Knots and contact structures Vera Vértesi Contact structures Knots and contact structures Knots in contact 3manifolds Appendix Vera Vértesi [email protected] 2009 Thanks for P. Massot and S. Schönenberger for some of the pictures Knots and contact structures Thermodynamics E T S P V Vera Vértesi Contact structures origin denition tight vs. overtwisted internal energy temperature entropy Knots in contact 3manifolds pressure Appendix volume First Law of Thermodynamics (Q d E = πΏQ β πΏW = processed heat, W = work on its surroundings) πΏ Q = T d S , πΏ W = Pd V for a reversible process Since E, S and V d E = T d S β Pd V are thermodynamical functions of a state, the above is true non-reversible processes too. Thermodynamics geometric setup Dene the 1form: πΌ = d E β T d S + Pd V States of the gas are on integrals of ker πΌ. How many independent variables are there? = What is the maximal dimension of an integral submanifold? πΌ β§ (d πΌ)2 = d E β§ d T β§ d S β§ d P β§ d V (d πΌ = βd T β§ d S + d P β§ d V β= 0) β Max dimensional integral manifolds are 2 dimensional. Deduce state equations for . . . βΆ . . . ideal gases; βΆ . . . van der Waals gases. Knots and contact structures Vera Vértesi Contact structures origin denition tight vs. overtwisted Knots in contact 3manifolds Appendix Knots and contact structures Contact structures denition Vera Vértesi Contact structure (2n + 1)manifold M is a maximally nonintegrable hyperplane distribution π in the tangent space of M . on a Locally: π = kerπΌ β Ξ©1 (M )) β πΌ β§ (d πΌ)n β= 0 (πΌ maximally nonintegrable Standard contact structure 2n+1 = {(x , . . . , x , y , . . . , y , z )} On β 1 n 1 n πΌst = d z + πΌ is contact: (d πΌ = n β i =0 πst = ker πΌ xi d yi βn i =0 d xi β§ d yi ) πΌ β§ (d πΌ)n = 2n d z β§ d x1 β§ d y1 β§ β β β β§ β§d xn β§ d yn β= 0 Darboux's theorem Every contact manifold locally looks like (β2n+1 , πst ). Contact structures origin denition tight vs. overtwisted Knots in contact 3manifolds Appendix Contact structures origin Knots and contact structures Vera Vértesi Contact structures origin denition tight vs. overtwisted βΆ thermodynamics; Knots in contact 3manifolds Appendix βΆ odd dimensional counterparts of symplectic manifolds; βΆ classical mechanics, contact element (Sophus Lie, Elie Cartan, Darboux); βΆ Hamiltonian dynamics; βΆ geometric optics, wave propagation (Huygens, Hamilton, Jacobi); βΆ Natural boundaries of symplectic manifolds. Contact structures applications βΆ (Eliashberg) New proof for Cerf 's Theorem: Di(S 3 )/Di(D 4 ) = 0; HF detects the genus of a knot; βΆ (Ghiggini, Juhász, Ni) HFK detects bered knots; βΆ (KronheimerMrowka) First step to the Poincaré conjecture: Every nontrivial knot in Vera Vértesi Contact structures origin denition tight vs. overtwisted Knots in contact 3manifolds βΆ (AkbulutGompf ) Topological description of Stein domains; βΆ (OzsváthSzabó) Knots and contact structures S3 has property P; βΆ (KronheimerMrowka) Knots are determined by their complement; βΆ (KronheimerMrowka, OzsváthSzabó) The unknot, trefoil and the gure eight knot are determined by their surgery. Appendix Contact structures on 3manifolds Contact structure (n=1) on a 3manifold distribution π Y is a maximally nonintegrable plane in the tangent space of maximally nonintegrable curve tangent to π β it rotates (positively) along any Vera Vértesi Contact structures origin denition tight vs. overtwisted Knots in contact 3manifolds Appendix Standard contact structure on π = ker(d z + xd y ) = β¨ ββx , x ββz β ββy β© Y. Knots and contact structures 3 β z y x Darboux's theorem Contact structures locally look like (β3 , πst ). Examples πsym = dz + r d π = β¨ ββr , r 2 Knots and contact structures 2 β βz β β β© βπ Vera Vértesi Contact structures origin denition tight vs. overtwisted Knots in contact 3manifolds Appendix β πOT = ker(cos rdz + r sin rd π) = β¨ ββr , r sin r ββz β cos r βπ β© Knots and contact structures Equivalence of contact structures Vera Vértesi Contact isotopy Two contact structures are isotopic if one can be deformed to the other with an isotopy of the underlying space. with πsym Ξ¨0 = id and and πst (Ξ¨1 )β (π0 ) = π1 are isotopic: βΞ¨t : Y β Y , Contact structures origin denition tight vs. overtwisted Knots in contact 3manifolds Appendix Classication of contact structures Do all 3manifolds admit contact structures? Knots and contact structures Vera Vértesi Contact structures origin denition tight vs. overtwisted Knots in contact 3manifolds Appendix Classication of contact structures Do all 3manifolds admit contact structures? Yes (Martinet) several proof using dierent technics: βΆ (Martinet, 1971) surgery along transverse knots; βΆ (ThurstonWinkelnkemper, 1975) open books; βΆ (Gonzalo, 1978) branched cover; βΆ (DingGeigesStipsicz) surgery along Legendrian knots. How many contact structures does a 3manifold admit? Knots and contact structures Vera Vértesi Contact structures origin denition tight vs. overtwisted Knots in contact 3manifolds Appendix Classication of contact structures Do all 3manifolds admit contact structures? Yes (Martinet) several proof using dierent technics: βΆ (Martinet, 1971) surgery along transverse knots; βΆ (ThurstonWinkelnkemper, 1975) open books; βΆ (Gonzalo, 1978) branched cover; βΆ (DingGeigesStipsicz) surgery along Legendrian knots. How many contact structures does a 3manifold admit? β Lutz twist: Once a contact structure is found we can modify it in the neighborhood of an embedded torus. Knots and contact structures Vera Vértesi Contact structures origin denition tight vs. overtwisted Knots in contact 3manifolds Appendix Knots and contact structures Tight vs. overtwisted contact structures Vera Vértesi overtwisted disc D ,β Y to π such that D Contact structures origin denition tight vs. overtwisted is tangent Knots in contact 3manifolds πOT = ker(cos r d z + r sin r d π) β β© = β¨ ββr , r sin r ββz β cos r βπ Appendix ccccc1 overtwisted ( β overtwisted cccccccc c c c c c c π [[[[[[[[[[[ [[[[[[[[ - tight (not overtwisted) disc) Theorem (Eliashberg): {overtwisted β ctct structures}/isotopy β {2plane elds}/homotopy tight contact structures are interesting geometric meaning: boundaries of complex/symplectic 4manifolds Knots and contact structures Tight vs. overtwisted contact structures Vera Vértesi overtwisted disc D ,β Y such that D to π along β D Contact structures origin denition tight vs. overtwisted is tangent Knots in contact 3manifolds πOT = ker(cos r d z + r sin r d π) β β© = β¨ ββr , r sin r ββz β cos r βπ Appendix ccccc1 overtwisted ( β overtwisted cccccccc c c c c c c π [[[[[[[[[[[ [[[[[[[[ - tight (not overtwisted) disc) Theorem (Eliashberg): {overtwisted β ctct structures}/isotopy β {2plane elds}/homotopy tight contact structures are interesting geometric meaning: boundaries of complex/symplectic 4manifolds Classication of tight contact structures Do all 3manifolds admit tight contact structures? Knots and contact structures Vera Vértesi Contact structures origin denition tight vs. overtwisted Knots in contact 3manifolds Appendix Knots and contact structures Classication of tight contact structures Vera Vértesi Do all 3manifolds admit tight contact structures? βΆ (Eliashberg) S3 admits a unique tight contact structure: πst ; Contact structures origin denition tight vs. overtwisted Knots in contact 3manifolds Appendix Knots and contact structures Classication of tight contact structures Vera Vértesi Do all 3manifolds admit tight contact structures? βΆ (Eliashberg) S3 admits a unique tight contact structure: but: πst ; βΆ (EtnyreHonda)βΞ£(2, 3, 5), the Poincaré homology sphere with reverse orientation does not admit tight contact structure; βΆ (LiscaStipsicz) there exist tight contact structure. β many 3manifold with no Contact structures origin denition tight vs. overtwisted Knots in contact 3manifolds Appendix Knots and contact structures Classication of tight contact structures Vera Vértesi Do all 3manifolds admit tight contact structures? βΆ (Eliashberg) S3 admits a unique tight contact structure: but: πst ; βΆ (EtnyreHonda)βΞ£(2, 3, 5), the Poincaré homology sphere with reverse orientation does not admit tight contact structure; βΆ (LiscaStipsicz) there exist tight contact structure. β many 3manifold with no How many tight contact structures does a 3manifold admit? Contact structures origin denition tight vs. overtwisted Knots in contact 3manifolds Appendix Knots and contact structures Classication of tight contact structures Vera Vértesi Do all 3manifolds admit tight contact structures? βΆ (Eliashberg) S3 admits a unique tight contact structure: but: πst ; βΆ (EtnyreHonda)βΞ£(2, 3, 5), the Poincaré homology sphere with reverse orientation does not admit tight contact structure; βΆ (LiscaStipsicz) there exist tight contact structure. β many 3manifold with no How many tight contact structures does a 3manifold admit? Characterization done on: βΆ (Giroux, Honda) Lens spaces; βΆ (Honda) circle bundles over surfaces; βΆ (Ghiggini, GhigginiLiscaStipsicz, Wu, Massot) some Seifert bered 3manifolds. Contact structures origin denition tight vs. overtwisted Knots in contact 3manifolds Appendix Methods for classication How can we prove tightness? βΆ llability; βΆ Legendrian knots. Knots and contact structures Vera Vértesi Contact structures origin denition tight vs. overtwisted Knots in contact 3manifolds Appendix How can we distinguish contact structures? βΆ homotopical data; βΆ contact invariant from HF-homologies (Seifert bered 3manifolds); βΆ embedded surfaces; if we know the contact structure on the surface, then it is also known in a neighborhood of the surface How the contact structure on a surface can be encoded? βΆ characteristic foliation; βΆ on convex surfaces a multicurve is enough. βΆ embedded curves. Knots and contact structures Knots in contact 3manifolds Vera Vértesi Legendrian knot is a knot L whose tangents lie in the contact planes: or TL β π πΌ(TL) = 0 Contact structures Knots in contact 3manifolds invariants standard contact structure classication Legendrian simple knots ... Appendix We have already seen Legendrian knots: βΆ an oriented plane eld is a contact structure, if it rotates along Legendrian foliations.. βΆ the boundary of an OT disc is Legendrian. π is tangent to D along βD β π does not twist as we move along βD Knots and contact structures Classical Invariants Vera Vértesi Thurston-Bennequin number Contact structures Knots in contact 3manifolds invariants standard contact structure classication Legendrian simple knots ... tb (L) = lk(L, Lβ² ) where Lβ² is a push o Lβ² of L in the transverse direction; If Ξ£ (i.e. a Seifert surface of βΞ£ = L), Appendix L then: tb Ξ£ (L) = lk(L, Lβ² ) = #(L β© Ξ£) Note: D is OT β tb D = 0 Jump back to the proof Rotation number rot (L) is a relative Euler number of π on Ξ£ w.r.t. TL Knots in (S 3 , πst ) Recall: π = ker(d z + xd y ) dz β= β TK β π ββ x = β dy Knots and contact structures Vera Vértesi Contact structures Knots in contact 3manifolds invariants standard contact structure classication Legendrian simple knots ... Appendix Claim: Any knot can be put in Legendrian position. Proof: Knots in (S 3 , πst ) Recall: π = ker(d z + xd y ) dz β= β TK β π ββ x = β dy Knots and contact structures Vera Vértesi Contact structures Knots in contact 3manifolds invariants standard contact structure classication Legendrian simple knots ... Appendix Claim: Any knot can be put in Legendrian position. Proof: Legendrian isotopy Legendrian isotopy Isotopy through Legendrian knots. Legendrian Reidemeister moves Legendrian isotopic knots are related by the following moves: Knots and contact structures Vera Vértesi Contact structures Knots in contact 3manifolds invariants standard contact structure classication Legendrian simple knots ... Appendix Knots and contact structures Are they Legendrian isotopic? Vera Vértesi Legendrian unknots A B Contact structures C Which ones are Legendrian isotopic? Remember: D Knots in contact 3manifolds invariants standard contact structure classication Legendrian simple knots ... Appendix Knots and contact structures Are they Legendrian isotopic? Vera Vértesi Legendrian unknots A B Contact structures C Which ones are Legendrian isotopic? Remember: AβΌ =B β D Knots in contact 3manifolds invariants standard contact structure classication Legendrian simple knots ... Appendix Knots and contact structures Are they Legendrian isotopic? Vera Vértesi Legendrian unknots A B Contact structures C Which ones are Legendrian isotopic? Remember: AβΌ =B β and CβΌ = D: D Knots in contact 3manifolds invariants standard contact structure classication Legendrian simple knots ... Appendix Classical invariants in (S 3 , πst ) Knots and contact structures Vera Vértesi Contact structures Thurston-Bennequin invariant: ) β #( tb(L) = (#( )) β #( ) rotation number: rot(L) = #( ) β #( )) Appendix for the Legendrian unknots: tb(A) = β1 tb(B) = β1 rot(A) = 0 rot(B) = 0 tb(C) = β2 rot(C) = β2 Knots in contact 3manifolds invariants standard contact structure classication Legendrian simple knots ... tb(D) = β2 rot(D) = β2 The two denition agree in (S 3 , πst = dz + xdy ) Knots and contact structures Vera Vértesi Contact structures The new denition tb(L) = (#( =3-0-2=1 ) β #( )) β #( ) Knots in contact 3manifolds invariants standard contact structure classication Legendrian simple knots ... Appendix The two denition agree in (S 3 , πst = dz + xdy ) Knots and contact structures Vera Vértesi Contact structures The new denition ) β #( tb(L) = (#( )) β #( =3-0-2=1 The old denition Appendix tb Ξ£ (L) = lk(L, L ) = #(L β© Ξ£) β² where Lβ² is a push o in the transverse direction and Ξ£ is a Seifert surface of ) Knots in contact 3manifolds invariants standard contact structure classication Legendrian simple knots ... L. The two denition agree in (S 3 , πst = dz + xdy ) ) β #( )) β #( =3-0-2=1 The old denition tb Ξ£ (L) = lk(L, Lβ² ) = #(L β© Ξ£) where Lβ² Ξ£ is a Seifert surface of β is a transverse direction: βz ) Knots in contact 3manifolds invariants standard contact structure classication Legendrian simple knots ... Appendix is a push o in the transverse direction and Vera Vértesi Contact structures The new denition tb(L) = (#( Knots and contact structures L. β βz The two denition agree in (S 3 , πst = dz + xdy ) ) β #( )) β #( =3-0-2=1 The old denition tb Ξ£ (L) = lk(L, Lβ² ) = #(L β© Ξ£) where Lβ² Ξ£ is a Seifert surface of β β z is a transverse direction: tb = 1 + 1 β 1 = 1 ) Knots in contact 3manifolds invariants standard contact structure classication Legendrian simple knots ... Appendix is a push o in the transverse direction and Vera Vértesi Contact structures The new denition tb(L) = (#( Knots and contact structures L. β The two denitions agree in general. Knots and contact structures A new smooth knot invariant tb Vera Vértesi can be decreased: Contact structures + Knots in contact 3manifolds invariants standard contact structure classication Legendrian simple knots ... tb (L± ) = tb (L) β 1 rot (L± ) = rot (L) ± 1 β Denition: K Appendix is a smooth knot type, then: tb (K ) = max{tb (L) : L is Legendrian repr. of K } Bennequin inequality: Ξ£ is a (genus g) Seifert surface for a Legendrian knot L, then: tb (L) + β£rot (L)β£ β€ βπ(Ξ£) (S , πst ) 3 is tight. L is the unknot then tb (L) + β£rot (L)β£ β€ βπ(D ) = β1, tb (L) β€ β1. So tb (L) β= 0 If thus tb distinguishes mirrors. Right and left-handed-trefoils Knots and contact structures Vera Vértesi Contact structures Knots in contact 3manifolds invariants standard contact structure classication Legendrian simple knots ... Appendix Bounds for the Thurston Bennequin number The Seifert surface of the trefoil is a punctured torus, thus tb (L) + β£rot (L)β£ β€ βπ(Ξ£) = 1 the right handed trefoil realizes this bound (tb = 1): but the left handed trefoil does not. (tb = β6) Knots and contact structures Classication of Legendrian unknots Legendrian isotopy β Vera Vértesi βΆ smoothly isotopy; βΆ rot βΆ tb Contact structures =; =. We have seen: tb (L) β€ β1 ··· tb = β1 rot = 0 tb = β2 rot = β2 Knots in contact 3manifolds invariants standard contact structure classication Legendrian simple knots ... Appendix tb = β2 rot = 2 Theorem (Eliashberg-Fraser): For any pair {(t , r ) : t + β£r β£ β€ β1 & r β‘ t mod 2} l k there is exactly one Legendrian unknot with tb = t and rot = r Proof on the Algebraic Geometry and Dierential Topology Seminar this Friday Knots and contact structures Classication of Legendrian knots Vera Vértesi For the unknot we had: tb = & rot Denition: = β A knot type is called Legendrian isotopic Legendrian simple if Contact structures tb and rot classify its Legendrian representations. Legendrian simple knots: βΆ (Eliashberg-Fraser) unknot; βΆ (Etnyre-Honda) torus knots, gure eight knot; βΆ ... Chekanov's example: First example for a Legendrian nonsimple knot: the 52 knot is enough to Knots in contact 3manifolds invariants standard contact structure classication Legendrian simple knots ... Appendix Some nonsimple knot types Checkanov's example: Knots and contact structures Vera Vértesi Contact structures Knots in contact 3manifolds invariants standard contact structure classication Legendrian simple knots ... Appendix tb = 1 and rot = 0 Other nonsimple knot types: βΆ (EpsteinFuchsMeyer) twist knots; βΆ Ng βΆ OzsváthSzabóThurston Knots and contact structures Further classication results Vera Vértesi Contact structures Classication of nonsimple knot types Knots in contact 3manifolds invariants standard contact structure classication Legendrian simple knots ... βΆ (EtnyreHonda) Can classify Legendrian realizations of K1 #K2 in terms of the classication of the Legendrian realizations of L1 βΆ (EtnyreHonda) and L2 (2, 3)-cable of the (2, 3) torus-knot; Appendix βΆ (EtnyreNg-V) Classication of Legendrian twist knots (work in progress). Legendrian Twist knots with maximal tb βΆ (Chekanov, EpsteinFuchsMeyer): βΌn are known to be dierent βΆ (EtnyreNg-V) There are exactly β ( (2n +1) 2 2 +1)2 β dierent Legendrian representations (work in progress). k l Knots and contact structures Vera Vértesi Contact structures Thanks for your attention! Knots in contact 3manifolds invariants standard contact structure classication Legendrian simple knots ... Appendix Equivalent characterizations of contact structures (Y , π) is contact i locally: Knots in contact 3manifolds π βΆ π = ker πΌ, βΆ π rotates (positively) along a Legendrian foliation; βΆ π π rotates (positively) along any Legendrian foliation; βΆ βΆ π is isotopic to is totally nonintegrable; is isotopic to πΌ β Ξ© 1 (Y ) (β3 , πst ); (β3 , πsym ); Jump back to 3-dimensional contact structures Vera Vértesi Contact structures βΆ where Knots and contact structures and πΌ β§ d πΌ β= 0; Appendix Knots and contact structures Property P Vera Vértesi Surgery cut out a tubular neighborhood of glue back along a dieomorphism such a map is determined by The surgery is then called Contact structures K Knots in contact 3manifolds K π : T2 β T2 Appendix π(π) = p πβ² + q πβ² p -surgery q µ Ξ» Property P K has Property P if surgery along K cannot give a counterexample for the Poincaré Conjecture. Fact (Lickorish, Wallace) Any 3manifold can be obtained from S 3 by surgery along a link. Jump back to the applications Knots and contact structures Lutz twist Vera Vértesi Contact structures Knots in contact 3manifolds Lutz twist Appendix We can change π π along a knot T: is standard along on T β π: π(T ) π π π = ker(cos( r )d t +r sin( r )d π) 2 Change π on 2 π(T ) π β² = ker(cos(πβ to: 3π 2 r )d t + r sin(πβ Jump back to the classication 3π 2 r )d π) Poincaré homology sphere from the dodecahedron: Knots and contact structures Vera Vértesi Contact structures Glue each pair of opposite faces of the dodecahedron by using Knots in contact 3manifolds the minimal clockwise twist. Appendix a factor of SO (3) with the rotational symmetries of the dodecahedron (A5 ) Jump back to the applications Convex surfaces Knots and contact structures Vera Vértesi Contact structures Knots in contact 3manifolds Appendix
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