Legendrian and transverse
cables of positive torus knots
In this paper we classify Legendrian and transverse knots in the knot types obtained from positive torus knots by cabling. This classification allows us to demonstrate several new phenomena. Specifically, we show there are knot types that have non-destabilizable Legendrian representatives whose Thurston-Bennequin invariant is arbitrarily far from maximal. We also exhibit Legendrian knots requiring arbitrarily many stabilizations before they become Legendrian isotopic. Similar new phenomena are observed for transverse knots. To achieve these results we define and study “partially thickenable” tori, which allow us to completely classify solid tori representing positive torus knots.
An -curve on the boundary of a solid torus refers to the curve , where is the longitude-meridian basis for the homology of the torus, and we denote this by the fraction The -cable of a knot type , denoted , is the knot type obtained by taking the -curve on the boundary of a tubular neighborhood of a representative of . Let be a positive -torus knot, where we may assume and and let be its -cable, also with . This paper concerns the classification of Legendrian and transverse knots representing and solid tori representing . Though the proofs of our classification results are heavily dependent on the ambient contact manifold being , all the Legendrian and transversal classification results hold in any tight contact manifold, as can be seen by consulting .
Studying Legendrian and transverse knots in cabled knot types has been very fruitful. For example, in  cabling was used to better understand open book decompositions of contact structures; in particular, leading to non-positive monodromy maps supporting Stein fillable contact structures, monoids in the mapping class group associated to contact geometry and procedures to construct open books on manifolds after allowable transverse surgery (from an open book for the original contact manifold). Moreover, the first classification of a non-transversely simple knot type was done in  for the -cable of the -torus knot. In that paper it was also shown that studying solid tori with convex boundary that represent a given knot type (that is, their core curves are in a given knot type) is key to understanding cables; such an analysis for solid tori representing negative torus knots yielded simple Legendrian and transverse classifications for cables of negative torus knots. Tori representing iterated cables of torus knots were further studied in [11, 12] as well as . Building on these works we completely classify embeddings of solid tori representing positive torus knots and use this to give a complete classification of Legendrian and transverse knots in the knot types of cables of positive torus knots.
Before discussing the technical classification results we state qualitative versions that demonstrate new phenomena in the geography of Legendrian knots. We begin with some notation. Given a topological knot type and integers and we denote by the set of Legendrian knots (up to Legendrian isotopy) topologically isotopic to and by
We similarly denote the set of transverse knots isotopic to by and the ones having self-linking number by .
We first consider cables of the right handed trefoil, that is, the -torus knot.
Let be the positive trefoil knot in The knot formed by -cabling is Legendrian simple if and only if Furthermore, given positive integers , , and , where and , there exists a slope such that contains Legendrian knots for some pair of integers with ; moreover, one of these does not destabilize, and they remain distinct when stabilized fewer than times (and there are stabilizations that will make them isotopic).
This theorem gives the first example of a knot type with non-destabilizable Legendrian knots with Thurston-Bennequin invariant arbitrarily far from the maximal Thurston-Bennequin invariant. We note that in  it was shown there are knot types that have arbitrarily many Legendrian knots with fixed classical invariants, so the above theorem gives only the second family of knots known to have this property. We also observe that this theorem gives the first set of Legendrian knots with the same invariants that requires arbitrarily many stabilizations before becoming Legendrian isotopic.
Let be the positive trefoil knot in The knot formed by -cabling is transversely simple if and only if Furthermore, given positive integers , , and , where and , let . Then there is some such that contains distinct transverse knots with , of which are non-destabilizable, and such that there is another non-destabilizable knot with . Moreover, these non-destabilizable knots must be stabilized until their self-linking number is before they become transversely isotopic.
In  it was shown that there are knot types, specifically certain twist knots, that have arbitrarily many transverse knots with the same self-linking number. The above theorem also gives such examples but, in addition, demonstrates three new phenomena concerning transverse knots that were not previously known. Specifically it gives the first example of knot types that have transverse knots with the same self-linking number that require arbitrarily many stabilizations before they become transversely isotopic, and it also gives the first examples where there are non-destabilizable transverse knots whose self-linking number is arbitrarily far from maximal. Finally, the theorem also gives the first knot type where there are non-destabilizable knots with distinct self-linking numbers.
With all the interesting and complicated behavior exhibited by cables of the right handed trefoil knot, one would expect to see behavior at least as complicated for cables of other positive torus knots. Surprisingly, cables of such knots turn out to be relatively simple.
Let be a positive -torus knot with . Then for any rational number and any with odd, there are at most 3 Legendrian knots in and at most 2 for all but one pair .
Let be a positive -torus knot with . Then for any rational number there are at most two transverse knots isotopic to the -cable of with the same self-linking number. However, for any positive integers and with , there is a rational number for which there is a non-destabilizable transverse knot with self-linking number at most and it must be stabilized exactly times to become isotopic to the destabilizable transverse knot with the same self-linking number.
As indicated above the key to proving these classification results is classifying solid tori with convex boundary realizing positive torus knots. This classification, discussed below, is the first complete such classification and exhibits features not seen before, such as the existence of partially thickenable tori (see Subsection 1.2).
In the next two subsections we state the precise classification theorems that lead to the above qualitative results. In Subsection 1.1 we state knot classification theorems for cables; in Subsection 1.2 we state classification theorems for embeddings of solid tori.
1.1. Classification results for cable knots
We begin with cables of the right handed trefoil knot.
Let be the -torus knot. Then the -cable of , , is Legendrian simple if and only if , and the classification of Legendrian knots in the knot type is given as follows.
If then there is a unique Legendrian knot with Thurston-Bennequin invariant and rotation number All others are stabilizations of
If , then the maximal Thurston-Bennequin invariant for a Legendrian knot in is and the rotation numbers realized by Legendrian knots with this Thurston-Bennequin invariant are
where is the integer that satisfies
All other Legendrian knots are stabilizations of these. Two Legendrian knots with the same and are Legendrian isotopic.
Suppose for a positive integer ; then is not Legendrian simple and has the following classification (see also Figure 1).
The maximal Thurston-Bennequin number is .
There are Legendrian knots , with
If then there are two Legendrian knots that do not destabilize but have
All Legendrian knots in destabilize to one of the or .
Let . For any and the Legendrian is not isotopic to a stabilization of any of the other ’s the , or .
Let . For any and the Legendrian is not isotopic to a stabilization of any of the ’s or .
The transverse classification is now an immediate corollary.
Let be the -torus knot. If then is transversely simple and all transverse knots are stabilizations of the one with maximal self-linking number .
If for a positive integer then is not transversely simple and has the following classification.
The maximal self-linking number is and there is a unique transverse knot in with this self-linking number.
There are distinct transverse knots in that do not destabilize and have self-linking number
If then there is a unique transverse knot in that does not destabilize and has self-linking number .
All other transverse knots in destabilize to one of the ones listed above.
None of the transverse knots listed above become transversely isotopic until they have been stabilized to have self-linking number There is a unique transverse knot in with self-linking number less than or equal to
For the classification of cables of other positive torus knots we need some notation. Given a rational number let be the largest rational number with an edge in the Farey tessellation to See Figure 2. (The superscript stands for ”anti-clockwise”, as is anti-clockwise of in the Farey tessellation.) Similarly the smallest rational number with an edge in the Farey tessellation to will be denoted by A formula for computing these numbers will be given in Subsection 2.1. We will refer to the interval as the interval of influence for .
Given a positive -torus knot and a positive integer, define
We will see in Subsection 1.2 that such represent boundary slopes of non-thickenable solid tori, and that the half-intervals of influence will represent boundary slopes of partially thickenable solid tori when . We will refer to the as exceptional slopes. If we think of the fractions as representing curves on a torus, we denote the homological intersection of curves with the curves by
We can now state the precise classification theorems for cables of general positive -torus knots.
Let be a -torus knot with . Let
where is the interval of influence for the exceptional slope defined above. The are all disjoint.
The classification of Legendrian knots in the knot type is then given as follows.
If then is Legendrian simple. Moreover, in this case we have the following classification.
If then there is a unique Legendrian knot with Thurston-Bennequin invariant and rotation number All others are stabilizations of
If or , then the maximal Thurston-Bennequin invariant for a Legendrian knot in is and the rotation numbers realized by Legendrian knots with this Thurston-Bennequin invariant are
where is the least integer bigger than . All other Legendrian knots are stabilizations of these. Two Legendrian knots with the same and are Legendrian isotopic.
If then there is some such that and is not Legendrian simple. The classification of Legendrian knots in is as follows.
The maximal Thurston-Bennequin invariant of is .
For each integer in the set
there is a Legendrian with
There are two Legendrian knots satisfying
if ; however, if then
and is not destabilizable.
All Legendrian knots in destabilize to one of the or
For any and the Legendrian is not isotopic to a stabilization of any of the or .
Any two stabilizations of the non-destabilizable Thurston-Bennequin invariant Legendrian knots in , except those mentioned in item (2e), are Legendrian isotopic if they have the same and .
From this theorem we can easily derive the transverse classification.
Let be a -torus knot with . Using notation from Theorem 1.9 we have the following classification of transverse knots in .
If for any then is transversely simple and all transverse knots in this knot type are stabilizations of the one with self-linking number .
If for some then is not transversely simple. There is a unique transverse knot in this knot type with maximal self-linking number, which is . There is also a unique non-destabilizable knot in this knot type and it has self-linking number . All other transverse knots in destabilize to either or and the stabilizations of and stay non-isotopic until they are stabilized to the point that their self-linking numbers are
in the case of , and
in the case of .
We now turn from classification results for cables of positive torus knots, to classification results for embeddings of solid tori representing the positive torus knots themselves.
1.2. Classification results for solid tori
Let be a solid torus in a manifold We say is in the knot type , or represents , if the core curve of is in the knot type
We say a solid torus with convex boundary in a contact manifold thickens if there is a solid torus that contains such that has convex boundary with dividing slope different from The existence of non-thickenable tori was first observed in ; the following theorem shows that non-thickenable tori exist for all positive -torus knots.
Let be a solid torus in the knot type of a positive -torus knot. In the standard tight contact structure on suppose that is convex with two dividing curves of slope Then thickens unless is an exceptional slope
for some positive integer in which case it might or might not thicken.
Moreover for each positive integer there are, up to contact isotopy, exactly two solid tori with convex boundary having dividing curves of slope that do not thicken, where . For there is exactly one solid torus with convex boundary having two dividing curves of slope This solid torus is a standard neighborhood of a Legendrian -torus knots with maximal Thurston-Bennequin invariant and it does not thicken.
A key feature in the knot classification results above in Subsection 1.1 is a complete understanding of not only non-thickenable tori but also partially thickenable tori, that is tori with convex boundary that thicken, but not to a maximally thick torus in the given knot type. The existence of such tori has not been observed before, but it is clear that such tori will be key to future Legendrian classification results. In addition it is likely they will be important in understanding contact surgeries. The following theorem shows that partially thickenable tori exist for all positive -torus knots.
Let be a positive -torus knot and let be the exceptional slopes. Let and . All solid tori below will represent the knot type .
If then and we have the following.
The intervals so
Any solid torus with convex boundary thickens to or to (that is a neighborhood of the maximal Thurston-Bennequin invariant -torus knot).
Any solid torus inside with convex boundary having dividing slope greater than (that is in ) does not thicken past the slope
Any solid torus inside with convex boundary having negative (or infinite) dividing slope will thicken to a neighborhood of the maximal Thurston-Bennequin invariant -torus knot.
If then we have the following.
For any , any solid torus inside with either boundary slope different from , or less than dividing curves, thickens to .
All the with are disjoint.
Any solid torus with convex boundary having dividing slope in thickens to or to (that is a neighborhood of the maximal Thurston-Bennequin invariant -torus knot).
Any solid torus inside for some , and with convex boundary having dividing slope in , does not thicken past the slope
Any solid torus inside with convex boundary having dividing slope outside of (that is greater than or equal to or negative) will thicken to a neighborhood of the maximal Thurston-Bennequin invariant -torus knot.
From this theorem we can classify solid tori in the knot types of positive torus knots.
Let be a positive -torus knot and let be the exceptional slopes. Let and .
If , then given a slope there is some integer such that and there are exactly solid tori representing the knot type with convex boundary having dividing slope and two dividing curves, only two of which thicken to a standard neighborhood of a Legendrian knot.
If , then given any slope we have the following.
If there is some integer such that and for any , then there are exactly solid tori representing the knot type with convex boundary having dividing slope and two dividing curves each of which thickens to a standard neighborhood of a Legendrian knot with .
If there is some integer such that and for any , then there are exactly solid tori representing the knot type with convex boundary having dividing slope and two dividing curves, all but two of which thicken to a standard neighborhood of a Legendrian knot with .
If there is some such that , then there are exactly solid tori representing the knot type with convex boundary having dividing slope and two dividing curves and they each represent a standard neighborhood of a Legendrian knot with .
Given any negative slope there is some negative integer such that . A solid torus with convex boundary having dividing slope and two dividing curves will thicken to a solid torus that is a standard neighborhood of a Legendrian knot.
We conclude this introduction with an outline of what follows. In Section 2 we collect needed preliminaries, including facts about continued fractions and convex surfaces, and we outline a strategy for classifying Legendrian knots. In Section 3 we classify embeddings of solid tori representing positive torus knots. In Section 4 we provide classifications for all simple cables of positive torus knots, and in Sections 5 and 6 we establish classifications for all non-simple cables of positive torus knots.
Acknowledgments. The first and third authors were partially supported by NSF Grant DMS-0804820. The second author was partially supported by QGM (Centre for Quantum Geometry of Moduli Spaces) funded by the Danish National Research Foundation. Some of the work presented in this paper was carried out in the Spring of 2010 while the first and third author were at MSRI, we gratefully acknowledge their support for this work.
In this section we first prove some important facts about continued fractions in Subsection 2.1. The remaining sections recall various facts concerning the classification of Legendrian and transverse knots from . The reader is assumed to be familiar with the basic notions associated to convex surfaces and Legendrian and transverse knots, but these sections are included for the convenience of the reader and to make the paper as self-contained as possible. All this information can be found in [5, 9].
2.1. Continued fractions, the Farey tessellation, and intersection of curves on a torus
In this section we collect various facts about continued fractions and the Farey tessellation (see Figure 4) that will be needed throughout our work.
Given a rational number we may represent it as a continued fraction
with and the other . We will denote this as . If we know that then we define
with the convention that if then ; we also define
The number is the largest rational number bigger than with an edge to in the Farey tessellation and is the smallest rational number less than with an edge to in the Farey tessellation. Moreover there is an edge in the Farey tessellation between and and is the mediant of and , that is if and then
Define and One may easily verify using induction that
From this one can inductively deduce that
Thus there is an edge in the Farey tessellation between and Similarly, let and and notice that Now we see that
and a similar expression for and induction yield In particular, there is an edge in the Farey tessellation between and
Finally by setting and noticing that , we can use the above formulas, and analogous ones, to inductively prove that . This establishes an edge in the Farey tessellation between and . Since there is an edge in the Farey tessellation between each pair of numbers in the set the lemma is established by noticing that the numerators (and denominators) of and are both smaller than the numerator (and denominator) of ∎
We recall that if we choose a basis for then there is a one-to-one correspondence between embedded essential oriented curves on and rational numbers , written in lowest common terms. Moreover given two rational numbers and we denote their homological intersection (which also happens to be the signed minimal intersection number) between the corresponding curves on by and it can be computed by
Notice that this number is only well defined up to sign (since the orientation on the curve corresponding to a fraction is not determined). Throughout this work we will only be concerned with the absolute value of this number (if the exact number is ever needed we will specify the orientations on the homology class corresponding to a fraction).
Fix some positive integer and set for and . If then the intervals for are all disjoint. If then the intervals are nested .
If is a positive rational number less than or greater than then for any we have
with equality only if or .
If and then
If then it is clear that and one easily checks that and So .
If then we notice that any number in is a mediant of and and hence has denominator strictly bigger than (since the denominator of is ), thus cannot be in this interval for any . Similarly cannot be in the interval . Thus the intervals are disjoint.
For the second statement notice that and have the same sign and will be some non-negative integral linear combination of and . For the last statement note that will be some positive integral linear combination of and . ∎
2.2. Convex surfaces and bypasses
In this subsection we discuss the main tools we will be using throughout the paper — convex surfaces. We assume the reader is familiar with convex surfaces as used in [5, 9]; but, for the convenience of the reader, we recall the fundamental facts from the theory that we will use in this paper.
2.2.1. Convex surfaces
Recall a surface in a contact manifold is convex if it has a neighborhood , where is some interval, and is -invariant in this neighborhood. Any closed surface can be -perturbed to be convex. Moreover if is a Legendrian knot on for which the contact framing is non-positive with respect to the framing given by , then may be perturbed in a fashion near , but fixing , and then again in a fashion away from so that is convex.
Given a convex surface with -invariant neighborhood let be the multicurve where is tangent to the factor. This is called the dividing set of If is oriented it is easy to see that where is positively transverse to the factor along and negatively transverse along . If is a Legendrian curve on a then the framing of given by the contact planes, relative to the framing coming from , is given by . Moreover if then the rotation number of is given by .
2.2.2. Convex tori
A convex torus is said to be in standard form if can be identified with so that consists of vertical curves (note will always have an even number of curves and we can choose a parameterization to make them vertical) and the characteristic foliations consists of vertical lines of singularities ( lines of sources and lines of sinks) and the rest of the foliation is by non-singular lines of slope . See Figure 3.
2.2.3. Bypasses and tori
Let be a convex surface and a Legendrian arc in that intersects the dividing curves in 3 points (where are the end points of the arc). Then a bypass for (along ), see Figure 3, is a convex disk with Legendrian boundary such that
are corners of and elliptic singularities of
A surface locally separates the ambient manifold. If a bypass is contained in the (local) piece of that has as its oriented boundary then we say the bypass will be attached to the front of otherwise we say it is attached to the back of .
When a bypass is attached to a torus then either the dividing curves do not change, their number increases by two, or decreases by two, or the slope of the dividing curves changes. The slope of the dividing curves can change only when there are two dividing curves. (See  for more details.) If the bypass is attached to along a ruling curve then either the number of dividing curves decreases by two or the slope of the dividing curves changes. To understand the change in slope we need the following. Let be the unit disk in Recall the Farey tessellation of is constructed as follows. Label the point on by and the point with Now join them by a geodesic. If two points on with non-negative -coordinate have been labeled then label the point on half way between them (with non-negative -coordinate) by Then connect this point to by a geodesic and to by a geodesic. Continue this until all positive fractions have been assigned to points on with non-negative -coordinates. Now repeat this process for the points on with non-positive -coordinate except start with See Figure 4.
The key result we need to know about the Farey tessellation is given in the following theorem.
Theorem 2.3 (Honda 2000, ).
Let be a convex torus in standard form with dividing slope and ruling slope Let be a bypass for attached to the front of along a ruling curve. Let be the torus obtained from by attaching the bypass Then and the dividing slope of is determined as follows: let be the arc on running from counterclockwise to then is the point in closest to with an edge to
If the bypass is attached to the back of then the same algorithm works except one uses the interval on . ∎
2.2.4. The Imbalance Principle
As we see that bypasses are useful in changing dividing curves on a surface we mention a standard way to try to find them called the Imbalance Principle. Suppose that and are two disjoint convex surfaces and is a convex annulus whose interior is disjoint from and but its boundary is Legendrian with one component on each surface. If then there will be a dividing curve on that cuts a disk off of that has part of its boundary on . It is now easy to use the Giroux Flexibility Theorem to show that there is a bypass for on .
2.2.5. Discretization of Isotopy
We will frequently need to analyze what happens to the contact geometry when we have a topological isotopy between two convex surfaces and . This can be done by the technique of Isotopy Discretization  (see also  for its use in studying Legendrian knots). Given an isotopy between and one can find a sequence of convex surfaces such that
all the are convex and
and are disjoint and is obtained from by a bypass attachment.
Thus if one is trying to understand how the contact geometry of and relate, one just needs to analyze how the contact geometry of the pieces of changes under bypass attachment. In particular, many arguments can be reduced from understanding a general isotopy to understanding an isotopy between two surfaces that cobound a product region.
There is also a relative version of Isotopy Discretization where and are convex surfaces with Legendrian boundary consisting of ruling curves on a convex torus. If and there is a topological isotopy of to relative to the boundary then we can find a discrete isotopy as described above.
2.3. Classifying knots in a knot type
In this section we briefly recall the standard strategy for classifying Legendrian knots in a given knot type as laid out in [4, 5]. We begin by recalling the “normal form” for a neighborhood of a Legendrian or transverse knot and the relation between them.
2.3.1. Standard neighborhoods of knots
Given a Legendrian knot , a standard neighborhood of is a solid torus that has convex boundary with two dividing curves of slope (and of course we will usually take to be a convex torus in standard form). Conversely given any such solid torus it is a standard neighborhood of a unique Legendrian knot. Up to contactomorphism one can model a standard neighborhood as a neighborhood of the -axis in with contact structure Using this model we can see that is a -transverse curve. The image of in is called the transverse push-off of and is called the negative transverse push-off. One may easily check that is well-defined and compute that
One may understand stabilizations and destabilizations of a Legendrian knot in terms of the standard neighborhood. Specifically, inside the standard neighborhood of , can be positively stabilized to , or negatively stabilized to . Let be a neighborhood of the stabilization of inside As above we can assume that has convex boundary in standard form. It will have dividing slope Thus the region is diffeomorphic to and the contact structure on it is easily seen to be a basic slice, see . There are exactly two basic slices with given dividing curves on their boundary and as there are two types of stabilization of we see that the basic slice is determined by the type of stabilization done, and vice versa. Moreover if is a standard neighborhood of then destabilizes if the solid torus can be thickened to a solid torus with convex boundary in standard form with dividing slope Moreover the sign of the destabilization will be determined by the basic slice . Finally, we notice that using Theorem 2.3 we can destabilize by finding a bypass for attached along a ruling curve whose slope is clockwise of (and anti-clockwise of ).
A neighborhood of a transverse knot can be modeled by the solid torus for sufficiently small , where are polar coordinates on and is the angular coordinate on , with the contact structure . Notice that the tori inside of have linear characteristic foliations of slope Thus for all integers with we have tori with linear characteristic foliation of slope Let be a leaf of the characteristic foliation of Any Legendrian Legendrian isotopic to one of the so constructed will be called a Legendrian approximation of
Lemma 2.4 (Etnyre-Honda 2001, ).
If is a Legendrian approximation of the transverse knot then is transversely isotopic to Moreover, is Legendrian isotopic to the negative stabilization of ∎
This lemma is a key ingredient in the following result from which our transverse classification results will follow from our Legendrian classification results.
Theorem 2.5 (Etnyre-Honda 2001, ).
The classification of transverse knots up to transverse isotopy is equivalent to the classification of Legendrian knots up to negative stabilization and Legendrian isotopy.
2.3.2. Classification strategy
The classification of Legendrian knots in a given knot type can be done in a (roughly) three step process.
Step I — Identify the maximal Thurston-Bennequin invariant of and classify Legendrian knots realizing this.
Step II — Identify and classify the non-maximal Thurston-Bennequin Legendrian knots in that do not destabilize and prove that all other knots destabilize to one of these identified knots.
Step III — Determine which stabilizations of the maximal Thurston-Bennequin invariant knots and non-destabilizable knots are Legendrian isotopic.
As stabilization of a Legendrian knot is well defined and positive and negative stabilizations commute, it is clear that these steps will yield a classification of Legendrian knots in the knot type .
Step II is facilitated by the observation above that bypasses attached to appropriate ruling curves of a standard neighborhood of a Legendrian knot yield destabilizations. Similarly, if is a Legendrian knot contained in a convex surface (and the framing given to by is less than or equal to the framing given by a Seifert surface) and there is a bypass for on then this leads to a destabilization of . Moreover one can find such a bypass in some cases by the Imbalance Principle discussed above.
2.3.3. Contact isotopy and contactomorphism
We begin by recalling a result of Eliashberg concerning the contactomorphism group of the standard contact structure on . Fix a point in and let be the group of orientation-preserving diffeomorphisms of that fix the plane and let be the group of diffeomorphisms of that preserve .
Theorem 2.6 (Eliashberg 1992, ).
The natural inclusion of