1 Introduction

Embeddability in the 3-sphere is decidable

Abstract

We show that the following algorithmic problem is decidable: given a -dimensional simplicial complex, can it be embedded (topologically, or equivalently, piecewise linearly) in ? By a known reduction, it suffices to decide the embeddability of a given triangulated 3-manifold into the 3-sphere . The main step, which allows us to simplify and recurse, is in proving that if can be embedded in , then there is also an embedding in which has a short meridian, i.e., an essential curve in the boundary of bounding a disk in with length bounded by a computable function of the number of tetrahedra of .

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Embeddability in the 3-sphere is decidable

1 Introduction

The embeddability problem. Let EMBED be the following algorithmic problem: given a finite simplicial complex of dimension at most , does there exist a (piecewise linear) embedding of into ?

A systematic investigation of the computational complexity of this problem was initiated in [MTW11]; earlier it was known that EMBED (graph planarity) is solvable in linear time, so is EMBED [GR79], and for every fixed, EMBED can be decided in polynomial time (this is based on the work of Van Kampen, Wu, and Shapiro; see [MTW11]).

For dimension , there is now a reasonably good understanding of the computational complexity of EMBED: for all with it is NP-hard (and even undecidable if ) [MTW11], while for it is polynomial-time solvable, assuming fixed, as was shown in a series of papers on computational homotopy theory [ČKM13, ČKM12, KMS13, ČKV13]. (However, the cases with known to be NP-hard but not proved undecidable are still intriguing.)

Thus, the most significant gap up until now has been the cases and , and in particular, after graph planarity (EMBED), the problem EMBED can be regarded as the most intuitive and probably practically most relevant case.

Embeddability in . Here we close this gap, at least as far as decidability is concerned.

Theorem 1.1.

The problem EMBED is algorithmically decidable. That is, there is an algorithm that, given a -dimensional simplicial complex , decides whether can be embedded (piecewise linearly, or equivalently, topologically) in .

Let us remark that one can naturally consider (at least) three different kinds of embeddings of a simplicial complex in , illustrated in the next picture for a -dimensional complex (graph):

For linear embeddings, also referred to as geometric realizations, each simplex of should be mapped affinely to a (straight) geometric simplex in . This kind of embeddability is decidable in PSPACE regardless of the dimensions, and it is not what we consider here.

For piecewise linear, or PL, embeddings, one seeks a linear embedding of some (arbitrarily fine) subdivision of . Finally, for a topological embedding, is embedded by an arbitrary injective continuous map.

While topological and PL embeddability need not coincide for some ranges of dimensions, for ambient dimension , they do,1 and this is the notion of embeddability considered here.

An algorithm for EMBED can be obtained from Theorem 1.1 by a simple reduction, given in Section 12.

Corollary 1.2.

The problem EMBED is decidable as well.

Thickening to 3-manifolds. For a -complex , (PL) embeddability in is easily seen to be equivalent to embeddability in , and from now on, we work with as the target.

The first step in our proof of Theorem 1.1 is testing whether a given simplicial -complex embeds in any 3-dimensional manifold at all.

Let us suppose that there is an embedding for some -manifold (without boundary), and take a sufficiently small closed neighborhood of the image in —the technical term here is a regular neighborhood. Then is a -manifold with boundary, called a -thickening of .

There is an algorithm, due to Neuwirth [Neu68] (see also [Sko95] for an exposition) that, given , tests whether it has any -thickening, and if yes, produces a finite list of all possible -thickenings, up to homeomorphism, as triangulated 3-manifolds with boundary (without the knowledge of ). Then embeds in iff one of its 3-thickenings does. Hence it suffices to prove the following.

Theorem 1.3.

There is an algorithm that, given a triangulated -manifold with boundary, decides whether can be embedded in .

Concerning the running time. Our proof does provide an explicit running time bound for the algorithm, but currently a rather high one, certainly primitive recursive but even larger than an iterated exponential tower. Thus, we prefer to keep the bounds unspecified, in the interest of simplicity of the presentation.

By refining our techniques, it might be possible to show the problem to lie in the class NP. Going beyond that may be quite challenging: indeed, as observed in [MTW11], EMBED is at least as hard as the problem of recognizing (that is, given a simplicial complex, decide whether it is homeomorphic to ). The latter problem is in NP [Iva08, Sch04], but nothing more seems to be known about its computational complexity (e.g., polynomiality or NP-completeness).

Related work. There is a vast amount of literature on computational problems for -manifolds and knots. Here we give just a sample; further background and references can be found in the sources cited below and in [AHT06]. A classical result is Haken’s algorithm deciding whether a given polygonal knot in is trivial [Hak61]. More recently, this problem was shown to lie in NP [HLP99], and, assuming the Generalized Riemann Hypothesis, in coNP as well [Kup11]. The knot equivalence problem is also decidable [Hak61, Hem79, Mat97], but nothing seems to be known about its complexity status.

Closer in spirit to the problem investigated here are algorithms for deciding whether a given -manifold is homeomorphic to , already mentioned above [Rub95, Tho94, Iva08, Sch04].

An important special case of Theorem 1.3 is testing embeddability into for an whose boundary is a single torus; this amounts to recognizing knot complements and was solved in [JS03]. Some of the ideas in that work are used in our proof, but most of the argument is fairly different.

In a different direction, Tonkonog [Ton11] provided an algorithm for deciding whether a given 3-manifold with boundary embeds into some homology 3-sphere2 (which may depend on ). His methods are completely different from ours (except for using a 3-thickening to pass from 2-dimensional complexes to 3-manifolds), and it seems to be only loosely related to the problems investigated here.

Future directions. Besides the obvious questions of finding a more efficient algorithm, say one in NP, and/or proving hardness results, one may consider embeddability into other 3-manifolds besides . We believe that this may be within reach of the methods used here, but definitely a number of issues would have to be settled.

The main technical contribution. Our algorithm relies on a large body of work in 3-dimensional topology.

When we talk about a surface in , unless explicitly stated otherwise, we always mean a -dimensional manifold with boundary properly embedded in , that is, with . Similarly, curves are considered properly embedded in a surface, so a connected curve can be a loop in the interior of the surface or an arc connecting two points of the boundary. Two properly embedded surfaces and are isotopic if they are embeddings of the same surface and there is a continuous family of proper embeddings starting with and ending with . An similar definition of isotopy applies to curves embedded in surfaces.

As in almost all algorithms working with 3-manifolds, we use Haken’s method of normal curves and surfaces, actually in a slightly extended form. Here we recall them very briefly; we refer to [Hem76, JT95] for background, and in Section 5 below we will discuss a variant.

A normal curve in a triangulated 2-dimensional surface intersects every triangle in finitely many disjoint pieces, which we can think of as straight segments, as in the left picture:

The main point is that such a curve is described, up to isotopy, purely combinatorially: namely, for every triangle , there are just three types of segments of the curve inside, and it is enough to specify the number of segments for each type, for each . In the picture, the numbers are .

Similarly, a normal surface in a triangulated 3-manifold intersects each tetrahedron in finitely many of disjoint pieces, each of them a triangle or a quadrilateral, as in the right picture above. This time there are seven types of pieces, four triangular and three quadrilateral, per tetrahedron (although no two types of quadrilateral pieces may coexist in a single tetrahedron, since they would have to intersect, which is not allowed). So a normal surface in a 3-manifold with tetrahedra can be described by a vector of nonnegative integers. This vector is called the normal vector of .

A normal isotopy is an isotopy during which the intermediate curve or surface stays normal; in particular, it may not cross any vertex of the triangulation.

Going back to embeddings, we first simplify the situation using a result of Fox [Fox48], which allows us to assume that the complement of the supposed embedding of in is a union of handlebodies.3 (These handlebodies may be knotted or linked in , though, as in the picture at the beginning of the next section.) This assumption is quite important and nontrivial; for example, we note that if is a solid torus, it can also be embedded in in a knotted way, so that the complement is not homeomorphic to a solid torus.

Thus, now we ask if there is a way of “filling” each component of with a handlebody so that the resulting closed manifold is homeomorphic to . Spherical boundary components are easy, since there is only one way, up to homeomorphism, of filling a spherical boundary component with a ball. However, already for a toroidal component there are infinitely many nonequivalent ways of filling it with a solid torus. Indeed, the filling can be done in such a way that a circle on the toroidal component of , as in the left picture,

is identified with a curve on the boundary of the solid torus, shown in the right picture, where may wind around the solid torus as many times as desired. For boundary components of higher genus, there are also infinitely many ways of filling, and their description is still more complicated. For every specific way of filling the boundary components of with handlebodies we could test whether the resulting closed manifold is an , but we cannot test all of the infinitely many possibilities. This is the main difficulty we have to overcome to get an algorithm.

Next, by more or less standard considerations, we can make sure that there is no “way of simplifying by cutting along a sphere or disk”—in technical terms, we may assume that is irreducible, that is, every -sphere embedded in bounds a ball in , and that has an incompressible boundary, i.e., any curve in bounding a disk in also bounds a disk in .

For dealing with such an , the following result is the key:

Theorem 1.4.

Let be an irreducible -manifold, neither a ball nor an , with incompressible boundary and with a -efficient triangulation . If embeds in , then there is also an embedding for which has a short meridian , i.e., an essential4 normal curve bounding a disk in such that the length of , measured as the number of intersections of with the edges of , is bounded by a computable function of the number of tetrahedra in .

In this theorem, -efficient triangulation is a technical term introduced in [JR03], whose definition will be recalled later in Section 7. We are using -efficient triangulations in order to exclude non-trivial normal disks and 2-spheres in .

We should also mention that the triangulations commonly used in 3-dimensional topology, and also here, are not simplicial complexes in the usual sense—they are still made by gluing (finitely many) tetrahedra by their faces, but any set of gluings that produces a manifold is allowed, even those that identify faces of the same tetrahedron. As a result, a particular tetrahedron may not have four distinct faces, six distinct edges and four distinct vertices. In particular, 0-efficient triangulations of the manifolds we consider have a single vertex in each boundary component and none in the interior, all edges in the boundary form loops. This is the necessary result of modifying a triangulation by collapsing simplices, a triangular face to an edge or to a vertex, etc.; see [JR03, Sec. 2.1] for a thorough discussion. There is even a mind-boggling one-tetrahedron one-vertex triangulation of the solid torus obtained by gluing a pair of faces of a single tetrahedron, see [JS03].

Let us remark that as in the theorem need not have a short meridian for every possible embedding, even if we assume that the complement consists of handlebodies. For example, if is a thickened torus (a torus times an interval), we can embed it so that the curves bounding disks in are arbitrarily long w.r.t. a given triangulation of . We must sometimes change the embedding to get short meridians.

It is also worth mentioning that this problem does not occur if is a single torus, i.e., the knot complement case. Here a celebrated theorem of Gordon and Luecke [GL89] makes sure that there is only one embedding, up to a self-homeomorphism of , and the meridian is unique up to isotopy. This is why the single-torus boundary case solved in [JS03] is significantly easier than the general case.

2 An outline of the arguments

Our algorithm for Theorem 1.3, deciding the embeddability of a given -manifold in , for the case of irreducible and with incompressible boundary, consists in testing every possible normal curve of length bounded as in Theorem 1.4. For each such candidate , we construct a new manifold by adding a -handle to along , which means that we glue a disk bounded by to the outside of and thicken it slightly, as illustrated in the following picture:

Here is the complement of the union of two (linked) handlebodies, a knotted solid 3-torus and a solid torus, and for , the solid 3-torus in the complement has been changed to a solid double torus.

Then we test the embeddability of each recursively, and is embeddable iff at least one of the is. It is not hard to show that the algorithm terminates, using the vector of genera of the boundary components of ; see Section 3.

The proof of Theorem 1.4 occupies most of the paper and has many technical steps. In this section we give an outline.

We assume to be embedded in , the complement being a union of handlebodies, and we apply a result of Li [Li10] stating that there is a planar surface (i.e., a disk with holes) that is “stuck” in its position in a suitable sense (namely, is either essential,5 or strongly irreducible and boundary strongly irreducible) and is meridional or almost meridional.

Here an essential curve is a meridian in a given embedding of in if it bounds a disk in . The surface is meridional if each component of is a meridian, and it is almost meridional if all components of but one are meridians. (Actually, Li has yet another case in his statement, but as we will check, that case can be reduced to the ones given above; see Lemma 4.4.) The next picture illustrates a meridional in the case where is embedded in as the complement of a solid torus neighborhood of the figure ‘8’ knot:

Next, by choosing as above with suitable minimality properties, one can make sure that is normal or almost normal6 for the given triangulation. For the case of essential, this is an old result going back to Haken and Schubert (and for our notion of complexity of , a proof is given in Section 7), while for strongly irreducible and boundary strongly irreducible this follows from [BDTS12]; also see [Sto00] for the case of a strongly irreducible surface in a closed manifold. It remains to show that, in this setting, at least one of the meridians in must be short.

Here we apply an average length estimate, which is an idea of Jaco and Rubinstein appearing in [JS03, JRS09].

Let be the components of , and let be the boundary length of . We know that all the but at most one are meridians. The length of the shortest meridian is bounded by the average , and we want to bound this average by a (computable) function of , the number of tetrahedra in the triangulation of .

Now by the theory of normal surfaces, the (almost) normal surface can be written as a normal sum7 of fundamental surfaces in ,

(1)

where the are positive integers and the are surfaces from a finite collection; their number, as well as can be bounded by a (computable) function of alone, and does not depend on .

Since the boundary length is additive w.r.t. normal sum, we have , where is the number of fundamental summands in the expression for , and so it suffices to show that , with some computable function .

The basic version of the average-length estimate uses the Euler characteristic as an accounting device. Since is additive as well, . Since is a planar surface with boundary components, we have .

Now an ideal situation for the average-length estimate (which we cannot guarantee in our setting) is when for every ; in other words, none of the summands is a disk, 2-sphere, annulus, Möbius band, or torus (or projective plane or Klein bottle, but these cannot occur in embedded in ). Then we get , and we are done (even with ).

In our actual setting, the summands with , i.e., spheres and disks, are excluded by the -efficient triangulation of . We also need not worry about torus summands, since they have empty boundary and thus do not contribute to . The real problem are annuli (and Möbius bands, but since twice a Möbius band, in the sense of normal sum, is an annulus, Möbius bands can be handled easily once we deal with annuli).

There are two kinds of annuli, which need very different treatment: the essential ones, and the boundary parallel ones. Here an annulus is boundary parallel if it can be isotoped to an annulus with while keeping the annulus boundary fixed. Boundary parallel annuli do not occur for essential, but they might occur for the case of strongly irreducible and boundary strongly irreducible.

To deal with the annulus summands, we first construct what we call an annulus curve . This is the boundary of a maximal collection of essential annuli, maximal in the sense that each of the two boundary curves of every other essential annulus, after a suitable normalization, either intersects or is normally isotopic to a component of . We bound the length of by a computable function of , and , the number intersections of with , by , for some computable , again assuming minimal in a suitable sense. For obtaining this bound we may need to change the embedding of , and we also use results about “untangling” a system of curves on a surface by a boundary-fixing self-homeomorphism from [MSTW13].

Similarly, we construct a collection of curves that helps to deal with boundary parallel annuli: those that have minimal boundary in a suitable sense either intersect , or their boundaries are normally isotopic to components of or curves from .

Having constructed such an and , we work with normal curves and surfaces in a “marked” sense, which also takes into account the position of the curves and surfaces w.r.t.  and . This, in particular, makes the number of intersections with additive w.r.t. the marked normal sum, which in turn allows us to bound the number of annulus summands in (1), both boundary parallel and essential, that intersect by .

Then we might have boundary-parallel annulus summands that avoid , but we show that those do not occur at all, since they would contradict the minimality of .

Finally, there remain essential annuli that have a boundary component parallel to a component of . Here we show that if such an annulus had the coefficient in (1) at least , then there is a self-homeomorphism of , namely, a Dehn twist in the annulus, that makes simpler, contradicting its supposed minimality. (Here we may again modify the assumed embedding of in in order to get a short meridian—and, as we have remarked, some such modification is necessary in the proof, since some embeddings may not have short meridians.) Hence for these essential annuli, too, the coefficients are bounded by a linear function of . This concludes the proof.

3 The algorithm

If embeds in , then it is orientable, and orientability can easily be tested algorithmically (e.g., by a search in the dual graph of the triangulation, or by computing the relative homology group ). So from now on, we assume orientable. In this situation, the boundary of is a compact orientable 2-manifold, and thus each component is a 2-sphere with handles.

We describe a recursive procedure EMB(X) that accepts a triangulated orientable 3-manifold with boundary and returns TRUE or FALSE depending on the embeddability of in . (With some more effort, for the TRUE case, we could also recover a particular embedding, but we prefer simplicity of presentation.) The procedure works as follows.

  1. (Each component separately) Let be the connected components of . If , test if there is an among the (several algorithms are available for that [Rub95, Tho94, Iva08, Sch04]), and if yes, return FALSE. Otherwise, still for , return the conjunction .

  2. (Fill spherical holes) Now we have connected. If it is an , return TRUE. If there are components of that are ’s, form by attaching a 3-ball to each spherical component of , and return .

  3. (Connected sum) Form a decomposition of into a connected sum8 of prime manifolds9 that are not -spheres.10 If , i.e., is not prime, return . If is prime but not irreducible, i.e., contains an that does not bound a ball, then return FALSE.

  4. (Boundary compression) Test if there is a compressing disc for (i.e., does not bound a disk in ).11 If yes, cut along , obtaining a new manifold . Three cases may occur:

    1. If has two components, and , return . This case may occur, for example, for a handlebody with two handles (a “thickened 8”) when separates the two handles.

    2. If is connected and the two “scars” after cutting along lie in the same component of , return EMB. This case may occur, e.g., for a solid torus.

    3. If neither of the previous two cases occur, then is connected but the scars lie in different components of . Return FALSE. To get an example of fitting this case, we can start with a thickened torus (i.e., torus times ) and connect the two boundary components by a -handle—which cannot be done in , but it does give a 3-manifold (with double torus boundary).

  5. (Short meridian) Now is irreducible and with incompressible boundary. Using [JR03, Thm. 5.20], retriangulate with a -efficient triangulation. Then proceed as described at the beginning of Section 2: let be a list of all closed essential normal curves in up to the length bound as in Theorem 1.4, for each form by attaching a -handle along , and return the disjunction .

Lemma 3.1.

The above procedure always terminates and returns a correct answer, assuming the validity of Theorem 1.4.

Proof.

First we show that the algorithm always terminates. Let be the components of numbered so that , where stands for the genus, and let be the vector . We consider these vectors ordered lexicographically (if two vectors have a different length, we pad the shorter one with zeros on the right).

Let us think of the computation of the algorithm as a tree, with nodes corresponding to recursive calls. The branching degree is finite, so it suffices to check that every branch is finite.

It is easy to see that cannot increase by passing to a connected component or to a prime summand, and that it decreases strictly by a boundary compression and also by the short meridian step. Indeed, we observe that in the boundary compression step or the short meridian step, exactly one of the boundary components is affected, and it is either split into two components and of nonzero genus and with , or it remains in one piece but the genus decreases by one. Since after steps 13 we have a connected irreducible manifold without spherical boundary components, for which the next step either finishes the computation or reduces strictly, every branch is finite as needed.

It remains to show that the returned answer is correct. For Step 2, we need that there is a unique way of filling a spherical hole; this is well known and can be inferred, for example, from the fact that there is only one orientation-preserving self-homeomorphism of up to isotopy [FM11, Sec. 2.2].

For Step 3, it is easily checked that a connected sum embeds iff the summands do. Moreover, every embedded in separates it, and hence if contains a non-separating , then it is not embeddable.

For Step 4, it is clear that if is embeddable, then so is .

If, in case (4a), and are both embedded, then it is easy to construct an embedding of : Denote ’s scars by and . Then a regular neighborhood of is a ball with boundary , and that meets both and in balls. Think of each as embedded in its own copy of , and take a connected sum of these two ’s so that . Similarly, if is embedded in case (4b), then we can connect the scars by a thin handle in and obtain an embedding of .

In case (4c), let be the components of containing the scars. Since the disk does not separate , we can choose a loop X meeting in a single point, and such that also meets in a single point. But then, if were embedded in , would yield a nonseparating surface in —a contradiction.

Finally, if one of the is embeddable in Step 5, then so is (since in we have -handle that was added to , and we can just assign it to the complement of ), and if is embeddable, then at least one of the is by Theorem 1.4. ∎

4 Intersections of curves and surfaces

In this section we collect terminology, definitions and basic results concerning properly embedded curves in surfaces and properly embedded surfaces in 3-manifolds. In particular, for latter sections we need that any pair of properly embedded surfaces, each either essential, or, strongly irreducible and boundary strongly irreducible, can be isotoped to intersect essentially. There are few new results in this section. The reader is referred to Hempel [Hem76] and Jaco [Jac80] for more background.

We assume throughout that all curves and surfaces have been isotoped to have transverse intersection.

4.1 Curves

A curve is a properly embedded -dimensional manifold in a surface , each component either a loop, which is closed, or an arc, which has two endpoints in .

A loop is trivial if it bounds a disk in and an arc is trivial if it co-bounds a disk in with some arc in . A curve is essential if no component is trivial.

Pairs of curves are assumed to intersect transversally. If and are a pair of curves, then their geometric intersection number taken over all pairs of curves where and are isotopic to and within , respectively. (The isotopies are also allowed to move endpoints of arcs within the boundary.)

We say that and bound a bigon if there is a disk bounded by a pair of sub-arcs, one from each curve; see Figure 3 in Section 6 below. We say that they bound a half-bigon if there is a disk bounded by a pair of sub-arcs, one from each curve, along with an arc in . If and bound a bigon or half-bigon, then they can be isotoped to reduce their intersection.

We need this converse, a mild generalization of Farb and Margalit’s bigon criterion:

Lemma 4.1 (Bigon criterion [Fm11]).

A pair of curves and realize their geometric intersection number if and only if they do not bound a bigon or half-bigon.

Proof.

Farb and Margalit show that any pair of connected loops that intersect non-minimally form a bigon. They also note that this extends to disconnected curves consisting of loops.

If either curve has an arc component, then the doubled curves are properly embedded closed curves in the double12 of the surface. If they intersect non-minimally in the original, they intersect non-minimally in the double and hence bound a bigon there. Thus, they bound a half-bigon in the original. ∎

4.2 Essential surfaces

We will assume that our surfaces are properly embedded in a 3-manifold that is irreducible, i.e., every sphere embedded in bounds a ball in , and boundary incompressible, i.e., any curve in bounding a disk in also bounds a disk in (is trivial).

Let be a surface properly embedded in . A compressing disk for a is an embedded disk whose interior is disjoint from and whose boundary is an essential loop in . A boundary compressing disk is an embedded disk whose boundary, , is the union of , an essential arc properly embedded in , and , an arc properly embedded in . Here is an illustration:

A surface is compressible if it has a compressing disk, boundary compressible if it has a boundary compressing disk, and incompressible and boundary incompressible if not, respectively. A surface is essential if it is incompressible, boundary incompressible, and not a sphere bounding a ball, or a disk co-bounding a ball with a disk in .

We establish some basic facts about surfaces in .

Proposition 4.2.

The following statements hold for properly embedded surfaces in , an irreducible, orientable -manifold with non-empty incompressible boundary:

  1. Every disk co-bounds a ball with a disk in .

  2. Every connected surface with an inessential boundary curve is either compressible or a disk.

  3. The boundary curve of every compressible annulus is trivial in boundary.

  4. Every boundary compressible annulus is boundary parallel (parallel to an annulus in ).

  5. No surface is a projective plane.

  6. Every Möbius band is essential.

Proof.

Because has incompressible boundary, the boundary of a properly embedded disk bounds a disk in . The union of these disks is a sphere that, because is irreducible, bounds a ball, yielding (i).

For (ii), suppose that some boundary curve of connected bounds a disk in . Then, among disks in bounded by boundary curves of , an innermost such disk can be pushed slightly into the interior of while keeping its boundary in . The boundary curve is either trivial in , in which case is a disk, or essential in , in which case is compressible.

Concerning (iii), let be a compressing disk for an annulus . Then separates into two annuli and . So and are properly embedded disks, each with one boundary curve of . Because is incompressible, the boundary curves are both trivial in .

As for (iv), let be a boundary compressing disk for an annulus . Then , the boundary of a regular neighborhood of their union, has two components, an annulus isotopic to and a disk. By (i), the disk co-bounds a ball with a disk in . But then the union of with the ball is a solid torus, across which is parallel to an annulus in .

Concerning (v), if is a projective plane, then , the boundary of its regular neighborhood, is a sphere which separates from . Then is reducible, for the sphere cannot bound a ball—no ball has interior boundary or contains an embedded projective plane.

Finally, in (vi), let be a a Möbius band. Suppose first that is compressible and let be a compressing disk for . Then cannot meet in a core curve of for this would imply that the core curve is orientation reversing in . So is a 2-sided curve in and separates it into an annulus and a narrower Möbius band . Then the union is an embedded projective plane contradicting (v).

Suppose a Möbius band is boundary compressible. Then is a boundary compressible annulus . By (iv), is boundary parallel, and co-bounds an solid torus with an annulus in the boundary. But the parallel region cannot contain the Möbius band , and hence is the union of two solid tori, and the solid torus parallel region. ∎

We say that a a pair of surfaces, and , intersect essentially if each component of the curve is essential in both and (they are allowed to be disjoint). It is well known that essential surfaces can be arranged to intersect essentially:

Lemma 4.3.

Let and be properly embedded essential surfaces in an irreducible manifold with incompressible boundary. Then can be isotoped so that they intersect essentially.

Proof.

Assume that we have isotoped to minimize the number of curve components in . We will show by contradiction that and intersect essentially.

We first note that if there is an intersection curve that is inessential in , then there is an intersection curve that is inessential in and vice-versa: If an intersection curve bounds a disk in , choose one whose disk is innermost. Since is incompressible, this disk is not a compressing disk for and it follows that its boundary, an intersection loop, is inessential in . The same observation applies to inessential intersection arcs.

Then, assuming that some intersection loop is trivial, we can pass to one that is innermost on , i.e., choose to be an intersection loop that bounds a disk whose interior is disjoint from . Since is not compressible, also bounds a disk . The union is a sphere that, because is irreducible, bounds a ball. And there is an isotopy of that is restricted to a neighborhood of , and that pushes across the ball and past . This isotopy of eliminates and any other intersection curves in the interior of , and it does not introduce any new intersection curves since was innermost.

Now assume some intersection arc is trivial in one of the surfaces, and as noted, we can let denote such an arc that is outermost in . That is, cuts off a disk whose interior is disjoint from and whose boundary meets in an arc. And cuts off a, not necessarily outermost, disk that also meets in an arc.

The union is a disk with its boundary in that, because is incompressible, bounds a disk . Since is irreducible, is a sphere bounding a ball. Moreover, there is an isotopy of that pushes a neighborhood of past and outside the ball. ∎

4.3 Almost meridional surfaces

Suppose that is an irreducible manifold with incompressible boundary that is embedded in . We recall that an essential curve is a meridian if it bounds a disk in . A properly embedded surface is meridional if each of its boundary curves is a meridian, and almost meridional if all but exactly one of its boundary curves is a meridian.

Let be a boundary compressing disk for an orientable surface . Then is a surface with at least two components. One component is isotopic to ; let be the union of the other components. Then is said to be the result of boundary compressing along .

Lemma 4.4.

Suppose that a manifold is embedded in . If is a connected almost meridional planar surface properly embedded in , then any surface obtained by boundary compressing contains an almost meridional component.

Proof.

Let be obtained from by boundary compressing along the disk . What happens to ? The disk meets at most two boundary components of . Any component not met by has two parallel copies in , one for and one for , so those are unchanged. Let be the one or two loops meet by the arc . Since lies on one side of the the 2-sided planar surface , when is a single loop, approaches it twice from the same side. It follows that is a pair of pants, i.e., an with three holes bounded by loops. One of these loops belongs to and two to , or vice-versa.

If any two of these three loops are meridians, then so is the third, since it bounds a disk, namely the union of the pants and the two disks pushed slightly into .

We apply this “two meridians implies three meridians” principle to show that has an almost meridional component, regardless of how the boundary compressing disk meets the boundary components of .

If the boundary compressing disk meets the non-meridional component twice, then the compression eliminates the non-meridional curve, and creates two new curves, each belonging to a separate component of . At least one of the new curves is not meridional, and hence its component is almost meridional.

If the boundary compressing disk meets a meridian and the non-meridian, then the compression does not separate , and trades these curves for a new non-meridional curve. Thus is almost meridional.

If the boundary compressing disk meets two distinct meridians, then they are eliminated and a new one is created. The connected surface is almost meridional.

If the boundary compressing disk meets a single meridian twice, then has two components, each with one of the two new curves, either both meridional or both non-meridional. If both are meridional, then the component with the original non-meridian on its boundary is almost meridional. If both are non-meridional, then the component without the original non-meridian is almost meridional. One of the two components of is almost meridional. ∎

Lemma 4.5.

Suppose that , an irreducible manifold with incompressible boundary, is embedded in . If contains an incompressible, almost meridional planar surface, then contains an essential almost meridional planar surface.

Proof.

An incompressible almost meridional surface can be sequentially boundary compressed until it is incompressible and boundary incompressible. By the prior lemma, each surface in the sequence, hence the final one, has an almost meridional component. This final component is not a disk because is boundary incompressible. Hence it is an essential almost meridional planar surface. ∎

4.4 Strongly irreducible surfaces

A two-sided surface properly embedded in is bi-compressible if it has a compressing pair , a pair of disks, each a compressing or boundary compressing disk, one for each side of the surface. The pair is simultaneous if .

A surface is weakly reducible if it is simultaneously bi-compressible using compressing disks only. A strongly irreducible surface is one that is bi-compressible using compressing disks but not simultaneously so. A surface is boundary weakly reducible if it is simultaneously bi-compressible using any combination of compressing disks and boundary compressing disks. A surface is boundary strongly irreducible if it is bi-compressible, using any combination of compressing or boundary compressing disks, but not simultaneously so.

Some of our results assume that a surface is both strongly irreducible and boundary strongly irreducible. It may seem that the strongly irreducible hypothesis is vacuous. But this is not the case—it guarantees that the surface has at least one (non-boundary) compressing disk for each side.

Lemma 4.6 ([Bdts12], Lemma 3.8).

In an irreducible manifold with incompressible boundary, the boundary of a strongly irreducible surface is essential in the boundary of the manifold.

We state here a special case of Lemma 4.2 of [Bac13]. This

Lemma 4.7 (Lemma 4.2 of [Bac13]).

Let be an essential surface and a surface that is strongly irreducible and boundary strongly irreducible. Then may be isotoped so that and intersect essentially.

Let us remark that Bachman does not give a proof but claims it to be a direct generalization of [Bac09, Corol. 3.8]. He has also provided us with an unpublished manuscript with a proof.

5 Theory of normal curves and surfaces in a marked triangulation

In this section we introduce a mild generalization of the theory of normal curves and surfaces.

Definition 5.1.

A marked triangulation is a pair consisting of a triangulation of a - or -manifold along with a marking , a finite set of points along the edges of .

If , then is a triangulation in the usual sense and we will usually omit and refer directly to . Similarly, when , we will describe objects as being normal rather than -normal, and note that our definitions restrict to the standard ones.

An arc in a triangle is -normal if its endpoints lie in distinct edges of the face and it misses . A properly embedded curve in the boundary surface is -normal if it is the union of -normal arcs. The length of , , is its number of intersections with the 1-skeleton.

There are several types of elementary surfaces contained in a tetrahedron . An -normal disk is a disk in whose boundary is an -normal curve of length 3 or 4 in . We also consider two types of -exceptional pieces: An octagon is a disk in whose boundary is an -normal curve of length 8 in . A tube is an unknotted annulus in whose boundary consists of two -normal curves whose total length is at most 8.

An -normal surface is a properly embedded surface that is the union of -normal disks. An almost -normal surface is a properly embedded surface that is the union of a single -exceptional piece and a collection of -normal disks.

The weight of an (almost) -normal surface is , the number of intersections with the 1-skeleton. Its length is the length of its boundary: .

An -normal isotopy is a normal isotopy that does not pass through any point in . An -type is the equivalence class of an -normal arc in a face, or, an -normal disc or -exceptional piece in a tetrahedron. Two types are -compatible if they have disjoint representatives. A pair of curves or surfaces are -compatible if each pair of types they possess are -compatible. That is, a pair of curves is compatible if, for each face of the triangulation, their arcs in that face are pairwise disjoint after an -normal isotopy. An analogous statement holds for surfaces.

We note that -compatibility is a local condition; in general it may not be possibly to make -compatible curves or surfaces globally disjoint by an -normal isotopy.

The -normal vector or -normal coordinates of an -normal curve, surface, or almost -normal surface is a uniquely determined vector , indexed over the set of normal types and with each entry recording the number of -normal objects of the index type.

If are -normal surfaces such that , then is an -normal sum of and , and we write . The same definition applies if is -normal and and almost -normal, or if are -normal curves.

We note that not every two -normal surfaces, for example, can be normally added—this is possible exactly if they are -compatible.

If and are -compatible, then one can construct an -normal sum as follows. In each face or tetrahedron , the -normal pieces and can be -normally isotoped to be disjoint, and then attached across each facet of to the pieces in an adjacent face/tetrahedron. This produces a properly embedded -normal curve, -normal surface, or almost -normal surface, respectively, which is the -normal sum.

However, in our considerations, we will mostly use a different geometric construction of an -normal sum, where we assume that the curves or surfaces in question intersect minimally, in a suitable sense, but then we do not isotope them to be disjoint as above, but rather they stay in place and we deal with their intersections as well; see Section 5.1 below.

It is well known that Euler characteristic, weight and length are all additive with respect to normal sum, and this works without change for the -normal case. If and are compatible (almost) -normal curves or surfaces then the following hold:

  1. .

An (almost) -normal curve or surface is fundamental if it cannot be expressed as the sum of other (almost) -normal curves or surfaces. Every (almost) -normal curve/surface is a non-negative integer combination of fundamentals.

Figure 1: A length 8 curve that is not the sum of two length 4 curves in the marked triangulation.

Here -normal curve theory differs from standard normal curve theory. While all normal curves are compatible, -normal curves have distinct compatibility classes, and this increases the number of fundamentals. In Figure 1, we see the boundary of a tetrahedron with two marked points, one on each of a pair of opposite edges. Let be the length 8 -normal curve that meets each of the sub-edges once. As a normal curve is not fundamental—it is the sum of the two distinct length 4 curves and . But in the marked triangulation these curves are incompatible and is fundamental.

If is an (almost) -normal curve or surface, then is a solution to a set of matching equations: For a triangulated surface, this set consists of one equation for each sub-edge in the interior of the surface. It sets equal the sum of those coordinates meeting the sub-edge from one side to the sum of those meeting it on the other side. In a triangulated 3-manifold, the set of matching equations consists of one equation for each -normal arc type contained in an interior face. This equation sets equal the sum of the coordinates for elementary types using the arc type on one side to the sum of those using it on the other side.

We say that a vector of the correct dimension is -admissible if all its coordinates are non-negative, it satisfies the matching equations, and is self-compatible, i.e., it does not possess non-zero coordinates for any pair of non--compatible types. If is an -admissible vector, there is an (almost) -normal curve/surface for which .

The following proposition is a straightforward generalization of a well known fact from normal surface theory to -normal surfaces; see [HLP99] for a nice exposition.

Proposition 5.2.

Given a -manifold with a marked triangulation , the set of fundamental -normal surfaces is computable, and both and the maximum weight of an are bounded by a computable function of and .

The bound on and has the form , where is a suitable polynomial.

Proof.

It is well known that without the marking, there are normal disk types, exceptional octagons and exceptional tubed pairs of disks. Moreover, the presence of a tubed pair of disks may split one type of normal disks into two, but certainly we have no more than types in total.

The points of divide each edge into at most subarcs. In order to specify an -normal type of a triangle, for example, we need to specify the subarc containing each of the three vertices, which leads to the bound . The worst bound is obtained for tubes and octagons, with , so a rough bound for the total number of -types is .

A similar way of counting applies to the number of matching equations, which represent compatibility of the coordinates of the -normal vector across the pieces of the edges of delimited by the points of . Indeed, the matching equations correspond to -arc types. There are at most interior faces, each with underlying normal arc types. A given -arc type is thus determined by this normal type and by the sub-arcs it meets, and so there are at most matching equations.

Then, reasoning as in [HLP99, Sec. 6], using a Hilbert basis of the appropriate integral cone, we obtain the bounds of the claimed form. ∎

5.1 Snug pairs of curves and surfaces, Haken sums, and normal sums

The normal sum of a pair of (almost) -normal curves or surfaces and has been defined, if they are -compatible, to be an -normal surface whose -normal vector is the sum of the -normal vectors of and , .

It is desirable to show that qualities of the sum, such as essentiality or minimality, also apply to the summands. Here we describe a well known geometric interpretation of the sum that makes this possible; also see, for example, [JO84, JT95]. We also present some related material.

Snug pairs. We begin with a definition of a “placement with no unnecessary intersections” for a pair of curves or surfaces.

Definition 5.3.

A pair of properly embedded curves or surfaces is snug if it is transverse and the number of components of the intersection is minimized over pairs , where and are isotopic to and , respectively. The pair is locally snug if is disjoint from the 1-skeleton , and, they are snug in the interior of each simplex of the triangulation (here we only allow isotopies moving each intersection of or with a face only within that face).

If and are locally snug -normal surfaces then it follows that:

  1. each pair of -normal arcs, one from and one from , meets in 0 or 1 points;

  2. each pair of -normal disks, one from and one from , meets in 2 or fewer arcs, and the union of the arcs has at most one endpoint in any face;

  3. no loop of lies inside a tetrahedron.

Any pair of compatible -normal curves or surfaces can be made locally snug by -normal isotopies that first make their intersections with edges disjoint and then “straighten” them so that: normal arcs are straight, normal triangles are flat, and normal quads are the union of two flat triangles. We do not define locally snug when is an almost normal surface and is a normal surface, for in that case we require only the definition of the normal sum and not its geometric interpretation.

Haken sum and normal sum of curves. Now, for a while, we deal only with curves, and we develop a geometric interpretation of their normal sum. Here we consider only unmarked triangulations, i.e., .

Let be a regular neighborhood of an intersection point of a pair of transverse curves and . We can remove the intersection by deleting the arcs in the interior of the disk and then attaching to along a pair of antipodal sub-arcs of . Thus, we replace the “” in with either “” or “”. This is called an exchange or a switch at . A curve is said to be a Haken sum of and if it is obtained by an exchange at each of their intersection points. Of course, is dependent on the direction of the switches and is therefore not well determined.

If, however, and are locally snug normal curves, then each intersection point is of the form where and are normal arcs in some face. Then and meet at least one common edge of the face. The regular exchange is the exchange that does not produce an abnormal arc, a non-normal arc with both endpoints attached to ; see Fig. 2 top.

Figure 2: Switches for curves and surfaces

As we will see, the normal sum of of locally snug curves can be obtained by doing all the regular exchanges.

Lemma 5.4.

Let and be locally snug normal curves. Then the Haken sum obtained by making all the regular exchanges is the normal sum ; i.e., .

Proof.

We show that the result holds in each face of the triangulation. In an abuse of notation, let and be restriction of the curves to a particular face. For contradiction suppose that they are a counterexample that minimizes . Then and are not disjoint, for in that case, the union is normal and normal vectors add.

Since they intersect in a face, we can identify an outermost half-bigon bounded by a sub-arcs of and and an edge of the face; see Figure 3. The regular exchange trades these sub-arcs and results in a pair of normal curves, normally isotopic to and normally isotopic to , that are locally snug but with fewer intersections. By assumption, these and satisfy the conclusion, hence so do and . ∎

Lemma 5.5.

Let be a Haken sum of locally snug properly embedded curves. Then is normal if and only if and are normal and all switches are regular, i.e., . In addition, if and are normal and contains at least one irregular switch, then contains an abnormal arc.

Proof.

() This is by Lemma 5.4. () We show that if either or is not normal then neither is for any Haken sum of the curves.

Suppose then that is an outermost non-normal arc, one that co-bounds a disk with a sub-arc of an edge . If meets the disk, it meets it in a collection of arcs, each with one endpoint in and one endpoint in , because is outermost and and are snug. Let be a regular neighborhood of the disk. Then, regardless of the switches, meets in a collection of arcs that have endpoints along the edge and endpoints not on the edge. It follows that at least one arc meets the edge in 2 points and is not normal. A symmetric argument applies if the outermost non-normal arc belongs to . Nor can either or possess a loop in a face. Local snugness implies that any loop is disjoint from the other curve and survives any Haken sum.

We now know that , are normal. To conclude the proof, it is sufficient to show that contains an abnormal arc if at least one switch is irregular (this contradicts the normality of , and thus proves the last claim of the lemma). In an abuse of notation, let and refer to the collection of normal arcs in a particular face.

We perform the specified switches in order according to the following scheme: If , then and form an outermost half bigon with an edge as in Figure 3. The regular switch produces collections and that are normally isotopic to and , but with one fewer intersections. An irregular switch produces a disjoint abnormal arc that survives any and all additional exchanges. If each exchange is regular, we can continue and the process produces a disjoint union If any exchange is not regular, the resulting curve contains an abnormal arc. ∎

Figure 3: Exchanges at the corners of a bigon and a half-bigon.

Normal sign. When and lie in an oriented surface, for example the boundary of an oriented manifold, we can define the normal sign of each point of . Viewing as horizontal and as vertical, the regular exchange at the point connects a pair of quadrants. The point has positive sign if the exchange connects the southwest quadrant to the northeast quadrant, and it has negative sign if it connects the northwest to the southeast; see Figure 4. This is equivalent to the definition given in [BDTS12]. The definition depends on the ordering of the pair of curves and on an orientation on the surface: reversing the order or the orientation reverses every sign.

Figure 4: Exchanges with positive and negative sign.

Normal sum of surfaces. Similar to the case of curves above, one can also construct the normal sum of normal surfaces geometrically, using suitable switches. We assume that and are locally snug.

We construct by specifying its intersection with the 1-, 2-, and 3-skeleta of the triangulation, respectively. First, we let the intersection of with the 1-skeleton to be the union of the intersection points from and those from .

Second, in each face we perform regular switches on all intersecting pairs of arcs , where comes from and from .

Finally, we construct the normal sum in the interior of each tetrahedron . As discussed earlier, each normal disk is either a flat triangle or a quadrilateral made of 2 flat triangles. It follows that every intersection between normal disks from compatible surfaces is either 1 or 2 arcs, not necessarily straight. Compatibility ensures that the regular switches prescribed at the endpoints of each arc are consistent with each other and can be extended across the entire arc of intersection. The normal sum and is the result of performing such regular switches along every such arc of intersection.

Note that any intersection arc between normal disks can be extended from a tetrahedron through a face to a neighboring tetrahedron. In its entirety this intersection curve between and is either a loop, or an arc with both endpoints in . Compatibility ensures that the regular switches in each face and through the interior of each tetrahedron agree. Thus we can regard the switch as a regular switch along the entire intersection curve.

Exchange arcs and surfaces, trace curves. Here we introduce some additional terminology. First, we consider a regular switch of two curves. Inside the neighborhood where the regular switch was performed, we identify an exchange arc that connects the points of the newly formed arcs corresponding to the former intersection points; see Figure 2.

Next, we consider two locally snug normal surfaces and . A patch is a component of . Regular switches reconnect the patches, and trace curves are the seams between patches after performing regular switches along all intersection curves; see Figures 2 and 5.

Figure 5: The exchange between surfaces along an intersection curve.

If the intersection curve is an arc, then, after performing a regular switch, we can identify an exchange rectangle, a rectangle whose top and bottom, say, are bounded by trace arcs and whose left and right sides are exchange arcs lying in .

If is a loop, then our assumption that is orientable means that a regular neighborhood of is a solid torus, not a solid Klein bottle. Again, since is orientable, is either orientation preserving in both and , or, orientation reversing in both and . In the former case, there is an exchange annulus, a zero-weight annulus bounded by the trace curves and with core . In the latter case, there is a single trace curve which bounds an exchange Möbius band (we will be able to exclude this case in our proofs, though).

As observed in [Hat82] and, in the context of normal surfaces, in [JS03], every intersection arc between surfaces connects intersection points of the boundary curves that have opposite normal sign:

Lemma 5.6.

Let be a normal sum of surfaces in an orientable manifold with an induced orientation on . Then every arc in joins a pair of points in with opposite normal sign.

6 Complexity and tight curves

In this section, we consider properly properly embedded curves in a triangulated surface. We assume that they are transverse to the 1-skeleton but, a priori, they are not assumed to be normal.

Fix, once and for all, an ordering of all normal arc types of the triangulated surface. For this purpose we do not take into account any marking present. As in the previous section, a normal curve determines a vector which records the number of normal arcs of the indexed type. Order these normal vectors lexicographically.

Recall that the length of a properly embedded curve is the number of intersections with the 1-skeleton, . We say that a curve is least length if it minimizes length over all curves to which it is isotopic.

Lemma 6.1.

A least length essential curve is normal.

Proof.

A loop in face demonstrates that the curve is not essential and any abnormal arc is either inessential or yields an isotopy reducing the length. ∎

If is a normal curve, then we define its complexity to be the pair consisting of its length and its normal vector,

We reiterate that we do not take into account any marking in the definition of complexity. If is not normal, we define its complexity to be . Complexities will also be ordered lexicographically.

Definition 6.2.

A curve is tight if it minimizes the complexity over all curves to which it is isotopic.

The interior of a connected inessential curve can be made disjoint from the 1-skeleton, so a tight inessential loop has and a tight inessential arc has .

Lemma 6.3.

A tight essential curve is normal and unique up to normal isotopy.

Proof.

Indeed, the complexities of two normal curves are equal if and only if their normal vectors are identical. ∎

Lemma 6.4.

Let be a Haken sum of locally snug properly embedded curves. Then with equality holding if and only if and are all normal, or, all not normal.

Proof.

The curve is constructed by performing an exchange at every intersection point of . This is done away from the 1-skeleton, so we have . Thus any difference in complexity is determined solely by the normal vectors of the curves. If is normal, then by the previous two lemmas and equality holds. If is not normal, then its normal vector is . Then complexity is additive when both or are not normal, and sub-additive otherwise. ∎

If a tight curve is written as a sum, then the exchange arcs are essential in the complement of the curve:

Lemma 6.5.

Suppose that a tight normal curve is written as a sum of two normal curves. Then no exchange arc co-bounds a disk with a sub-arc of the curve.

Proof.

Perform an irregular exchange only at the intersection corresponding to this exchange arc. The new curve is a Haken sum with one component a trivial loop, the rest isotopic to , and the same total length. It follows that the trivial loop has zero length otherwise would not be tight. Therefore and the second component of are normally isotopic. However, by Lemma 5.5 there is an abnormal arc in the second component of , a contradiction. ∎

Lemma 6.6.

Let be a bigon or half-bigon bounded by a pair of locally snug normal curves and ; see Figure 3. Let and be the pair of isotopic curves obtained by corner exchange(s) that trade the sides of . Then one of the following holds:

  1. , thus and are normally isotopic to and , respectively, and ;

  2. ;

  3. .

Proof.

Let and be as indicated in Figure 3. Note that the exchange doesn’t add or remove intersections with the 1-skeleton, and so the total length is unchanged. If the traded arcs differ in length then one curve increases and the other decreases in length, hence in complexity. In this case, either (2) or (3) holds. So we continue assuming and .

If any exchange is irregular, then one of the curves, say , is not normal. Then its complexity has decreased, yielding conclusion (2). Conclusion (3) results when is not normal.

We are left in the case that the exchange trades length fairly and and are both normal. Because length and normal vectors are both additive with respect to normal addition, we have . If , then and by Lemma 6.3 the trade yields normally isotopic curves, conclusion (1). Otherwise, either (2) or (3) holds. ∎

Lemma 6.7.

Let be a tight essential curve and set of pairwise snug, tight essential curves. Then, after a normal isotopy of , is pairwise snug.

Proof.

Normally isotope to minimize the total of all intersections with . By way of contradiction, suppose some pair is not snug, that there is for which . Among all such take one that, together with , determines an innermost bigon; then any other curves from meeting that bigon meet it in arcs that run straight across.

Apply Lemma 6.6. Since all curves are tight, we must have the first conclusion. But, trading across the bigon reduces intersections between and without raising intersections of any other pair—a contradiction. ∎

Lemma 6.8.

Suppose that a tight essential normal curve is a normal sum . Then and are tight, essential, and after a normal isotopy, snug.

Proof.

Normally isotope and/or to minimize . This does not change their sum.

First we show that the pair is snug: If not, then some pair of sub-arcs of and bound a bigon or half-bigon . Apply Lemma 6.6. The first conclusion does not hold, so without loss of generality assume that . Isotope back slightly so that and still overlap and form a very thin bigon. Since and are isotopic as graphs, is isotopic to some Haken sum . But by Lemma 6.4, , a contradiction.

It follows that and are both essential. If either possesses a component that bounds a disk, then the fact that and are snug implies that this component misses the other curve, survives normal addition, and contains an inessential component, a contradiction.

It remains to show that each summand is tight. Without loss of generality, suppose that is not tight, that there is a tight curve with lower complexity, , that is isotopic to but not normally so. Isotope to intersect minimally.

Then any innermost (half-) bigon in the complement of is bounded by and , since it cannot be bounded by and , which are snug. And because any patch of is a sub-arc of , any innermost (half-) bigon bounded by and is also a (half-) bigon bounded by the tight curves and which, using Lemma 6.6 again, can be eliminated by a normal isotopy of . This contradicts the minimality of the intersection between and .

Then, sub-curves of and co-bound a product region as in Figure 6.

Figure 6: Possible product regions derived from isotopy of and .

If they are not snug, is a bigon or half-bigon. If they are snug, is a rectangle when is an arc, an annulus when is a two-sided loop, and a bigon with corners identified when is a one-sided loop.

In all of these cases, as observed above, no arc of forms a (half-) bigon inside , and must therefore run across and have an endpoint in both and .

In the non-snug case, let be the curve of less complexity obtained by routing along when it meets the bigon or half-bigon. In the snug case, let . In either case, . Moreover, the complex is isotopic to and because they are isotopic, there are exchanges, not necessarily regular, so that the Haken sum is a curve isotopic to . But by Lemma 6.4, . This contradicts the fact that is tight. ∎

Rails and fences. Now we again consider a triangulation with a marking , and auxiliary curves in it that, unlike -normal curves, go through the points of .

A rail is a normal arc with its endpoints in , and a fence is a normal curve that is the union of rails.

We note that if a face contains an -normal arc and a rail that are locally snug, then is either or , depending only on the endpoints of and the -normal type of .

The following lemma can also be considered obvious:

Lemma 6.9.

Intersection number with fences is additive with respect to normal addition of -normal curves: If is an fence and and are -compatible, -normal curves, then

Proposition 6.10.

Let be a fence that is a tight essential curve (w.r.t. the unmarked triangulation). Suppose that a sum of -normal curves is tight, essential and snug with . Then

  1. and are both snug with respect to ;

  2. where is the geometric intersection number;

  3. if is two-sided, connected and normally isotopic to then, after a normal isotopy, every point of has the same normal sign.

Proof.

Among counterexamples to conclusion (1) of the proposition, choose one that minimizes . Then and are pairwise locally snug, and we will show that they are in fact pairwise snug. Suppose not and let be an innermost (half-) bigon bounded by some pair of the curves.

If is bounded by and either of the other curves, say , then every sub-arc of in crosses and meets both and . Let be the result of rerouting around as in Lemma 6.6. Then is isotopic to some Haken sum that has fewer intersections with . This contradicts our assumption that and are snug.

If is bounded by and , then every sub-arc of in crosses and meets both and . Let and be the curves given by Lemma 6.6. Because is tight, and are tight and normally, but not necessarily -normally, isotopic to and , respectively. Because the normal sum was defined, and are locally snug. The isotopy doesn’t create intersections, and so and are also locally snug. Then for some generalized Haken sum of and . By Lemma 5.5 that sum is a normal sum, . Note that and are snug with if and only if and are, as the move did not change the number of times they meet . Since we obtain a contradiction and establish conclusion (1).

Since , and their sum are all snug with respect to and intersections with respect to are additive, we have additivity of geometric intersection number, conclusion (2).

We now prove the final statement of the proposition. Assume that is normally isotopic to . Then since is two-sided. By (2) and the fact that and are both snug with we have: .

Figure 7: Mixed normal signs imply that .

And since is normally isotopic to , there is a normal, not necessarily -normal, isotopy taking to . Now cuts across a thin regular neighborhood of in a collection of arcs that span (cut across) the neighborhood. Together they cut this neighborhood into rectangles; see Figure 7.

Each regular exchange connects a pair of rectangles at a corner of each. In fact, every rectangle that meets twice must be attached to another rectangle at one of its corners. Otherwise, an arc of bounds the unattached rectangle, showing that the arc it is trivial in the neighborhood of and can be isotoped out of it. This would imply , contradicting the equality shown earlier. The fact that each rectangle is attached at exactly one corner implies that as we follow every intersection with must have the same normal sign. Since is normally isotopic to we have our desired conclusion (3). ∎

7 Normal summands of incompressible annuli

We would like to apply two well known results from normal surface theory: (1) an essential surface is isotopic to a normal surface, and (2) every summand of a least weight essential normal surface is also least weight and essential (Theorem 6.5 of Jaco and Tollefson [JT95]). But, as will be seen shortly, our notion of surface complexity prioritizes the reduction of boundary complexity over the reduction of total surface weight. Thus the results (1) and (2) cannot be applied as stated.

Proposition 7.1 recovers the first result using our notion of complexity. Proposition 7.2 gives a weaker version of the second for incompressible annuli. While we expect the full version to hold with our notion of complexity, we prove a restricted version both to simplify the proof and to incorporate boundary parallel annuli which are non-essential. Our proof follows the strategy of [JO84] and [JT95].

The complexity of a properly embedded surface is the triple

We compare complexities lexicographically. Thus, the complexity of is measured first by the complexity of its boundary, then by the weight of , , and then by the number of components of the intersections with the -skeleton of .

A normal surface is least complexity if it minimizes complexity among normal surfaces to which it is isotopic (but not necessarily normally isotopic).

A surface is tight if it minimizes complexity, ranging over all those surfaces to which it is isotopic.

A tight normal surface is clearly least complexity, and as a consequence of Proposition 7.1, a normal essential surface of least complexity is tight. But, this does not hold in general for surfaces that are not essential: for example, a normal boundary parallel annulus may be least complexity but after tightening no longer normal.

We first recover normalization of an essential surface. We will apply this with surfaces whose boundaries are tight, hence least length.

Proposition 7.1.

Suppose that is a triangulated, irreducible manifold with incompressible boundary. If is a tight, properly embedded, essential surface, then is normal.

Proof.

To prove is normal we must show that it meets each tetrahedron in a collection of disks whose boundaries are normal curves of length 3 or 4. We adopt the view taken in [BDTS12], showing meets each tetrahedron in pieces that are incompressible and edge incompressible.

If any component of is compressible in , then, by an innermost disk argument, we obtain a compressing disk avoiding all other components of , and hence is compressible inside .

Because is essential, the boundary of any compressing disk for is trivial in . Because is irreducible, compressing along yields a surface <