On multiply twisted knots that are Seifert fibered or toroidal
Abstract.
We consider knots whose diagrams have a high amount of twisting of multiple strands. By encircling twists on multiple strands with unknotted curves, we obtain a link called a generalized augmented link. Dehn filling this link gives the original knot. We classify those generalized augmented links that are Seifert fibered, and give a torus decomposition for those that are toroidal. In particular, we find that each component of the torus decomposition is either “trivial”, in some sense, or homeomorphic to the complement of a generalized augmented link. We show this structure persists under high Dehn filling, giving results on the torus decomposition of knots with generalized twist regions and a high amount of twisting. As an application, we give lower bounds on the Gromov norms of these knot complements and of generalized augmented links.
1. Introduction
This paper continues a program to understand the geometry of knot and link complements in , given only a diagram of the knot or link. Each knot complement decomposes uniquely along incompressible tori into hyperbolic and Seifert fibered pieces, by work of Jaco–Shalen [15] and Johannson [16]. By Mostow–Prasad rigidity, the metric on the hyperbolic pieces is unique. Thus this geometric information on the complement is completely determined by a diagram of the knot. However, reading geometric information off of a diagram seems to be difficult.
In recent years, techniques have been developed to relate geometric properties to a diagram for classes of knots and links admitting particular types of diagrams, such as alternating [18], and highly twisted knots and links [22, 21, 12]. However, many links of interest to knot theorists and hyperbolic geometers do not admit these types of diagrams. These include Berge knots [6, 4, 5], twisted torus knots and Lorenz knots [7], which contain many of the smallest volume hyperbolic knots [9]. These knots admit diagrams that are highly nonalternating, that have few twists per twist region, but contain regions where multiple strands of the diagram twist around each other some number of times. The ideas of this paper and a companion paper [20] grew out of a desire to understand geometric properties of these “multiply twisted” knots and links, given only a diagram. The results here give a first step towards such an understanding.
In this paper, we consider a class of multiply twisted knots in , described below, and classify those which are not hyperbolic. We completely determine those which are Seifert fibered, and describe the unique torus decomposition (the JSJ decomposition) for those which are toroidal. We obtain our results by augmenting diagrams of the multiply twisted knots, that is, encircling regions of the diagram where multiple strands twist about each other by a simple closed curve, called a crossing circle. This generalizes a construction of Adams [1]. Since all knots are obtained by Dehn filling some generalized augmented link, the geometric properties of these links are also interesting. In [20], we describe geometric properties of those generalized augmented links which are known to be hyperbolic. Combining these results leads to the JSJ decomposition of certain multiply twisted knots.
To state our results carefully, we need some definitions.
1.1. Generalized augmented links
First, although our main results relate to knots in , in fact many results in this paper apply more generally to knots and links in a 3–manifold . We will assume is compact, with (possibly empty) boundary consisting of tori, and admits an orientation reversing involution fixing a surface . For example, is such a manifold, taking to be a separating 2–sphere. A solid torus is another example, with a Möbius band or annulus.
Let be a knot or link in that can be ambient isotoped into a neighborhood of . We define a diagram of the knot or link with respect to the surface to be a projection of to yielding a 4–valent graph on with over–under crossing information at each vertex.
Given a diagram, we may define twisting, twist regions, and generalized twist regions exactly as in [20], whether or not our link is in . We review the definitions briefly here. More precise statements are found in [20].
A twist region of a diagram is a region in which two strands twist about each other maximally, as in Figure 1(a). Note that the two strands bound a “ribbon surface” between them. A generalized twist region is a region of a diagram in which multiple strands twist about each other maximally, as in Figure 1(b). Note that all strands lie on the ribbon surface bounded between the outermost strands. A half–twist of a generalized twist region consists of a single crossing of the two outermost strands, which flips the ribbon surface over once. Figure 1(b) shows a single full–twist, or two half–twists of five strands.
We may group all crossings of a diagram into generalized twist regions, so that each crossing is contained in exactly one generalized twist region. This is called a maximal twist region selection, and is not necessarily unique. For example, in Figure 1(b), we could group all crossings into one generalized twist region on five strands, or group each crossing into its own twist region on two strands.
Given a maximal twist region selection, at each generalized twist region insert a crossing circle, i.e. a simple closed curve encircling the strands of the generalized twist region, bounding a disk perpendicular to the projection plane of the diagram. The complement of the resulting link is homeomorphic to the complement of the link whose diagram is obtained by removing all full–twists from each generalized twist region. This is illustrated in Figure 2. The resulting link, with crossing circles added and all full–twists removed, is defined to be a generalized augmented link. We will always assume such a link contains at least one crossing circle, to avoid trivial cases.
(a)  (b) 
Note that if is a generalized augmented link, obtained by augmenting a knot in , then is obtained from by Dehn filling. Let denote a small embedded tubular neighborhood of in . Let be (the isotopy class of) the meridian of (i.e. bounds a disk in ), and let be the longitude. Suppose full–twists were removed at to go from the diagram of to that of . Then Dehn filling along the slope on , for each , yields . See, for example, Rolfsen [25] for a more complete description of this process.
We refer to this type of Dehn filling as twisting along the disk , or along , or, when or are understood, simply as twisting. Note any link in is obtained by twisting some generalized augmented link.
Finally, we wish to use diagrams of knots that do not involve unnecessary twisting. That is, we wish them to be reduced in the sense of the following definition, which we will use to generalize Lackenby’s definition of twist reduced [18], and Menasco–Thistlethwaite’s definition of standard [19].
Definition 1.1.
A generalized augmented link with knot strands and crossing circles is said to be reduced if the following hold:

(Minimality of twisting disks) The twisting disk with boundary intersects in points, where , and is minimal over all disks in bounded by . That is, if is another disk embedded in with boundary , disjoint from all other crossing circles, then .

(No redundant twisting) There is no annulus embedded in with one boundary component isotopic to on and the other isotopic to on , .

(No trivial twisting) There is no annulus embedded in with one boundary component isotopic to on and the other boundary component on .
These conditions allow us to rule out unnecessary crossing circles and generalized twist regions. For example, condition (1) prohibits “nugatory” twist regions, such as those shown in Figure 3. Condition (2) rules out redundant generalized twist regions and their associated crossing circles, such as shown in Figure 4, where two crossing circles encircle the same generalized twist region.
1.2. Results
We are now ready to state the main results of this paper.
In Section 3, we classify all reduced augmented links which are Seifert fibered. This is the content of Theorem 3.8. As a consequence, we obtain the following result.
Corollary 3.9.
Let be a knot in which has a diagram whose augmentation is a Seifert fibered reduced augmented link. Then is a torus knot.
In Section 4, we describe a torus decomposition of generalized augmented links. In particular, in Theorem 4.1, we show that components of such a torus decomposition are either in some sense trivial, or atoroidal reduced augmented links. Applying this to links in , using results on hyperbolic generalized augmented links of [20], we obtain a torus decomposition for all links in obtained by high Dehn fillings of toroidal generalized augmented links. This is Theorem 4.7. In particular, if at least 6 half–twists are inserted when we twist along , then the torus decomposition of will agree with that of , aside from “trivial” pieces.
We wish to apply these results to as many knots in as possible. In Section 5, we show that any knot in admits a diagram whose augmentation is reduced. Thus the above results will apply at least to their augmentations. This is Theorem 5.1.
In Section 6, we apply these results to knots in , to determine the JSJ decomposition of multiply twisted knots. Our main result is the following.
Theorem 6.4.
Let be a knot in which is toroidal, with a twist–reduced diagram and a maximal twist region selection with at least half–twists in each generalized twist region. Let denote the corresponding augmentation. Then there exists a sublink of , possibly containing fewer crossing circles, such that:

The essential tori of the JSJ decomposition of are in one–to–one correspondence with those of .

Corresponding components of the torus decompositions have the same geometric type, i.e. are hyperbolic or Seifert fibered.

Essential tori of and form a collection of nested tori, each bounding a solid torus in which contains , and is fixed under a reflection of .
Putting this theorem with the results on hyperbolic geometry of generalized augmented links in [20], we obtain as an application a lower bound on the Gromov norm of such knots.
1.3. Comments and additional questions
The results of this paper give geometric information based purely on diagrammatical properties of extensive classes of knots. However, because of the high amount of twisting required, for example in Theorem 6.4, these classes still do not include examples of many knots. Considering knots which are not included leads to two interesting remaining questions.
First, can the results of Theorem 6.4 be sharpened to require fewer half–twists? We can construct examples of atoroidal knots whose geometric type (hyperbolic or Seifert fibered) does not agree with that of the corresponding reduced augmented link . However, in all these examples, at least one generalized twist region contains fewer than half–twists. In [3], AïtNouh, Matignon, and Motegi, working on a related question, show that when exactly one crossing circle is inserted into the diagram of an unknot, and then the unknot is twisted, inserting at least half–twists, the geometric type of the resulting knot (Seifert fibered, toroidal, or hyperbolic) agrees with that of the unknot union the crossing circle. While these results do not apply to generalized augmented links, the result requiring only half–twists is intriguing.
Secondly, given an arbitrary diagram of a knot, is there a way to optimize the maximal twist region selection? There are many ways to choose a maximal twist region selection. For example, Figure 1(b) could either be seen as a single full–twist of strands, or as half–twists, each of strands. To apply the results of this paper, it seems we would want to select generalized twist regions to maximize the number of half–twists in each twist region. Is there an algorithm that, given a diagram of a knot , produces a reduced diagram and a maximal twist region selection with the highest number of half–twists per generalized twist region possible for ? Results along these lines would be interesting.
1.4. Acknowledgements
This research was supported in part by NSF grant DMS–0704359. We thank John Luecke for helpful conversations.
2. Reflection
A generalized augmented link in a manifold admits a reflection through a surface fixed pointwise, just as in the case of links in [23, Proposition 3.1]. This reflection is necessary for many of the results that follow, and so we state it first.
Proposition 2.1.
Let be a 3–manifold with torus boundary which admits an orientation reversing involution fixing a surface pointwise. Let be a link in which may be isotoped to lie in a neighborhood of . Finally, let be an augmentation of a diagram of , with crossing circles and knot strands . Then admits an orientation reversing involution which fixes a surface pointwise, and each is embedded in . In particular, admits an orientation reversing involution .
Proof.
Isotope crossing circles to be orthogonal to , preserved by the reflection of through . If there are no half–twists in the diagram of , then all components of are embedded in . Hence the reflection in preserves each as well as each , so and the involution is the restriction of the involution of to .
If there are half–twists in the diagram of , then the reflection of through the surface gives a new link in which all the directions of the crossings at each half–twist have been reversed. Let be the homeomorphism of which twists exactly one full time in the opposite direction of these half–twists at each corresponding crossing circle. Applying changes the diagram to one in which the crossings of half–twists have been reversed again, hence to the diagram of . So is homeomorphic to , and the orientation reversing involution of is given by reflection of in followed by the homeomorphism .
Finally, we describe the surface fixed pointwise by in the case of half–twists. In this case, is equal to outside a neighborhood of those crossing disks for which the corresponding crossing circle bounds a half–twist. Inside such a neighborhood, the surface follows the ribbon surface of the half–twist between the outermost knot strands. Between and the outermost knot strands, runs over the overcrossing, under the undercrossing, and meets up with the surface on the opposite side of the link. Its boundary runs twice along the meridian of , once along the longitude, as in Figure 5. ∎
In [20, Lemma 3.1], we showed that in this setting, the slopes on will meet the surface of Proposition 2.1 exactly twice. We will use this fact here, and so we state it as a lemma.
Lemma 2.2.
Let be a generalized augmented link in , with reflective surface from Proposition 2.1 and twisting disks . Then for each , meets exactly twice on .∎
3. The Seifert fibered case
In this section, we classify Seifert fibered augmented links.
3.1. Incompressibility of surfaces
Let be the complement of a generalized augmented link. By Proposition 2.1, admits an involution which fixes a surface . We show that and the (punctured) twisting disks are incompressible.
Lemma 3.1.
Let be an irreducible –manifold with torus boundary components which admits an orientation reversing involution with fixes a surface . Then is incompressible.
Proof.
Suppose not. Suppose is a compressing disk for . Then lies on , so is fixed by , and is a disk whose boundary agrees with that of . We first show we can assume and are disjoint except on their boundaries. If not, consider the intersections of and . Note these intersections consist of closed curves on and on — not arcs, since is contained on , and acts as a reflection in a small neighborhood of .
Let be a circle of intersection of and which is innermost on . Then lies on and , hence lies on and , and is innermost on . Surger: Replace by replacing the disk bounded by on with the disk bounded by on , and push off slightly. Call this new disk .
We claim that the number of intersections is now less than . Outside a neighborhood of the disk bounded by , agrees with . Hence outside a neighborhood of the disk bounded by , agrees with .
There are two cases to consider. First, if is outside the disk bounded by on , then and the disk it bounds in are still contained in (since agrees with outside ). Similarly, will be outside the disk bounded by on , so and the disk bounded by on will still remain on . An example of this is illustrated in Figure 6.
When we push off to form , we may do so equivariantly. So doesn’t intersect in a neighborhood of the disk bounded by on , and doesn’t intersect in a neighborhood of the disk bounded by on . Elsewhere, and agree, so other intersections have not changed. Thus the number of intersections has decreased under this operation.
If instead is inside the disk bounded by on , then when we surger, even before pushing off, now bounds a disk on , whose image under is bounded by in , hence is contained in . But now is disjoint from . Since and agree elsewhere, and have fewer intersections.
Repeating this process a finite number of times, we obtain a compressing disk such that and are disjoint.
Then is a sphere in . Since is irreducible, bounds a ball in whose boundary is invariant under . Since meets , so does , and hence must be preserved by the involution .
Now we have an orientation reversing involution of a ball which fixes a circle on the boundary of the ball and swaps the disks on the boundary. Double the ball across its boundary and extend . This gives an orientation reversing involution of with fixed point set a surface. It follows from work of Smith in the 1930s that the fixed point set must be a –sphere containing , and intersected with this fixed point set must therefore be a disk. But the fixed point set of is , so contains a disk with boundary . This contradicts the fact that was a compressing disk for . ∎
We now use this result to show incompressibility of .
Lemma 3.2.
Let be a reduced generalized augmented link in such that is irreducible, with twisting disks . Then each is incompressible in .
Proof.
Let be the surface of the reflection provided by Proposition 2.1, and suppose, by way of contradiction, that the punctured disk , say, is compressible. Let be a compressing disk in . Consider . Note , and therefore , cannot be empty, else bounds a disk on . So consists of a nonempty collection of arcs and closed curves on .
We may assume does not contain any closed curves on , using the incompressibility of , Lemma 3.1. For if there is a closed curve component of , it bounds a disk on , hence can be isotoped off.
Next we show that we can take to be invariant under . Consider an outermost arc of . This bounds a disk whose boundary consists of the arc and a portion of . Consider . This is a disk embedded in with boundary on , which is invariant under . If is a compressing disk for , then replace by , and we have a compressing disk invariant under . So suppose is not a compressing disk for . Then bounds a disk on . So is a 2–sphere, which must bound a 3–ball by irreducibility. Isotope through this ball, pushing along through , reducing the number of intersections of with . Since there are only finitely many intersections of , repeat this process only a finite number of times, and eventually obtain a compressing disk for invariant under , or we reduce to the case , contradicting the fact that is a compressing disk.
Now, does not bound a disk on . Hence it must bound a disk in punctured by . Replace by replacing with . Then we have a new disk which is invariant under , which meets the fewer times. But the generalized augmented link is reduced, which means, by Definition 1.1, that the number of intersections is minimal over all disks in bounded by . Since is another such disk, this is a contradiction. ∎
3.2. Annuli
Next, we show a series of results on annuli that are admitted in a generalized augmented link.
Lemma 3.3 (Lemma 2.5.3 of [10]).
Let be an irreducible –manifold with torus boundary components, not homeomorphic to . Let and be boundary components which are incompressible. Suppose and are properly embedded annuli in with , with on . Then is isotopic to on , .
Lemma 3.3 is actually not as general as [10, Lemma 2.5.3]. Because the statement of that lemma is a little different from Lemma 3.3, we reproduce the proof here for convenience.
Proof.
Assume and are in general position. Let , denote the number of intersections of and . Suppose one is nonzero.
There is an isotopy of the such that , and any arc of intersection of runs from one torus to the other. Otherwise, an arc of intersection runs from one torus back to itself. By an innermost arc argument, using the irreducibility of and the incompressibility of , we can isotope and to remove this arc of intersection.
Now, we claim we can replace , if necessary, so that and intersect just once. Suppose and intersect at least twice. Choose two arcs adjacent to each other on , say and . That is, and bound a disk on whose interior is disjoint from . The arcs and will also bound a disk on , whose interior is not necessarily disjoint from , but must be disjoint from by choice of .
The boundaries of are , an arc on which we denote , , and an arc on , which we denote . Similarly, the boundaries of are , on , , and on . Then gives an annulus embedded in with boundary components on and on . Since (because is disjoint from ), has slope , say, on , where . Hence after isotopy, will consist of a single arc (see [10, Figure 2.3]). Replace with .
Now, is homeomorphic to , where is the union of two simple loops on which intersect transversely in a single point. It has a regular neighborhood homeomorphic to , where is a regular neighborhood of on . But now bounds a disk on , bounds a disk on , so is a 2–sphere, which bounds a 3–ball in since is irreducible. Then is homeomorphic to . ∎
Corollary 3.4.
Suppose is an irreducible –manifold with torus boundary components, not homeomorphic to , which admits an orientation reversing involution which fixes a surface meeting incompressible components and of . Suppose is an annulus embedded in with boundary components lying on and . Then the slopes of on , , are preserved by the involution .
Proof.
If does not preserve one of the slopes on , then and are two distinct annuli embedded in with nonisotopic boundary components. This contradicts Lemma 3.3. ∎
We now apply these results to generalized augmented links. First, we need to rule out the case that a generalized augmented link might have complement in homeomorphic to .
Lemma 3.5.
Suppose is a generalized augmented link in and is irreducible. Then is not homeomorphic to .
Proof.
Suppose not. Since has just two boundary components, and since we assume any generalized augmented link has at least one crossing circle , one boundary component of corresponds to and the other to a knot strand. Since the punctured is incompressible by Lemma 3.2 and –sided, it must be either horizontal or vertical in a Seifert fibering of . But then must be an annulus, contradicting the fact that . ∎
Using this fact, we can rule out annuli embedded in .
Lemma 3.6.
Let be a generalized augmented link in such that is irreducible. Then there is no annulus embedded in with one boundary component on parallel to , and the other boundary component disjoint from .
Proof.
Suppose is an annulus embedded in with boundary component on parallel to .
Consider the intersection of with . Since is parallel to , and meets twice by Lemma 2.2, we may isotope so that consists of one or two points, depending on whether consists of two or one components, as illustrated on the left and right of Figure 5, respectively.
If consists of one point, then has two meridional components, and is a meridian on . But now consider . This consists of arcs and curves of intersection. Any arc has two endpoints on , so there must be an even number of points of intersection of . However, we are assuming there is just one such point. This is a contradiction.
Thus must consist of two points, and must have a single arc component running from to . This arc bounds a disk in , and a disk in , since the other boundary component of is disjoint from . We may assume the interiors of and are disjoint by an innermost curve argument, for intersections must be simple closed curves in both, since has just one arc component, and hence if and are not disjoint we may isotope them off of each other using irreducibility of . So is a disk with boundary on . This must bound a disk on . By irreducibility of , we may therefore isotope to have no intersections with , contradicting the fact that is parallel to . ∎
Lemma 3.7.
If is a reduced generalized augmented link in such that is irreducible, then there is no annulus embedded in with boundary components on , , for .
3.3. Seifert fibered augmented links
We may now classify all Seifert fibered reduced generalized augmented links.
Theorem 3.8.
If is irreducible and Seifert fibered, where is a reduced generalized augmented link in , then has just one crossing circle component , is a solid torus, is an embedded annulus or Möbius band in , and the knot strands are parallel to the boundary of . In particular, if is an annulus, there are at least two knot strand components.
Proof.
Suppose is Seifert fibered. First, by Lemma 3.7, there can be no annuli between link components and . This implies that there cannot be more than one link component .
Now, is incompressible by Lemma 3.2. Since is –sided, it is horizontal or vertical in . If vertical, it must be an annulus, contradicting the fact that . So is horizontal. Then the meridians of the knot strands (i.e. the curves which bound disks in ) cannot be Seifert fibers, so the Seifert fibering of extends to . The base orbifold of is branch covered by the horizontal surface , hence it is a disk with one singular point. Thus must be a solid torus.
Since is incompressible by Lemma 3.1, if it is orientable, then it is an annulus. Because the knot strands are embedded in and nontrivial, they must be parallel to the core of the solid torus . In particular, if there is just one knot strand component, then is homeomorphic to , contradicting Lemma 3.5. So in this case there are at least knot strand components.
If is nonorientable, then by work of Frohman [11] and Rannard [24], is pseudovertical in a solid torus, meaning, in this case, it is a punctured nonorientable surface in the solid torus . These were classified by Tsau [27], and have boundary of the form , where is a meridian of the solid torus, is a longitude, , and is an odd integer.
In our case, we know which boundary slopes can occur, because of the existence of the involution . In particular, since intersects exactly twice, by Lemma 2.2, . Then by untwisting, we may assume , and so is the slope , and is a Möbius band in the solid torus .
We obtain the following immediate consequence of this result.
Corollary 3.9.
Let be a knot in which has a diagram whose augmentation is a Seifert fibered reduced generalized augmented link. Then is a torus knot.
Proof.
Let denote the augmentation of . First we show is irreducible. The manifold is homeomorphic to , where are crossing circles encircling generalized twist regions. Let be a sphere in , and assume it does not bound a ball in . Then the image of under the homeomorphism is a sphere in which does not bound a ball. Since is irreducible, must bound a ball in . Hence some lies in . But is unknotted, hence bounds a disk in . This contradicts property (1) of the definition of reduced, Definition 1.1.
By Theorem 3.8, the diagram can have only one generalized twist region. The augmentation is the link shown in Figure 7, since there is just one component . Thus when we twist to obtain , we remove from the diagram and add an even number of crossings at the twist region it bounds. This is a torus knot. ∎
4. Essential tori and augmented links
In this section, we consider reduced generalized augmented links such that is toroidal. We show that the torus decomposition of satisfies some nice properties.
Recall that by work of Jaco and Shalen [15] and Johannson [16], every irreducible 3–manifold with (possibly empty) torus boundary contains a pairwise disjoint collection of embedded essential tori , unique up to isotopy, such that the closure of a component of is either atoroidal or Seifert fibered. If admits an orientation reversing involution , then by the equivariant torus theorem, first proved by Holzmann [14], each incompressible torus in is isotopic to one which is preserved by or taken off itself. Then applied to gives a new torus decomposition of , which by uniqueness must agree with . Thus the closure of each component of is either fixed by , or taken off itself. This is the equivariant torus decomposition of Bonahon and Siebenmann [8]. We refer to this equivariant torus decomposition as the JSJ decomposition.
Theorem 4.1.
Let be a reduced generalized augmented link in , such that is irreducible. Then there exists a collection of tori, incompressible in , such that if is a component of , satisfies one of the following:

does not meet any crossing circles of .

meets exactly one crossing circle , is homeomorphic to , and is isotopic to a simple closed curve on .

is the complement of a reduced generalized augmented link.
The collection of tori may be larger than the minimal collection of the JSJ decomposition. However, we find by adding incompressible tori to the JSJ decomposition.
Lemma 4.2.
Let be a reduced generalized augmented link in such that is irreducible. Let denote the tori of the JSJ decomposition for . Suppose there is a component of that contains a crossing circle and an embedded annulus with one boundary component on some and one on . Let be the torus obtained by taking the boundary of a small regular neighborhood of the union of , , and . Then is incompressible in .
The collection of Theorem 4.1 will be obtained by adding to any of Lemma 4.2 which are not isotopic to tori in .
Before proving the lemma, we illustrate by example a situation in which the lemma will apply. Consider the link in Figure 8. The heavy line in that figure shows the location of an incompressible torus . When we cut along , we obtain a hyperbolic generalized augmented link on the outside, homeomorphic to the complement of the Borromean rings. On the inside, and bound an annulus. Take the boundary of a regular neighborhood of the union of this annulus with and , and we obtain an incompressible torus as in Lemma 4.2. Cutting along , we split the inside into two components, one homeomorphic to , and the other (in this example) another copy of the Borromean rings complement.
Recall that we are interested in links obtained by twisting generalized augmented links. When we do twisting along in the above example, the component becomes the manifold . Thus the two incompressible tori and become isotopic to each other after twisting. By the results in [20], sufficiently high twisting along the remaining crossing circles in each gives a manifold which remains hyperbolic. Thus the torus decomposition of the twisted link contains just one of and . We will see this in Theorem 4.7.
Proof of Lemma 4.2.
Suppose by way of contradiction that is compressible. A compressing disk can be isotoped to lie in , else we obtain a compressing disk for the torus in , which is impossible.
Surger along a compressing disk for in to obtain a sphere embedded in . Note that is irreducible, since is irreducible and is incompressible. Thus bounds a ball in . Now, cannot be on the side of containing since this side contains boundary components and of . Thus bounds a solid torus in . Recall that was formed by taking the boundary of a regular neighborhood of the union of , an annulus , and in . Since bounds a solid torus in , it must be the case that is homeomorphic to union a regular neighborhood of the annulus . This has just two boundary components: and . We claim this is impossible.
Since contains , the surface of Proposition 2.1 meets , and the torus boundary components of are preserved by . Hence is preserved by , where the union is over in . The annulus has one boundary component, , say, on , taken by to . Hence it is isotopic to an annulus which meets in two arcs and is preserved under . Since tori and are also preserved under , the solid torus is preserved under . Then must be an annulus, a Möbius band, or two meridional disks in .
We form by attaching a thickened annulus to . This thickened annulus is attached along some slope on . Since is taken to itself with reversed orientation by , the slope must be taken to by . There are very few possibilities for .
In case is an annulus or Möbius band, must bound a disk in . Attaching an annulus to along two meridians gives a manifold with compressible boundary, but neither nor is compressible.
Thus consists of two meridional disks, and must be some longitude of . When we attach a thickened annulus to longitude slopes, the resulting manifold is homeomorphic to . Then is parallel to , contradicting the fact that is essential in . ∎
Lemma 4.3.
Let and , , be crossing circles that satisfy the hypotheses of Lemma 4.2. Then the tori and are not isotopic.
Proof.
is formed by taking the boundary of a regular neighborhood of the union of an incompressible torus, an annulus with boundary on the torus and on , and . Thus there is an annulus with boundary on and on . Similarly for there is an annulus with boundary on and on . If and are isotopic, then the annulus may be isotoped to have boundary on and on . Then we form a new annulus between and by taking the union of , , and an annulus running along between these two. This contradicts the fact that is reduced. ∎
Form the collection of Theorem 4.1 by adding to all the tori of Lemma 4.2 which are not isotopic to a torus in . By Lemma 4.3, these are not isotopic to each other. The closure of each component of is still either atoroidal or Seifert fibered, but may no longer be the minimal such collection. By construction, components of either do not contain crossing circles of ; contain a single crossing circle sandwiched between incompressible tori and of Lemma 4.2, in which case the second possibility of Theorem 4.1 holds; or contains crossing circles of , but none of these bound annuli with boundary on . For the proof of Theorem 4.1, we need to examine these components, and show that any such is homeomorphic to the complement of a reduced generalized augmented link.
Generalized augmented links are defined to lie in a 3–manifold admitting a reflection through a surface . First, we define a manifold which will play the role of thus underlying 3–manifold.
Let be a component of which contains at least one crossing circle , but does not contain an embedded annulus with boundary on and on , for any . Consider . In the manifold , may intersect essential tori of . Then will meet boundary components of . Let be the boundary components of that component of which meets . Replace by the manifold obtained by Dehn filling along the slopes , . Let denote the solid tori attached in the Dehn filling.
Do this Dehn filling for each contained in . We obtain a new manifold . Note is well–defined because the are disjoint; thus if is met by and , then and must give the same slope, so the Dehn fillings along that slope are the same. Similarly, might meet several times, but again along the same slope.
Now, let be the link in consisting of components and , as well as the cores of each distinct solid torus in the set . We will abuse notation slightly and continue to refer to these cores of solid tori by . The following is immediate.
Lemma 4.4.
For the manifold and the link in constructed as above, is homeomorphic to . ∎
We claim that is a reduced generalized augmented link in , with components taking the role of the crossing circles, and components and taking the role of the knot strands.
By assumption, there is at least one , bounding , meeting at least one or . So the link contains at least the minimal number of necessary link components to be a generalized augmented link.
Lemma 4.5.
The manifold admits an involution through a surface , a link is contained in a neighborhood of , has diagram and maximal twist region selection such that when we encircle generalized twist regions of by crossing circles and untwist, the result is a link isotopic to . That is, is a generalized augmented link, given by augmenting a diagram of in .
Proof.
The involution of Proposition 2.1 preserves , where the union is over crossing circles in . It has fixed point set , and components in are embedded in . We show that the involution extends to the solid tori , and that the cores are embedded in the surface .
Recall that has boundary which is an incompressible torus in , and is preserved by . Moreover, some meets in a meridian of . The slope cannot bound a disk in by incompressibility of . Since it does bound a disk in , this disk must be punctured by some component in . Therefore the slope must be taken by to . This means a meridian of the solid torus is inverted by the involution . Since the boundary is preserved by , it follows that the involution extends to give an involution of the solid torus . Also, must be a longitude of , in the sense that it intersects the meridian exactly once. Therefore the core, , is embedded in .
Now note that and satisfy the conclusion of Proposition 2.1. Since acts on as an extension of the involution of Proposition 2.1 acting on , we know that meets the in in the same way it meets the in . Namely, either meets in two meridian components, as on the left in Figure 5, or in a single component as on the right in Figure 5. Form the surface by taking to be outside of a neighborhood of those crossing disks that meet half–twists. Inside a neighborhood of a half–twist, will appear as on the right of Figure 5. The surface , however, should run straight through the crossing circle, meeting in two meridians, as on the left of Figure 5. Note there is a reflection in which still preserves , although it reverses crossings at half–twists. Moreover, the reflection preserves a meridian of each crossing circle, hence extends to a reflection of through the surface , where is obtained from by capping off boundary components on the by disks.
Notice that the strands and lie in a neighborhood of , as do the crossing circles . Hence when we twist along crossing circles, we form a link which still lies in a neighborhood of . Moreover, we may project to such that the twists obtained by twisting along the form distinct generalized twist regions, and so that is an augmentation of a diagram of . ∎
We now need to show that is reduced in . To do so, we find disjoint embedded disks bounded by the crossing circles in , and show these are minimal in the sense of part (1) of Definition 1.1.
Lemma 4.6.
Let be the crossing circles of . Each bounds a disk in such that the collection is embedded in , the involution restricts to an orientation reversing involution of each , and the disks meet and in points, where and is minimal over all disks in bounded by .
Proof.
Let be the crossing circles in . Each bounds a disk in . By construction of , the collection extends to an embedded collection of disks in , such that restricts to an orientation reversing involution of each .
If the is not minimal, then we can find a collection of disjoint, embedded punctured disks which meet fewer times. Replace the collection with this new collection. Each must meet and in at least points, for otherwise we would have an annulus between some and a component or . The first cannot happen by definition of a reduced generalized augmented link in . The second cannot happen by assumption: is assumed to be a component of which does not contain an embedded annulus with boundary components on and on , and is a component of . ∎
We are now ready to complete the proof of Theorem 4.1.
Proof of Theorem 4.1.
Let be the collection of tori described after the proof of Lemma 4.3. Let be a component of . If does not contain any crossing circle , then we are done. If contains a crossing circle and an embedded annulus with boundary and boundary on , then by construction of , is homeomorphic to , is the only crossing circle contained in , and the curve is boundary parallel in . This is the second case of the theorem.
So assume contains a crossing circle, but does not contain any embedded annuli with boundary on and on . Then by Lemma 4.4, is homeomorphic to the manifold , which, by Lemma 4.5 is a generalized augmented link complement. By Lemma 4.6, satisfies condition (1) of the definition of reduced, Definition 1.1. It satisfies condition (2) as well, since any annulus embedded in with boundary components on and is embedded in with boundary components on and . Since is reduced, no such annulus exists. The link satisfies condition (3) of Definition 1.1 by assumption, given the definition of . ∎
Theorem 4.7.
Let be a reduced generalized augmented link in with irreducible, and let be the tori of Theorem 4.1. Let be the link formed by twisting along all the crossing circles of , subject to the restriction that if is contained in a component of which is not homeomorphic to , then at least half–twists are inserted when we twist along . Then there is a torus decomposition of for which components of the decomposition

are either atoroidal or Seifert fibered,

are in onetoone correspondence with the components of which are not homeomorphic to ,

and have the same geometric type (hyperbolic or Seifert fibered) as the corresponding component of .
Remark 4.8.
The decomposition of Theorem 4.7 may not be the JSJ decomposition. In particular, there may be two Seifert fibered components which our decomposition separates, but which are considered as one in the minimal JSJ decomposition. We will see in Section 6 that this does not happen when comes from the augmentation of a knot in , but it could happen more generally.
Proof.
Let be a component of . If contains no crossing circles, then twisting does not affect and so the result holds.
Similarly, if is homeomorphic to , and contains just a single boundary parallel , then twisting yields a manifold homeomorphic to , which will not be a component of a torus decomposition.
If contains a crossing circle, but is not , then is either hyperbolic or Seifert fibered.
If it is hyperbolic, by [20, Proposition 3.5], the slope of the twisting on a horoball neighborhood of a crossing circle has length at least , where is the number of half–twists inserted. Since , this length is greater than , hence by the 6–Theorem [2, 17], the result of Dehn filling is hyperbolic.
If is Seifert fibered, Theorem 3.8 implies is homeomorphic to the complement of parallel strands embedded in an annulus or Möbius band in a solid torus. After twisting, we obtain a torus link, or a torus link, where is the number of knot strands of . These torus links are still Seifert fibered. ∎
5. Reducing knot diagrams
We wish to apply the previous results to as many knots and links as possible.
In this section, we prove that all knots in admit a diagram such that the augmentation is reduced, as in Definition 1.1. We say the diagram of a link is twist reduced if there exists a maximal twist region selection such that the corresponding augmentation of gives a reduced generalized augmented link.
Theorem 5.1.
Let be a knot in with diagram and a maximal twist region selection. Then there exists a twist reduced diagram for .
We will find the diagram of Theorem 5.1 by forming the augmentation of the given diagram, and then removing unnecessary crossing circles and extracting unnecessary knot strands from crossing disks. When we do twisting on remaining crossing circles, projecting twists to the projection plane in in the usual way, we will obtain the desired diagram of the theorem.
Definition 5.2.
A standard diagram of a generalized augmented link is a diagram such that all knot strands lie on the projection plane except at half–twists, which are contained in a neighborhood of the corresponding crossing circle. Crossing circles are perpendicular to the projection plane, and crossing disks project to straight lines running directly under the crossing circles of the diagram. For example, the portions of the diagrams in Figures 2(b), 3, and 4 are standard.
The next few lemmas ensure part (1) of Definition 1.1 will hold.
Lemma 5.3.
Let be a generalized augmented link in with standard diagram. Suppose there exists a disk embedded in , with boundary some crossing circle , disjoint from the other crossing circles, and suppose that meets the knot strands fewer times than does . Then there exists such a disk such that in addition, .
Proof.
Suppose meets the interior of some . We may assume the intersection is transverse and consists of simple closed curve components. There is some innermost disk on whose boundary is a curve on . Consider the disk constructed by taking the disk outside , replacing inside by . Push off slightly, so and do not intersect.
Suppose first that meets the knot strands fewer times than does . Then replace by , replacing by . This disk has fewer intersections with than does .
Suppose instead meets the knot strands the same number of times as does . Replace by replacing with the portion of bounded by , and push off . We have decreased the number of intersections of with without increasing the number of intersections of with knot strands.
In either case, we have a new disk which meets the interiors of the fewer times. Repeat a finite number of times, and we obtain a disk as in the statement of the lemma. ∎
Lemma 5.4.
Let be a generalized augmented link in with standard diagram. Suppose there is a disk embedded in such that has boundary some crossing circle , is disjoint from the other crossing circles, and