipyramids are denoted . The construction puts an upper bound on volume

 vol(L)<∑ivol(Bidealni)

where denotes the number of edges in the -th face of the projection of . For link projections whose faces have many edges, this gives a dramatically better upper bound on volume than is generated directly from Thurston’s octahedra.

Much of this paper is devoted to generalizing the bipyramid decomposition to links in and , and then exploiting the symmetry of alternating -uniform tiling links to precisely determine the complete hyperbolic structure. In Section 2, we present relevant background on generalized hyperbolic polyhedra. Section 3 deals with the maximal volumes for certain classes of generalized polyhedra which appear in the generalized octahedral and bipyramidal decompositions. In Section 4, we prove Theorem 4.2, which shows that in the face-centered bipyramid decomposition of the complement of an alternating -uniform tiling link, the complete hyperbolic structure is realized by choosing the dihedral angles along edges incident to the apex in a given bipyramid are equal to the planar angles of the corresponding polygon in the equilateral realization of that -uniform tiling, and they determine the dihedral angles on the equatorial edges.

The explicit construction of hyperbolic structures on alternating -uniform tiling links allows us to calculate exact volume densities for all links associated to 3- or 4-regular -uniform tilings and many -uniform tilings with a mix of 3-valent and 4-valent vertices. Section 4 also contains the aforementioned results on the set of volume densities for all links in thickened surfaces, showing they are a dense subset of the interval .

We are grateful to Yair Minsky for helpful comments and to the referee, who gave us particularly detailed and helpful feedback on an earlier version of the paper. We were also greatly aided by Xinyi Jiang, Alexander Kastner, Greg Kehne and Mia Smith.

## 2. Geometric structures

A generalized hyperbolic tetrahedron is the convex hull of four points which may be finite (within ), ideal (on ), or ultra-ideal (outside ). Ultra-ideal points fit most naturally into the Klein ball model of . Given an ultra-ideal point outside the unit sphere in , consider the cone of lines through tangent to the sphere. The canonical truncation plane associated to is the plane containing the circle where that cone intersects the sphere. Geodesic lines and planes through the ultra-ideal point are computed as Euclidean lines and planes through the point as usual, cut off at the truncation plane (see Figure 3). Note that all edges of the tetrahedron that passed through before truncation are perpendicular to the corresponding truncation plane. For more background on ultra-ideal points, see, e.g., [Ush06].

We will often prefer to work in the Poincaré ball model for computing lengths and angles. Since the two models agree on the sphere at infinity, we can do this by using the Klein model to locate the ideal boundaries of geodesic lines and planes, and then construct the geodesics corresponding to those boundaries in the Poincaré model.

A generalized hyperbolic tetrahedron is fully determined by its six dihedral angles ([Ush06]). We can thus specify a tetrahedron with a vector , with the dihedral angles labelled as in Figure 4. We restrict our attention to mildly truncated tetrahedra, those in which truncation planes for distinct ultra-ideal vertices do not intersect within . It should be understood that all truncated tetrahedra in this paper are mildly truncated. In this case there is a formula for the volume of a generalized tetrahedron in terms of its dihedral angles.

###### Theorem 2.1 ([Ush06]).

Given a generalized hyperbolic tetrahedron with dihedral angles as in Figure 4, let

 a=eiA,b=eiB,...,f=eiF,
 G=⎛⎜ ⎜ ⎜⎝1−cosA−cosB−cosF−cosA1−cosC−cosE−cosB−cosC1−cosD−cosF−cosE−cosD1⎞⎟ ⎟ ⎟⎠,
 U(z,Δ)=12(Li2(z)+Li2(abdez)+Li2(acdfz)+Li2(bcefz)−Li2(−abcz)−Li2(−aefz)−Li% 2(−bdfz)−Li2(−cdez)),

where is the dilogarithm function, defined by the analytic continuation of the integral

 Li2(x)=−∫x0log(1−t)tdt

for . Then the volume of is given by

 \emphvol(Δ)=12Im(U(z1,Δ)−U(z2,Δ)).

The generalized volume formula in terms of the angles is unwieldy, but the total differential takes an elegant form in terms of the edge lengths. This is Schläfli’s differential formula, in the three-dimensional case (see, e.g. [Sch50], [Kel89]).

###### Theorem 2.2 (Schläfli’s differential formula).

Given a generalized hyperbolic tetrahedron with dihedral angles and corresponding edge lengths , the differential of the volume function is given by

 d\emphvol(Δ)=−126∑i=0ℓidθi.

Let be an anannular finite volume hyperbolic 3-manifold with boundary components of genus greater than 1. By doubling the manifold along its boundary, we obtain a finite volume manifold. The Mostow-Prasad Rigidity Theorem and the fact the orientation-reversing homeomorphism fixing the boundary components is realized as an isometry yields the following.

###### Theorem 2.3.

A finite volume anannular hyperbolic 3-manifold with boundary has exactly one finite-volume complete hyperbolic structure with totally geodesic boundary.

## 3. Maximal generalized tetrahedra and bipyramids

On a given link complement in any of the cases we have discussed, both Thurston’s octahedral construction at the crossings and the bipyramid construction in the faces yield octahedra and bipyramids that decompose into generalized tetrahedra with two ideal vertices.

###### Lemma 3.1.

Let be a tetrahedron labelled as in Figure 4 with and ideal vertices. For fixed dihedral angle on the edge between and , the maximal volume for such a tetrahedron is achieved when the other angles are

 D=arccos(12(cosA−1)),
 B=C=E=F=π−D2.

In this case and are ultra-ideal.

###### Proof.

The condition that and are ideal vertices imposes the constraints

 B+F+D=C+E+D=π.

We use Lagrange multipliers to maximize the volume subject to those constraints. Let

 g1(B,C,D,E,F)=B+F+D−πg2(B,C,D,E,F)=C+E+D−π

Then the method of Lagrange multipliers with multiple constraints makes us consider solutions of

 ∇B,…,F,λ,μ(vol(A,…,F)−λg1(B,…,F)−μg2(B,…,F))=0

which yields

 ∂vol∂B=∂vol∂F=λ
 ∂vol∂C=∂vol∂E=μ
 ∂vol∂D=λ+μ

By Schläfli’s differential formula, those equations become

 ℓB =ℓF ℓC =ℓE ℓD =ℓB+ℓC.

However, all these edges have at least one ideal endpoint, so the lengths are infinite. To recover useful information from Schläfli’s formula, we replace the ideal vertices with finite vertices and and consider the limiting behavior as they approach the ideal points and . That is, we require

 lim¯v2→v2(ℓB−ℓF)=lim¯v1→v1(ℓC−ℓE)=lim¯v1→v1¯v2→v2(ℓD−ℓB−ℓC)=0.

Up to isometries of in the Poincaré ball model, we may let , , and , as in Figure 5. Then lies at some point

 v3=(r3cosDsinϕ,r3sinDsinϕ,r3cosϕ).

Let be the endpoints of edges B, C, E, and F respectively, opposite from and . That is, if is finite or ideal then , but if is ultra-ideal then and are the intersections of their respective edges with the truncation plane for , and similarly for and . The point lies on a unique horosphere centered at , given by

 x2+y2+(z−(1−RB))2=R2B

for some . By symmetry, lies on an opposite horosphere of the same radius centered at . Similarly and lie on horospheres of radii and around and respectively. As and approach and , limits to the finite distance between the concentric horospheres of radii and , and similarly limits to the distance between concentric horospheres of radii and . Furthermore, since we can split edge D at the origin, tends to twice the distance between concentric horospheres of radii and . Thus the optimization conditions become

 RE=RF=RB=12.

This is satisfied if and lie at the origin, but then is degenerate with volume 0. Alternatively the truncation planes for and must both be mutually tangent to the two horospheres of radius centered at and . Therefore must be equidistant from the Euclidean centers of the horospheres, which lie at , so it must lie on the -plane. Thus . At this point we can see by symmetry of the tetrahedron and the ideal vertex constraints that

 B=C=E=F=π−D2.

The Euclidean radius of the truncation plane for is , so by the Pythagorean theorem we have

 (√r23−1+12)2=r23+(12)2

which implies . By the same reasoning . Thus and .

Finally we compute an explicit relationship between and . The top and bottom faces of this tetrahedron are given by

 (x−1√2)2+(y−1−cosD√2sinD)2+(z±1)2=1−cosDsin2D,

and the angle between them is

 A=arccos(1+2cosD).

Inverting the function, we arrive at

 D=arccos(12(cosA−1)).

###### Corollary 3.2.

The maximal volume generalized tetrahedron with two ideal vertices has volume , with angles

 A=0,D=π2,B=C=E=F=π4.
###### Proof.

We see immediately from the Schläfli formula that decreasing any one angle increases the volume. Thus among tetrahedra with two ideal vertices as described in Lemma 3.1, the maximal volume occurs when . The other angles follow from the formulas in the lemma. The volume of this tetrahedron is . This can be seen by gluing four copies around the central edge with angle to obtain the polyhedron in Figure 6. By chopping off eight tetrahedra from this polyhedron, each with one finite vertex and three ideal vertices, we are left with a single ideal regular octahedron. The eight tetrahedra we chopped off can be reassembled around the finite vertex to create a second ideal regular octahedron. Hence the polyhedron has volume and the original tetrahedron has volume . ∎

###### Corollary 3.3.

The maximal volume of a generalized hyperbolic octahedron with two opposite ideal vertices is .

###### Proof.

As mentioned, a generalized hyperbolic octahedron with two opposite ideal vertices can be cut into four tetrahedral wedges around the core line connecting the ideal vertices. All four of these wedges may be the maximal tetrahedron described in Corollary 3.2, yielding the generalized octahedron as shown in Figure 6. By decomposing as in Figure 7 and then recomposing, we can turn this into two ideal regular octahedra with volume . ∎

Lemma 3.1 also allows us to determine the maximal bipyramids with ideal vertices around the central polygon and ultra-ideal apexes.

###### Corollary 3.4.

The maximal volume generalized -bipyramid with ideal vertices around the central polygon is made of identical maximal volume tetrahedra of the type described in Lemma 3.1, with angles .

###### Proof.

A generalized -bipyramid with ideal vertices around the central polygon can be cut along its core line into tetrahedra , each with two ideal vertices. Label the edges of each as in Figure 5, with subscripts to distinguish the tetrahedron to which they belong. The ideal vertices on each tetrahedron dictate the constraints

 Bi+Fi+Di=Ci+Ei+Di=π,

and the fact that the tetrahedra are all glued together along the core line additionally requires

 n∑i=1Ai=2π.

We again maximize using Lagrange multipliers and substitute from Schläfli’s formula, obtaining the equations

 ℓAi =ℓAj ℓBi =ℓFi ℓCi =ℓEi ℓDi =ℓBi+ℓCi

for all . By the same logic used in the proof of Lemma 3.1, the last three equations imply that is isometric to a tetrahedron in the Poincaré ball model with vertices

 vi1=(0,0,−1),vi2=(0,0,1),vi4=(√2,0,0),vi3=(√2cosDi,√2sinDi,0).

In this model it is apparent that increases monotonically with . Thus implies , and consequently . The other angles depend on as in the lemma because the tetrahedra are maximal. ∎

We refer to the bipyramids described in Corollary 3.4 as maximal doubly truncated -bipyramids, denoted . Some volumes of these bipyramids are shown in Figure 7(b). We note that the ratio

 vol(Btruncn)n

is strictly increasing with , asymptotically approaching as the tetrahedral wedges making up the bipyramids approach the maximal wedge (with angle ) discussed in Corollary 3.2. This contrasts with the case of ideal bipyramids, where

 limn→∞vol(Bidealn)n=0,

peaking when at , the volume of a regular ideal tetrahedron (see [Ada17]).

## 4. Alternating k-Uniform Tiling Links

A tiling of , or is -uniform if it consists of regular polygonal tiles meeting edge-to-edge such that there are transitivity classes of vertices and a compact fundamental domain for the symmetry group of the tiling.

We can associate an alternating linkÊ to a -uniform tiling as follows. Since there is a compact fundamental domain, in a Euclidean tiling, we can quotient to get a torus. In a hyperbolic tiling, by Selberg’s Lemma, there is a finite index normal subgroup of the symmetry group of the tiling. Thus, we can project the tiling of or to a closed surface . (For a spherical tiling, the sphere already plays the role of ). In the case of a 3-regular tiling, the graph corresponding to the edges of the tiling satisfies the hypotheses of Petersen’s Theorem (see, e.g. [Pet91], [Bra17]),Ê which therefore implies there is a perfect matching. That is to say, there is a subcollection of the edges of such that each vertex is an endpoint of exactly one such edge. Each edge in the subcollection is then replaced with a bigon.Ê Thus, we may now assume we have a 4-regular tiling of , allowing bigon tiles.

It may be the case that there are nontrivial simple closed curves avoiding the vertices that cross Ê an odd number of times. But because is 4-regular, any isotopy of a simple closed curve will preserve the parity of the number of intersections. Take a cover of corresponding to the finite index subgroup of generated by the squares of the generators.Ê Then, lifting the graph to a graph in , all simple closed curves on cross an even number of times. (In the case of being the sphere, all simple closed curves already cross an even number of times.) Hence, we can checkerboard shade the tiling corresponding to . We can now choose the crossings at the vertices so that for the white regions, the strands on the boundary go from under a crossing to over a crossing as we travel clockwise around the boundary. For the black regions, the strands on the boundary go from under a crossing to over a crossing as we travel counterclockwise around the boundary. This yields a consistent way to chose crossings so the resultant link is alternating. In the case of a spherical tiling, the link lives in .

In the case of a Euclidean tiling, the link lives in . In the case of a hyperbolic tiling, has genus at least 2 and the link lives in . We call such a link an alternating -uniform tiling link. We can then lift the choice of crossings back to the link in or to obtain the corresponding infinite alternating -uniform tiling link.

Note that this process can result in multiple associated infinite alternating -uniform tiling links to a -uniform tiling, defined both by the first choice of how to put in a crossing at a vertex, by where we choose to add bigons and by the size of the fundamental domain we choose. By taking larger and larger fundamental domains, the number of possible links grows exponentially. For example, the 6.6.6 tiling has an infinite number of ways we may add bigons to make it a -regular graph, all of which respect some of the symmetry group of the tiling. (The notation means that around every vertex, we have in consecutive order a -gon, a -gon, an -gon, and an -gon.) Our results hold independently of the choice of such bigons.

The alternating -uniform tiling links derived from spherical tilings correspond precisely to Archimedean polyhedra. Although we include that case here, note that in [AR02], the authors classified exactly the alternating hyperbolic links corresponding to the 3-regular and 4-regular Archimedean solids.

For link complements in , with genus , we concern ourselves exclusively with the case where and are totally geodesic. This satisfies the hypothesis of Theorem 2.3 to ensure that a complete hyperbolic structure of finite volume is unique if it exists. Throughout this section, we often repress the genus and refer to any thickened surface of genus by .

Just as a link in can be decomposed into bipyramids with some finite vertices, from which a volume bound can be obtained, link complements in can be decomposed into bipyramids with all ideal vertices. Similarly, link complements in can be decomposed into bipyramids with all ideal vertices except the apexes, each of which is truncated (i.e. treated as an ultra-ideal vertex).

The generalized octahedral and bipyramid decompositions apply in this context to give upper bounds on the volumes of any link complements in in terms of its crossing number . In the octahedral upper bound, the maximal ideal octahedron with volume is replaced with the maximal octahedron with two opposite ideal vertices, with volume by Corollary 3.3.

###### Corollary 4.1.

If is a hyperbolic link in , then .

We also obtain an upper bound on volume in terms of the face-centered bipyramids by replacing the regular ideal bipyramids that yielded the upper bound for a link in by the maximal doubly truncated bipyramids .

### 4.1. Computing volumes of tiling links

In this section we provide a method to compute the volumes for all alternating -uniform tiling links in appropriate thickened surfaces.

###### Theorem 4.2.

An alternating -uniform tiling link has a hyperbolic complement realized by maximally symmetric face-centered bipyramids. Moreover, the angles between edges of the regular polygons in the equilateral realization of that -uniform tiling determine the dihedral angles along edges incident to the apexes for the face-centered bipyramids, and those angles determine the dihedral angles on the equatorial edges.

The statement about an equilateral realization of the tiling is irrelevant in the Euclidean case, as the angles in a -uniform Euclidean tiling are independent of the side lengths. For non-Euclidean tilings, the angles may vary significantly. For instance, a spherical projection of a truncated cube with small triangular faces has very different angles than one with large triangular faces.

###### Proof.

We prove the theorem by explicitly demonstrating that the corresponding face-centered bipyramids glue together along their faces so that around each edge, they fit together correctly via isometries of the faces to yield an angle of . Moreover, the cusps inherit a Euclidean structure. We ultimately see that this implies that there is a decomposition into positive volume ideal tetrahedra that satisfy the gluing equations of Thurston as in [Thu80]. Hence the link complements have complete hyperbolic structures as described.

In the three cases of spherical, Euclidean and hyperbolic alternating -uniform tiling links, the top and bottom apexes of the bipyramids are finite, ideal and ultra-ideal, respectively. In all three cases, vertical edges are defined to be the edges that have an endpoint at either the top or bottom apex (or, in the hyperbolic case, they have a vertex on the truncation plane). The remaining edges, always with ideal endpoints, are called equatorial edges.

We now describe the explicit geometric realization of the bipyramids that will make up the link complements. We choose them to have the maximum possible symmetry. We first consider tilings that are 4-regular and therefore require no insertions of bigons.

For each -gon face in the projection of a tiling link to or , the vertical edges of the corresponding -bipyramid will all have the same dihedral angle , which is the angle at the corresponding vertex of in the equilateral realization of the tiling of or . The equatorial edges will all have dihedral angle . Note this must occur to realize maximal symmetry for the bipyramids while ensuring that the sum of the four dihedral angles for the edges sharing each ideal equatorial vertex add up to .

In the Euclidean case, where the top and bottom vertex of the bipyramids are ideal, this completely determines each -bipyramid up to isometry. In the upper-half-space model of , we can choose one apex at and one apex at the origin. Then the fact all vertical edges have the same angle forces the equatorial vertices to form the vertices of a regular polygon centered at the origin, thereby determining the bipyramid up to isometry (see Figure 10).

But in the spherical case, we still have the freedom to determine how far the finite vertices are from the equatorial plane. Let be the length of the edge of the equilateral spherical tiling on a unit sphere, and let be the angle at the origin between rays to the endpoints of that edge of length on the sphere. Then for each , the corresponding -bipyramid with two finite vertices is chosen to be maximally symmetric so that the angle between the edges that share a face at the finite vertex is . Hence, all faces of all bipyramids are isometric to one another and can be glued together in pairs according to the necessary gluings. Each small sphere in the link complement centered at the finite vertex will be tiled by its intersections with the bipyramids as a scaled up or down version of this tiling of the sphere. The bipyramids then fit together at each of the two finite vertices so that the link of each of the two finite vertices is a sphere.

In the case of a hyperbolic alternating -uniform tiling link, we choose each generalized bipyramid so that the edge length of each edge in its two totally geodesic faces is exactly the length of the corresponding edge in the equilateral -uniform tiling of . This ensures that all non-boundary faces of the bipyramids are isometric and can be glued together as necessary. When they are glued together, the totally geodesic faces fit together to form a tiling of the two surfaces that form the boundary of .

Note that in the case of the monohedral tilings of (only one such) or by right-angled -gons, all the -bipyramids are isometric, and all their dihedral angles are equal to .

Around a crossing, the edge labels on the adjacent bipyramids appear as in Figure 11. We first verify that the sum of the dihedral angles in each edge class add up to , and the bipyramids around the edge can be glued together along their faces. For the edge classes of vertical edges, this is automatic, since there are four bipyramids, each with one edge contributing to an edge class of vertical edges. The four edges each have dihedral angle equal to the angle of the corresponding polygon’s vertex in the tiling that generates the bipyramids. Since the links of the apexes fit together to form a tiling, the four angles of the polygonal tiles around the vertex must add up to . As we glue the bipyramids on, one at a time, around a given edge, the final gluing fits together a face on the last bipyramid with a face on the first, since their crossections on the totally geodesic boundary tile the boundary.

Each edge class for the equatorial edges corresponds to an edge in the link complement at one of the crossings, going from the undercrossing to the overcrossing. As in edge 1 of Figure 11, there are four bipyramids, each with one edge in the edge class, with dihedral angles and . But we know from vertical edge 2 that , and since , we have that , and the four bipyramids glue together via isometries on their faces. Becasue of the maximal symmetry of the bipyramids that we are using, the link of each ideal vertex is a rhombus. Hence, when we glue the rhombi around a vertex corresponding to the crossection of an edge, the edge of the last rhombus matches in length with the length of the edge of the first rhombus, to which it is to be glued, as in Figure 12.

We now check that the gluings as described yield a complete structure on the cusps corresponding to the link complement. As mentioned, the link of each ideal equatorial vertex of a bipyramid is a rhombus, and those rhombi glue together around each vertex to yield exactly of angle.

But then, as in Figure 12, the eight rhombi fit together in groups of four each around the top strand and around the bottom strand of the crossing. In each case, the top edge of the resultant diagrams glues to the bottom edge by Euclidean translation. This Euclidean transformation corresponds to the holonomy of the meridian of the link. Since the rhombi fit together nicely around each vertex corresponding to the edges, we see that the holonomy of the longitude is also forced to correspond to a Euclidean transformation. Therefore the cusp has a complete Euclidean structure.

We now consider the case when there are bigons. Thus, the original graph was 3-regular, and every vertex appears on exactly one bigon. Since, in the -uniform tiling, the sum of the angles around each vertex is , there are two polygons meeting along the bigon edge, with associated bipyramids and and there are two polygons, each at each end of the bigon edge, which must have the same angles and therefore they are -bipyramids with identical . Call their associated bipyramids and . (See Figure 13.) Again, for each , the vertical edges have dihedral angles , where is the angle on the corresponding regular polygon in the tiling. And the equatorial edges have dihedral angles where, as before, it is still the case .

Then it is immediate from the tiling that and . So again, the vertical edges glue up to provide an angle of for the corresponding edge classes. Also note that = and .

The vertical edges inside the two crossings at each end of the bigon are isotopic to one another, so we collapse them down to a single edge as in [Men83] (see also [AR02]). This means that and each contribute two equatorial edges to this edge class, and and contribute one equatorial edge each to this edge class. So we need . But since , we know so the desired equation follows from the previous equations. Note in particular we know .

As in Figure 14, we see how the rhombi corresponding to each of the ideal equatorial vertices of the bipyramids glue together to create the cusp. Again, we see that the top and bottom edges of the resulting figure glue to one another via a Euclidean translation. Again, this implies that the cusp inherits a complete Euclidean structure.

We now show in each of the three cases of spherical, Euclidean and hyperbolic tilings, that from our bipyramids, we can obtain an ideal tetrahedral decomposition into explicit tetrahedra that satisfy Thurston’s gluing equations. Hence the bipyramids do yield a complete hyperbolic structure to the manifolds.

We first consider the spherical case. In [Men83] and implicitly in [Thu80], a method is given to decompose an alternating link complement in into two combinatorially equivalent polyhedra, with the gluings on the faces provided to reproduce the link complement. Although here, we use the face-centered bipyramids to decompose the link complements, in the case of links in , one need only cut each bipyramid open along its equatorial polygon, and glue all the top halves of bipyramids together and the bottom halves together to attain Menasco’s decomposition into two polyhedra. Note that these two polyhedra are both combinatorially equivalent and isometric, with all totally geodesic faces. The polyhedra are then easily decomposed into ideal tetrahedra (cone all vertices to one vertex for instance), which because of how the bipyramids glue up, will satisfy Thurston’s gluing equations.

In then Euclidean case, each ideal regular -bipyramid decomposes into ideal tetrahedra glued around the central edge from one apex to the other. The collection of resulting ideal tetrahedra satisfy the Thurston gluing equations and thus give a complete hyperbolic structure to the manifold.

In the case of a link in , we can take the decomposition into face-centered bipyramids and then double along the two totally geodesic boundaries. Each bipyramid doubles to a pair of bipyramids, glued to each other along their two totally geodesic faces. If we cut each of the bipyramids open along their equatorial -gons, we obtain two regular ideal -prisms. Thus the double of decomposes into regular prisms. Since the tiling of each totally geodesic boundary by its intersections with each generalized bipyramid is equilateral, each of these prisms must have the property that the perpendicular length between any of the edges that run from the top to the bottom and that share a face must be the same. Call that length . Note that we can define a regular prism up to isometry by , which is the number of edges on its top or bottom face, and , this perpendicular distance between the vertical edges on a shared face. Let be such a prism. Each such prism can then be decomposed into ideal tetrahedra, which, because of how the bipyramids and hence prisms fit together, will again will satisfy the Thurston gluing equations and generate a valid complete hyperbolic structure on the manifold.

Note that in all cases of a hyperbolic alternating -uniform tiling link, the two totally geodesic boundaries of are isometric, both being tiled by the same -uniform tiling.

Theorem 4.2 allows us to compute the volumes of alternating -uniform tiling links by knowing the volumes of the corresponding -bipyramids. As in the proof of Theorem 4.2, splitting the bipyramidal decomposition of a spherical alternating -uniform tiling link into top and bottom pyramids along their equatorial planes and gluing the tops and bottoms around the vertex links of and respectively, we recover Menasco’s decomposition into two ideal polyhedra. Thus for an alternating link derived from a spherical -uniform tiling we obtain a decomposition of into two copies of the polyhedron associated to the spherical tiling.

###### Corollary 4.3.

A alternating -uniform link corresponding to a spherical tiling has volume exactly twice that of the ideal hyperbolic polyhedron of the same combinatorial description with ideal vertices on the sphere at infinity placed according to the equilateral tiling on the sphere.

Note that this fact is implicit in the work of [AR02], where explicit hyperbolic structures for links corresponding to Archimedean solids were determined. In [Riv94], it is proved that that a maximally symmetric hyperbolic polyhedron has maximal volume for the polyhedra of the same combinatorial type, so these polyhedra are maximal volume of the combinatorial type.

The volumes of these polyhedra are enumerated in Figure 15 along with the angles of their associated spherical tilings.

In Figure 16 we give explicit values for the volume densities of links in appropriate thickened surfaces.

### 4.2. Volume density in S×I

In addition to finding exact values for the volumes of links in and , we can also ask about the set of their volume densities . The generalized octahedral construction with one octahedron per crossing and and ideal immediately gives the following lemma.

###### Lemma 4.4.

The volume density of a link in a thickened torus is at most .

The upper bound is realized by a link in by taking the quotient of the infinite square weave link via a subgroup of the symmetry group of the square tiling.

The generalized bipyramid construction with and ultra-ideal, along with Corollaries 3.2 and 3.4 immediately gives the equivalent result for .

###### Lemma 4.5.

The volume density of a link in a thickened surface is strictly less than .

###### Lemma 4.6.

There exists a sequence of links in thickened surfaces with volume densities approaching .

###### Proof.

For , let be an infinite alternating -uniform tiling link derived from the tiling of and be a closed surface such that its fundamental group is a subgroup of the symmetries of . Suppose a fundamental domain for is composed of -gons, and let denote the tiling link in after quotienting by the appropriate surface group. Note that there will be -gons in a projection of onto . By Theorem 4.2 the volume of is where denotes the generalized -bipyramid with all dihedral angles equal to . The crossing number of the resulting link in is seen to be by results in [AFLT02] and [Fle03]. Hence the volume density of in is . Observe that breaks up into isometric generalized tetahedra with dihedral angles and . Sending to infinity makes these wedges tend to the maximal tetrahedron from Corollary 3.4, so just as with , we see as in the discussion following Corollary 3.4,

 limn→∞4vol(B□n)n=2voct.

###### Theorem 4.7.

The set of volume densities of links in across all genera , assuming totally geodesic boundary, is a dense subset of the interval .

###### Proof.

Let and We will find a link such that its density is within of . By Lemma 4.6 and its proof, there exists an alternating link in a thickened surface coming from a tiling of with , such that . By taking an -fold cyclic cover of around one handle, we obtain a link in , where has higher genus than , such that both the volume and crossing number of have been scaled up by a factor of .

As in Figure 17(a) and (b), we choose two strands in the projection on the same complementary region and we add a trivial component so that the resulting projection of is still alternating. In [AARH17], it was proved that a link in that projects to an alternating projection on such that the closure of each complementary region is a disk must be hyperbolic. So the complement of in is hyperbolic.

Drilling out raises the volume of by some amount which may depend on and the particular choice of drilling. We rearrange the projection as in Figure 17(c). Then we perform -Dehn filling on , replacing the two original strands with a region of crossings to create a link , as for example appears in Figure 17(d). The resulting link is alternating, and hence by [AARH17] is hyperbolic. Moreover, its volume is at most . The crossing number of is since a reduced alternating projection in is known to realize the minimal crossing number (see [AFLT02]).

To show that the density of the resulting link is less than , we set

 tn=⌊nvol(L)+bn2x⌋−nc(L)2

for each . From the discussion above, we can calculate that

 D(L′′′)≤nvol(L)+bnnc(L)+2tn+1=x+nvol(L)+bn−x(2⌊nvol(L)+bn2x⌋+1)2⌊nvol(L)+bn2x⌋+1

and by increasing the numerator by and decreasing the denominator by on the right hand side we see

where the last inequality follows from the fact that for any ,

 A−B⌊AB⌋=B(AB−⌊AB⌋)

Now we can make the denominator on the right hand side arbitrarily large and so by setting large enough we get

 D(L′′′)

To prove that the volume density does not drop too much throughout this procedure, we employ a result of Futer, Kalfagianni, and Purcell [FKP08], which provides a lower bound for volume change under Dehn filling along a slope :

 vol(L′′′)≥(1−(2πℓ)2)3/2vol(L′′)

where is the length of . Since (and thus ) increases without bound as does, we can choose sufficiently large such that

 (1−(2πℓ)2)3/2>xx+ϵ.

Now observe that by our choice of we have

and hence we get that

 vol(L′′)nc(L)+2tn+1≥x−xnc(L)+2tn+1>x−εx

for large enough . Therefore putting all of our estimates together, we see that for sufficiently large ,

 D(L′′′)≥(1−(2πℓ)2)3/2vol(L′′)nc(L)+2tn+1≥(xx+ϵ)(x−εx)=x2−εx+ε=x−ε.

Note that cyclic covers of tori remain tori and hence the following result can be deduced through arguments similar to those above.

###### Corollary 4.8.

The set of volume densities for links in is a dense subset of the interval .

Taking cyclic covers of a surface of genus increases the number of handles, and so it is natural to ask about the set of volume densities in a surface of a fixed genus. As mentioned in the introduction, we define the genus of a non-orientable surface in terms of the Euler characteristic as , so that “fixed genus” directly translates to “fixed Euler characteristic.”

Suppose we fix a surface of genus . Then the maximal volume density of a link in is achieved when the average volume per crossing is highest. As the average volume per crossing increases as we increase the number of edges in our bipyramid, we are trying to tile using as few polygons as possible with the largest number of sides. By Euler characteristic considerations, the minimum number of crossings comes from a tiling by a single -gon.

From that logic, we propose the following conjecture:

###### Conjecture 4.9.

The maximum volume density for links in thickened surfaces of fixed genus is given by

 βg:=\emphvol(B□8g−4)2g−1

and is realized by an alternating -uniform tiling link derived from a tiling by right-angled -gons in a closed non-orientable surface of genus .

As the proof of Theorem 4.7 involves increases in genus, it cannot be employed to show density of volume densities in . However, augmentation and taking a belted sum with links in leaves genus intact, and thus the set of volume densities of links in can be shown to be dense over at least the interval by employing results of, e.g., [Bur15] and [ACJ17].

We conclude with a few open questions.

###### Question 1.

Is the set of volume densities of links in for fixed genus dense in ?

###### Question 2.

Can bipyramidal decompositions be used to compute the volume of non-alternating tiling links?

###### Question 3.

Can bipyramidal decompositions be used to compute the volumes of links derived from non--uniform tilings?

In both of the preceding questions, the bipyramids will not remain regular, and some skewing factor should be taken into account to determine the complete hyperbolic structure and calculate exact volumes. But bipyramids can substantially cut down the number of equations produced as compared to when one tries to find the hyperbolic structure by cutting into ideal tetrahedra and satisfying the gluing and completeness equations.

### References

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