Problems in Low Dimensional Contact Topology. were held concerning contact geometry. The rst was run by Emmanuel

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1 Topology Geometric Georgia ternational Topology Conference 2001 Studies in Advanced Mathematics AMS/IP 35, 2003 Volume the 2001 Georgia ternational Topology Conference, two problem sessions During were held concerning contact geometry. The rst was run by Emmanuel and focused on problems in three dimensional contact topology; the second Giroux run by John Etnyre and focused on knots and contact homology. was article collects problems from the sessions, and adds some background. Sections This 1 through 5 deal with three dimensional contact geometry, and Sections 6 10 deal with knots. through would like to thank the organizers the Georgia Conference for running We problem sessions and encouraging us to write this problem list. We would also the to thank Ko Honda and Will Kazez, who took exceptional notes during the like sessions, and Josh Sablo, who compiled a list some the problems problem here. discussed Three Dimensional Contact Geometry I. this section the reader is assumed to be familiar with the basic notions in 1. Which 3-manifolds have tight contact structures, and what types tight contact structures do they admit? question breaks down into several subquestions, but rst note that a tight This structure may be: contact weakly llable (or weakly semi-llable), 1. strongly llable (or strongly semi-llable), 2. holomorphically llable (or holomorphically semi-llable), 3. universally tight, or 4. virtually overtwisted. 5. that a holomorphically llable contact structure is also a strongly llable Recall Mathematics Subject Classication. Primary 57R17; secondary 53D35, 57M words and phrases. Contact structure, tight, knot, convex surface. Key supported in part by NSF Grant # DMS JE supported by the American stitute Mathematics and NSF Grant # DMS LN s in Low Dimensional Contact Topology John B. Etnyre and Lenhard L. Ng contact geometry, see [31], and 3-manifold topology, see [49]. 1. Existence and types contact structures contact structure, and strongly llable implies weakly llable. Moreover, a weakly c2003 American Mathematical Society and ternational Press 335

2 contact structure is tight [19, 48]. It is known that there are tight contact llable that are not weakly llable [34] and weakly llable contact structures structures are not strongly llable (the rst such examples were found by Eliashberg that on the 3-torus, see also [15]). However, on homology spheres we know that [22] llable structures must also be strongly llable [69]. For a more thorough weakly the notions llability and their relations, see [28]. discussion Tight 6 j [ [? [? symplectically llable % Strongly symplectically llable Weakly [? llable Holomorphically a contact structure is llable, it is interesting to consider the number and type If llings. particular: 2. Do there exist only nitely many (strong or weak) llings (X 4 ;!) a xed (M 3 ; ), up to blowup and blowdown? it is not too hard to show there are always innitely many strong Actually, if we allow the fundamental group X to vary [74]. So it is most interesting llings consider this question when X is required to be simply connected. to main technique used to show a contact structure is not llable is Seiberg- The with positive curvature. It would be exciting to nd other methods for manifolds that a contact structure is not llable. showing is obvious that universally tight and virtually overtwisted contact structures It disjoint sets, but it is not clear if all tight contact structures must fall into form these sets. For any 3-manifold with a residually nite fundamental group, one is known (and not hard to prove) that any tight contact structure is either it tight or virtually overtwisted. Recall that a group G is residually nite universally every nontrivial element G is in the complement some nite index normal if G: Of course, it is conjectured that all 3-manifolds have residually nite subgroup groups. fundamental are no known connections between a tight structure being llable (in any There and being universally tight or virtually overtwisted. sense) 5. If is a universally tight contact structure on M, is there some cover M in which the pulled-back contact structure is llable? nite recall that the Poincare homology sphere with the nonstandard orientation We does not admit a tight contact structure [32]. Moreover, any tight contact P 336 JOHN B. ETNYRE AND LENHARD L. NG Weakly symplectically semi-llable % Strongly symplectically semi-llable Witten theory. This technique was initiated by Lisca [58] and only applies to 3. How can one determine if a contact structure is llable? 4. Are universally tight contact structures llable? It is known that the converse is not true. An easier question might be the following. structure on the connected sum M#N can be decomposed into a tight contact

3 on M and a tight contact structure on N [8]. Thus any 3-manifold structure form M#( P ) does not admit any tight contact structure, so when discussing the tight contact structures, we will always consider irreducible 3-manifolds. existence one believes the conjectured picture 3-manifolds coming from Thurston's If 6. Do all hyperbolic manifolds admit a tight contact structure? type tight structures do they support? What 7. Which Seifert bered spaces admit tight contact structure? Which llable? Which are universally tight or virtually overtwisted? are and Thurston [26] have shown that a Reebless foliation may be Eliashberg into a tight (in fact, weakly semi-llable and universally tight) contact perturbed Moreover, Gabai [40] can construct such a foliation on any irreducible structure. with b 1 1: Thus, when considering the above questions, one can focus 3-manifold rational homology spheres, or consider strong llability or virtually overtwisted on structures. contact 3 ; then there is a strongly llable contact structure on ( 1 ; 2 1 ; where e 0 = P 3 b i Gompf can also construct llable contact structures on many Seifert Moreover, spaces with e 0 = 1; see [46]. bered only manifold currently known not to admit a tight contact structure is The, which is a Seifert manifold over S 2 with invariants 1 2 ; 1 3 ; 1 5 : P are several other manifolds that do not admit weakly llable contact There [58, 59], Lisca has shown that the Seifert manifolds with invariants structures. ( 1 2 ; 1 3 ; 1 5 ); ( 1 2 ; 1 3 ; 1 4 ); ( 1 2 ; 1 3 ; 1 3 is believed, at least by some, that these manifolds do admit tight contact It This result would be particularly interesting, because then these would structures. the rst known manifolds admitting tight contact structures but not admitting be llable structures. any for hyperbolic manifolds, very little is known about the existence tight As structures. There are no known constructions contact structures in terms contact taut foliations. If this were true, we could perturb them into tight contact admit but it has recently been shown that there are hyperbolic 3-manifolds structures, taut foliations [6, 70]. It is still possible, however, that all hyperbolic without admit a tight contact structure. manifolds PROBLEMS IN LOW DIMENSIONAL CONTACT TOPOLOGY 337 Geometrization conjecture, then the main question breaks into two: Gompf [46] has constructed strongly llable contact structures on many Seifert spaces. the above results, here is our main concern: Let ( 1 bered ; 2 ; Given 3 ) Seifert bered space over S 2 with three singular bers with invariants 1 be ; 2 a 1 2, 3 ) if e 0 6= 1; i=1 i c and bxc is the greatest integer less than or equal to x. ) do not admit weakly llable contact structures. It would be very interesting to know: 8. Do the Seifert manifolds with invariants ( 1 2 ; 1 3 ; 1 5 ); ( 1 2 ; 1 3 ; 1 4 ); ( 1 2 ; 1 3 ; 1 3 ) admit tight contact structures at all? hyperbolic structures. It was once conjectured that all hyperbolic manifolds Another important question is: 9. Do all Haken manifolds admit a tight contact structure?

4 manifold with b 1 1 is known to be Haken, and from the work Eliashberg- Any [26] and Gabai [40], it is known that these manifolds admit semi-llable, Thurston tight contact structures; see also [54] for a pro that avoids foliation universally Thus our main interest here again is rational homology spheres. theory. S 3 ; S 2 S 1 [20] 1. Lens spaces [30, 43, 50] 2. Torus bundles [43, 51] 3. Circle bundles [44, 51]. 4. ((2; 3; 5)) = 0 or 1, depending on orientation, where Tight (M) denotes #Tight set tight contact structures up to isotopy. See [41] for further results on the 11. Classify tight contact structures on [0; 1]; where is a surface genus g > 1: [55], Honda, Kazez, and Matic classify tight contact structures on [0; 1]; the relative Euler class the contact structure is extremal, e()() = (2g when and some mild conditions hold on the dividing curves. It seems much more 2), to understand the situation when the Euler class is not extremal. dicult surface bundles over circles, 1. 3-manifolds with Haken decompositions, and 2. application 11 to surface bundles is obvious: given a tight contact The on a surface bundle, split it along a convex ber; the result is [0; 1], and structure dividing curves (or characteristic foliation) on the boundary components are the by the monodromy the bundle. Now if one could normalize the dividing related on a surface ber, then one could apply the solution to 11 to obtain curves upper bound on the number tight contact structures on the surface bundle. an get a lower bound, one could use surgery and \state transition," see To [52]. Haken decomposition a 3-manifold is a way successively cutting the manifold A along incompressible surfaces until one arrives at a disjoint union three-balls. decompositions have been very useful in studying manifolds that support Haken [40, 75]. Honda, Kazez, and Matic [54] have used these decompositions to them tight contact structures on manifolds with b 1 1: The hardest step in construct process is understanding the rst cut. This would be greatly facilitated by this 338 JOHN B. ETNYRE AND LENHARD L. NG 2. Uniqueness and classication Of course the main problem here is: 10. Classify tight contact structures on all 3-manifolds. Currently contact structures are classied on the following manifolds: The classication is also known on some Seifert bered spaces. For example, bered spaces. Seifert specic problem whose resolution would have great importance for the clas- A sication contact structures is: This problem has relevance to the classication tight contact structures on: 3. general 3-manifolds. answering 11; it might then be possible to generalize the results [54].

5 can be obtained by gluing two handlebodies together; such a representation 3-manifold a 3-manifold is called a Heegaard decomposition. These decompositions been very useful in studying 3-manifold topology (e.g., branched covers, Casson have invariants, : : : ). So to understand tight contact structures on a 3-manifold, handlebodies genus 1, this has been done [43, 50]. Using techniques from For it is not hard to give an upper bound on the number tight contact struc- [50], on a handlebody with xed dividing curves on its boundary, but an actual tures seems dicult. If one could solve this problem, then to understand classication structures on an arbitrary 3-manifold, one would have to understand how to tight together tight structures on the handlebodies. Honda has developed a method glue \state transition" for gluing tight contact structures together. trying to called this method, a solution to 11 would be helpful. implement 13. Which manifolds admit only nitely many tight contact structures? [9, 53], it has been shown that any closed oriented irreducible toroidal 3-manifold innitely many universally tight contact structures. Moreover, it has re- admits been announced, by Colin, Giroux, and Honda, that any closed oriented cently 3-manifold admits only nitely many tight contact structures (for this atoroidal it is important to keep in mind that 0 is a nite number!). light statement, results, the above question, as stated, has been answered; however, there are, these 14. Do all 3-manifolds admit only nitely many virtually overtwisted contact structures? [9, 53], the innity universally tight contact structures were constructed rst constructing one special universally tight contact structure, in which the by torus T has a neighborhood contactomorphic to (T ( 1; 1); 1 ), incompressible where n = ker(sin(2nz)dx + cos(2nz)dy); y are coordinates on T, and z is the coordinate on ( 1; 1). Then one ob- x; innitely many tight contact structures by replacing (T ( 1; 1); 1 ) with tains ( 1; 1); n ): tuitively, one is making the contact structure twist more near (T When doing this, one must prove that all the contact structures obtained are universally T: tight, and that they are dierent. To prove tightness, it is important that begin with a special tight contact structure. To conclude that there are innitely we many dierent tight contact structures, one needs to know that the \twisting T is nite." This \twisting" is measured in [9, 11] using an invariant called along torsion T (see below), and in [53] using the twisting various the transverse to T. any event, it is not clear that a similar construction can knots performed on virtually overtwisted contact structures. be torsion a torus T in a contact manifold (M; ) is the largest n for which The ( 1; 1); n ) can be contact embedded in M so that T f0g is isotopic to T ; (T PROBLEMS IN LOW DIMENSIONAL CONTACT TOPOLOGY 339 To study a general 3-manifold, one can use Heegaard decompositions. Any one might rst want to consider: 12. Classify tight contact structures on handlebodies. Short a complete classication, we can ask more specic questions. For example, we can start with the following motivational question. course, many interesting renements this question. Foremost among them is:

6 no such n can be embedded, the torsion is dened to be 0. We would like to if the following questions. answer 15. Are there only nitely many contact structures with minimal torsion? example, if we drop the hypothesis that the torus is normal in Colin's theorem For does an arbitrary tight have nite torsion? Note that for the inniteness [9], Colin and Honda-Kazez-Matic, which rely, more or less, on torsion theorems nite, the contact structures constructed are very special. We can also ask: being many specic examples (e.g., the tight contact structures on T 3 ), we know that the tight contact structures with minimal torsion are strongly llable, while only studying 3{manifolds one usually decomposes them in various ways. When since any orientable 3{manifold has a unique prime decomposition, one usu- First, restricts to irreducible manifolds. One may also do this when considering tight ally structures since for a connected sum M 1 #M 2, we know Tight (M 1 #M 2 ) = contact (M 1 ) Tight (M 2 ); see [8]. Next, a 3{manifold is usually decomposed along Tight tori. Specically, given an irreducible orientable 3{manifold M, incompressible is a collection incompressible tori T 1 ; : : : ; T k such that the components there n [T i are atoroidal or Seifert bered; moreover, this collection is unique if it is M with respect to this property. These tori give the Johannson-Shalen-Jaco minimal decomposition M: (JSJ) explicitly, suppose that T 1 ; : : : ; T k is a minimal system tori given by the JSJ More Can we classify contact structures on M if we have a classication decomposition. the pieces M n ([T i )? on related question is: A 20. Suppose two embeddings (T 2 I; i ) (M; ) are smoothly isotopic and maximal. Are they necessarily contact isotopic? vector eld v on a contact manifold (M; ) is called a contact vector eld A the ow v preserves : The contact structure is called convex if there is a if like contact vector eld on M: Recently Giroux has proved (using ideas gradient [42]) that any contact structure on a closed 3-manifold is convex. This gives from 21. How are properties like llability and virtual overtwistedness in convex vector elds associated to a given contact structure? reected 22. Does the dynamics the ow a contact vector eld say anything about the contact structure? 340 JOHN B. ETNYRE AND LENHARD L. NG 16. Is torsion always nite in a tight contact structure? 17. How are torsion and llability related? the rest are only weakly llable [22]. Is this always the case? Specically, we ask: 18. Can large torsion sometimes imply no lling? 19. Is there a contact JSJ decomposition? 3. Convex contact structures and open book decompositions us another tool to study contact structures. For example:

7 [45] has provided an interesting approach to 21. Recall that an Giroux book decomposition a closed 3-manifold M is a bration the complement open a link L; called the binding the open book. If F is the ber the bration M n L, then M n L is obtained from F [0; 1] by gluing F f1g to F f0g a dieomorphism : F! F called the monodromy. Following Thurston and by [76], one can nd a 1-form on F such that d is a volume form Winkelnkemper F: Then = dt+t+(1 t), t 2 [0; 1]; is a contact form on F [0; 1] that on to a contact form on M n L, and it is not hard to extend over M: The descends structure = ker is said to be adapted to the open book decomposition. contact has shown how to construct a contact vector eld on (M; ) that is transverse Giroux the bers the open book. Thus from an open book decomposition, one gets to adapted convex contact structure. Moreover, given a convex contact structure, an can construct an open book decomposition to which the contact structure Giroux adapted. Hence there is a one-to-one correspondence between convex contact is and open book decompositions. structures work Loi and Piergallini [62] (see also [2, 63, 66]), Giroux can Generalizing that a contact structure is Stein llable if and only if it is adapted to an open show decomposition whose monodromy is a product positive Dehn twists. Thus book 23. What conditions on the link or monodromy map an open decomposition imply that the adapted contact structure is tight? (Weakly or book M and a contact structure : Let B() be the set all open book decompositions Fix to which is adapted. Let g() = minfgenus(f )g, where the minimum is 24. Can g() be eectively computed for any class contact structures? If so, is it an eective invariant the contact structure? Is it related to 25. What is the minimum g() over all tight structures (or over universally tight structures, etc.) on M? all 26. Can the contact homology or symplectic eld theory [25] a structure be computed in terms an adapted open book decomposition? contact a contact structure can be reconstructed from an adapted open book decomposition, Since it is clear that contact homology can be computed from the open book We would like to see an eective algorithm that takes dynamical information data. the monodromy (and/or topological information about the binding) the about book decomposition and outputs the contact homology. open book decompositions have been used to study the topology 3-manifolds Open 27. Find purely topological applications Giroux's correspondence convex contact structures and open book decompositions. between answering this problem might seem overly optimistic, Giroux and Goodman Though have been able to answer some longstanding questions about bered links using [45] PROBLEMS IN LOW DIMENSIONAL CONTACT TOPOLOGY 341 we can rene 21 to: strongly) symplectically llable? Universally tight? Virtually overtwisted? taken over all bers F brations in B(): any properties? We also ask: for quite some time now. So we end this section with a very open ended problem. this machinery.

8 1. surgery on the lighter curves on the left can Figure a virtually overtwisted contact structure. The darker curve yield an immersed disk, as shown on the right, with respect to bounds its twisting number is 0. which is very little known concerning the group contactomorphisms a There contact structure. given 28. Given a contact structure on M; understand the group Di (M, contactomorphisms : ) 30. Study the inclusion map from Di (M; ) to Di + (M), the group orientation-preserving dieomorphisms. S 3 with the standard tight contact structure, this map is an isomorphism on For 0 level. What can one say about higher homotopy groups? The inclusion is the surjective on 0 for tight contact structures on T 3, since a preferred direction not \twisting" is picked out by a contact structure. We also know that for (T 3 ; n ), work in [17] should be very helpful here, but there are technical hypotheses Eliashberg's in most the theorems in [17] which make it dicult to address this directly. [16], it has been shown that 0 for an overtwisted contact problem contains at least two elements. order to address 31 using structure 32. Does there exist an overtwisted contact structure with two overtwisted disks which are not contact isotopic? that Eliashberg's theorem does not state that homotopy classes plane elds Note in one-to-one correspondence with overtwisted contact structures; this is only are [46] has shown that if a contact structure is obtained from Gompf on a knot as in Figure 1, and the immersed disk on the left surgery 1 lifts to an embedded disk in a nite cover, then the contact structure is Figure overtwisted. The reason is that there is a Hopf-like unknot bounding an virtually 342 JOHN B. ETNYRE AND LENHARD L. NG 4. Contactomorphisms Several interesting subproblems are: 29. Compute 0 (Di (M; )): the kernel the inclusion on 0 contains at least n elements. 31. Compute Di (M; ) when is overtwisted. [17], one might rst want to consider the following. true at the 0 level. 5. Miscellaneous immersed overtwisted disk that can be unwound in a cover.

9 33. Can we always nd such an immersed disk in a virtually overtwisted contact structure? the virtually overtwisted phenomenon could be suciently well understood, If could try to address the following question. one 34. Given a virtually overtwisted contact structure, is there some to nd an associated universally tight structure? procedure another way, are all contact structures obtained from universally tight contact Said via some surgery-type construction? structures we perform a Lutz twist on a contact structure, a tube overtwisted When is produced. This raises the natural question: disks 35. Can an overtwisted disk always be completed to an overtwisted tube? is unlikely, given the fact that P does not support a tight contact structure. This However, perhaps the putative overtwisted tube is only immersed, while each 36. If a contact structure has an overtwisted tube, is there an un- which yields a tight structure? Lutz-twist Knots II. refer to the article by Etnyre [31], elsewhere in this volume, for intro- Please denitions concerning and transverse knots. Unless otherwise ductory the ambient contact manifold is assumed to be (S 3 ; std ). Sections 6 through stated, deal with the problem classifying and transverse knots; Sections address applications these knots to contact geometry and topology. and K is a topological knot type in any contact 3-manifold, let L(K) denote the If knots type K. The main problem in knot theory set 37. Given K, classify the knots in L(K) up to isotopy. the case (S 3 ; std ), recall that there are two \classical invariants" knots, tb and r. A knot type is called simple if two this type with equal classical invariants must be isotopic. For knots simple knots, classication is concluded by calculating precisely which (tb; r) can be achieved. values the prime knot types which we can presently classify are All simple: the unknot in any tight contact structure [24] 1. torus knots in any tight contact structure [33] 2. the gure eight knot in (S 3 ; std ) [33]. 3. can also classify knots in a non-prime knot type if we know the We for the prime summands [35]; see Section 7. Thus, in our discussions classication PROBLEMS IN LOW DIMENSIONAL CONTACT TOPOLOGY 343 overtwisted disk is actually embedded. 6. Examples is the following. below, we will usually restrict attention to prime knot types. addition, for all

10 2. Stabilization a knot replaces a segment Figure the knot's front (yz) projection with a zigzag as shown. There above knot types, any knot which does not maximize tb in its knot the can be obtained from a maximal-tb knot through the operation stabilization, type adds zigzags to the front (yz) projection the knot; see Figure 2. (It is which to show that the stabilizations S are well-dened operations up straightforward isotopy.) The classication results [33, 35], and probably future to rely on the powerful theory convex surfaces developed by Giroux; classications, Etnyre's article [31]. see all knot types are simple, as we will see in Section 8. Not the 5 2 knot shown in Figure 4 (a few pages hence) is not instance, (Here we use the usual knot terminology from, e.g., [71].) simple. digress for a moment to discuss the situation transverse knots. Call a We type transversely simple if all transverse knots in this knot type are completely knot by their self-linking number. Using characteristic foliation techniques, determined [21] showed that the unknot is transversely simple; similarly, Etnyre Eliashberg and Etnyre-Honda [33] have shown that torus knots and the gure eight knot [29] transversely simple. Through a completely dierent approach, using braid are Birman and Wrinkle [5] have proved that all so-called exchange reducible foliations, are transversely simple; by work Menasco [64], these include all iterated knots knots. Birman and Menasco [4] have adapted dicult results in braid theory torus construct families knots which are not transversely simple. to results about transverse knots seem likely to arise from their connection Further knots. Given an oriented knot K, we obtain two with knots K + and K by pushing the knot o itself in a direction tangent transverse the contact eld so that the orientation induced on K + and K as transverse to agrees/disagrees with the orientation on K: (Note there is some discrepancy knots the literature as to the sign convention. The one given here seems natural in many perspectives, but the other choice is also frequently used, e.g. in [3].) from any transverse knot is transverse isotopic to a knot thus obtained. Conversely, if we denote by S the operations positive and negative stabilization on Now knots, then two transverse knots (K 1 ) + ; (K 2 ) + are transverse isotopic and only if the knots K 1 ; K 2 are stably isotopic, i.e., S n K 1 ; S n K 2 if isotopic for some n 0; see, e.g., [27, 33]. It follows that classifying transverse are up to transverse isotopy is equivalent to classifying knots up to knots isotopy. stable trying to attack the main problem, we would like to mention particularly important subproblems. some 344 JOHN B. ETNYRE AND LENHARD L. NG S + S are two dierent stabilizations S, which change r by 1. For 38. Classify knots in L(K), where

11 K is a bered knot; 1. K is a hyperbolic knot; 2. K is not simple; 3. K is any knot type in any overtwisted contact structure. 4. are some remarks about these subproblems. Here 38 (1): It is currently possible (though not likely) that prime bered are always simple (for non-prime knots see below). We note that knots only bered knots genus 1 are the gure eight knot and the right- and left- the trefoil knots, and knots in all these knot types have been handed The next step is to look at genus 2 bered knots. It is probable that the classied. in [33] can be used in this situation (at least sometimes), but it seems techniques some new ideas will be needed as the genus the knot grows. that 38 (2): We currently know very little about the classication hyperbolic knots; so far, we only understand gure eight knots. mentioned in Section 1, it would be very helpful to understand interactions between As hyperbolic and contact geometry, and some clues to this interaction could from understanding some more knots in hyperbolic knot types. come particular, it would be great to have a classication in a non-bered knot type. However, it might be easier to try knot types that are both hyperbolic and bered. W. Kazez has suggested looking at the knot 8 20 ; this is a hyperbolic hyperbolic knot whose holonomy seems to be interesting. bered 38 (3): There are currently no prime knot types K that are not Leg- simple for which we can classify L(K): The most obvious one to consider endrian the 5 2 knot type. is 39. There are exactly two knots realizing the 5 2 knot Conjecture with tb = 1 and r = 0: All other knots realizing 5 2 are stabilizations type 38 (4): It is a surprising fact that much progress has been made towards classifying knots in tight contact structures, but little is known knots in overtwisted contact structures. Call a knot about in an overtwisted contact structure loose if the contact structure on the com- K L is overtwisted. It seems that most knots in overtwisted plement structures are loose, but there are also non-loose knots [16]. If contact loose knots have the same knot type and tb and r, then they are two isotopic [16, 24]. Thus it would be interesting to know: 40. For what contact manifolds and knot types K are all in L(K) loose? knots example, let 0 be the contact structure on S 3 obtained from the tight contact For by applying one Lutz twist. structure 41. If is any overtwisted contact structure on S 3 besides 0, Conjecture all unknots are loose. then 42. Any unknot in 0 with tb 6= 1 or r 6= 0 is loose, Conjecture there are exactly two unknots with tb = 1 and r = 0, one loose, and a nal remark, we note that any knot that violates the Bennequin As inequality is automatically loose. PROBLEMS IN LOW DIMENSIONAL CONTACT TOPOLOGY 345 these, and are determined by their tb and r: one not.

12 prototypical structure result for knots concerns connected sums. A two knots L 1 2 L(K 1 ) and L 2 2 L(K 2 ), we can form their Given sum L 1 #L 2 in a standard way (see, e.g., [12]). [35] it is shown that connected sum knots gives a bijection connected L(K 2 ) L(K1 ) L(K 1 #K 2 );! the equivalence relation is two types: where (L 1 ; S (L 2 )) (S (L 1 ); L 2 ); 1. (L 1 ; L 2 ) (L 2 ; L 1 ) if K 1 = K 2 : 2. we can classify L(K 1 #K 2 ) in terms L(K 1 ) and L(K 2 ): Using this result Thus the classication negative torus knots, one may easily nd non- and bered knot types K for which we can completely classify L(K). -simple nothing is known about the structure L(K) for general knot types Virtually One important notion here is destabilizing a knot, i.e., isotoping K. to become a stabilization another knot, and then removing the it this raises tb by 1. stabilization; 43. Do all knots destabilize until they reach the maximal tb for their knot type? possible seems unlikely that the answer to this question is yes in general, but it is yes It all knot types for which the question has been answered [33]. 43 for important since the method used in all current classication results is to show is knots destabilize to maximal tb knots, and then to classify these. This 44. Is there a knot type with representatives that do not but have arbitrarily negative tb? destabilize a knot type K, let K (p;q) be the (p; q)-cable K; that is, a representative Give K (p;q) sits on the boundary a tubular neighborhood a representative K addition to connected sum and cabling, there are many other operations one perform on knots. can 46. What can one say about knots that are in a knot obtained from type Whitehead doubling or some other satellite construction? 1. Murasugi sums? 2. background on satellites, see, e.g., [68]. It is plausible that Whitehead For doubling two non-isotopic knots leads to non-isotopic if the original knots have the same tb, then the doubles have the same tb knots; r. Examples similar to the non-isotopic knots Figure 4 can be constructed and doubling unknots with the same tb but dierent r. by are some other interesting structural questions. Here 346 JOHN B. ETNYRE AND LENHARD L. NG 7. Structure Results scheme would have real diculties if the answer to the following question were yes. as a (p; q) curve. The techniques in [33] should allow one to address the following. 45. Classify knots in L(K (p;q) ) in terms L(K): 3. your other favorite topological abuse knots?

13 3. Basic move preserving tb and topological class, but not Figure isotopy class. necessarily 47. How are the knots in a knot type aected by crossing the knot type? changes example, is there a formula for how the maximal value tb changes under For changes? crossing 48. If K 1 and K 2 are topologically isotopic with the same tb, is a set local moves which, along with isotopy, will send K 1 to there basic move suggested by L. Ng is given in Figure 3. This move seems to suce A many examples. However, there are examples in which this is not sucient, as in S n1 + S n2 K 1 and S n1 + S n2 both positive and negative stabilizations.) By considering knots need are topologically connected sums negative torus knots, one sees that the that 49. Are there bounds on the number stabilizations necessary to two knots isotopic, in terms the topology the knot type? make rst example a knot type which is not simple was produced The by Chekanov [12] and Eliashberg and Her [23]. As mentioned independently the knot type is 5 2 ; the two non-isotopic 5 2 knots with equal before, invariants are shown in Figure 4. classical method used to distinguish between the two 5 2 knots is a new, powerful The knots, motivated by the development contact homology invariant which applies Gromov's holomorphic-curve techniques to contact manifolds. [23], invariant takes the form a dierential graded algebra (DGA), a free non- This unital algebra with a grading and dierential; the homology the commutative is precisely the contact homology S 3 relative to the knot (see algebra [36]). contact homology is dicult to compute in general, this relative Although homology has a simple combinatorial formulation due to Chekanov [12]; contact addition, it can be treated completely combinatorially, without reference to the in theory contact homology. Chekanov's denition allows us to write down general DGA invariant explicitly, given the knot diagram a knot. Two the knots are then isotopic only if they have equivalent DGAs, under an equivalence called \stable tame isomorphism." The formulation the algebraic PROBLEMS IN LOW DIMENSIONAL CONTACT TOPOLOGY 347 K 2? can show using the characteristic algebra (see Section 8). one result Fuchs and Tabachnikov [39] states that two topologically iso- A topic knots K 1 and K 2, with identical classical invariants, must have K 2 isotopic for some n 1 ; n 2 0. (Note that we number necessary stabilizations can be arbitrarily large [35], but one can ask: 8. invariants and contact homology DGA has subsequently been rened in [36, 65, 67].

14 4. Example, due to Chekanov and Eliashberg-Her, Figure knots, both topological type 5 2 and with tb = 1 two practice, because the diculty manipulating the DGA directly, applications have used simpler invariants derived from the DGA. The easiest are polynomials which calculate the graded dimension linearizations Poincare-type the DGA. Chekanov used these polynomials to show that the knots in Figure 4 not isotopic. Along the same lines, it can be shown [12, 27] that are exist knot types and tb and r for which there are arbitrarily many non-isotopic there representatives. more involved DGA-derived invariant is the characteristic algebra [67]. This A be used to answer (in the negative) the mirror question Fuchs can Tabachnikov [39], which asks whether a knot with r = 0 must nec- and be isotopic to its mirror, its image under the contactomorphism essarily y; z) 7! (x; y; z). Further applications the characteristic algebra are given (x; [67], including knots distinguished without using any sort grading. It seems in that the characteristic algebra is also linked to topology. possible 50 (Ng). The abelianized characteristic algebra is a topological Conjecture More precisely, it depends only on K and tb. invariant. 51. Fully understand the space DGAs modulo stable tame isomorphism. particular, are there other useful invariants which can be derived 52. If two DGAs have the same homology, are they stable tame isomorphic? and Pushkar [13] have discovered another non-classical invariant Chekanov knots, using admissible decompositions fronts. No applications this are currently known, besides examples in which the Poincare-polynomial invariant can already be used. fact, the following seems possible; see [38] for a technique 348 JOHN B. ETNYRE AND LENHARD L. NG and r = 0, which are not isotopic. from the DGA? Along these lines, it would be particularly nice to answer: result in this direction.

15 53. The Chekanov-Pushkar invariant can be deduced from Conjecture polynomials for the corresponding DGA. Poincare the DGA theory can be derived for contact manifolds other than Analogues 3 ; std ). (S 54. Construct combinatorial relative contact homology theories for knots in other contact manifolds. has been done for the solid torus S 1 R 2 with the standard tight contact This by Ng and Traynor, and, with more diculty, for circle bundles over structure orientable surfaces with a contact structure transverse to the bers [73]. closed note that for S 1 R 2, Traynor [77] has dened another non-classical invariant We a particular class links, using generating functions and algebraic topology; for for these links, all current calculations indicate that this invariant mysteriously, with a link-graded Poincare polynomial derived from the DGA. coincides now return to (S 3 ; std ). One major remaining diculty with the DGA We 55. Are there useful contact-homology-type invariants for stabilizations? problem is particular importance in trying to understand transverse knots; This Section 6. see possible approach to understanding invariants for stabilizations is to use One constructions, which in many cases produce knots with nontrivial DGAs. satellite simplest possible satellite, the Whitehead double, yields no interesting invariants The for stabilizations [68], but more complicated satellites may encode useful information. possible approach is to apply the symplectic eld theory Eliashberg- Another 56. Combinatorially understand the full symplectic eld theory for knots. algebraic structure behind absolute symplectic eld theory is somewhat well The [25], but this is far from the case for the relative symplectic eld theory understood for the case. Here, the algebraic formalism has resisted several needed at a reasonable denition. attempts a knot K in (M; ), there is a natural (tb(k) 1) surgery Given K, called surgery, which produces another contact manifold; see on 57. Let K be a knot inside (S 3 ; std ), and let (M; ) be Stein llable contact structure obtained by performing surgery on K. the there a way to calculate the contact homology (M; ) from the DGA invariant Is to K? associated nding such a relation would be great, it seems unlikely that one can Though the contact homology (M; ) merely from the DGA K; however, one determine almost certainly compute the contact homology (M; ) from the information can would dene the symplectic eld theory K. Thus it is probably more which to try to solve 56 rst before attacking this problem. As a possibly realistic rst step, we ask the following. simpler PROBLEMS IN LOW DIMENSIONAL CONTACT TOPOLOGY 349 invariant is that it vanishes for stabilized knots. Givental-Her [25], which generalizes contact homology, to knots. [18, 46, 78] for background.

16 58. If (M; ) in 57 is obtained from a stabilized does the contact homology (M; ) have a simple form? (Is is almost trivial? knot, now turn from intrinsic properties knots to applications We knots to contact geometry. Bennequin [3] famously established the an exotic contact R 3 by studying transverse unknots, and existence have played a similar role in distinguishing between contact structures on the knots [56] and homology spheres [1]. each case, one investigates what sorts 3-torus (or transverse) knots can be carried by the contact structure. 60. Can tight contact structures 1 and 2 on the same manifold M be distinguished by nding a knot a certain tb (or twisting always among curves isotopic to x = y = const [56]. More generally, tight contact number on T 2 bundles over S 1 are determined by their homotopy class plane structures sphere (p; q; pqn 1), obtained by surgery on distinct knots. Brieskorn Seiberg-Witten theory, the contact structures are distinguished by the Chern Via Stein 4-manifolds which bound (p; q; pqn 1). Could one use classes theory instead Seiberg-Witten theory to distinguish these examples? knot known [18] that surgery on a knot in a llable contact structure is another llable structure. produces we allow manifolds with boundary, the answer is no [52] (see also [10]). Nothing If known for the closed case. is 62. Consider two knots K 1, K 2 with the same tb and which are not isotopic. Let (M; 1 ) and (M; 2 ) be the contact manifolds ob- r by performing surgery on K 1 and K 2, respectively. Are 1 and tained 2 necessarily isotopic? might like to begin working on this problem by considering the maximal tb One the 5 2 knot. examples 63. K be a maximal tb knot in (S 3 ; std ). Is (S 3 n Let std j 3 S nk) universally tight? K; 64. K has non-maximal tb but is not a stabilization (cf. If is (S 3 n K; std j 3 S nk) universally tight? 43), 350 JOHN B. ETNYRE AND LENHARD L. NG Does it equal the base eld?) 9. Relation to contact geometry 59. Distinguish other contact structures using knots. A more precise version this problem might be given by the following question. number) and r which exists in one, but not in the other? example, the tight contact manifolds (T 3 = R 3 =Z 3 ; n ), n 2 Z +, given by the 1- For n = sin(2nz)dx + cos(2nz)dy, are distinguished by the maximal twisting forms and by information about knots [51]. eld would be interesting to see if the Lisca-Matic examples [60] contact struc- It tures can also be distinguished in this way. These are tight structures on the We next address the eect surgery on contact structures. It 61. Does surgery preserve tightness? Here are some nal questions.

17 it is possible for the answers these questions to be yes, that would be Though amazing. More realistically, we can ask: quite 65. If K is a maximal tb knot in (S 3 ; std ); is the Z- S 3 n K tight? cover these questions could be helpful in addressing 61, as well Understanding providing a connection to the topology the knot. as knot theory has deep connections with topology as well as with geometry, principally through the information encoded in the Thurston- contact denote the maximal tb over all knots type K. This number is tb(k) see the next paragraph. nite; pro the existence an exotic contact structure on S 3 reduces, Bennequin's modern terminology, to establishing that the unknot in (S 3 ; std ) satises tb = in His method extends to an upper bound on tb(k) for all K, in terms the 1. K. Since then, upper bounds for tb have been obtained in terms the genus genus, HOMFLY polynomial, and Kauman polynomial; see [37] for a survey slice results. particular, Rudolph [72] (q.v. [57, 60]) has shown that if tb(k) 0, K is not slice. then this vein, consider the smooth knot concordance group C; this is a direct sum countably many Z, Z=2Z, and possibly Z=4Zfactors. It is currently unknown there are any, or even countably many, Z=4Zfactors. One can see (at least whether ) the Z=2Zfactors from the fact that amphichiral knots have order 2. The some Rudolph allows us to see some the Zfactors as well, through result 66. Can we use knots to improve or reprove other results slice knots? concerning 67 (Etnyre). A knot is either order 2, or concordant to a knot Conjecture which either it or its mirror has tb 0: for 63, 64, and 65 are relevant to this problem. We also have the following s problem; see [14] for a start. related 69. Find relationships between knot invariants and the norm or Thurston norm. Alexander 70. Understand more deeply the relation between L(K) and tb(k) the one hand, and the HOMFLY and Kauman polynomials on the other. on to Fuchs-Tabachnikov, there are no nite-type invariants According knots besides tb, r, and standard topological nite-type invariants. Can one non-nite-type topological knot invariants by looking at L(K)? particular: extract PROBLEMS IN LOW DIMENSIONAL CONTACT TOPOLOGY Relation to topology Bennequin invariant a knot in (S 3 ; std ). For a knot type K, let knot theory; if tb(k) 0, then K is innite order in C. See, e.g., [7, 61]. One might be able to prove results such as the following. We can consider topological properties besides sliceness, as well. 68. Relate knot invariants to the Alexander module. It would also be illuminating to consider:

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