The Surgery Unknotting Number of Legendrian Links

The Surgery Unknotting Number of Legendrian Links

A. Bianca Boranda Bryn Mawr College, Bryn Mawr, PA 19010 Lisa Traynor Bryn Mawr College, Bryn Mawr, PA 19010 ltraynor@brynmawr.edu  and  Shuning Yan Bryn Mawr College, Bryn Mawr, PA 19010
Abstract.

The surgery unknotting number of a Legendrian link is defined as the minimal number of particular oriented surgeries that are required to convert the link into a Legendrian unknot. Lower bounds for the surgery unknotting number are given in terms of classical invariants of the Legendrian link. The surgery unknotting number is calculated for every Legendrian link that is topologically a twist knot or a torus link and for every positive, Legendrian rational link. In addition, the surgery unknotting number is calculated for every Legendrian knot in the Legendrian knot atlas of Chongchitmate and Ng whose underlying smooth knot has crossing number or less. In all these calculations, as long as the Legendrian link of components is not topologically a slice knot, its surgery unknotting number is equal to the sum of and twice the smooth -ball genus of the underlying smooth link.

LT is partially supported by NSF grant DMS-0909021. ABB and SY completed this work as Bryn Mawr College undergraduates; they received support from NSF grant DMS-0909021 to pursue this research.

1. Introduction

A classical invariant for smooth knots is the unknotting number: the unknotting number of a diagram of a knot is the minimum number of crossing changes required to change the diagram into a diagram of the unknot; the unknotting number of is the minimum of the unknotting numbers of all diagrams of . In the following, we will define a sugery unknotting number for Legendrian knots and links.

Legendrian links are smooth links that satisfy an additional geometric condition imposed by a contact structure. We will focus on Legendrian links in with its standard contact structure. The notion of Legendrian equivalence is more refined than smooth equivalence: there is only one smooth unknot, but there are an infinite number of Legendrian unknots. Figure 4 shows the front projections of three different Legendrian unknots; the entire infinite “tree” representing all Legendrian unknots can be found in Figure 7.

The act of changing a crossing (smoothly passing a knot through itself) is not a natural operation in a contact manifold. Instead, given a Legendrian link, we will attempt to arrive at a Legendrian unknot through a Legendrian “surgery” operation in which two oppositely oriented strands in a Legendrian -tangle are replaced by an oriented, Legendrian -tangle as illustrated in Figure 1. It is shown in Proposition 3.5 that every Legendrian link can become a Legendrian unknot after a finite number of surgeries. The surgery unknotting number of a Legendrian link , , measures the minimal number of these surgeries that are required to convert to a Legendrian unknot; see Definitions 3.1 and 3.6.

Figure 1. Oriented Legendrian surgeries: a basic, compatibly oriented -tangle is replaced by a basic, compatibly oriented -tangle.

In the following, our goal is to study and calculate this Legendrian invariant .

1.1. Main Results

Lower bounds on exist in terms of the classical invariants of . These invariants include invariants of the underlying smooth link type and the classical Legendrian invariants of : the Thurston-Bennequin, , and rotation number, , as defined in Section 2.

Theorem 1.1.

Let be a Legendrian link. Then:

  1. if has components, denotes the underlying smooth link type of , and denotes the smooth -ball genus of 111 denotes the minimal genus of a smooth, compact, connected, oriented surface with ., then

Remark 1.2.

In parallel to Theorem 1.1 (1), when is a Legendrian knot with underlying smooth knot type , the well-known Slice-Bennequin Inequality says that:

(1.1)

There are now a number of proofs of this result, but all use deep theory. In [15], Lisca and Matić prove this using their adjunction inequality obtained by Seiberg-Witten theory. See also [2] and [18]. In contrast, the proof of Theorem 1.1 is elementary and is given in Lemmas 3.8 and 3.9.

When is a knot, combining Theorem 1.1(2) and the Slice-Bennequin Inequality (1.1), we find:

Corollary 1.3.

For any Legendrian knot , if denotes the smooth knot type of then

Thus when , .

As we will see below, this corollary sometimes allows us to calculate the smooth -ball genus of a knot.

Using the established lower bounds, we can calculate when the underlying smooth link type of falls within some important families.

Theorem 1.4.
  1. If is a Legendrian knot that is topologically a non-trivial twist knot, then .

  2. If is a -component Legendrian link that is topologically a -torus link, and , then

Theorem 1.4 is proved in Section 4 as Theorems 4.1 and 4.2. The proof of this theorem relies heavily on the classification of Legendrian twist knots given by Etnyre, Ng, and Vértesi, [13], and the classification of Legendrian torus knots by Etnyre and Honda, [12], which was extended to a classification of Legendrian torus links by Dalton, [9]. When is topologically a positive torus link, , of maximal Thurston-Bennequin invariant, the calculation of is obtained realizing the lower bound given in Theorem 1.1 by the Legendrian invariants of . Thus by Corollary 1.3, we are able to to reprove the Milnor conjecture about torus knots, originally proved by Kronheimer and Mrowka:

Corollary 1.5 ([14]).

If is a -torus knot, , then

By comparing of the Legendrian and of the underlying smooth link type, we can rephrase the conclusions of Theorem 1.4 as:

Corollary 1.6.

If is a Legendrian link that is topologically a non-slice twist knot 222Casson and Gordon proved that the only twist knots that are slice are the unknot, , and ; [4]. or a -component torus link, , then

As an additional family of Legendrian links, we consider “positive, Legendrian rational links”. These links are defined as Legendrian numerator closures of the Legendrian rational tangles studied, for example, by the second author, [21], and Schneider, [19]. These links are positive in the sense that an orientation is chosen on the components so that all the crossings have a positive sign. Such Legendrian links are specified by a vector of positive integers; see Definition 4.4 and Figure 18. Lemma 4.5 gives conditions on the that guarantee that the link is positive.

Theorem 1.7.

If is a positive, Legendrian rational link, then

where equals when is odd and equals when is even.

This is proved in Section 4; see Theorem 4.6.

Remark 1.8.

When is a positive, Legendrian rational link, the calculation of is obtained realizing the lower bound given in Theorem 1.1 given by the classical Legendrian invariants of . Thus by Corollary 1.3, when is a positive, Legendrian rational knot, Theorem 1.7 gives a formula for twice the smooth -ball genus of the underlying smooth knot. This can be used to get formulas for the smooth -ball genus of a knot in terms of its rational notation. In particular,

This is an alternate to formulas for calculating the smooth -ball genus in terms of crossings and Seifert circles as given by Nakamura in [17]. In turn, using Nakamura’s formula, we see that when the underlying link type of is a -component link ,

see Remark 4.7.

Given the above calculations, it is natural to ask:

Question 1.9.

If is a Legendrian knot that is topologically a non-slice knot , is More generally, if is a Legendrian link of components that is topologically the link , is

To investigate the knot portion of this question, we examined Legendrian representatives of knots with crossing number or less. There is not yet a Legendrian classification of all these knot types, but a conjectured classification is given by Chongchitmate and Ng in [7].

Proposition 1.10.

Assuming the conjectured classification of Legendrian knots in [7] 333Potential duplications in their atlas will not affect the statement., if is a Legendrian knot that is topologically a non-slice knot with crossing number or less,

The only non-torus and non-twist knots with crossing number at most are , , , , , , , , , , , and . While doing the calculations for Legendrians with these knot types, in general we found that for a Legendrian whose underlying smooth knot type satisfied , where denotes the (-dimensional) genus of the knot, it is fairly straight forward to show that . Legendrians that are topologically , and fall into this category. For the remaining knot types under consideration, the calculation of the smooth -ball genus follows from the fact that the topological unknotting number of these knots is equal to . We show that in a front projection of a Legendrian knot, it is possible to locally change any negative crossing to a positive one by surgeries; see Lemma 5.2. This allowed us to prove Proposition 1.10 in the cases where is topologically , or . For the remaining cases of , , and , results of [20] show that it is not possible to find a front projection that can be unknotted at a negative crossing. However, we found front projections that could be unknotted at a positive crossing in a special “S” or “hooked-X” form: a positive crossing in one of these special forms can be locally changed to a negative crossing by surgeries; see Lemma 5.5.

1.2. The Lagrangian Motivation and Discussion

All of our calculations indicate that is measuring an invariant of the underlying smooth link type and that this invariant will be the same for and when they represent smooth knots that differ by the topological mirror operation. Below is an explanation for why this may be true.

Although the definition of the surgery unknotting number has been formulated above combinatorially, the motivation comes from trying to understand the flexibility and rigidity of Lagrangian submanifolds of a symplectic manifold. From theory developed by Bourgeois, Sabloff, and the second author in [3], the existence of an unknotting surgery string , as defined in Definition  3.1, implies the existence of an oriented Lagrangian cobordism in so that , for . Furthermore, if is the Legendrian unknot with maximal Thurston-Bennequin invariant, this cobordism can be “filled in” with a Lagrangian so that . In fact, it is shown by Chantraine in [6] that if is not the Legendrian unknot with maximal Thurston-Bennequin invariant, then the cobordism cannot be filled in to ; moreover, when there does exist the filling to and the smooth underlying knot type of is , then the genus of agrees with the smooth -ball genus of .

From this Lagrangian perspective, it is a bit more natural to consider surgery strings where is a Legendrian unlink (a trivial link of Legendrian unknots), and define a corresponding “surgery unlinking number”; this is a project that the second author has begun to pursue with other undergraduates. A Lagrangian analogue of Question 1.9 is:

Question 1.11.

If is a Legendrian knot with underlying smooth knot type , does there exist a Lagrangian cobordism constructed from Legendrian isotopy and oriented Legendrian surgeries between and , a Legendrian that is a smooth unlink, that realizes ?

Any Lagrangian constructed from Legendrian isotopy and oriented Legendrian surgeries would be in ribbon form; this means that the restriction of the height function, given by the coordinate, to the cobordism would not have any local maxima in the interior of the cobordism. So a positive answer to Question 1.11 would imply that the slice genus agrees with the ribbon genus; for some background on this and related problems, see, for example, [16].

Acknowledgements

The ideas of this project were inspired by joint work of Josh Sabloff and the second author. We are extremely grateful for the many fruitful discussions we have had with Sabloff throughout this project. We also gained much by discussions with Chuck Livingston about the smooth -ball genus; we are very thankful for his clear explanations. We also thank Paul Melvin and other members of our PACT (Philadelphia Area Contact/Topology) seminar for useful comments during a series of presentations on this work.

2. Background Information on Legendrian Links

Below is some basic background on Legendrian links. More information can be found, for example, in [11].

The standard contact structure on is the field of hyperplanes where . A Legendrian link is a submanifold, , of diffeomorphic to a disjoint union of circles so that for all , . It is common to examine Legendrian links from their -projections, known as their front projections. A Legendrian link will generically have an immersed front projection with semi-cubical cusps and no vertical tangents; conversely, any such projection can be uniquely lifted to a Legendrian link using . Figure 2 shows Legendrian versions of the trefoils and .

Figure 2. Front projections of (a) a Legendrian knot that is topologically the (negative/left) trefoil and (b) a Legendrian knot that is topologically the mirror trefoil . At crossings, it is not necessary to specify which strand is the overstrand: the strand with lesser slope will always be on top.

and are equivalent Legendrian links if there exists a -parameter family of Legendrian links joining and . In fact, Legendrian links are equivalent if and only if their front projections are equivalent by planar isotopies that do not introduce vertical tangents and the Legendrian Reidemeister moves as shown in Figure 3.

Figure 3. The three Legendrian Reidemeister moves: there is another type move obtained by flipping the planar figure about a horizontal line, and there are three additional type moves obtained by flipping the planar figure about a vertical, a horizontal, and both a vertical and horizontal line.

Every Legendrian knot and link has a Legendrian representative. In fact, every Legendrian knot and link has an infinite number of different Legendrian representatives. For example, Figure 4 shows three different Legendrians that are all topologically the unknot. These unknots can be distinguished by classical Legendrian invariants, the Thurston-Bennequin and rotation number. These invariants can easily be computed from a front projection of the Legendrian link once we understand how to assign a sign to each crossing and an up/down direction to each cusp.

Figure 4. Three different Legendrian knots that are all topologically the unknot.

A positive (negative) crossing of a front projection of an oriented Legendrian link is a crossing where the strands point to the same side (opposite sides) of a vertical line passing through the crossing point; see figure 5. Each cusp can also be assigned an up or down direction; see Figure 6. Then for an oriented Legendrian link , we have the following formulas for the Thurston-Bennequin, , and rotation number, , invariants:

(2.1)

where is the number of positive crossings, is the number of negative crossings, is the number of right cusps, is the number of down cusps, and is the number of up cusps in a front projection of . Given that two front projections of equivalent Legendrian links differ by the Legendrian Reidemeister moves described in Figure 3, it is easy to verify that and are Legendrian link invariants.

Figure 5. (a) Negative crossings; (b) Positive crossings.

Figure 6. (a) Right and left down cusps; (b) Right and left up cusps.

The two unknots in the second line of Figure 4 are obtained from the one at the top by adding an up or down zig-zag (also known as a stabilization). In general, this stabilization procedure will not change the underlying smooth knot type but will will decrease the Thurston-Bennequin number by ; adding an up (down) zig-zag will decrease (increase) the rotation number by . If is a Legendrian knot, we will use the notation to denote the double stabilization of , the Legendrian knot obtained by adding both a positive and negative zig-zag.

In fact, as discovered by Eliashberg and Fraser, all Legendrian unknots are classified by their Thurston-Bennequin and rotation numbers:

Theorem 2.1 ([10], [12]).

Suppose and are oriented Legendrian knots that are both topologically the unknot. Then is equivalent to if and only if and .

Figure 7. The tree of all Legendrian unknots.

Figure 7 describes all the Legendrian unknots. Notice that any Legendrian unknot is equivalent to one that is obtained by adding up and/or down zig-zags to the unknot with Thurston-Bennequin number equal to and rotation number equal to shown in Figure 4.

In general, it is an important question to understand the “geography” of other knot types. By work of Etnyre and Honda, [12] and Etnyre, Ng, and Vértesi, [13], we understand the trees/mountain ranges for all torus and twist knots. The Legendrian knot atlas of Chongchitmate and Ng, [7], gives the known and conjectured mountain ranges for all Legendrian knots with arc index at most ; this includes all knot types with crossing number at most and all non-alternating knots with crossing number at most .

3. The Surgery Unknotting Number

In this section, we define the surgery operation, show that every Legendrian link can be unknotted by surgeries, define the surgery unknotting number, and give some basic properties of the surgery unknotting number.

The surgery operation can be viewed as a “tangle surgery”: the replacement of one Legendrian tangle by another. A basic, compatibly-oriented Legendrian -tangle is a Legendrian tangle that is topologically the 0-tangle where the strands are oppositely oriented and each strand has neither crossings nor cusps; the two basic, compatibly-oreinted Legendrian -tangles can be seen on the left side of Figure 1. A basic, compatibly-oriented Legendrian -tangle is Legendrian tangle that is topologically the -tangle where the strands are oppositely oriented and each strand has precisely cusp and no crossings; the two basic, compatibly-oriented Legendrian -tangles can be seen on the right side of Figure 1.

Definition 3.1.

An oriented, Legendrian surgery of an oriented, Legendrian link is the Legendrian link obtained by replacing a basic, compatibly-oriented Legendrian -tangle with a basic, compatibly-oriented Legendrian -tangle; see Figure 1. An oriented surgery string consists of a vector of oriented, Legendrian links where, for all , is obtained from by an oriented, Legendrian surgery. An oriented, unknotting surgery string of length for consists of an oriented surgery string where and is topologically an unknot.

To start, we have the following relationships between the classic invariants of two Legendrian links related by surgery:

Lemma 3.2.

If is an oriented, Legendrian link and is obtained from by an oriented, Legendrian surgery, then:

  1. the parity of the number of components of and differ;

  2. , and .

Proof.

The statements about the Thurston-Bennequin and rotation numbers are easily verified using Equation (2.1). Regarding the parity, one surgery to a knot will always produce a link of two components, while doing a surgery to a link will increase or decrease the number of components by depending on whether or not the strands in the -tangle belong to the same component of the link. ∎

Recall that for any Legendrian knot , the Legendrian knot obtained as the double stabilization of will have and . Thus it is potentially possible that can be obtained from by two oriented Legendrian surgeries. In fact, it is possible.

Lemma 3.3.

For any oriented, Legendrian knot there exists an oriented surgery string with and .

Proof.

These surgeries are illustrated in Figure 8. Every Legendrian link must have a right cusp. By a Legendrian isotopy, we call pull a right cusp far to the right and perform one surgery near this right cusp. This produces a link consisting of the original link and a Legendrian unknot. After a Legendrian isotopy, a second surgery can be done using one strand near the same cusp of the original link and a strand from the unknot. The result is . ∎

Figure 8. Two oriented, Legendrian surgeries produce from .

In the chart of Legendrian unknots given in Figure 7, we see that any two unknots with the same rotation number are related by a sequence of double stabilizations. Thus we get:

Corollary 3.4.

If and are oriented, Legendrian unknots with and , then there exists an oriented surgery string
, where , and .

Thus if we can reach a Legendrian unknot by surgeries, then we can reach an infinite number of Legendrian unknots by surgery. The basis for our new invariant is the fact that every Legendrian link can be “unknotted” by a string of surgeries:

Proposition 3.5.

For any oriented, Legendrian link , there exists an oriented, unknotting surgery string . Moreover, if has components and there exists a front projection of with crossings, then .

Proof.

Assume that there is a front projection of with crossings. We will first show that there is an oriented surgery string , where and is a trivial link of Legendrian unknots. If has components, we will then show that it is possible to do an additional surgeries to get this into a single unknot.

Given the initial Legendrian link having a projection with crossings, assume that of these crossings are negative. It is then possible to construct a surgery string where and has a front projection with crossings, all of which are positive. This surgery string is obtained by doing a surgery to the right of each negative crossing as shown in Figure 9, and then doing a Legendrian isotopy to remove the positive crossing introduced by the surgery. Next, by applying a planar Legendrian isotopy, it is possible to assume that all the crossings of have distinct -coordinates. The left cusps associated to the leftmost positive crossing are either nested or stacked and fall into one of the cases listed in Figure 10. For each case, it is possible to do a surgery immediately to the right of this leftmost crossing. After Legendrian Reidemeister moves, the crossing is eliminated and the number of crossings of the projection of the resulting link has decreased by 1; see Figure 10. What was the second leftmost positive crossing is now the leftmost positive crossing and the procedure can be repeated. In this way, we obtain a surgery string of Legendrian links where has a front projection with no crossings. It follows that is topologically a trivial link of unknots. By applying a Legendrian isotopy, we can assume that consists of Legendrian unknots which are vertically stacked and where each unknot is oriented “clockwise”; an example of this is shown in Figure 11. It is then easy to see that after applying additional surgeries, we can obtain a Legendrian unknot. Thus there is a length unknotting surgery sequence for . By Lemma 3.2, if has components, has at most components. Thus we see that , as claimed. ∎

Figure 9. A negative crossing can be removed by an oriented Legendrian surgery and then Legendrian isotopy.

Figure 10. Three cases for the leftmost positive crossing and their associated left cusps; three additional cases are obtained by reversing the orientations on both strands.

Figure 11. After all crossings are eliminated, a Legendrian isotopy can be applied so that is a stack of Legendrian unknots oriented clockwise. After additional surgeries, a Legendrian unknot is obtained.
Definition 3.6.

Given a Legendrian link , the (oriented) Legendrian surgery unknotting number of , , is defined as the minimal length of an oriented, unknotting surgery string for .

Remark 3.7.

Here are some basic properties of :

  1. By Lemma 3.2, for any Legendrian link , the parity of is opposite the parity of the number of components of ;

  2. For any oriented, Legendrian link with components, , with iff is topologically an unknot;

  3. If is a topolopologically non-trivial Legendrian knot and there exists an oriented unknotting surgery string for of length , then ;

  4. If is obtained from by stabilization(s), then .

Propositon 3.5 and, more importantly, explicit calculations will give upper bounds for . Now we turn to examining some lower bounds for .

First, by Theorem 2.1, if is a Legendrian unknot, then . Thus if is a Legendrian link with a “large” Thurston-Bennequin and/or rotation number, one is forced to do a certain number of Legendrian surgeries. More precisely, Lemma 3.2 implies:

Lemma 3.8.

For any Legendrian link ,

Lemma 3.8 gives us improved lower bounds over those given in Remark 3.7 when . 444The parity of agrees with the parity of the number of components of , so for knots, we get interesting new bounds when . For example, there exists a Legendrian whose underlying smooth knot type is and whose classical invariants satisfy ; see, for example, [7]. Thus Lemma 3.8 implies . However for many links, . For example, for any Legendrian that is topologically the knot, . Although Lemma 3.8 will not help us, in this case we can make use of:

Lemma 3.9.

For a Legendrian link with components, let denote the underlying smooth link type of , and let denote the smooth -ball genus of . Then

Proof.

From a Legendrian surgery string of length that ends at an unknot, one can construct a smooth, orientable, compact, and connected -dimensional surface in with boundary equal to and Euler characteristic equal to ; the genus, , of this surface satisfies . Thus, by definitions of the smooth -ball genus,

Since is the minimum length of a surgery unknotting string, the claim follows. ∎

A convenient table of smooth -ball genera of knots can be found at Cha and Livingston’s KnotInfo website, [5].

4. The Surgery Unknotting Number for Families of Knots

In this section we will calculate the surgery unknotting numbers for Legendrian twist knots, Legendrian torus links, and positive, Legendrian rational links. The fact that we can precisely calculate these numbers for the first two families rests upon classification results of [13], [12], and [9].

4.1. Legendrian twist knots

A twist knot is a knot that is smoothly equivalent to a knot in the form of Figure 12. In other words, a twist knot is a twisted Whitehead double of the unknot.

Figure 12. The twist knot ; the box contains right-handed half twists if , and left-handed twists if . Notice that and are unknots.
Theorem 4.1.

If is a Legendrian knot that is topologically a non-trivial twist knot then .

Proof.

Etnyre, Ng, and Vértesi, have classified all Legendrian twist knots, [13]. In particular, every Legendrian knot with maximal Thurston-Bennequin invariant that is topologically , for some , is Legendrian isotopic to one of the form in Figure 13, and every Legendrian knot with maximal Thurston-Bennequin invariant that is topologically , for with maximal Thurston-Bennequin invariant is Legendrian isotopic to one of the form in Figure 14. 555We omit since those corresponds to the unknot. Every Legendrian knot that is topologically a non-trivial twist knot is obtained by stabilization of one of these with maximal Thurston-Bennequin invariant. By Remark 3.7, it suffices to show for any Legendrian knot that is topologically a non-trivial twist knot and has maximal Thurston-Bennequin invariant, . For , we can do the two unknotting surgeries near the “clasp”. The sign of the crossings in the clasp will depend on whether is even or odd: Figure 15 shows the positions of two surgeries that result in an unknot. ∎

Figure 13. Any Legendrian knot that is topologically a negative twist knot, with , and has maximal Thurston-Bennequin invariant is Legendrian isotopic to one of the form in (a) where the box contains half twists, each of form as shown in (b) or of form as shown in (c).

Figure 14. Any Legendrian knot that is topologically a positive twist knot, with , and has maximal Thurston-Bennequin invariant is Legendrian isotopic to one of the form in (a) where the box contains half twists, each of form as shown in (b).

Figure 15. For a Legendrian knot with maximal Thurston-Bennequin invariant that is topologically , (a) gives the surgery points when is even, and (b) gives the surgery points when is odd.

4.2. Legendrian Torus Links

A torus link is a link that can be smoothly isotoped so that it lies on the surface of an unknotted torus in . Every torus knot can be specified by a pair of coprime integers: we will use the convention that the -torus knot, , winds times around a meridonal curve of the torus and times in the longitudinal direction. See, for example, [1]. In fact, is equivalent to and to . So we will always assume that ; in addition we will assume since we are interested in non-trivial torus knots. For , , with and , will be a -component link where each component is a torus link. We will only consider torus links of non-trivial components.

Theorem 4.2.

If is a -component Legendrian link that is topologically the -torus link, , then .

Proof.

First consider the case where is topologically a positive torus knot, with . As shown by Etnyre and Honda, [12], the list of different Legendrian representations of a positive torus knot can be represented as a “tree” in parallel to the tree of unknots shown in Figure 7. Namely, for fixed , there is a unique Legendrian knot that is topologically with maximal Thurston-Bennequin invariant and ; any Legendrian knot that is topologically is obtained by stabilizations of . By Remark 3.7, it suffices to show that if is a Legendrian knot that is topologically and has maximal Thurston-Bennequin invariant, then . By Lemma 3.8,

In fact, it is possible to unknot with surgeries. Starting from the left most string of crossings, do successive surgeries as illustrated for the -torus knot in Figure 16. In general, this take the -torus knot to the -torus link. Repeating this times results in the -torus knot, which is an unknot. 666By Corollary 1.3, we can now reprove the Milnor conjecture as mentioned in Corollary 1.5.

The above proof easily generalizes to positive torus links of non-trivial components. Dalton showed in [9] that there is a unique Legendrian link that is topologically with maximal Thurston-Bennequin invariant . The construction of this one exactly parallels the construction in Figure 16, and so the same pattern of surgeries of will produce a Legendrian unknot.

Figure 16. The Legendrian -torus knot with maximal tb invariant. The general, positive Legendrian -torus knot with maximal Thurston-Bennequin invariant is constructed using strands and a length string of crossings. Shown are the oriented Legendrian surgeries that unknot the Legendrian positive -torus knot with maximal tb.

Next consider the case where is topologically a negative torus knot, with . In this case, Etnyre and Honda have shown that the list of different Legendrian representations of a negative torus knots, for and , can be represented as a “mountain range” where the number of representatives with maximal Thurston-Bennequin invariant depends on the divisibility of by . Namely, if , , then there will be Legendrian representatives of with maximal Thurston-Bennequin invariant of . Half of these different representatives with maximal Thurston-Bennequin invariant are obtained by writing , where , and then is constructed using the form shown in Figure 17 with and copies of the tangle inserted as indicated: . The other Legendrian versions of with maximal Thurston-Bennequin invariant are obtained by reversing the orientation. For negative torus knots, Lemma 3.8 will not be a useful lower bound. However, since the calculation of the -ball genus is the same for both the knot and its mirror, the calculations in the positive torus knot case and Corolloary 1.3, (or [14]), show that for a negative torus knot , . Thus, by Lemma 3.9

In fact, it is possible to arrive at an unknot with surgeries. Figure 17 shows the claimed surgeries: a surgery is done to the right of all crossings in the , , and regions (contributing surgeries), and between each “” in the string one does successive surgeries (contributing surgeries). Thus the total number of surgeries is:

Figure 17. The oriented Legendrian surgeries that unknot a Legendrian negative -torus knot with maximal Thurston-Bennequin invariant.

The above proof easily generalizes to negative torus links. It follows from [17] that ; see Remark 4.3. Dalton showed in [9] that there are Legendrian links that are topologically with maximal Thurston-Bennequin invariant, and all Legendrians that are topologically are obtained by stabilizations of one of these. Each of these with maximal Thurston-Bennequin invariant can be constructed as in Figure 17, and so the same pattern of surgeries will produce a Legendrian unknot. ∎

Remark 4.3.

Nakamura’s formula, [17], for the smooth -ball genus of a -component positive link is that

where is the number of Seifert circles and is the number of crossings in a non-split positive diagram for . It is straightforward to see that when is the positive torus link , using the diagram corresponding to Figure 16, and . So,

Thus for any Legendrian link that is topologically , for either positive or negative,

4.3. Positive, Legendrian Rational Links

Definition 4.4.

Given a vector of integers , where , and implies for , we construct the rational Legendrian link to be the Legendrian numerator closure of the Legendrian tangle as demonstrated in Figure 18; see also [1], [21], [19]. The rational Legendrian link is positive if all crossings are positive.

This Legendrian link is topologically the numerator closure of the rational tangle associated to the rational number with continued fraction expansion ; [8].

Figure 18. (a) The general form of , (b) the general form of , (c) the general form of , the general form of .

The “even” entries of the vector denote the strings of vertical crossings. It is straightforward to verify that the parity of these vertical entries determine when is a positive link:

Lemma 4.5.
  1. When is odd, there exists an orientation on the components of so that it is a positive link if and only if is even, for all even. Moreover, is a knot when is odd.

  2. When is even, there exists an orientation on the components of so it is a positive link if and only if is odd and are all even. Moreover, is a knot when is even.

The Legendrian surgery unknotting number of a positive link has a convenient formula in terms of the “odd” entries, which correspond to the strings of horizontal crossings. There will be some differences in following formulas depending on whether is constructed from an odd or an even length vector. Define

measures the parity of the “length” of the vector .

Theorem 4.6.

If is a positive, Legendrian rational link, then

Proof.

This will be proved using the lower bound on provided by Lemma 3.8, and explicit calculations.

We will first show that

It is easy to verify that when all the crossings are positive, the up and down cusps cancel in pairs and thus the rotation number vanishes. To calculate , notice that when is odd the number of right cusps is more than the number of vertical crossings, , while when is even, the number of rights cusps is more than the number of vertical crossings. Thus:

Thus, by Lemma 3.8,

In fact, it is possible to unknot by doing surgeries in each horizontal component and surgery in each vertical segment; Figure 19 illustrates some examples of this. When , there are no vertical segments; for other odd , the number of vertical components is one less than the number of horizontal components, and when is even, the number of vertical components agrees with the number of horizontal components. Thus

and the desired calculation of follows.

Figure 19. Two positive, Legendrian rational knots of odd and even lengths. In both cases, it is possible to unknot by doing surgeries in each horizontal segment ( odd) and surgery in each vertical segment.

Remark 4.7.

In the above proof, is obtained by realizing the lower bound given by the classical Legendrian invariants. Thus, by Corollary 1.3, we see that when has an underlying topological type of the knot , . Moreover, when has an underlying topological type of a -component link , we can compare to the smooth -ball genus of using Nakamura’s formula (see Remark 4.3) for the smooth -ball genus of a positive link . When is odd, the number of Seifert circles is , while when is even, . Thus we find that for a -component, positive, Legendrian rational link ,

5. The Surgery Unknotting Number for Small Crossing Knots

Given the calculations of the previous section, it is natural to ask Queston 1.9 in the Introduction. To investigate the knot portion of this question, we examined Legendrian representatives of low-crossing knots. There is not a Legendrian classification of all these knot types, but a conjectured classification of these knot types can be found in [7]. In the following, we prove Proposition 1.10, which says that the surgery unknotting number of the Legendrian agrees with twice the smooth -ball genus of the underlying smooth knot for all Legendrians that are topologically a non-slice knot with crossing number at most .

In Section 4, Proposition 1.10 is verified for all torus and twist knots. The only non-torus and non-twist knots with or fewer crossings are , , , , , , , , , , , and . The needed calculations fall into three categories as described below.

Example 5.1.

For the smooth knots , and , the genus, , agrees with the smooth -ball genus . 777This is also the situation for the torus and non-slice twist knots studied in Section 4. In general, we find that for a Legendrian whose underlying knot type satisfied , it is fairly straight forward to show that . For example, Figure 20 shows all conjectured representatives of , , , , , and with maximal Thurston-Bennequin invariant (after perhaps selecting alternate orientations and/or performing the Legendrian mirror operation). For each of these with maximal Thurston-Bennequin invariant, it is possible to unknot with surgeries as indicated.

Figure 20. Front projections representing all conjectured Legendrian representatives of , , , and with maximal Thurston-Bennequin invariant. For all of these knot types, ; the indicated surgery points realize .

In general, we found that for a Legendrian whose underlying knot type satisfied , it is more difficult to calculate . To do calculations for our remaining cases, we made use of the well-known fact that the unknotting number of a knot, , gives an upper bound to the smooth -ball genus:

(5.1)

Figure 21 demonstrates two topological surgeries that produce a crossing change; an argument as in the proof of Lemma 3.9 then proves Inequaltiy 5.1. Notice that the topological Reidemeister moves used in the equivalence are not Legendrian Reidemeister moves. However, near a negative crossing, it is possible to “Legendrify” this construction:

Figure 21. A sequence of two topological surgeries in a neighborhood of a negative crossing that toplogically change the crossing. An analogous picture shows that a positive crossing can be changed into a negative crossing by two topological surgeries.
Lemma 5.2.

If the Legendrian knot has a front projection that can be topologically unknotted by changing a negative crossing, then

Proof.

Figure 22 demonstrates how two surgeries can locally produce a topological crossing change.

Figure 22. A sequence of two oriented surgeries in a neighborhood of a negative crossing that toplogically change the crossing.

Example 5.3.

Using Lemma 5.2, it is possible to show that for any conjectured Legendrian representative of , , , or , . Figure 23 shows the conjectured Legendrian representatives of these knot types with maximal Thurston-Bennequin invariant (after perhaps selecting alternate orientations and/or performing a mirror operation) and the negative crossing that when topologically changed produces an unknot.

Figure 23. Front projections representing all conjectured Legendrian representatives of , , , and with maximal Thurston-Bennequin invariant. These projections can be topologically unknotted at the indicated negative crossing.

We were not able to find front projections of the conjectured maximal Thurston-Bennequin representatives of , , or that could be topologically unknotted by changing a negative crossing; in fact, by [20], it is not possible to do this even in the smooth setting. Luckily, sometimes we can topologically change a positive crossing when it has a special form.

Definition 5.4.

A positive crossing is of S form, Z form, or hooked-X form if it takes the form as shown in Figure 24.

Figure 24. A positive crossing of (a) S form, (b) Z form, and (c) Hooked-X form. Reversing the orientations on both strands keeps the respective forms. Also reflecting the planar figure in (c) about a horizontal line produces another Hooked-X form.

Figure 25. A positive crossing of S form can be transformed into a negative crossing with surgeries. Similarly, a positive crossing of form can be transformed into a negative crossing with surgeries.

Figure 26. A positive crossing of hooked-X form can be transformed into a negative crossing with Legendrian surgeries.
Lemma 5.5.

If is a non-trivial Legendrian knot that has a projection that can be topologically unknotted by changing a positive crossing in S, Z, or hooked-X form, then

Example 5.6.

Using Lemma 5.5, it is possible to show that for any conjectured Legendrian representative of , , or , . Figure 27 shows the conjectured Legendrian representatives of these knot types with maximal Thurston-Bennequin invariant (after perhaps selecting alternate orientations and/or performing a mirror operation). These projections differ from those in [7] by Legendrian Reidemeister moves of type II and III. The black dot indicates a positive crossing that when topologically changed produces an unknot.

Figure 27. Front projections representing all conjectured Legendrian representatives of , and with maximal Thurston-Bennequin invariant. Each of these can be topologically unknotted by changing the indicated positive crossing in S or Hooked-X form.

The proofs of Lemmas 5.2 and  5.5 in fact show that if the Legendrian knot has a front projection that can be topologically unknotted by changing negative crossings and crossings in S, Z, or hooked-X form, then However, for our calculations we did not need this more general form.

References

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