INVARIANTS OF TOPOLOGICAL AND LEGENDRIAN LINKS IN LENS SPACES WITH A UNIVERSALLY TIGHT CONTACT STRUCTURE. Christopher R. Cornwell

Transcription

1 INVARIANTS OF TOPOLOGICAL AND LEGENDRIAN LINKS IN LENS SPACES WITH A UNIVERSALLY TIGHT CONTACT STRUCTURE By Christpher R. Crnwell A DISSERTATION Submitted t Michigan State University in partial fulfillment f the requirements fr the degree f DOCTOR OF PHILOSOPHY Mathematics 2011

2 ABSTRACT INVARIANTS OF TOPOLOGICAL AND LEGENDRIAN LINKS IN LENS SPACES WITH A UNIVERSALLY TIGHT CONTACT STRUCTURE By Christpher R. Crnwell In this thesis a HOMFLY plynmial is fund fr knts and links in a lens space L(p, q). Further study f this plynmial invariant finds a relatinship with the classical invariants f Legendrian and transverse links, when L(p, q) is endwed with a universally tight cntact structure. In fact certain criteria are fund which, if satisfied by any numerical invariant f links in L(p, q), guarantee that the invariant fits int a Bennequin type inequality. A linear functin f the degree f the HOMFLY plynmial is then shwn t satisfy these criteria. A crllary is that certain simple Legendrian and transverse realizatins f knts admitting grid number ne diagrams maimize the classical invariants in their knt type. In rder t btain the abve results, frmulae are fund fr cmputing the classical invariants f Legendrian and transverse links frm a tridal frnt prjectin. Having these frmulae, and knwn results abut fibered links that supprt a given cntact structure, it is fund whether the duals f sme families f Berge knts supprt the universally tight cntact structure.

3 ACKNOWLEDGMENT I first epress my gratitude t my dissertatin advisr, Effie Kalfagianni, fr her part in my prgress thrugh the mathematics PhD prgram at Michigan State University. Her guidance and supprt have been an invaluable resurce t me in my studies and my develpment as a mathematician. I d like t thank Matt Hedden fr many enlightening cnversatins n many tpics, and I want t thank Ken Baker fr an illustrated cnversatin which related t sme f the results in this thesis. Als, I thank Lenny Ng and Dan Rutherfrd fr cnversatins abut my thesis and related prblems; in particular, Dan pinted ut hw the prf f Therem implied Crllary I d als like t thank my fellw graduate students at MSU. I ve learned a great deal frm many f yu. I appreciate the atmsphere f mathematical learning, as well the wnderful friendship, that we ve been able t enjy during my years here. Finally, I want t thank my wife, Whitney. Yur unfailing lve, encuragement, and supprt have nt nly helped me succeed, but have made me happy. iii

4 TABLE OF CONTENTS List f Figures v 1 Intrductin Summary f results Definitins and Backgrund Skein Plynmials Cntact structures n 3-dimensinal maniflds Review f knwn results Pwer series invariants in Q-hmlgy 3-spheres Bennequin type inequalities Links and grid diagrams in L(p, q) Hmtpies and the HOMFLY plynmial Hmlgy classes Skein Thery The HOMFLY plynmial in L(p, q) Trivial Links in L(p, q) Every link is hmtpic t a trivial link The plynmial invariant Classical invariants f Q-nullhmlgus Legendrian and transverse links Definitins Frmulae fr tb Q (K), rt Q (K), and sl Q (K) frm a grid prjectin in L(p, q) Bennequin-type bunds in L(p, q) Eamples and Cmputatins Fibered knts and the universally tight cntact structure Backgrund Surgeries n fibered knts, cntact structures and Berge knts Cmputatins iv

5 LIST OF FIGURES 2.1 Prjectins in a skein relatin The frnt prjectin f a Legendrian figure 8 knt Psitive and negative stabilizatin f K (A) shws a grid diagram (with grid number 2) in L(7, 2) n a fundamental dmain f T. In (B) we alter the fundamental dmain. (C) is the straightening f (B). Fr interpretatin f the references t clr in this and all ther figures, the reader is referred t the electrnic versin f this dissertatin A grid diagram fr L(5, 3) with crrespnding grid prjectin Stabilizatin f type X: NW (tp) and f type O: SW (bttm) A nn-interleaving cmmutatin in L(7, 2) A mve which is neither an interleaving nr nn-interleaving cmmutatin D: the lift t S 3 f the grid diagram D in L(5, 1) A crssing change n a grid prjectin : crssing change; 2 3: rw cmmutatin; 3 4: clumn cmmutatin; 4 5: tw rw cmmutatins; 5 6: destabilizatin; 6 7: many clumn cmmutatins; 7 8: destabilizatin Psitive and negative skein crssings Reslutin at a psitive skein crssing Reslutin at a negative skein crssing The interleaving regins f p, a pint in the image f H : C (D) R v

6 3.7 The grid number 1 knt K 3 in L(7, 2) The trivial link diagram D(I) in L(5, 2) with I = (0, 1, 2, 0, 3) Decreasing Grid Number Decreasing Grid Number: Case 2a Decreasing Grid Number: Case 2b The interleaving f unaltered clumns in D The interleaving f altered clumns in D with a cmmn gd chice f α The interleaving f altered clumns in D withut a cmmn gd chice f α Cnstructing a ratinal Seifert surface Hw tb changes with a disk slide On an X : NE destabilizatin c d (P ) increases A skein crssing change and reslutin at a psitive skein crssing A skein crssing change and reslutin at a negative skein crssing Eample: A link B in L(5, 1) and its skein tree Eample: A link L in L(5, 1) and its skein tree Grid diagram assciated t the link Ln Berge knts n a Seifert surface f the trefil Berge knts n a Seifert surface f the figure 8 knt In surface (A) the embedded curve btained frm the weighted edges as indicated is already a negative braid. In surface (B), this is nt the case. By istpy f the surface, we get the tp-left surface in Figure 5.4, where it shws hw t get a negative braid clsure vi

7 5.4 The istpy f Σ abve starts at the tp-left figure and ends at the bttmright figure. The istpy frm the tp-right figure t the bttm-left figure is a Reidemeister I mve f ne f the bands in the clsure, which mve cancels the twisting. Frm the bttm-left figure t the bttm-middle figure, ne perfrms a finger mve thrugh the full twist vii

8 Chapter 1 Intrductin In this thesis we study knts and links in lens spaces thrugh the tl f tridal grid diagrams. Grid diagrams f knts and links in S 3 have garnered recent attentin, due t applicatins in Legendrian knt thery and knt Fler hmlgy. Their cunterparts that describe links in a lens space are f similar interest in this setting. In [5] tridal grid diagrams were cnsidered in rder t cnstruct cmbinatrial knt Fler hmlgy in lens spaces. Later, in [4], the crrespndence between Legendrian links in universally tight cntact lens spaces and tridal grid diagrams was develped. The cnstructin in [5] is f particular relevance t an pen prblem knwn as the Berge Cnjecture, which psits a cmplete descriptin f knts in S 3 that admit Dehn surgery resulting in a lens space. In fact, the cnjecture is nly still pen fr integer surgeries, and thus can be refrmulated in terms f which knts in a given lens space yield an integer surgery that gives S 3. In this setting, the Berge Cnjecture states that such a knt must admit a tridal grid diagram with grid number 1. In the wrk f Baker and Grigsby [4], that indicates the crrespndence between Legen- 1

9 drian links in universally tight cntact lens spaces and tridal grid diagrams, a relatinship is fund between the Thurstn Bennequin number f a knt, that f its mirrr, and the grid number f the knt. This relatinship suggests that finding bunds fr the Thurstn Bennequin number f a knt might be useful in cntrlling the grid number. Our wrk here uses skein plynmials (specifically the HOMFLY plynmial) t find such a bund n the Thurstn Bennequin number. In fact, previus t ur wrk, the eistence f a HOMFLY plynmial fr links in lens spaces was an pen prblem. In the wrk belw we reslve this prblem, and tridal grid diagrams are fundamental t the prf f the eistence f the HOMFLY plynmial. In additin, we develp a skein thery f tridal grid diagrams. We then give sufficient criteria fr an invariant f links in lens spaces t bund frm abve the self-linking number f a transverse link in the universally tight cntact structure. Equivalently, this means that the invariant bunds the sum f the Thurstn-Bennequin number and abslute value f the rtatin number, f a Legendrian representative f the link. We then derive an invariant frm the HOMFLY plynmial that satisfies these criteria. 1.1 Summary f results Thrughut what fllws we let p, q be cprime integers with 0 q < p. The lens space which results as p q surgery n the unknt in S3 is dented by L(p, q) (nte that this ecludes the case S 1 S 2 ). Fr each L(p, q) we cnstruct a cllectin f links called trivial links. There is eactly ne trivial link in each hmtpy class f links. We prve the fllwing. Therem Let L be the set f istpy classes f riented links in L(p, q) and let T L L dente the set f istpy classes f trivial links. Define T L T L t be 2

10 thse trivial links with n null-hmtpic cmpnents. Let U be the istpy class f the standard unknt, a lcal knt in L(p, q) that bunds an embedded disk. Suppse we are given a value Jp,q(τ) Z[a ±1, z ±1 ] fr every τ T L. Then there is a unique map Jp,q : L Z[a ±1, z ±1 ] such that (i) Jp,q satisfies the skein relatin a p Jp,q(L+) a p Jp,q(L ) = zjp,q(l 0 ). (ii) Jp,q(U) = a p+1. (iii) Jp,q (U L) = a p a p z Jp,q(L). As usual, the links L+, L, and L 0 differ nly in a small neighbrhd. The eact cnstructin f these links in L(p, q) is made clear in the subsequent tet. Remark In a large class f Q-hmlgy spheres, Kalfagianni has fund a pwer series valued invariant f framed links that satisfies the Kauffman skein relatin [32]. The ideas in the prf f Therem shuld als be capable f shwing that this invariant prvides a Kauffman plynmial fr links in L(p, q). We als give criteria fr a Q-valued invariant f links in L(p, q) t bund the classical invariants in (L(p, q), ξ UT ), where ξ UT is a universally tight cntact structure n L(p, q) defined by the pushfrward f the standard cntact structure. Our result is a lens space analgue f a therem f Lenny Ng [41]. T a given trivial link τ let T (τ) be a particular transverse representative f τ defined belw, in Remark Therem Let i be a Q-valued invariant f riented links in L(p, q) such that i(l+) + 1 ma ( i(l ) 1, i(l 0 ) ) and i(l ) 1 ma ( i(l+) + 1, i(l 0 ) ), 3

11 where L+, L, and L 0 are riented links that differ as in the skein relatin. If sl Q (T (τ)) i(τ) fr every trivial link τ in L(p, q), then fr every link L in L(p, q), sl Q (L t ) i(l), where L t is any transverse representative in (L(p, q), ξ UT ) f L. Mrever, if L l is a Legendrian representative f L then tb Q (L l ) + rt Q (L l ) i(l). In rder t prve Therem we prvide eplicit frmulae t calculate the invariants tb Q, rt Q, and sl Q frm a prjectin f the link t a Heegaard trus. These frmulas are in the spirit f thse used t cmpute the classical invariants in (S 3, ξ st ) frm a frnt prjectin (see [16]). Mrever, cmbining the frmula fr tb Q with a result f Baker and Grigsby [4], we find a very shrt prf f a result f Fintushel and Stern [18], that if integral surgery n a knt in L(p, q) yields S 3, then ±q is a quadratic residue md p. Given a link K in L(p, q), the plynmial Jp,q(K) is a tw-variable plynmial in variables a and z, and its definitin depends n a nrmalizatin n the set f trivial links. The fllwing is a crllary f Therem Crllary Let Jp,q dente the HOMFLY plynmial invariant in L(p, q), nrmalized s that if τ is a trivial link with n nullhmtpic cmpnents, r is the unknt, then Jp,q(τ) = a p sl Q (T (τ))+1. Given an riented link L in L(p, q), set e(l) t be the lwest 4

12 degree in a f Jp,q(L). If L t is a transverse representative f L in (L(p, q), ξ UT ), then sl Q (L t ) e(l) 1. p The Franks-Williams-Mrtn (FWM) inequality is a celebrated result that relates the braid inde and algebraic crssing number f a braid t the HOMFLY plynmial f its clsure (which is a link in S 3 ). We pint ut that Crllary is a generalizatin f this result t lens spaces. Fr mre details, see the discussin f the FWM inequality belw. Finally, we pinpint the maimum self-linking and maimum Thurstn-Bennequin numbers f trivial knts as a crllary f Crllary Crllary If τ is a trivial link in L(p, q), then T (τ) has maimal self-linking number amng all transverse representatives f τ. If τ is a trivial knt, then the Legendrian knt assciated t its grid number ne diagram has maimal Thurstn-Bennequin number. The thesis is rganized as fllws: In Chapter 2 we give definitins and backgrund that are fundamental t the study f plynmial invariants, and f Legendrian and transverse links. We als review the wrk f Kalfagianni and Lin that paves the way fr Therem We end the chapter with a discussin f tridal grid diagrams and their crrespndence t Legendrian links. In Chapter 3 we cnsider hmtpies f links in L(p, q) and develp a skein thery f tridal grid diagrams. Using these tls, we prve the eistence f the HOMFLY-PT link plynmial fr links in L(p, q). In Chapter 4 we review cnstructins f [7],[4] that etend classical invariants f Legendrian and transverse knts t the ratinally null-hmlgus setting. We then give frmulae fr the Thurstn-Bennequin, rtatin, and 5

13 self-linking numbers frm data given by a prjectin adapted t a grid diagram. In that same chapter we prve Therem and Crllaries and We end the chapter with a cmputatin, giving a sequence f Legendrian knts and links in (L(5, 1), ξ UT ) n which the FWM inequality is sharp and arbitrarily negative. Finally, in Chapter 5 we discuss fibered knts in certain lens spaces L(p, q) that are the cres f surgery n tw families f Berge knts. In particular, we shw that many f these knts d nt supprt the universally tight cntact structure n L(p, q). 6

14 Chapter 2 Definitins and Backgrund 2.1 Skein Plynmials Fr ur study f riented knts and links in a lens space it is requisite that we cnsider n-singular links. Let M be an riented 3-dimensinal manifld, and let S be a disjint unin f riented circles. A piecewise-linear (r alternatively, smth) map K : S M is an n-singular link if it has eactly n transverse duble pints, and is an embedding away frm these pints. We ften write K fr the image K(S). Tw n-singular links K 1, K 2 are equivalent if there is an ambient istpy M I M, taking K 1 t K 2, such that the duble pints remain transverse thrughut the istpy. We smetimes say that K 1 and K 2 are smthly istpic, t distinguish such istpy classes frm mre restrictive ntins f equivalence, such as Legendrian and transverse istpy (defined belw). Nte that a 0-singular link is a link in M, and is a knt if the dmain S is cnnected. Much f classical knt thery (the study f knts and links in S 3 ) benefits frm the ability t discretize istpies. That is, tw links K 1, K 2 in S 3 are istpic if and nly if any 7

15 regular prjectin f K 1 t the equatrial plane can be taken t a regular prjectin f K 2 via a finite sequence f three types f lcal changes t the prjectin, knwn as the three Reidemeister mves. Given any n-singular link K, give S the structure f a simplicial cmple s that, fr each duble pint in K, there are distinct 1-simplees σ 1, σ 2 S with the interir f σ i cntaining ne f the pints in K 1 (). We may assume that K(σ 1 ) K(σ 2 ) is cntained in an embedded disk D M with K( σ 1 ) K( σ 2 ) D. Mrever, since σ i inherits an rientatin frm that f S, we may refer t the initial and terminal pints f σ i. Let M be a duble pint f K. Subdividing the simplicial structure n S if necessary, there is a small ball neighbrhd B M f such that D is prperly embedded in B and K B = K(σ 1 ) K(σ 2 ). Define a 1 and a 2 t be tw simple arcs in distinct cmpnents f B \ D, ging frm the initial pint t the terminal pint f K(σ 1 ). Define b 1 and b 2 t be simple arcs n D with b 1 ging frm the initial pint f K(σ 1 ) t the terminal pint f K(σ 2 ) and b 2 ging frm the initial pint f K(σ 2 ) t the terminal pint f K(σ 1 ). We define three (n 1)-singular links frm K in the fllwing manner: K+ = K(S \ σ 1 ) a 1 ; K = K(S \ σ 1 ) a 2 ; (2.1.1) K 0 = K(S \ (σ 1 σ 2 )) (b 1 b 2 ). Remark Nte that a different chice f a 1 and a 2 wuld interchange K+ and K. If M is riented, this ambiguity can be dealt with. Since is a transverse duble pint, cnsider the rdered pair f tangent vectrs K (σ 1 ), K (σ 2 ) at. There is a unique vectr nrmal t D at that cmpletes this rdered pair t an riented frame that agrees with the 8

16 rientatin n B at. Such a vectr pints int ne f the cmpnents f B \ D. Take a 1 t be in the same cmpnent. The first mtivatin fr discussing n-singular links is t define a skein triple f links. Definitin Let K 1, K 2, K 3 be any three links in M that are istpic t K+, K, and K 0 respectively, fr sme 1-singular link K. Then (K 1, K 2, K 3 ) is called an riented skein triple in M. Given a 3-manifld M, let L dente the set f istpy classes f (riented) links in M. A (Laurent) plynmial invariant f riented links in the variables 1,..., m is a well-defined map J : L Z[ ±1 1,..., ±1 m ]. A plynmial invariant f riented links J is called a skein plynmial if there is a hmgeneus degree 1 plynmial F (X, Y, Z) with cefficients in Z[ ±1 1,..., ±1 m ] such that fr any skein triple (K+, K, K 0 ), we have F (J(K+), J(K ), J(K 0 )) = 0. Skein plynmials in the setting f S 3 include the Aleander, Jnes, and HOMFLY-PT plynmials. We nte that the Aleander and Jnes plynmials are ne-variable specializatins f the tw-variable HOMFLY-PT plynmial. Furthermre, all mentined skein plynmials can be cmputed frm regular prjectins. The linear plynmial F (X, Y, Z) in the definitin f a skein plynmial is ften called the skein relatin fr that plynmial invariant. As ur fcus is n the HOMFLY-PT plynmial, we will use this skein relatin as an eample. Nte that the HOMFLY plynmial was intrduced in [20] (see als [50]), and is determined by its skein relatin, alng with a chice f nrmalizatin n the unknt. As bserved by Ocneanu, the HOMFLY plynmial can als be epressed using a trace f representatins int Hecke algebras (see als [28]). Let K+, K, K 0 be three links in S 3 which admit regular prjectins that are identical utside a neighbrhd f sme crssing, and that differ as in Figure 2.1 near the crssing. 9

17 L+ L L 0 Figure 2.1: Prjectins in a skein relatin Then (K+, K, K 0 ) is a skein triple. The HOMFLY-PT riented link plynmial J : L Z[v ±1, z ±1 ] satisfies the skein relatin v 1 J(K+) vj(k ) = zj(k 0 ), (2.1.2) (that is, F (X, Y, Z) = v 1 X vy zz) and is usually nrmalized s that J(unknt) = 1. In fact, using this nrmalizatin n the unknt, we can use inductin t see that any riented invariant f links satisfying (2.1.2) must be Laurent plynmial-valued. Indeed, cnsider a regular prjectin D f a link K in S 3 and let c(d) be the set f crssings f D. There is a subset f c(d) such that if we change which strand is ver-crssing at each crssing in this subset, we btain a prjectin f the standard unlink. Let u(d) be the number f crssings in the smallest such subset. Suppse D has a psitive crssing in this subset, as n the left in Figure 2.1 (the case where the subset has nly negative crssings being similar). Then we may cnsider K as K+ in a skein triple (K+, K, K 0 ). Further, c(d 0 ) < c(d+) and u(d ) < u(d+), where D = D+ and D, D 0 are diagrams f K, K 0 as shwn in Figure 2.1. Thus we may inductively assume that J(K ), J(K 0 ) are Laurent plynmials, nting that (2.1.2) implies J ( L ) = v 1 v J (L). z 10

18 Our prf that there eists a HOMFLY-PT plynmial fr links in a lens space uses a cmpleity functin argument, as we did in the previus paragraph. We define a cmpleity functin that behaves nicely with respect t skein triples. Then we deduce that an invariant f links J M fund by Kalfagiannin and Lin [33], which satisfies the HOMFLY-PT skein relatin, takes Laurent plynmial values in the setting f a lens space. By cnstructin, J M takes values in a ring f pwer series, but it was nt knwn t cnverge t a plynmial invariant. It is still an pen prblem whether J M is Laurent plynmial-valued when M is nt a lens space (see [31]). The cnstructin f J M, which we will return t in Sectin 2.3, uses Vassiliev (r finitetype) link invariants. Nte that, given any invariant f (n 1)-singular links (in particular, beginning with an hnest link invariant), we can define an invariant f n-singular links by setting F (L ) = F (L+) F (L ). Thus any invariant f n-singular links gives an invariant f m-singular links fr m > n. A link invariant is called a Vassiliev invariant f rder n if the derived invariant vanishes n any n + 1-singular link, but des nt n sme n-singular link. 2.2 Cntact structures n 3-dimensinal maniflds Cntact structures are a natural gemetric structure that may be studied n any dddimensinal manifld. We fcus n the setting f a 3-dimensinal manifld Y. In this case, a cntact structure ξ n Y is a rank 2 subbundle f T Y that des nt agree lcally (as subbundles) with the tangent bundle f any embedded surface in Y. If ξ is riented then there is a glbal 1-frm α satisfying α dα > 0 such that ξ = ker α. The pair (Y, ξ) is called a cntact 3-manifld. Tw cntact 3-maniflds (Y 1, ξ 1 ), (Y 2, ξ 2 ) are cntactmrphic 11

19 if there is a diffemrphism Y 1 Y 2 which carries ξ 1 t ξ 2. An embedded disk in (Y, ξ) is vertwisted if its tangent bundle restricted t its bundary agrees with ξ. A cntact structure ξ is vertwisted if (Y, ξ) cntains an vertwisted disk and ξ is tight therwise. The prperty f ξ being tight r vertwisted is clearly invariant under cntactmrphism. Finally given any map f 3-maniflds, ϕ : Ỹ Y, and a cntact structure ξ n Y, ne can take the pullback bundle ϕ ξ, and this will be a cntact structure n Ỹ. If Ỹ is the universal cver f Y and ϕ ξ is tight, then ξ is called universally tight. If K Y is a link and, fr sme cntact structure ξ n Y, K is tangent t ξ at each pint, then K is called a Legendrian link in (Y, ξ). Tw Legendrian links are Legendrian istpic if there is an istpy frm ne t the ther, thrugh Legendrian links. A link K in Y is called transverse if T K is transverse t ξ at each pint f the image f K, and tw such links are transversely istpic if there is an istpy thrugh transverse embeddings frm ne t the ther. Nte that (assuming ξ is riented) a transverse link cmes with a natural rientatin. A fundamental eample f a cntact 3-manifld is (R 3, ξ st ), where R 3 has crdinates (, y, z) and ξ st = ker(dz yd). Prjectin t the z-plane in (R 3, ξ st ) is called frnt prjectin, and the frnt prjectin f a Legendrian link has a particularly nice frm since the y-crdinate can be recvered frm the slpe in the z-plane f the frnt prjectin. As a cnsequence, a diagram f a link in the z-plane is the frnt prjectin f sme Legendrian link in (R 3, ξ st ) if and nly if the nly singular pints f the prjectin are semi-cubical cusps and transverse duble pints, the prjectin has n vertical tangencies, and at a duble pint the strand with mre negative slpe is the ver-crssing strand. An eample is given in Figure

20 Figure 2.2: The frnt prjectin f a Legendrian figure 8 knt In additin t the underlying smth istpy class f a Legendrian link K, there are tw invariants f Legendrian links called the classical invariants f K. These invariants are traditinally defined when K is nullhmlgus in Y. The first is the Thurstn-Bennequin number tb(k), which measures the framing induced frm the cntact planes, relative t the framing given by a Seifert surface fr K (a cnnected riented surface, prperly embedded in the link cmplement, with bundary equal t K). That is, cnsider the simple clsed curve K n the bundary f a regular neighbrhd f K which is determined by a nnzer sectin in the (real) line bundle ξ K ν, where ν is the nrmal bundle f K. Chse a Seifert surface Σ fr K. Then tb(k) is the algebraic intersectin Σ K. The secnd classical invariant is the rtatin number rt(k), which is defined as the winding number f T K after a trivializatin f ξ Σ, given a Seifert surface Σ fr K. We nte that rt(k) actually depends n the hmlgy class f Σ, but nt n the chsen trivializatin. Hwever, as this thesis will nly wrk in the setting f lens spaces, rt(k) will nly depend n K. In (R 3, ξ st ), ne can cmpute the Thurstn-Bennequin number and rtatin number f an riented Legendrian link frm its frnt prjectin in the fllwing way. Let P be a frnt prjectin f a Legendrian link K, let w(p ) be the writhe f P, and c(p ) the number f cusps in the prjectin. Then, using the fact that the vectr field / z is transverse t ξ st everywhere, ne can shw that tb(k) = w(p ) 1 2 c(p ). Als, given an rientatin, 13

21 let c d (P ) (resp. cu(p )) be the dwnward (resp. upward) riented cusps, then rt(k) = 1 2 (c d (P ) c u(p )). If K is the Legendrian knt with frnt prjectin shwn in Figure 2.2, then tb(k) = 3 and, given either rientatin, rt(k) = 0. Transverse links have ne classical invariant ther than the underlying smth link type. If K is a transverse link, this invariant is called the self-linking number f K and written sl(k). The self-linking number is defined as fllws: given a Seifert surface Σ fr K, since ne can trivialize ξ Σ, we can chse a nn-vanishing vectr field v in ξ Σ. Nrmalizing v determines a simple clsed curve K n the bundary f a regular neighbrhd f K, since K is transverse t ξ. Define sl(k) t be the linking f K with K, i.e. Σ K. Our study is that f links in a lens space, which f curse need nt be nullhmlgus. Hwever, the definitins given abve can be etended t the setting f a ratinally nullhmlgus link with the help f ratinal Seifert surfaces (defined belw), which are a generalizatin f a Seifert surface and are embedded in the eterir f the link. There is an peratin ne can perfrm n a Legendrian knt K in any cntact 3-manifld called psitive (resp. negative) stabilizatin n K that gives a different Legendrian knt dented S + (K) (resp. S (K)). T describe the peratin, we pint ut that a therem f Darbu states that given a pint p in a cntact 3-manifld (Y, ξ), there eists a neighbrhd U f p, such that there is a cntactmrphism ϕ : (U, ξ U, p) (R 3, ξ st, 0). If K is Legendrian in (Y, ξ) then suppse p is a pint n K. Suppse that Figure 2.3(a) is the frnt prjectin f ϕ(u K). Then S + (K) is defined by replacing U K with the image f Figure 2.3(b) under ϕ 1, and S (K) is defined by replacing U K with the image f Figure 2.3(c) under ϕ 1. The Legendrian istpy type f S ± (K) des nt depend n the chice f pint p r cntactmrphism ϕ. 14

22 (a) (b) (c) Figure 2.3: Psitive and negative stabilizatin f K 2.3 Review f knwn results In this sectin we review what is knwn abut the prblem f finding skein plynmials in maniflds ther than S 3, in additin t results that relate invariants f smth istpy f knts t the classical invariants f their Legendrian and transverse realizatins. We will als intrduce ne f the main tls fr ur study f links in L(p, q), tridal grid diagrams, and their crrespndence t Legendrian links in L(p, q) with a universally tight cntact structure Pwer series invariants in Q-hmlgy 3-spheres Original cnstructins f the HOMFLY plynmial have been difficult t reprduce in maniflds ther than S 3, as they rely n planar link prjectins. Hwever, it was shwn in [12] that the HOMFLY and Kauffman plynmials are generating functins fr particular sequences f Vassiliev (r finite-type) link invariants (see als [8],[9], and [34]). In [30] and [33], Kalfagianni and Lin cnsidered Vassiliev invariants f links in (certain) Q-hmlgy spheres in their effrt t find a HOMFLY plynmial. They succeeded in finding an invariant that takes values in a ring f pwer series. Kalfagianni and Lin begin by cnsidering hmtpies f links in Q-hmlgy spheres. In particular, they shw that any free hmtpy f a link in M can be perturbed slightly t a particularly nice frm called almst general psitin. Amng ther prperties, a hmtpy in almst general psitin nly fails t be istpy at finitely many mments, at which times ne 15

23 has a 1-singular link, and near these times ne passes frm ne reslutin f the 1-singular link t anther (e.g. L+ t L ). We described abve hw singular link invariants may be derived frm a link invariant. Kalfagianni and Lin give an integrability cnditin in [33], determining when a given 1-singular link invariant is derived frm a link invariant. Their therem placed certain restrictins n the ambient manifld. Recently Kalfagianni [32] was able t remve sme f these restrictins, and we cite her therem here, adapted t the setting f unframed links. Recall that an atridal 3-manifld M is ne such that π 2 (M) = 0 and M has n essential tri. Let L (1) dente the istpy classes f 1-singular links in M. Therem ([32]). Suppse that M is a Q-hmlgy sphere with π 2 (M) = 0, such that if H 1 (M) 0 then M is atridal. Als assume that R is a ring which is trsin free as an abelian grup and let f : L (1) R be an invariant f 1-singular links. Then there eists a link invariant F : L R s that f(l ) = F (L+) F (L ) fr any 1-singular link L if and nly if f( ) = 0 f(l +) f(l ) = f(l+ ) f(l ), where L ± and L± dente the fur 1-singular reslutins f any 2-singular link, and dentes a 1-singular link with a small lp near sme duble pint. Almst general psitin f hmtpies was an essential ingredient in prving Therem 16

24 One easily checks that sme f derived frm a link invariant F will satisfy the tw cnditins f the therem. Given a singular link invariant satisfying these cnditins, ne takes a hmtpy in almst general psitin frm a trivial link in T L t the given link and sums the values f the singular invariant at the finite singular mments in the hmtpy. T see this is well-defined ne must shw that the definitin is independent f the hmtpy used, r that a glbal integrability cnditin hlds. It is knwn [29] that the 2-variable HOMFLY plynmial fr links in S 3 is equivalent t a sequence f 1-variable Laurent plynmials {Jn(t)}, and that replacing t by e gives a pwer series invariant Jn() with cefficients that are Vassiliev invariants [8, 12]. Kalfagianni and Lin use these ideas in [33] t arrive at Therem belw. They define their invariant J M by defining inductively a sequence f Vassiliev invariants whse generating functin is a pwer series satisfying the HOMFLY-PT skein relatin. The i th cefficient f this pwer series is defined frm a singular link invariant, which they shw integrates t a link invariant by shwing it satisfies the cnditins f Therem The fllwing therem is then btained. Let U be the istpy class f a knt in M that is the standard unknt in a small ball neighbrhd f a pint f M. Fi T L, a cllectin f links in M such that there is eactly ne representative in T L fr each hmtpy class f links in M, and furthermre, if T L T L has k cmpnents that are hmtpically trivial then T L = L k U fr sme link L that has n hmtpically trivial cmpnents. Define R := C[[, y]] t be the ring f frmal pwer series in the variables and y ver C. Define v R by v := e y = 1 + y + y2 2 + y3 6 + and let z R be defined by z := e e = Therem ([33]). Let M be a Q-hmlgy 3-sphere with π 2 (M) = 0, such that if 17

25 H 1 (M) 0 then M is atridal. Let L be the set f istpy classes f links in M. Then, given values J M (T ) fr each T T L such that J M (T U) = v 1 v z J M (T ), there is a unique map J M : L R such that v 1 J M (L+) vj M (L ) = zj M (L 0 ). Given M as abve, it is unknwn whether J M can be made t take values that are Laurent plynmials. In this cnsideratin, Kalfagianni asked the fllwing questin [31] which encapsulates the primary difficulty f the prblem, and which we answer affirmatively when M is a lens space: Questin 1. Is there a chice f T L such that every link L M can be reduced t disjint unins f unlinks and elements in T L by a series f finitely many skein mves? Bennequin type inequalities Much effrt has gne int finding upper bunds fr the classical invariants in (S 3, ξ st ), where ξ st is the standard tight cntact structure n the 3-sphere, e.g. [10], [19, 37], [51], [52], [48], [49], [55], [40], [58]. Much f this wrk benefits frm the fact that the classical invariants f Legendrian/transverse knts in (S 3, ξ st ) can be cmputed easily frm a frnt prjectin. Less is knwn abut these invariants f Legendrian knts in ther cntact maniflds. A therem f Eliashberg [15] generalizes the Bennequin inequality fr null-hmlgus Legendrian knts in any 3-manifld with a tight cntact structure: Therem (Eliashberg-Bennequin inequality). Let ξ be a tight cntact structure n a 18

26 3-manifld, Y. If K is a null-hmlgus knt in Y and F is a Seifert surface fr K, then tb(k l ) + rtf (K l ) 2g(F ) 1 (2.3.1) fr any K l, a Legendrian representative f K. This bund can be imprved in sme settings. Lisca and Matic imprved the bund in the case that the cntact structure is Stein fillable [36], and this imprvement was etended t the setting f a tight cntact structure with nn-vanishing Seiberg-Witten cntact invariant by Mrwka and Rllin [39]. An analgus therem was prved by Wu [57] fr the Ozsváth- Szabó cntact invariant. These imprvements invlved replacing the Seifert genus in the Eliashberg-Bennequin inequality with the genus f a surface which is prperly embedded in a 4-manifld bunded by Y. As such bunds invlve the negative Euler characteristic f a surface with bundary K, they must be n less than -1. In [25], Hedden intrduced an integer τ ξ (K) that is defined via the filtratin n knt Fler hmlgy assciated t (Y, [F ], K), where [F ] is the hmlgy class f a Seifert surface fr K. He shwed that in the case that the Ozsváth-Szabó cntact invariant is nn-zer, the right side f (2.3.1) can be replaced by 2τ ξ (K) 1. With such a bund he shwed that fr any cntact manifld with nn-zer cntact invariant, there eist prime Legendrian knts with arbitrarily negative classical invariants. In anther directin, ne culd cnsider ratinally null-hmlgus knts in a cntact manifld (Y, ξ). In such a setting there is a ntin f ratinal Seifert surface and crrespnding classical invariants tb Q, rt Q, and sl Q (see Definitin belw). Baker and Etnyre [7] etend the Eliashberg-Bennequin inequality t this setting: 19

27 Therem Let (Y, ξ) be a cntact 3-manifld with ξ a tight cntact structure. Let K be a knt in Y with rder r > 0 in hmlgy and let Σ be a ratinal Seifert surface fr K. Then fr K t, a transverse representative f K, sl Q (K t ) 1 r χ(σ). Mrever, if K l is a Legendrian representative f K then tb Q (K l ) + rt Q (K l ) 1 r χ(σ). There is an inequality fund by Franks and Williams [19], and independently by Mrtn [37], that relates the inde and algebraic crssing number f a braid t a degree f the HOM- FLY plynmial f its clsure. Later, using the Bennequin s wrk, Fuchs and Tabachnikv reinterpreted the result in terms f the self-linking number f a transverse knt in (S 3, ξ st ) [21]. This inequality has cme t be knwn as the Franks-Williams-Mrtn (FWM) inequality. Mre precisely, we describe the FWM inequality as fllws. The HOMFLY plynmial J(K) is a plynmial invariant f links in the variables v, z such that if U S 3 is the unknt then J(U) = 1, and J satisfies v 1 J(K+) vj(k ) = zj(k 0 ), (2.3.2) where K+, K, and K 0 frm a skein triple. 20

28 Therem (Franks-Williams-Mrtn (FWM) inequality). Let e(k) dente the minimum degree f v in J(K). Then fr any transverse representative K t f K, sl(k t ) e(k) 1. Mrever, if K l is a Legendrian representative f K then tb(k l ) + rt(kl ) e(k) 1. As discussed in the intrductin, Therem gives a HOMFLY plynmial fr links in lens spaces, and Crllary etends the FWM inequality t universally tight cntact lens spaces Links and grid diagrams in L(p, q) In this sectin we review tridal grid diagrams in L(p, q), which are central t ur discussin and view f links in lens spaces, and were fully studied in [4] (see als [5]). We will als (briefly) review the crrespndence between Legendrian links in L(p, q) with a universally tight cntact structure and tridal grid diagrams. Let us begin with the definitin f that cntact structure. The manifld L(p, q), which is the result f p q surgery n the unknt, is als btained as a qutient f S 3 C 2 by the equivalence relatin (u 1, u 2 ) (ωpu 1, ωpu q 2πi 2 ), where ωp = e p. Let π : S 3 L(p, q) be the qutient map. Represent pints (u 1, u 2 ) f S 3 in plar crdinates, letting u i = (r i, θ i ). The kernel ξ st 21

29 f the 1-frm α = r 1 2dθ 1 + r2 2 dθ 2 is the unique (up t rientatin) tight cntact structure n S 3 [22]. The 1-frm α is cnstant alng any trus in S 3 determined by a fied r 1. Since such a trus is fied (nt pintwise) under the actin (u 1, u 2 ) (ωpu 1, ω q pu 2 ), the pushfrward ξ UT = π (ξ st ) is a well-defined, and is clearly a cntact structure n L(p, q). Since ξ UT pulls back t the standard cntact structure n S 3, which is tight, ξ UT is universally tight. The pints f L(p, q) can be identified with pints in a fundamental dmain f the cyclic actin n S 3. Thus, since r 2 is determined by r 1 in S 3, we can describe L(p, q) by L(p, q) = { [ (r 1, θ 1, θ 2 ) r 1 [0, 1], θ 1 [0, 2π), θ 2 0, 2π p )}. Analgus t the crrespndence between planar grid diagrams and Legendrian links in (S 3, ξ st ) ([42],[53]), we can define tridal grid diagrams in L(p, q) t get a crrespndence between grid diagrams and Legendrian links in (L(p, q), ξ UT ). T be precise we define grid diagrams in L(p, q) as fllws (cf. [4]). Definitin A grid diagram D with grid number n in L(p, q) is a set f data (T, α, β, O, X), where: T is the riented trus btained via the qutient f R 2 by the Z 2 lattice generated by (1, 0) and (0, 1). α = { α 0,..., α n 1 }, with αi the image f the line y = i n in T. Call the n annular cmpnents f T α the rws f the grid diagram. β = { } β 0,..., β n 1, with βi the image f the line y = p q ( pn i ) in T. Call the n annular cmpnents f T β the clumns f the grid diagram. 22

30 O = { O 0,..., O n 1 } is a set f n pints in T ( α β) such that n tw Oi s lie in the same rw r clumn. X = { X 0,..., X n 1 } is a set f n pints in T ( α β) such that n tw Xi s lie in the same rw r clumn. The cmpnents f T α β are called the fundamental parallelgrams f D and the pints O X are called the markings f D. Tw grid diagrams with crrespnding tri T 1, T 2 are cnsidered equivalent if there eists an rientatin-preserving diffemrphism T 1 T 2 respecting the markings (up t cyclic permutatin f their labels). Such a grid diagram has slanted β curves. Fr cnsideratins f bth cnvenience and aesthetics, we alter the fundamental dmain f T and straighten ur pictures s that the β curves are vertical. Figure 2.4 shws hw this straightening is accmplished. z α 0 fundamental parallelgram z α 1 (A) (B) z β 0 β 1 (C) z z z Figure 2.4: (A) shws a grid diagram (with grid number 2) in L(7, 2) n a fundamental dmain f T. In (B) we alter the fundamental dmain. (C) is the straightening f (B). Fr interpretatin f the references t clr in this and all ther figures, the reader is referred t the electrnic versin f this dissertatin. A link K L(p, q) is assciated t a grid diagram D in L(p, q) in the fllwing manner. 23

31 Let Σ be the trus in L(p, q) f cnstant radius r 1 = 1/ 2 which splits L(p, q) int tw slid tri V α and V β. Identify T with Σ such that the α-curves f D are negatively-riented meridians f V α and the β-curves are meridians f V β. Net cnnect each X t the O in its rw by an hrizntal riented arc (frm X t O) that is embedded in T and disjint frm α. Likewise, cnnect each O t the X in its clumn by a vertical riented arc embedded in T and disjint frm β. The unin f the 2n arcs makes a multicurve γ. Remve selfintersectins f γ by pushing the interirs f hrizntal arcs up int V α and the interirs f vertical arcs dwn int V β. We nte that in the assciatin f a link t a grid diagram in L(p, q) the authr used the ppsite cnventin as that adpted in ther places in the literature [4, 5, 45, 47]. The cnventin used in ther places in the literature was adpted t fit cnventins cming frm knt Fler hmlgy. Hwever, fr the purpses f this thesis, it is mre clear t use the apprach presented belw (as in [13]) as there is n reference t Fler hmlgy theries. Definitin Let K be a link assciated t a grid diagram D in L(p, q) with grid number n. Fr sme 0 < m < n, suppse D is a subcllectin f m rws and m clumns f D such that the 2m markings cntained in the rws f D are eactly the 2m markings cntained in the clumns f D. Then D is a grid diagram fr sme sublink f K. If this sublink has ne cmpnent then D is called a cmpnent f D. Remark N part f Definitin prhibits a marking in X and a marking in O frm being in the same fundamental parallelgram. T a grid diagram that has grid number ne (and s, nly ne marking in X and ne marking in O) and its tw markings in the same fundamental parallelgram, we assciate a knt in L(p, q) that is cntained in a small ball neighbrhd and bunds an embedded disk. 24

32 Remark Ecept fr the case described in Remark 2.3.8, we assume that each marking f D is the center pint f the fundamental parallelgram that cntains it. Let the straightened fundamental dmain f T have nrmalized crdinates {(θ 1, θ 2 ) θ 1 [0, p), θ 2 [0, 1)}, s that each O and X sharing the same clumn have the same θ 1 -crdinate md 1. Under the requirements f Remark 2.3.9, the prjectin f K t Σ (with vertical arcs crssing under hrizntal arcs) is called a grid prjectin assciated t D K (the authrs f [4] call this a rectilinear prjectin). Nte that K has an rientatin given by cnstructin and s the grid prjectin is als riented. Figure 2.5 shws an eample f a grid diagram with a crrespnding grid prjectin. Given a grid diagram in a lens space L(p, q) (with the identificatin f T t Σ), the basis f vectrs given by parallel translates f tangent vectrs t {α 0, β 0 } is cherently riented with the glbal frame {d/dθ 1, d/dθ 2 }. The analgue t frnt prjectins in this setting, called tridal frnt prjectin, prjects radially thrugh the tri V α and V β t Σ. A slight perturbatin f a grid prjectin gives a tridal frnt prjectin, which determines a Legendrian link in (L(p, q), ξ UT ) since the slpe dθ 2 /dθ 1 n the prjectin determines the radial crdinate r 1. The cusps f the frnt prjectin crrespnd t lwer-left and upper-right crners f the grid prjectin, and we call these crners the cusps f a grid prjectin Figure 2.5: A grid diagram fr L(5, 3) with crrespnding grid prjectin. 25

33 If D has grid number n then there are 2 2n different grid prjectins as there are tw chices f vertical arc fr each clumn, and tw chices f hrizntal arc fr each rw. In a given rw (resp. clumn), the difference in chice f hrizntal (resp. vertical) arc crrespnds t a Legendrian istpy acrss a meridinal disk f V α (resp. V β ) (see [4]). S the Legendrian istpy class f the link is independent f the chice f grid prjectin. In view f the crrespndence abve, Legendrian links in (L(p, q), ξ UT ) can be discussed via grid diagrams. There is a set f grid mves such that tw grid diagrams crrespnd t the same Legendrian link if and nly if there is a sequence f such grid mves taking ne grid diagram t the ther [4]. These mves cme in tw flavrs: grid (de)stabilizatins and cmmutatins. X O X O Figure 2.6: Stabilizatin f type X: NW (tp) and f type O: SW (bttm) Grid Stabilizatins and Destabilizatins: Grid stabilizatins increase the grid number by ne and shuld be thught f as adding a lcal kink t the knt. They are named with an X r O, depending n the type f marking at which stabilizatin ccurs, and with NW, NE, SW, r SE, depending n the psitining f the new markings. Figure 2.6 shws an X:NW stabilizatin and an O:SW stabilizatin. Destabilizatins are the inverse f a stabilizatin. Any (de)stabilizatin is a grid mve that preserves the istpy type. Hwever, the crrespndence between ur grid diagrams and tridal frnt prjectins is such that cusps crrespnd t upper-right and lwer-left crners f a grid prjectin. Only 26

34 (de)stabilizatins f types NW and SE preserve Legendrian istpy type Figure 2.7: A nn-interleaving cmmutatin in L(7, 2). Cmmutatins: A cmmutatin interchanges tw adjacent clumns (r rws) f the grid diagram. Let A be the annulus cnsisting f the tw adjacent clumns c 1, c 2 (resp. rws r 1, r 2 ) invlved in the cmmutatin. This annulus is sectined int pn segments f the n rws (resp. clumns) f the grid diagram. Let s 1, s 1 be the tw segments in A cntaining the markings f c 1 (resp. r 1 ). If the markings f c 2 (resp. r 2 ) are cntained in separate cmpnents f A s 1 s 1, the cmmutatin is called interleaving. If they are in the same cmpnent f A s 1 s 1 the cmmutatin is called nn-interleaving. We nte that in the literature a cmmutatin typically refers nly t what we call a nn-interleaving cmmutatin. We have etended the terminlgy t include the interleaving case. A nninterleaving cmmutatin f clumns (resp. rws) is a grid mve that preserves Legendrian istpy type [4]. An interleaving cmmutatin crrespnds t a crssing change (see Lemma 3.2.1). An eample f nn-interleaving cmmutatin is shwn in Figure 2.7. Nte that a cmmutatin (interleaving r nn-interleaving) des nt include a clumn echange f the type illustrated in Figure 2.8, where there is a rw cntaining markings f bth c 1 and c 2. We end the sectin with a cnstructin that plays an imprtant rle in what fllws. 27

35 Figure 2.8: A mve which is neither an interleaving nr nn-interleaving cmmutatin. Given a grid diagram D in L(p, q) define a grid diagram D f a link in S 3 as fllws: cut T alng α 0 t get an annulus A. The bundary f A is a disjint unin f tw cpies f α 0. Let ne f these cpies be α + 0 and the ther be α 0. Take p cpies f A, say A 0,..., A p 1 and glue α + 0 n A i t α 0 n A i+1(md p) fr i = 0,..., p 1 by the identity map. Nte that the trus cnstructed frm A 0,..., A p 1 cvers T under the cver π : S 3 L(p, q) and s the link assciated t D cvers the link assciated t D. Call D the lift f D t S 3. (An eample is shwn in Figure 2.9, where the grid diagram n the right is the lift f the grid diagram in L(5, 1) n the left. The link crrespnding t the lift is a Hpf link.) D D Figure 2.9: D: the lift t S 3 f the grid diagram D in L(5, 1). 28

36 Chapter 3 Hmtpies and the HOMFLY plynmial 3.1 Hmlgy classes Definitin Fr an riented link K in L(p, q), define µ(k) t be the hmlgy class f K in H 1 (L(p, q)). Remark Let K, K be riented links. We nte that µ(k) = µ(k ) if and nly if K and K are freely hmtpic, since the free hmtpy class f an riented knt in L(p, q) is determined by its hmlgy class. Let C be the cre f the handlebdy Vα in L(p, q) (dual t a meridinal disk f Vα). Then H 1 (L(p, q)) = Z/p is cyclically generated by γ = [C] H 1 (L(p, q)). We identify µ(k) with the multiple f γ it represents. Suppse D K = (T, α, β, O, X) is a grid diagram with assciated link K in L(p, q). Orient the α and β curves s that the algebraic intersectin α i β j is psitive n T, fr all pairs 29

37 i, j. Let R K be a grid prjectin fr D K with its given rientatin. We can cmpute µ(k) frm the grid diagram as fllws. Lemma Given D K and R K as abve, chse an α-curve α i in the grid diagram. Then µ(k) is equal (md p) t α i R K, the algebraic intersectin f α i with R K. Prf. Push the interirs f hrizntal arcs n R K slightly int Vα t get a knt K cntained in Vα. The winding number f K in Vα is clearly cunted by α i R K and K is istpic t K, s µ(k ) = µ(k). W W O X O X Figure 3.1: A crssing change n a grid prjectin. 3.2 Skein Thery We nw develp a skein thery f grid diagrams in L(p, q). Nte that grid diagrams have been discussed in the setting f singular links in S 3, and used t etend link Fler hmlgy t singular links [2]. What we describe here as grid diagrams that differ by a skein crssing 30

38 change are eactly the grid diagrams in [2] that crrespnd t the tw reslutins f the duble pint f a 1-singular link (see, fr eample, Figure 10 in [2]). If tw links L+ and L differ as described in (2.1.1), they are istpic t links with assciated grid diagrams differing by the grid mve in Figure 3.1. The net lemma refines this. In a figure f a grid diagram, slanted gray bars indicate sme number f clumns (and their markings) which may be between the clumns f interest. Lemma Let L, L be links crrespnding t grid diagrams that differ by an interleaving cmmutatin. Then L, L differ by a crssing change. Prf. We prve the statement fr rw cmmutatin. The case with clumns is similar. The prf is given by the sequence f grid mves detailed in Figure 3.2, where arrws indicate cmmutatins t be perfrmed, and a triple f markings that is t be destabilized is circled. Referring t Figure 3.2, there are 8 steps. Frm step 1 t 2 we perfrm a crssing change. Frm 2 t 5 we perfrm a number f nn-interleaving cmmutatins. Ging frm 5 t 6 is destabilizatin. Frm 6 t 7 invlves several cmmutatins. Each f these cmmutatins is nn-interleaving since the markings f the clumn being mved t the right are in adjacent rws. Finally we destabilize frm 7 t 8. Nte that all grid mves abve crrespnd t istpy f the link, ecept the first which, as in Figure 3.1, crrespnds t interchanging the tw reslutins f a singular link defined by (2.1.1). Lemma shws that if tw grid diagrams differ nly by the interleaving cmmutatin f tw adjacent clumns (r rws), then the crrespnding links are sme pair L+, L. Mtivated by this result, we make the fllwing definitin. 31