Handlebody decompositions of three-manifolds and polycontinuous patterns.

We introduce the concept of a handlebody decomposition of a three-manifold, a generalization of a Heegaard splitting, or a trisection. We show that two handlebody decompositions of a closed orientable three-manifold are stably equivalent. As an application to materials science, we consider a mathematical model of polycontinuous patterns and discuss a topological study of microphase separation of a block copolymer melt.

Introduction

A Heegaard splitting is a decomposition of a closed orientable three-manifold into two handlebodies of the same genus. It is well known that every closed orientable three-manifold admits a Heegaard splitting. By the Reidemeister–Singer theorem [1,2], two Heegaard splittings of a given three-manifold are stably equivalent, i.e. isotopic after a finite number of stabilizations.

Many generalizations of Heegaard splittings have been investigated. Gómez-Larrañaga [3] studied orientable three-manifolds decomposed into three solid tori. Coffey and Rubinstein analysed orientable three-manifolds built from three -injective handlebodies [4]. In [5], Koenig considered a trisection of a closed orientable three-manifold, which is an embedded branched surface decomposing the manifold into three handlebodies with connected pairwise intersections. Koenig introduced the notion of stabilization for a trisection and showed an analogue of the Reidemeister–Singer theorem for trisections of three-manifolds.

In this paper, we consider a generalization of all of the above. We define a handlebody decomposition to be a decomposition of a closed orientable three-manifold into a finite number of handlebodies (see definition 2.1 for the detailed definition). We will also introduce stabilizations for handlebody decompositions and show an analogue of the Reidemeister–Singer theorem for handlebody decompositions (see theorem 3.5).

The primary motivation of this study comes from materials science. We are interested in the characterization of bicontinuous patterns, tricontinuous patterns and polycontinuous patterns of microphase separation of a block copolymer melt (see §6b). See [6,7] for related research. A mathematical model of a bicontinuous (resp. tricontinuous or polycontinuous) pattern is a triply periodic non-compact surface (resp. tribranched surface or polyhedron) embedded in that divides it into two (resp. three or a finite number of) possibly disconnected submanifolds as shown in figure 1 (see definition 5.8 for more details). We are particularly interested in the case where the submanifolds are the open neighbourhood of networks.

Figure 1

A tricontinuous pattern and an entangled network. (Online version in colour.)

If a bicontinuous pattern is triply periodic, then by considering the quotient of the action, the pattern induces a Heegaard splitting of the three-dimensional torus (see remark 7.1). If a polycontinuous pattern is triply periodic and satisfies suitable conditions, then it corresponds to a handlebody decomposition of (corollary 5.12). Hence a characterization of handlebody decompositions of gives that of triply periodic polycontinuous patterns. The Reidemeister–Singer-type theorem of polycontinuous patterns (corollary 6.3) follows from that of handlebody decompositions of . This point of view allows us to explain how two polycontinuous patterns are related, which will be discussed in §6b.

This paper is organized as follows. In §2, we define a handlebody decomposition of a three-manifold. In §3, we introduce several types of stabilization operations of handlebody decompositions and prove an analogue of the Reidemeister–Singer theorem for them. In §4, we particularly focus on decompositions of three-manifolds into three handlebodies. In §5, we study a mathematical model of polycontinuous patterns. We define polycontinuous patterns and, more generally, net-like patterns. The correspondence between triply periodic net-like patterns and handlebody decompositions of is given. In §6, we discuss stabilizations of net-like patterns. We also present how this research relates to the subject of materials science. In §7, we give characterizations of net-like patterns.

Handlebody decompositions of three-manifolds

We work in the piecewise linear category throughout this paper.

By a two-dimensional polyhedron , we mean the underlying space of a non-collapsible locally finite two-dimensional complex such that the link of each vertex contains no isolated vertices. A connected component of the set of points of having neighbourhoods homeomorphic to discs is called a sector. The set of all points not contained in the sectors is called its singular graph. A two-dimensional polyhedron is said to be simple if, after giving a structure of a complex in a suitable way, the link of each point in is homeomorphic to one of the three models shown in figure 2. A point whose link is homeomorphic to the model in figure 2c is called a vertex of its singular graph. See Matveev [8] for more details.

Figure 2

A neighbourhood of each point of a simple polyhedron. (a) A non-singular point, (b) a triple point and (c) a vertex. (Online version in colour.)

Definition 2.1 (Handlebody decomposition).

Let be a closed, connected, orientable three-manifold and a connected compact two-dimensional polyhedron embedded in . We call a type- of if , where is the interior of a handlebody of genus . The polyhedron is called a partition for the decomposition. A handlebody decomposition is said to be proper if there is no simple closed curve in that intersects a sector of transversely once, where is the singular graph of . A handlebody decomposition is said to be simple if its partition is a simple polyhedron.

Remark 2.2.

(1) Let be a type- handlebody decomposition of , and a handlebody of genus for . Then there exists a continuous map such that the restriction of to the interior of is a homeomorphism to . Then we have . Suppose that the handlebody decomposition is proper. Then for the closure of each sector, there exists a pair of handlebodies such that . We denote the union of all such surfaces by . (Note that .)

(2) In general, may not be injective on the singular graph of . If the decomposition is simple and proper, then is a homeomorphism.

The notion of handlebody decompositions generalizes both Heegaard splittings [9] and trisections [5] of closed orientable three-manifolds. In fact, a simple proper handlebody decomposition with is nothing but a Heegaard splitting, while that with , where each is connected, is a trisection. By [10], any closed, connected, three-manifold admits a simple (non-proper) type- handlebody decomposition. Therefore, it is easily seen that for any sequence of non-negative integers, there exists a simple (possibly non-proper) type- handlebody decomposition of .

Stable equivalence

This section discusses the stable equivalence of simple proper handlebody decompositions of a three-manifold. We assume that a handlebody decomposition is simple and proper throughout this section. By remark 2.2, for a handlebody decomposition , there exist handlebodies and continuous maps such that the restriction of each to the interior of is an embedding . For simplicity, we regard as and as . Then the intersection of and is a possibly disconnected surface with boundary. We denote it by .

Stabilizations and destabilizations of handlebody decompositions

The following operations for handlebody decompositions are a generalization of the ‘stabilization’ for Heegaard splittings.

Definition 3.1.

Let be a simple proper type- handlebody decomposition of a closed, connected, orientable three-manifold .

Take a properly embedded arc in , and an arc in such that the endpoints of lie in the interior of , and is parallel to in relative to the endpoints, i.e. the endpoints of are equal to that of , and bounds a disc in . Then we get a type- handlebody decomposition of with where and are a regular neighbourhood of and its interior in , respectively. We call this operation a type- (along ). Conversely, we assume that there exist properly embedded discs of and in such that the boundary of is in , and the boundary of intersects that of transversely exactly one point. Then we can perform the inverse operation of a type- stabilization. We call this operation a type- (along ). See figure 3a.

Figure 3

Stabilizations and destabilizations. Red curves represent the singular graphs. Both ends of the arc of a type-0 stabilization are contained in , whereas one end of the arc of a type-2 stabilization is contained in and the other is in with . A type-0 stabilization connects two parts of by the 1-handle . On the other hand, a new branch locus and a new component of appear after a type-2 stabilization. (a) Type-0, (b) type-1 and (c) type-2. (Online version in colour.)

Take a properly embedded arc on such that the endpoints of lie in the boundary of for . Then we get a type- handlebody decomposition of with We call this operation a type- (along ). Conversely, if there exists a non-separating disc of such that the boundary of intersects the singular graph of the partition transversely exactly two points, then we can perform the inverse operation of a type- stabilization. We call this operation a type- (along ). See figure 3b.

Take two points on the interior of and that of for , and we connect the points by a properly embedded arc in . Let be an arc in such that is parallel to . Then we get a type- handlebody decomposition of with We call this operation a type- (along ). Conversely, if there exists a disc component of whose boundary intersects a properly embedded non-separating disc in transversely once, then we can perform the inverse operation of a type- stabilization. We call this operation a type- (along ). See figure 3c.

Remark 3.2.

Consider a type- handlebody decomposition of a closed, connected, orientable three-manifold with . For every , we can obtain a type- handlebody decomposition of by performing type- stabilizations repeatedly in a suitable way.

Definition 3.3.

A handlebody decomposition is said to be stabilized if it is obtained from another handlebody decomposition by a stabilization.

When , a type-0 stabilization is nothing but a stabilization of Heegaard splittings. In electronic supplementary material, we discuss the independence of these stabilizations.

Stable equivalence theorem

This subsection will generalize Koenig’s argument [5] on the stable equivalence of decompositions. We first recall the following operations on simple polyhedra embedded in a closed orientable three-manifold introduced by Matveev [8] and Piergallini [11] under our setting.

Definition 3.4.

Let be the partition of a handlebody decomposition of .

Let be a properly embedded arc in . A modification of in a neighbourhood of , as in figure 4a, is called a 0-2 move along . By this operation, the number of vertices of increases by two, and a new disc component appears in . Conversely, we can perform the inverse operation of 0-2 move along a disc component, , of . We call the operation a . By this operation, the number of vertices of decreases by two, and the disc component is removed from .

Figure 4

Moves on a handlebody decomposition. As the arc of a 0-2 (resp. 2-3) move connects and , the 0-2 (resp. 2-3) move produces a new component of . The boundary is contained in the singular graph and has two (resp. three) vertices. A 0-2 move increases the number of vertices by two and a 2-3 move increases by one. (a) 0-2 move and 2-0 move and (b) 2-3 move and 3-2 move. (Online version in colour.)

Let be an edge of the singular graph of . A modification of in a neighbourhood of , as in figure 4b, is called a 2-3 move along . By this operation, the number of vertices of increases by one, and a new disc component appears in . Conversely, we can perform the inverse operation of 2-3 move along a disc component, , of . We call the operation a . By this operation, the number of vertices of decreases by one, and the disc component is removed from .

We note that the above moves do not change the topological type of each handlebody of a decomposition.

We say that two handlebody decompositions and of a closed orientable three-manifold are equivalent if there exists an ambient isotopy of that moves to and each to simultaneously.

Theorem 3.5.

Let and be simple proper handlebody decompositions of a closed orientable three-manifold . Then and are equivalent after applying -, -, - moves and types- and - stabilizations finitely many times.

Proof.

Set and as in remark 2.2. We will prove the theorem in the following steps.

In the case of , we perform 0-2, 2-0, 2-3 moves and type-1 stabilization appropriately until it holds for any . Then becomes a simple polyhedron without vertices.

For each , we deform into a disc by type- stabilizations. Then, is a Heegaard splitting. By applying the same process for , becomes a Heegaard splitting, so by the Reidemeister–Singer theorem, we have after applying type- stabilizations.

For each , we deform into a disc by type- stabilizations. We denote by the surface at this stage, and keep it throughout the steps hereafter.

We cover along with by type-1 stabilizations. Then it holds that after handle slides.

() We cover along with by 0-2, 2-0 moves and type- stabilizations. Then it holds that after handle slides.

If , after performing the operations described in the first half of Step 0, the decompositions and become trisections. Then, they are equivalent by using Koenig’s theorem. Hence, in this proof, we assume that .

Step 0. Put . Let be the minimum element of in the lexicographical order. First, we change to be connected if it is disconnected as follows. Take an arc properly embedded in the closure of that connects different components of . If the arc is contained in some , a type-1 stabilization along the arc decreases the number of components by one. Otherwise, we can perform a type-1 stabilization after - moves along the arc to decrease the number of components. Hence, by repeatedly applying this process finitely many times, we may assume that is connected.

Next, take mutually disjoint arcs properly embedded in so that they cut open into a disc. We perform either a type- stabilization or a 0-2 move along each of the arcs according to whether both ends of the arc lie in for or not. Then becomes a disc. Since gives the simple proper handlebody decomposition , the boundary has either at least two vertices or no vertex of .

Suppose is a disc and has more than two vertices. Let be a sub-arc of cut off by the vertices, a properly embedded arc in parallel to , and (, or ) the handlebody with (figure 5a). We perform either a 2-0 move along after a type- stabilization along or a 2-3 move along according to whether there exists a different handlebody ( or ) from with or not (figure 5b,c). Each operation reduces the number of vertices in by two or one. By continuing this process, the number of vertices in can be reduced to two. Then we have after performing a - move on .

Figure 5

(a) The disc sector . (b) Performing a type-1 stabilization along . We can perform a 2-0 move along the greyed region. (c) Performing 2-3 move along . (Online version in colour.)

Suppose is a disc and has no vertices. There is a handlebody (, or ) with . In other words, and share their boundary components in . Since is connected, at least one of and has another boundary component. If shares another boundary component with (and ), we take an arc in which connects and . Then we can remove by a 2-0 move after applying a 0-2 move along the arc. A disc component of arises in this operation. It follows that or , as is greater than in the lexicographical order by the minimality of . If shares another boundary component with (and ), similarly, we can remove by a - move after applying a - move. In this case, there is a possibility that changes to a disc from the empty set by this operation with . This implies that the minimal element of varies from to . In such a case, we take an oriented arc in from a point in to a point in or . Then we can remove , and the minimal element of increases after successively applying - and - moves along the arc from the start to the end.

By repeating the same process, we have . Namely, for . Since each vertex of is contained in four different handlebodies, this condition implies that has no vertex.

Step 1. For each , we will deform into a disc by applying similar operations in Step 0. Since , we may assume that is connected, if necessary, by performing type-1 stabilizations along arcs in . Take a maximal set of non-separating arcs properly embedded in . By performing type- stabilizations along the arcs, becomes a disc. Then is a handlebody. By applying the same process for , becomes a handlebody. Hence and are Heegaard splittings of . By the Reidemeister–Singer theorem, these two Heegaard splittings become equivalent after performing a finite sequence of type- stabilizations. In particular, we can assume .

Step 2. Similarly to Step 1, for each , we can deform into a disc by performing type- stabilizations along suitable arcs properly embedded in .

Claim 3.6.

For , let be a complete meridian disc system of such that for . Then there exist disjoint meridian discs (, ) of such that , , and intersects transversely in a single point, where denotes the union of all .

Proof of claim 3.6.

According to the deformation of at this step, there exist mutually disjoint separating discs in such that each cuts off a handlebody from so that is homeomorphic to . (The union can be regarded as the handlebody at the end of Step 1.) We can take mutually disjoint arcs properly embedded in so that , and intersects transversely in a single point. See figure 6. Let be a disc corresponding to such that for each . Then the assertion holds since . ▪

Figure 6

A deformation at Step 2. (Online version in colour.)

Let denote the surfaces at this stage. Claim 3.6 implies that any one-handle of each handlebody () can be a local one-handle after a handle slide on .

Step 3. For handle slide of , we will cover along with by type- stabilizations. Take a maximal set of mutually non-parallel, non-boundary parallel arcs properly embedded in whose endpoints lie in . We perform type- stabilizations along those arcs. The surface becomes the union of annuli such that for each . Since all spines of are covered by , becomes a local unknotted handlebody after performing handle slides by claim 3.6 (figure 7). Applying the same process for and arranging genera of and by performing type- stabilizations if necessary, we can assume that .

Figure 7

(a) Before performing the operation in Step 3. (b) After performing the operation. The handlebody is a local unknotted handlebody. (Online version in colour.)

According to the deformation of at this step, there exists a separating disc in that cuts off a handlebody from so that is homeomorphic to . ( can be regarded as the previous at the end of Step 2.) Let be the surface , which is a subsurface of and homeomorphic to .

Step (). At the beginning of Step , we may have as subsurfaces of , respectively, that are homeomorphic to , and annuli between and a component of , where for each , , and for each .

Similar to Step 3, we will cover along with by performing handle slides of . By a 0-2 move and a 2-0 move on , extends to , and an annulus arises. Continuing the same operation on , () becomes the annulus . By the same operation as Step 3 on , becomes the union of annuli including . (In the case of , includes at Step 4.) By the same argument at Step 3, then, can be a local unknotted handlebody after handle slide, and we can assume that .

When we finish Step , we have . Since this automatically implies that , the proof is completed. ▪

Handlebody decomposition consisting of three handlebodies

In this section, we provide several results of handlebody decompositions consisting of three handlebodies. We keep assuming that all handlebody decompositions are simple and proper unless otherwise specified.

Section 4a will consider stabilizability on handlebody decompositions containing a three-ball. In [12], Waldhausen showed that any genus- Heegaard splitting of is stabilized for . On the other hand, Koenig found an infinite family of unstabilized type- handlebody decompositions of (see §6 in [5]). We will show that a closed connected orientable three-manifold not containing a non-separating sphere admits an unstabilized type- handlebody decomposition, where is the Heegaard genus of the manifold (proposition 4.2). Furthermore, we will see that almost all lens spaces admit a type- handlebody decomposition (proposition 4.9). In §4b, we will study handlebody decompositions of the three-dimensional torus . These decompositions play an important role in polycontinuous patterns (§5).

Handlebody decompositions containing a three-ball

We first introduce the result of Gómez-Larrañaga [3]. That result gave a complete classification of all closed connected three-manifolds that admit handlebody decompositions with small genera.

Theorem 4.1 ([3, propositions 1–3, theorem 1]).

Let be a type- handlebody decomposition of a closed connected orientable three-manifold with . We denote by the connected sum of some copies of , and denote by or a lens space with non-trivial finite fundamental group. Then the following hold:

If all are equal to , then is homeomorphic to or . Conversely, and admit such a handlebody decomposition.

If and , then is homeomorphic to , , or . Conversely, these manifolds admit such a handlebody decomposition.

If and , then is homeomorphic to , , , , or . Conversely, these manifolds admit such a handlebody decomposition.

If all are equal to , then is homeomorphic to , , , , , , , , or , where denotes a Seifert fibre space with at most three exceptional fibres. Conversely, these manifolds admit such a handlebody decomposition.

Let be a closed orientable three-manifold with a Heegaard splitting of genus . Then, we can take non-separating discs in so that they separate into two three-balls. Hence, admits a type- handlebody decomposition (see [13, example 1.2]). The following proposition classifies such a decomposition.

Proposition 4.2.

Let be a closed, connected, orientable three-manifold of Heegaard genus . Suppose that does not contain a non-separating sphere. Then admits a type- handlebody decomposition if and only if we have . In particular, a type- handlebody decomposition of is unstabilized.

To prove the above proposition, we first show the following lemma.

Lemma 4.3.

Let be a type- handlebody decomposition of a closed, connected, orientable three-manifold . Suppose that does not contain a non-separating sphere. Then is a genus- Heegaard splitting of .

Let denote a surface as in remark 2.2. We show that the surface consists of discs. Assume that contains a non-disc component . Then there exists an essential simple loop in such that each complementary region of in contains a connected component of . Since and are three-balls, the simple loop bounds a disc in each of and . Then the union of the two discs is a non-separating disc, which is a contradiction.

Thus, consists of only discs. It follows that the union of and is a handlebody, which implies the assertion. ▪

Proof of proposition 4.2.

Let be the Heegaard genus of a closed, connected, orientable three-manifold . Then, as explained above, admits a type- handlebody decomposition. Then, by remark 3.2, we can obtain a type- handlebody decomposition of for each . Conversely, assume that admits a type- handlebody decomposition. By lemma 4.3, this handlebody decomposition induces a Heegaard splitting of genus . Thus, we have . This particularly implies that a type- handlebody decomposition of is unstabilized. ▪

Example 4.4.

An unstabilized type- handlebody decomposition of is constructed as follows. Let be a genus-2 Heegaard splitting of . By using [14, §5 and fig. 4], we can take a non-primitive disc triple of , which separates into two three-balls. We denote the three-balls by and , and put . Then forms a type- handlebody decomposition of . Because each component of is a non-primitive disc in , we can see that the boundary of any properly embedded disc in transversely intersects the singular graph of the partition in at least six points. Hence we cannot perform a destabilization along any properly embedded discs in . Therefore, the decomposition is unstabilized.

Next, we will consider the stabilizability of type- handlebody decompositions.

Proposition 4.5.

Let be a closed, connected, orientable, irreducible three-manifold. Suppose that is not a lens space with non-trivial finite fundamental group. Then, for each , any type- handlebody decomposition of is stabilized. In fact, such a decomposition is obtained from a type- handlebody decomposition by performing a type- stabilization.

We first assume that consists of discs. Then there exists a meridian disc of whose boundary intersects transversely exactly twice. Hence we can perform a type- destabilization along the meridian disc.

In the remainder, we assume that has a non-disc component.

Claim 4.6.

We have for each component of .

Proof of claim 4.6.

Suppose that . Since , the boundary has at least three components. Then there exists a component of such that it is an inessential loop in . Hence bounds a properly embedded disc in . The closed curve also bounds a properly embedded disc in since is a three-ball. Because each complementary region of in contains a component of , the two properly embedded discs in and form a non-separating sphere. This contradicts the irreducibility of . ▪

Claim 4.7.

A core curve of each annulus component of is essential in .

Proof of claim 4.7.

Assume that a core curve of an annulus component of is inessential in . Then bounds a properly embedded disc in , and each complementary region of intersects . Since is a three-ball, also bounds a properly embedded disc in . Hence the two discs form a non-separating sphere. This is a contradiction. ▪

Claim 4.8.

The surface contains precisely one annulus component.

Proof of claim 4.8.

We assume that contains two annulus components. Then, by claim 4.7, their core curves, and , are parallel essential loops in . Thus, cobounds a properly embedded annulus in . Since is a three-ball, each of and bounds a disc in . Because each complementary region of in intersects , the union of the annulus and the discs is a non-separating sphere. This is a contradiction.

Let be a core curve of the annulus component of . Then we have , where , and and denote homology classes of a meridian loop and a longitude loop, respectively. Since is a three-ball, each component of bounds a properly embedded disc in . Then the union of and the two discs is a separating sphere in . Thus, if and , then has a lens space as a connected summand. However, is irreducible and not a lens space. Hence we have or . If , then bounds a properly embedded disc in . The curve also bounds a properly embedded disc in since is a three-ball. So, the discs form a non-separating sphere, which is a contradiction. Hence we have . Thus, we can take a meridian disc of that intersects the boundary of transversely exactly two points. Therefore, we can perform a type-1 destabilization along the meridian disc. ▪

The following proposition implies that the assumption that is not a lens space in proposition 4.5 is essential.

Proposition 4.9.

Any lens space with non-trivial finite fundamental group admits an unstabilized type- handlebody decomposition.

Let be a genus-2 Heegaard splitting of . Then there exists a pair of non-primitive discs and in as in example 4.4. Note that is a three-ball, where and are regular neighbourhoods of and , respectively. We take an unknotted arc in the interior of that joins and and intersects them at only its endpoints. Let be a one-handle attached to sides of each and along . Then, is a solid torus (figure 8). Hence, for any lens space , there exists a homeomorphism from to the boundary of a solid torus such that is homeomorphic to the manifold pasted by and along . We put , . Thus, is a type- handlebody decomposition of . By the construction, each meridian disc of and intersects the singular graph at least four and six times, respectively. Hence, the handlebody decomposition is unstabilized. ▪

Figure 8

An unstabilized type- handlebody decomposition of a lens space with non-trivial finite fundamental group. The decomposition consists of three handlebodies , and .

Examples: the three-dimensional torus

We will show some examples of handlebody decompositions of the three-dimensional torus .

First, we consider handlebody decompositions consisting of one ball and two handlebodies. By proposition 4.2, admits a unstabilized type- handlebody decomposition (figure 9a). Thus, for and , admits a type- handlebody decomposition by remark 3.2. Figure 9b illustrates a type- handlebody decomposition of . On the other hand, by theorem 4.1, admits neither type-, type-, nor type- handlebody decompositions. In addition, by propositions 4.2 and 4.5, there is no type- handlebody decomposition of . Hence, any type- handlebody decomposition of is unstabilized. In summary, we have the following proposition.

Figure 9

(a) A type- handlebody decomposition of . Coloured edges illustrate the singular graph, and greyed discs are all components of as in remark 2.2. (b) A type- handlebody decomposition of . (c) A decomposition of into three hexagons. (d) The hexagonal honeycomb decomposition of . (Online version in colour.)

Proposition 4.10.

Let be a pair of non-negative integers with . The 3-dimensional torus admits a type- handlebody decomposition if and only if the pair is not in .

Theorem 4.1 guarantees that admits a type- handlebody decomposition. Figure 9c shows a decomposition of into three hexagons. By taking the product with , we have a decomposition of into three solid tori. We call this handlebody decomposition the hexagonal honeycomb decomposition (figure 9d). In general, a three-manifold admits a lot of handlebody decompositions of the same type. The next proposition asserts that the hexagonal honeycomb decomposition is the unique type- handlebody decomposition of up to self-homeomorphism of .

Proposition 4.11.

For a simple and proper type- handlebody decomposition of , there exists a self-homeomorphism of that maps the partition of the decomposition to that of the hexagonal honeycomb decomposition.

Let be a simple and proper type- handlebody decomposition of . Let denote a surface as in remark 2.2.

Claim 4.12.

For any , there is no disc component in .

Proof of claim 4.12.

Suppose there is a disc component in some . Without loss of generality, we can assume that . If is essential in , is a punctured lens space. Since is prime, is a three-ball. Thus, a triple gives a simple and proper type- handlebody decomposition of . However, from proposition 4.10, does not admit such a decomposition, which is a contradiction.

Suppose is inessential in . Then we can take a disc in such that . Put . Since does not contain non-separating spheres, is separating. Thus, consists of only the disc . Hence, is a genus- handlebody. This is impossible because is a torus. ▪

By claim 4.12 and [3, lemma 1], each component of is an annulus. Suppose the core of an annulus of is meridional in . Then, is longitudinal in ; otherwise, we can find a punctured lens space or a punctured in , which is a contradiction. Then, by removing the neighbourhood of a meridian disc in and attaching it to , we have a decomposition of with two three-balls and a solid torus, i.e. a type- decomposition, which contradicts proposition 4.10. Therefore, , and are fibre tori of a Seifert fibration of . As the Seifert fibre structure of is unique up to self-homeomorphism of , we can assume that (), where , and are discs in satisfying . By the Euler characteristic, the intersection of two of the discs consists of precisely three arcs. Hence, this structure corresponds to the hexagonal honeycomb decomposition in .

When it comes to the case of type- handlebody decompositions, the uniqueness no longer holds, as we see in the following example.

Example 4.13.

By performing type- stabilizations to the honeycomb decomposition three times, we have the two type- handlebody decompositions shown in figure 10. In figure 10a, each is homeomorphic to the disjoint union of a two-holed torus and a disc. On the other hand, in figure 10b, the surface is homeomorphic to the disjoint union of a one-holed torus and an annulus, and each is homeomorphic to a three-holed sphere. Hence, they are different decompositions. ▪

Figure 10

(b,d) A pair of different handlebody decompositions of . The decomposition (b) (resp. (d)) is of type- obtained from the hexagonal honeycomb decomposition by type- stabilizations along , and in (a) (resp. , and in (c)). (Online version in colour.)

Topological study of polycontinuous patterns

In this section, we will consider ‘polycontinuous patterns’, which are roughly three-periodic structures assembled by polymers. See, for example, [6,15] for studies on polycontinuous patterns. We will suggest a mathematical model of polycontinuous patterns (definition 5.8).

Polycontinuous patterns and net-like patterns

First, we define ‘net-like patterns’ that satisfy the essential properties of polycontinuous patterns.

Definition 5.1.

We denote by the -dimensional torus. Let be a graph embedded in such that each component of is unbounded. If there exists a covering map such that all covering transformations of preserve , then is called a net.

In this paper, we mainly discuss the case where .

Remark 5.2.

In crystal chemistry (e.g. [16]), the term ‘net’ means a periodic, connected, simple, abstract graph. In this paper, we allow a net to be disconnected. Furthermore, all nets are embedded in Euclidean space.

Definition 5.3.

Let be a non-compact connected two-dimensional polyhedron embedded in . The polyhedron is called a net-like pattern if there exist a covering map and a net such that the following conditions hold: We call the pair a framed net-like pattern, and its frame. We say that a connected component of (resp. ) is a labyrinthine domain (resp. labyrinthine net) of .

All covering transformations of preserve both and .

The polyhedron divides into unbounded open components , where is a finite or countable set.

There exists a strong deformation retraction of onto .

A net-like pattern is said to be proper if there is no simple closed curve in that does not cross the singular graph and intersects a sector of transversely once. A net-like pattern is said to be simple if is a simple polyhedron.

More generally, we can define net-like patterns for any closed prime three-manifold with a (possibly non-Euclidean) crystallographic group and its covering space, but this paper will not deal with it.

Remark 5.4.

Consider two net-like patterns that satisfy the following conditions: Then, they can be transformed to each other by a (possibly infinite) sequence of IX-moves, XI-moves and isotopies (see [17, theorem 3.1]).

They have the same labyrinthine net.

They do not have a disc sector.

The singular graphs of them have no vertices.

By using [11,18], if two net-like patterns with the same labyrinthine net are simple, then the patterns can be transformed to each other by a (possibly infinite) sequence of 0-2 moves, 2-0 moves, 2-3 moves, 3-2 moves and isotopies.

The following two propositions state a relationship between (framed) net-like patterns and handlebody decompositions of .

Proposition 5.5.

Let be a handlebody decomposition of , and the preimage of under the universal covering map of . Suppose that, for each , the induced homomorphism is not trivial, where is the inclusion map. Then the pair is a framed net-like pattern. Furthermore, if is simple (resp. proper), then the net-like pattern is also simple (resp. proper).

Let be the connected components of the preimage of under . Since the homomorphism is not trivial, each open component is unbounded. Each open handlebody contains a simple finite graph that is a strong deformation retract of . Then the preimage, , of under is a net. Furthermore, each connected component of is a strong deformation retract of some . Hence, is a framed net-like pattern.

Since is a local homeomorphism, if is simple, then the preimage is also simple. Next, assume that the handlebody decomposition of is proper, whereas the net-like pattern is not proper. Then, there exists a simple loop in that transversely intersects at a single point only in a sector. Thus, there exists a simple loop in isotopic to that intersects a sector of transversely once. Hence, the handlebody decomposition is not proper, which is a contradiction. Therefore, the net-like pattern is proper. ▪

Proposition 5.6.

Let be a framed net-like pattern.

The image gives a handlebody decomposition of . If is simple, then the handlebody decomposition is also simple.

Let be the set of labyrinthine domains of , where is a finite or countable set. Suppose that is proper. Suppose further that for any , with , where is the image of under some covering transformation, and are not adjacent to the same sector. Then the handlebody decomposition given by is proper.

Let be the covering transformation group of . Set .

(1) Since is a connected two-dimensional polyhedron, its projection image is also a connected two-dimensional polyhedron. Furthermore, if is simple, then is also simple.

The complement consists of finite open components because is compact and is the underlying space of a locally finite complex. We show that each open component is an open handlebody. There exists a labyrinthine domain such that . Furthermore, since is a net-like pattern, there exists a labyrinthine net such that is a strong deformation retract of . Put . Then, is an embedding of a graph in . So, the fundamental group is free. Since is a covering map, and is a strong deformation retract of , the inclusion map induces an isomorphism from to . Hence, is the interior of a handlebody because is free. Therefore, gives a handlebody decomposition of .

(2) We suppose that is proper and is not proper. Then there exists a simple loop such that it transversely intersects at a single point only in a sector. Let be a lift of . Since is proper, is an arc (not a loop) whose initial point and terminal point are contained in different labyrinthine domains and , respectively, of . Note that and are adjacent. This is impossible because there exists a covering transformation that takes to , so to . ▪

Example 5.7.

Figure 11a illustrates a simple proper net-like pattern that comes from the hexagonal honeycomb tessellation of . A yellow polygon illustrates a fundamental domain of its frame. We call the pattern the hexagonal honeycomb pattern. By proposition 5.6, the hexagonal honeycomb pattern with the frame induces the hexagonal honeycomb decomposition (see figure 9d). Note that a tessellation of induces a net-like pattern in general. The meanings of colours except yellow in figure 11a will be explained in definition 5.10.

Figure 11

(a) The hexagonal honeycomb pattern and (b) a non-simple net-like pattern. (Online version in colour.)

We now suggest a strict mathematical definition of polycontinuous patterns.

Definition 5.8.

We say that a net-like pattern is an (or a polycontinuous pattern) if the following conditions hold:

has precisely labyrinthine domains.

is proper.

Any sector of is not a disc.

Note that for any positive integer , there exists an -continuous pattern. In the remainder, we call a two-continuous (resp. three-continuous) pattern a bicontinuous (resp. tricontinuous) pattern, according to the conventions of soft materials [6,15].

The following corollary is a polycontinuous pattern version of proposition 5.5.

Corollary 5.9.

Let be a proper handlebody decomposition of , and the preimage of under the universal covering map of . Suppose that the following two conditions hold: Then, is a polycontinuous pattern. In particular, if for each , then is an -continuous pattern. Furthermore, if is simple (resp. proper), then is also simple (resp. proper).

For each , we have .

Any sector of is not a disc.

Colourings of patterns

In this subsection, we will define colourings of net-like patterns. Each labyrinthine domain of a net-like pattern is a mathematical model of polymers assembled in one kind of block. In materials science, one kind of block may form many domains of a net-like pattern in general. To describe such a situation, we introduce ‘colours’ of net-like patterns, of which each colour corresponds to one kind of block of polymers.

Definition 5.10.

Let be a net-like pattern. Set . Let denote the set of all labyrinthine domains of . A surjection is called an if there exists a frame of such that the following conditions hold: The image is called the colour of . The frame of is said to be compatible with the colouring . A net-like pattern together with a fixed (-) colouring is called an ( net-like pattern (figure 11b). We say that two coloured net-like patterns, and , are equivalent if there exists an ambient isotopy of that moves to , and each pair of corresponding labyrinthine domains has the same colour after permuting the colours. If a surjection satisfies only the condition (1), then we call a non-effective , and is said to be non-effectively .

For each covering transformation of and for any , we have .

Two sides of a local part of each sector have different colours. Namely, for each point in a sector , the two labyrinthine domains, and , that have non-trivial intersection with satisfy .

Let be a (possibly non-effectively) -coloured net-like pattern with a colouring , where , and a frame of compatible with . By proposition 5.6, the image gives a handlebody decomposition of . Denote by the set of all handlebodies of the decomposition. Then we say that is of type , where is a sequence of the genera of the handlebodies in coloured by . (For simplicity, if the length of is equal to , then we put .)

Note that, as opposed to colourings of graphs on surfaces, for any integers , with and , there is an -coloured framed net-like pattern that does not admit -colouring.

Remark 5.11.

If a net-like pattern admits a colouring, then it is necessarily proper.

In fact, a coloured net-like pattern with its frame compatible with the colouring satisfies the condition that induces a proper handlebody decomposition of since any two labyrinthine domains sharing a sector have different colours (see proposition 5.6). Hence, the following holds.

Corollary 5.12.

Let be a framed net-like pattern and let be a sequence of positive integers for . Suppose that admits an -colouring and is of type . Then gives a proper type- handlebody decomposition such that for .

The converse of the above corollary is clear by proposition 5.5.

Corollary 5.13.

Let be a proper type- handlebody decomposition of and let be the universal covering map . We assume that for . We further assume that, for each handlebody , the induced homomorphism is not trivial, where is the inclusion map. Then is a coloured net-like pattern of type , where .

A sufficient condition for the equivalence of patterns

Corollaries 5.12 and 5.13 say there is a nice relationship between coloured net-like patterns and proper handlebody decompositions. This subsection gives a sufficient condition for two coloured net-like patterns to be equivalent.

To this end, we first consider adjusting a framed net-like pattern to another frame. Let be a framed net-like pattern, and let be a covering map . Since the two covering maps are equivalent, there exists a self-homeomorphism of such that . If is orientation-preserving, we say that and have the same orientation. Otherwise, we say that and have different orientations.

If and have the same orientation, then is isotopic to , and is a framed net-like pattern. Consider the case that and have different orientations. Let be an orientation-reversing self-homeomorphism of . Then, is also a covering map . Hence, there exists an orientation-preserving homeomorphism such that . So, is isotopic to , and we have . Furthermore, is a framed net-like pattern because each covering transformation of is also that of .

To summarize, we obtain the following lemma.

Lemma 5.14.

Let be a framed net-like pattern, and let be a covering map . Then, there exists a net-like pattern such that the following three conditions hold: In particular, if is (non-effectively) -coloured, then so is . Furthermore, and have the same type.

The covering map is a frame of .

The pattern is isotopic to .

Either or there exists an orientation-reversing self-homeomorphism of with .

By lemma 5.14, we can assume that any two net-like patterns have the same frame. Proposition 5.6 says the two framed net-like patterns induce two handlebody decompositions of , respectively. If the two decompositions are isotopic, then the two patterns are also isotopic. In fact, we can say more as follows.

Lemma 5.15.

Let and be framed net-like patterns. Suppose that there exists an orientation-preserving self-homeomorphism of that maps to . Then, is isotopic to .

By the assumption, we have an orientation-preserving self-homeomorphism of that maps to . Let be the unique lift of . Then is a homeomorphism of , and we have . Therefore, and are isotopic. ▪

By the above lemmas, we have the following proposition.

Proposition 5.16.

Let and be -coloured framed net-like patterns. We assume that and are homeomorphic under an orientation-preserving or orientation-reversing self-homeomorphism of according to whether the covering maps and have the same orientation or different orientations. Suppose that any two corresponding handlebodies under are the images of labyrinthine domains with the same colour (after permuting the colours). Then and are equivalent.

Stabilizations on net-like patterns

In this section, we will discuss (de)stabilizations of net-like patterns and introduce some examples.

Stabilization theorem for net-like patterns

Section 3 showed an analogue of the Reidemeister–Singer theorem for handlebody decompositions (theorem 3.5). This subsection shows a net-like pattern version of the theorem. To do so, we define (de)stabilizations of net-like patterns. First, we will use an example to explain how to define it.

In example 5.7, we introduced the hexagonal honeycomb pattern that is a three-coloured framed net-like pattern of type . We consider a type- stabilization on the pattern. Let be a properly embedded arc in a sector of the pattern (figure 12a). We assume that is lifted by , i.e. the restriction of to is injective, where is the frame of the pattern. By corollary 5.12, gives a simple proper handlebody decomposition. As noted in example 5.7, is a simple proper type- handlebody decomposition (figure 9d). Since is lifted and contained in a sector, the image is a properly embedded arc in a sector of . Furthermore, connects two labyrinthine domains mapped to the same handlebody by . So, the arc connects the same handlebody. Thus, we can perform a type- stabilization along . Hence, a type- handlebody decomposition is obtained by performing a type- stabilization along . By corollary 5.13, the preimage of under gives a three-coloured framed net-like pattern of type (figure 12b). Hence, we obtain the new net-like pattern  from . We will call such an operation a type-.

Figure 12

(a) The hexagonal honeycomb pattern introduced in example 5.7. The parallelepiped illustrates a fundamental domain of the pattern. (b) The net-like pattern obtained by performing a type- stabilization on the honeycomb pattern along the properly embedded lifted arc . (Online version in colour.)

Based on the above example, we give a strict definition of stabilizations for patterns as follows.

Definition 6.1.

Let be a simple, -coloured, framed net-like pattern of type , where is a sequence of positive integers for . Put . By corollary 5.12, gives a simple proper handlebody decomposition of such that is a genus- handlebody coloured by . Let , , , and be labyrinthine domains. We assume that, for each pair of the labyrinthine domains except for , and , the two domains are different and share a sector. (There is a possibility that or .) We further assume that , and . Here, , and are distinct handlebodies, and , and .

Depending on the type of stabilization, we take an arc as follows. Then, we can obtain a new handlebody decomposition performed by a suitable stabilization on along . We can see by corollary 5.13 that the preimage of the partition of is a simple, coloured, framed net-like pattern of type . Here, each is equal to except for the following sequences:

The arc is a properly embedded lifted arc in that connects and . We assume that a lifted disc in contains as a part of its boundary, and the other part is contained in .

The arc is a properly embedded lifted arc in a sector of that connects and .

The arc is a properly embedded lifted arc in that connects and . We assume that a lifted disc in contains as a part of its boundary, and the other part is contained in .

We call this operation a type- and its inverse operation a type- for each . In electronic supplementary material, we discuss sufficient conditions for performing a destabilization.

Note that the result of a (de)stabilization of a polycontinuous pattern is not necessarily a polycontinuous pattern. Further note that in a type- stabilization along an arc for net-like patterns, even if the arc connects different labyrinthine domains, we cannot perform the operation if they are the same colour.

Definition 3.4 introduced some operations for handlebody decompositions called moves. We next consider a net-like pattern version of them. Of course, we can perform the original operations on simple coloured net-like patterns, but they generally lose periodicity after performing them. Thus, we give adapted ‘moves’ to net-like patterns in a similar way to the stabilizations.

Definition 6.2.

Let be a simple, -coloured, framed net-like pattern. Take a properly embedded lifted arc in a sector (resp. an edge of the singular graph of ) so that it connects labyrinthine domains and of different colours. By corollary 5.12, gives a simple proper handlebody decomposition . Then, we can obtain a new handlebody decomposition performed by a 0-2 (resp. 2-3) move on along . By corollary 5.13 the preimage of the partition of is a simple, coloured, framed net-like pattern. We call such an operation a 0-2 (resp. 2-3) move (along and its inverse operation a 2-0 (resp. 3-2) move.

Note that, similar to type-2 stabilizations of net-like patterns, even if we can perform a move on a handlebody decomposition corresponding to a pattern, it does not necessarily mean that we can perform the corresponding move on the pattern.

An analogue of the Reidemeister–Singer theorem for net-like patterns is as follows.

Corollary 6.3.

Let and be simple, -coloured, framed net-like patterns of type and , respectively, where and are positive integers. Then and are equivalent after applying 0-2, 2-0 and 2-3 moves, and type- and type- stabilizations finitely many times.

We assume that and have the same orientation. The proof of the other case is similar. By corollary 5.12, the images and give type- and type- simple proper handlebody decompositions of , respectively. Hence, by theorem 3.5, there exists a simple proper handlebody decomposition such that and are isotopic to the partition of the decomposition after applying 0-2, 2-0 and 2-3 moves, and type-0 and type-1 stabilizations to them finitely many times. By corollary 5.13, is a simple -coloured net-like pattern. Therefore, by proposition 5.16, each of and is equivalent to after applying 0-2, 2-0 and 2-3 moves, and type-0 and type-1 stabilizations finitely many times. ▪

In the above corollary, we assume that, for each colour, all labyrinthine domains coloured by it are mapped to the same handlebody because moves performed in the proof of theorem 3.5 generally do not preserve the colouring. On the concept of colourings, we can regard single-coloured domains as composed of the same kind of blocks, so connecting these parts is a natural idea.

Definition 6.4.

Let be a simple, -coloured, framed net-like pattern. Take a properly embedded lifted arc in a sector so that it connects labyrinthine domains, and , of the same colour. We assume that and are different handlebodies of the simple proper handlebody decomposition induced by . By performing a 0-2 move on along , the modified and are intersected, and by corollary 5.12, their intersection consists of only the disc created by the operation. So, is also a handlebody. Hence, we have a new handlebody decomposition by replacing and with . By corollary 5.13, the decomposition induces a new simple, coloured, framed net-like pattern . We call such an operation a domain-connection (along and its inverse operation a domain-disconnection.

Remark 6.5.

We can obtain the type of in definition 6.4 as follows. Let be the sequence of positive integers in the type of corresponding to the colour of the labyrinthine domains and . We remove the integers corresponding to and from and append their sum. Then, we denote a new sequence by . By replacing with , we obtain the type of .

By applying the following to a coloured net-like pattern, it satisfies the assumption of corollary 6.3.

Lemma 6.6.

Let be a simple, -coloured, framed net-like pattern of type , where is a sequence of positive integers for . Set . Then, we have a simple, -coloured, framed net-like pattern of type by applying 0-2 moves and domain-connections with respect to finitely many times to .

Since is connected, for each colour , there exist labyrinthine domains, and , with colour and an embedded lifted arc joining and in such that the following hold: Then, by cutting at its intersection with the singular graph, we obtain the sequence of sub-arcs . Thus, we can perform 0-2 moves along , , , and we can finally apply domain-connection along . By repeating the above process, all labyrinthine domains with colour are joined. Then, the type corresponding to colour is by remark 6.5. Therefore, we have a net-like pattern of type . ▪

The images and are distinct handlebodies.

The arc does not cross any labyrinthine domains with colour .

The arc intersects the singular graph of transversely.

Microphase separation of a block copolymer melt

One motivation for this research comes from materials science. We are interested in the characterization of polycontinuous patterns that appear as microphase separation of a block copolymer melt [6,7].

In this subsection, we discuss block copolymers and phase separation of a block copolymer melt. One reference of this subject is [19]. A polymer is a molecule of high molecular weight created by chemically coupling large numbers of small reactive molecules, called monomers. If a polymer is made of one type of monomer, it is called a homopolymer. A polymer containing two or more chemically distinct monomers is referred to as a copolymer. A block copolymer is an important type of copolymer, in which monomers of a given type form polymerized sequences called blocks. If a block copolymer contains two (respectively three) blocks, it is called a diblock (resp. triblock) copolymer. If a linear diblock copolymer is made of blocks of monomers A and B, it is called an AB diblock copolymer. An ABA triblock copolymer is a linear triblock copolymer consisting of a sequence of a block of monomer A, a block of monomer B, and a block of monomer A. See figure 13a. SBS (styrene–butadiene–styrene) triblock and SIS (styrene–isoprene–styrene) triblock copolymers are examples of linear triblock copolymers. Polymers with more complex architecture have been synthesized. For example, a star polymer has one branched point connecting several linear polymers. An ABC triblock-arm star-shaped molecule (3-star polymer) as in figure 13a is an example of triblock copolymer with a star architecture, where A, B and C blocks are mutually immiscible.

Figure 13

(a) A red dot indicates the monomer A, green monomer B, blue monomer C and yellow monomer D. Left: AB diblock copolymer, ABA triblock copolymer and ABC star-shaped block copolymer. Right: ABCD star-shaped block copolymer. (b) A double gyroid that is a famous bicontinuous pattern. (Online version in colour.)

A block copolymer melt is a solvent-free viscoelastic liquid composed of block copolymers. Due to the chemical distinction of monomers, we can observe phase separation in a block copolymer melt. A domain of phase separation consists of monomers of one type. Microphase separation of a block copolymer melt is phase separation with domains of the mesoscopic size scale. Sphere, cylinder, bicontinuous and lamellar structures appear as microphase separation of AB diblock or ABA triblock copolymers [19,20].

An example of bicontinuous patterns is the Gyroid surface. In materials science, in the bicontinuous pattern of an AB diblock copolymer melt, the domain of the A monomer is the neighbourhood of the partition surface, and that of the B monomer forms two labyrinths (figure 13b). A tricontinuous pattern is a mathematical model of microphase separation of an ABC star-shaped block copolymer melt. The branch line of a tricontinuous pattern consists of the connection points of the A, B and C blocks in the block copolymers [6]. See [21-23] for studies on geometric phases of star polymer melts. Note that a sector of a tricontinuous pattern is the interface of two domains.

Next, we discuss a mathematical model of microphase separation with four phases. Let A, B, C and D be four chemically distinct monomers. We consider the polycontinuous pattern of melts of four types of three-star block copolymers of ABC, ABD, ACD and BCD. In this case, four different branched lines appear. The interface of domains and these branched lines form a simple polyhedron. The vertex of the simple polyhedron of the polycontinuous pattern corresponds to the point where four domains A, B, C and D meet. The four edges corresponding to the connecting points of the ABC, ABD, ACD and BCD triblock star polymers are placed around a vertex. Also, ABCD four-star polymers are synthesized [24,25], and their morphologies have been discussed in [26,27]. The joining point of four blocks of the block copolymer corresponds to a vertex of the simple polycontinuous pattern.

We want to analyse the property of materials with this structure via a topological study of these polycontinuous patterns. We hope the characterization and the classification of polycontinuous patterns will lead to the design of polymeric materials with the desired properties.

As an application of corollary 6.3, we can discuss the relation between two microphase-separated structures of the same type. Here, we discuss the polymer science implications of stabilization and destabilization operation of patterns.

Observation 6.7.

The type-0 destabilization for a bicontinuous pattern can be considered as the model of the cancelling of an unstable local one-handle structure of the pattern of the microphase separation. The type-1 destabilization (resp. stabilization) for a polycontinuous pattern can be considered as the model of the separation (resp. amalgamation) of the domains during the uniaxial elongation of polymeric materials.

Example: a 3srs pattern

A 3srs pattern is an example of a tricontinuous pattern. In this subsection, we will show the pattern can be destabilized to the hexagonal honeycomb pattern.

First, we introduce a 3srs net. An srs net is a 3-periodic ‘minimal’ net in (see [28] and figure 14a). Figure 14a illustrates an srs net with a cubical fundamental domain, of which the length of each edge is 8. The net is an infinite trivalent graph, and the space group of it is (see [15,29]). Note that a rotation around the cube diagonal (shown in figure 14a) generates an action of order 3 and preserves the cube. A 3srs net is the union of the images of the srs net under the action (figure 14b).

Figure 14

(a) An srs net. The orange line passes through the points and . Note that this net is topologically the same as the srs-b net (see [30]). (b) A 3srs net. The rotation around preserves the net. (Online version in colour.)

Figure 15a illustrates a branched surface in with a cubical fundamental domain. The branched surface is the union of precisely three surfaces with the boundary (figure 15b–d). It is clear that the branched surface is a simple three-coloured tricontinuous pattern, and each component of the srs net is a labyrinthine net of the pattern. We call the tricontinuous pattern the 3srs pattern. The 3srs pattern is of type as illustrated in figures 14 and 15.

Figure 15

(a) The tricontinuous pattern corresponding to the 3srs net with a cubical fundamental domain. (b–d) Surfaces with boundary, each of which is shared by exactly two labyrinthine domains. (Online version in colour.)

Theorem 6.8.

The 3srs pattern can be destabilized to the hexagonal honeycomb pattern, i.e. the 3srs pattern can be obtained from the hexagonal honeycomb pattern by a finite sequence of type-1 stabilizations.

Let be the srs pattern, and its frame obtained from a cubic fundamental domain as shown in figure 15. Put . Figure 16 shows a simple proper type- handlebody decomposition of induced by . We denote by , and surfaces with boundary as in remark 2.2. By definition 6.1, if we destabilize the decomposition to the hexagonal honeycomb decomposition by performing a finite sequence of type- destabilizations, then we can also destabilize to the hexagonal honeycomb pattern by corresponding destabilizations.

Figure 16

A handlebody decomposition of induced by the srs pattern. The ‘cores’ of handlebodies are the quotient of the 3srs net. The bold curves on the boundaries of handlebodies make up the singular graph. Each denotes a surface defined in remark 2.2.

First, for each , we take three meridian discs , and of the handlebody as shown in figure 16a–c. Each disc intersects the singular graph of transversely exactly two points. Furthermore, any two different discs are disjoint. Hence, we can perform type-1 destabilizations along them. By this operation, we obtain a type- handlebody decomposition of (see figure 17a). For simplicity, we denote each handlebody and the partition of the destabilized handlebody decomposition by the same symbol , , and , respectively. Note that the preimage of the union of spines of , and is isotopic to a net as shown in figure 17b. See [31] for examples of materials with this chemical framework. See also [32]. The destabilized net-like pattern is also a simple coloured tricontinuous pattern.

Figure 17

(a) A type- handlebody decomposition of . (b) A 3hcb net that is the preimage by the universal covering map of the core of the handlebodies , and .

For the type- handlebody decomposition, we can perform a type- destabilization along a meridian disc of (figures 17a and 18a). The type of resulting decomposition is . Figure 18a–e illustrates a destabilization to the type- handlebody decomposition, which produces a type- handlebody decomposition. The type- handlebody decomposition illustrated in figure 18f is the hexagonal honeycomb decomposition (figure 9d). ▪

Figure 18

A sequence of type- destabilizations from the type- handlebody decomposition to the type- handlebody decomposition. (a) Type-(2, 2, 1), (b) a meridian disc D of H, (c) type-(2, 1, 1), (d) a meridian disc D of H1, (e) type-(, 1, 1), (f) the hexagonal honeycomb decomposition. (Online version in colour.)

Characterization of patterns

In this section, we will prove that bicontinuous patterns are unique. We will also show that simple, coloured, framed net-like patterns of type are unique. On the other hand, we will provide two different simple coloured net-like patterns of type .

Bicontinuous patterns and Heegaard splittings of

By definition, an -continuous pattern consists of precisely labyrinthine domains, and it is proper. Hence, by assigning a different colour to each domain, the pattern admits an -colouring. In general, a frame of the pattern is not compatible with the colouring. However, by expanding the fundamental domain, we can obtain a frame compatible with the colouring. Then, by corollary 5.12, the pattern with the frame gives a proper type- handlebody decomposition of . Hence, the pattern is a framed net-like pattern of type . In particular, we note the following for each simple bicontinuous pattern and such a frame.

Remark 7.1.

Any simple bicontinuous pattern and its frame compatible with a colouring induce a Heegaard splitting of .

By [33,34], Heegaard splittings of are determined by their Heegaard genera. Hence, we can prove the uniqueness of bicontinuous patterns.

Theorem 7.2.

Any two simple bicontinuous patterns are equivalent.

Let and be bicontinuous patterns of types and , respectively. For the frame , there exists a basis of such that the translations defined by the vectors generate the covering transformation group. We denote by a group generated by translations , and . Hence we have a covering map . Since the Euler characteristic of is -times that of , the surface gives a Heegaard splitting of of genus . Similarly, we can take a covering map so that also gives a Heegaard splitting of genus . Therefore, by using [34, Théorème] and proposition 5.16, the two simple bicontinuous patterns are equivalent. ▪

The Gyroid, the Schwartz D surface and the Schwartz P surface are famous triply periodic minimal surfaces that decompose into precisely two open components (see [35]), i.e. the surfaces are simple bicontinuous patterns. In [36, appendix], Squires et al. gave an isotopy from the Gyroid to the Schwartz D surface and the Schwartz D surface to the Schwartz P surface by an explicit formula. Note that theorem 7.2 is a generalization of the result but does not give a formula for transformation between patterns.

The uniqueness of framed patterns of type (1, 1, 1)

We consider the hexagonal honeycomb pattern introduced in example 5.7. Recall that its pattern admits a colouring and a frame compatible with it, as in figure 11a. The pattern induces the hexagonal honeycomb decomposition of . Hence, the hexagonal honeycomb pattern with the frame is of type . By propositions 4.11 and 5.16, the hexagonal honeycomb pattern is a canonical model of simple coloured net-like patterns of type . Therefore, we have the following.

Theorem 7.3.

Any simple, coloured, framed net-like pattern of type is equivalent to the hexagonal honeycomb pattern.

Note that a simple three-coloured net-like pattern whose labyrinthine nets consist of lines is not necessarily equivalent to the hexagonal honeycomb pattern in general (see example 7.4). Also, there are distinct simple coloured net-like patterns of type (see example 7.5).

Example 7.4.

We consider a tessellation of the plane by three kinds of tiles: square, hexagon and eight-sided polygon. Figure 19 shows a net-like pattern induced by the tessellation. The left side (figure 19a) illustrates a framed net-like pattern of type that is not coloured since eight-sided components are assigned to the same colour, and they are adjacent. On the other hand, the pattern admits a four-colouring (figure 19b). However, it is no longer type . This pattern is called in [23, fig. 8(k)] and the colouring given there corresponds to a coloured net-like pattern of type .

Figure 19

(a) A simple non-effectively coloured net-like pattern of type . (b) A simple coloured net-like pattern of type . The handlebody decomposition induced by the pattern contains two blue solid tori and two red solid tori. (Online version in colour.)

Example 7.5.

Figure 20 illustrates two simple coloured net-like patterns, and , of type with a cubical fundamental domain, where and denote their frames compatible with the colourings, respectively. We can see the two patterns are not equivalent as follows. Let and be nets associated with and . We consider the image and , where and are the inclusion maps, respectively. By figure 20a is isomorphic to . On the other hand, is isomorphic to by figure 20b. Hence, is not equivalent to .

Figure 20

(a,b) Two framed simple coloured net-like patterns of type . (c) The labyrinthine nets of (b). (Online version in colour.)

By theorem 7.3, any two simple coloured framed net-like patterns of type are equivalent. However, simple coloured net-like patterns of type are not unique.

The labyrinthine nets of these types of patterns are called cubic rod (cylinder) packings [37] or weavings [38]. Many of those structures do not correspond to simple coloured net-like patterns.