On the Khovanov Homology of 2- and 3-Strand Braid Links

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1 Advaces i Pue Mathematics, 06, 6, Published Olie May 06 i SciRes O the Khovaov Homology of - ad -Stad Baid Lis Abdul Rauf Nizami, Mobee Mui, Tawee Sohail, Ammaa Usma Divisio of Sciece ad Techology, Uivesity of Educatio, Lahoe, Paista Uivesity of Sciece ad Techology of Chia, Hefei, Chia Received 9 Jauay 06; accepted 8 May 06; published May 06 Copyight 06 by authos ad Scietific Reseach Publishig Ic This wo is licesed ude the Ceative Commos Attibutio Iteatioal Licese (CC BY) Abstact Although computig the Khovaov homology of lis is commo i liteatue, o geeal fomulae have bee give fo all of them We give the gaded Eule chaacteistic ad the Khovaov homology of the -stad baid li, α = Keywods, ad the -stad baid ( ) Khovaov Homology, Khovaov Bacet, Gaded Eule Chaacteistic, Baid Li, Joes Polyomial Itoductio Khovaov homology is a ivaiat fo oieted lis which was itoduced by Mihail Khovaov i 000 as a categoificatio of the Joes polyomial [], Khovaov assiged a bigaded chai comple C i j ( L ) to the oieted li diagam L whose diffeetial was gaded of bidegee (,0 ) ad whose homotopy type depeded oly o the isotopy class of L The bigaded homology goup i, j, H ( D ) of the chai comple C i j ( D ) povides a ivaiat of oieted lis, ow ow as Khovaov homology Although Khovaov s costuctio is combiatoial fom which Khovaov homology is algoithmically computable, we shall follow athe a simple way of Ba-Nata s, which he itoduced i [] to compute the Khovaov homology Lis ad Li Ivaiats A li i is a fiite collectio of disjoit cicles smoothly embedded i These cicles ae called the How to cite this pape: Nizami, AR, Mui, M, Sohail, T ad Usma, A (06) O the Khovaov Homology of - ad -Stad Baid Lis Advaces i Pue Mathematics, 6,

2 compoets of the li If a oietatio of the compoets is specified, we say that the li is oieted A li cosistig of oly oe compoet is called a ot Lis ae usually studied via pojectig them o the plae A pojectio with ifomatio of ove- ad udecossig is called a li diagam Some li diagams ae give i Figue Two lis ae called isotopic (o euivalet) if oe of them ca be tasfomed to aothe by a diffeomophism of the ambiet space Rema By a li we shall mea a diagam of its isotopy class Reidemeiste gave i [] a fudametal esult about the euivalece of two lis: Two Lis ae euivalet if ad oly if oe ca be tasfomed ito the othe by a fiite seuece of ambiet isotopies of the plae ad the local Reidemeiste moves give i Figue To classify lis oe eeds a li ivaiat [4], a fuctios I: Lis {umbes o polyomials o colous, etc} that gives oe value fo all lis i a isotopy class of lis ad gives diffeet values, but ot always, fo diffeet classes of lis To chec whethe a fuctio is a li ivaiat oe has to show that it is ivaiat ude all the Reidemeiste moves This pape is coceed with the li ivaiats: the Khovaov homology ad the Joes polyomial Baids oto itself Two isotopic ots ae give i Figue A -stad baid is a set of o-itesectig smooth paths coectig poits o a hoizotal plae to poits eactly below them o aothe hoizotal plae i a abitay ode [5] The smooth paths ae called stads of the baid A -stad baid is give i Figue 4 The poduct ab of two -stad baids is defied by puttig the baid a above the baid b ad the gluig thei commo ed poits A baid with oly oe cossig is called elemetay baid The ith elemetay baid i o stads is give i Figue 5 A useful popety of elemetay baids is that evey baid ca be witte as a poduct of elemetay baids Fo istace, the above -stad baid is ( )( )( i i i i ) = The closue of a baid b is the li ˆb obtaied by coectig the lowe eds of b with the coespodig uppe eds, as you ca see i Figue 6 A impotat esult by Aleade coectig ots ad baids is: Tivial -compoet li Hopf li Tefoil ot Figue Li diagams Figue Isotopic ots Figue Reidemeiste moves 48

3 Figue 4 A -stad baid Figue 5 Elemetay baid i Figue 6 Closue of a baid Theoem [6] Each li ca be epeseted as the closue of a baid 4 The Kauffma Bacet ad the Joes Polyomial I 985 V F R Joes evolutioized ot theoy by defiig the Joes polyomial as a ot ivaiat via Vo Neuma algebas [7] Howeve, i 987 L H Kauffma itoduced a state-sum model costuctio of the Joes polyomial that was puely combiatoial ad emaably simple [8] A Kauffma state s of a li L is obtaied by eplacig each cossig ( ) of L with the 0-smoothig o the -smoothig (so that the esult is a disjoit uio of cicles embedded i the plae) We deote by ( L) the set of all Kauffma states of L A smoothig of tefoil ot is give i Figue 7 Let s be a state i ( L), γ ( s) the umbe of cicles i the state, ad α ( s) ad β ( s) the umbes of cossigs i states 0 ad The the Kauffma bacet fo L is defied by the elatio s α β γ ( s) ( s ) ( ) ( s ) L = It is well ow that the Kauffma bacet satisfies the elatios: L = L + L 0 ( ) L = L This bacet is ot ivaiat ude the fist Reidemeiste move [9], see, fo istace, [4] To ovecome this difficulty, oe eeds somethig moe: Let us coside that the li diagam L is ow oieted The each cossig appeas eithe as, which is called the positive cossig o as, which is called the egative cossig If we deote the umbe of positive cossigs by + ad the umbe of egative cossigs by, the the uomalized Joes polyomial is defied by the elatio = 48

4 Figue 7 0- ad -smoothigs ( ) ( ) ˆ + J L = L () ad its omalized vesio by the elatio J( L) = J ˆ ( L) () + Sice this polyomial is ivaiat ude all thee Reidemeiste moves, it is a ivaiat fo oieted lis Eample It is easy to chec that the omalized Joes polyomial of the li : is J ( ) = O the Way to Khovaov Homology Defiitio A gaded vecto space W is a decompositio of W ito a diect sum of the fom W =, m IWm whee each { W m} is a homogeeous compoet with degee m of the gaded vecto space W Defiitio Let V ad W be two homogeeous compoets of gaded vecto spaces The degee of the teso poduct V W is the sum of the degees of V ad W Defiitio Let W = mwm be a gaded vecto space with homogeeous compoets { W m} The gaded dimesio of W is the powe seies m dimw : = dimwm Defiitio 4 The degee shift { } l that dimw { l} = dimw l of a gaded vecto space W Wm m = is defied by ( { }) W l : = Wm l, so Defiitio 5 Ba-Nata discoveed i [] that Khovaov s idea was to eplace the Kauffma bacet what he called the Khovaov bacet L, which is a chai compleample of gaded vecto spaces whose gaded Eule chaacteistic is L Liewise the Kauffma bacet, the Khovaov bacet is defied by the aioms: ad L = L = V L 0 0,, d toemoveumbeig (befoeeacheuatio) { } Hee V is a gaded vecto space with gaded dimesio + Defiitio 6 The chai compleample C of gaded vecto spaces of a piece C of that compleample) is defied as: d d + d + C C C = 0 0 The height shift opeatio [ s ] o the chai compleample C is defied: if C C[ s] C (whee the gadig is the height s =, the C = C Defiitio 7 The gaded Eule chaacteistics of a chai compleample is defied to be the alteatig sum of the gaded dimesios of its homology goups, ie ( C) ( ) χ : = dimh m 484

5 Theoem [] If the degee of the diffeetial is zeo ad if all the chai goups ae fiite dimesioal, χ C is also eual to the alteatig sum of the gaded dimesios of the chai goups, ie ( ) χ ( C) = ( ) Theoem [] The gaded Eule chaacteistic of ( ) L, ie χ ( ( )) ˆ C L = J( L) : dimc C L is eual to the uomalized Joes polyomial of Now we give the gaded Eule chaacteistic of Fist, some temiology: By the symbols L,,, +, ad we shall mea the oieted li diagam, the set of cossigs i L, the umbe of cossigs i L, the umbe of positive cossigs ad the umbe of egative cossigs i L, espectively Let V be the gaded vecto space with two basis elemets v ± whose degees ae ± espectively, so that dimv = + With evey veteample α of the cube { 0,} we associate the gaded vecto space Vα ( L) : = V { }, whee is the umbe of cycles i the smoothig of L coespodig to α ad is the height α = Σ iαi of α We the set the th chai goup L (fo 0 ) to be the diect sum of all the vecto spaces at height : L : = V : α ( L α = α ) Befoe computig the Khovaov homology, we defie two gadigs, the homological gadig ad the uatum gadig The homological gadig of the chai compleample is defied as g ( ) = c ( v), whee C( L) ad c ( v ) is the umbe of -smoothigs i the coodiates of V I case of chai compleample, the uatum gadig of the chai goups is ( ) = p( ) + g( ) + + ad is, ( ) = p( ) + g( ) + + i case of co-chai compleample Now owad we shall use the otatio Kh fo the Khovaov homology, whee the fist ieample idicates the homological gadig ad the secod ideample idicates the uatum gadig We eed these gadigs to compute the Joes polyomial fom the Khovaov homology Eample Hee is the Khovaov homology of : ) The -cube: The -cube of the tefoil ot is give i Figue 8 ) Khovaov Bacet: The Khovaov bacets alog with thei -dimesios ae give i Table ) Uomalized Joes polyomial: The gaded Eule chaacteistic of is 6 χ ( ) = ( + ( + + ( + ( + = + + () 4) Khovaov Homology: I ode to compute the Khovaov homology of, we multiply the uomalized +, whee i ou case is (, ) ( 0, ) + = ˆ 9 5 J = Joes polyomial with the facto ( ) ( ) ( ) The Khovaov Homology of the li is peseted i Table Figue 8 The -cube of the tefoil ot 485

6 Table Khovaov Bacets Khovaov Bacet -dimesio 0 = V dim 0 = ( + ) = V { } = V { } = V { } dim = ( + ) dim = ( + ) dim = ( + ) Table Homology of Homology degee, Kh 0 Gadig Rema χ ( ) is actually the uomalized Joes polyomial of 6 The Mai Theoem This sectio cotais the chai comple, Khovaov bacet, gaded Eule chaacteistic, ad Khovaov homology of the baid li Popositio 4 The chai comple of the li is V V V V V 0 ( ) Poof We poof it by iductio o, usig the tic that istead of, we use + ad that istead of + V The epasio holds obviously fo =, that is we use just fo fist tem i the epasio of ( ) Now, suppose that the esult holds fo =, that is ( ) + V = + V 0 V, Fo = +, we have + V = + V + + V + V 0 ( ) ( ) 486

7 Now, eplacig by ( ) ( + V ) = ( + V)( + V) + V ( ) = + V + V + V + + V + V 0 0 = + + V V + V ( + ) = + V + + V + V + ad by +, we eceive the desied esult Theoem 5 The gaded Eule chaacteistic of is ( ) ( ) ( ) ( χ = = ( + ) Poof The poof is simple; just by followig the defiitio Popositio 6 The uomalized Joes polyomial of is ˆ J( ) = ( ) + + +, ad the omalized is J = ( ) ( ) ( ) ( ) Poof Sice the uomalized Joes polyomial is the alteative sum of Khovaov bacets, we have ˆ J( ) = ( + ( + + ( + ( + ( ) ( ) ( ) ( = + + ( + ) + ( + + ) ( ) ( ) + + ( ) + ( ) +! ( )( ) ( ) ! ( ) ( ) ( ) ( )( ) ( ) 4 + ( ) !! 4 4 Now afte cacelatio of tems, which behave diffeetly fo eve ad odd, we eceive the desied esult Fo istace, see the cases fo = 5, 6: ˆ J( ) = ( + ( ) ( ) ( + + ( + ( ( ) 0( ) 0( ) ( ( = =

8 ˆ J( ) = ( + ( ) ( ) ( ) ( + ( + + ( ( ) 5( ) 0( ) ( ( ) ( = = Theoem 7 (Mai theoem) a) If is eve, the, ( ) if = Kh = 0 if = b) If is odd, the c) If +, the if =, Kh ( ) = if = 0 if, Kh ( ) = < if = 0 Poof We pove it usig the elatio ˆ ( ) ( ) + J = (4) ad establishig a table with the help of the uatum ad homological gadigs The homological gadig appeas i a ow ad uatum gadig appeas i a colum The homological gadigs eceive alteatig sigs, statig positive sig fom 0; a tem with egative sig appeas at a odd, while the positive sig appeas at a eve The powes of i the elatio epeset the uatum gadig Coespodig to each tem i the elatio, a space appeas i the table at the (, ) positio th a) I case of eve umbe of cossigs we eceive a -compoet li; hece, at th homological gadig, two spaces appea, oe at uatum gadig ad oe at uatum gadig ( + ) Please see Table fo the homology of, whee is eve b) Howeve, i odd umbe of cossig we always eceive a ot; this cofims that at highest homological gadig thee eists a space agaist the uatum gadig th Moeove, at ( + ) th uatum gadig oe space should appea with positive coefficiet i the Euatio (4) Thus, a space actually appeas at the positio ( +, ) The homology of, whee is odd, is give i Table 4 c) Sice at height 0 we eceive the space V V, at 0 th homological level thee eist two spaces, oe at ( ) th ad oe at th uatum gadigs This completes the poof Now we give the gaded Eule chaacteistic of the -stad baid α ( ) = ( factos); this seuece cotais the powes of Gaside elemet = : α ( ) = We will use Table 5, whee X is the caoical fom of α ( ) (ie the smallest wod i the legth-leicogaphic ode with < ) ad Y is a cojugate of X, suitable fo computatios The umbe of factos i each of the si Y is + Theoem χ = ) ( ) ) ( ) ) ( ) 4) ( ) χ = χ = χ =

9 Table Homology of, whee is eve ( ) ( ) ( ) + +, Kh 0 + Table 4 Homology of, whee is odd ( ) ( ) ( ) + +, Kh 0 + α = Table 5 Classificatio of the baid ( ) α ( ) X Y ) ( ) 6) ( ) χ = χ = Poof (4) Sice thee ae 6 + cossigs i the li, thee ae The Khovaov bacets alog with thei -dimesios ae give i Table 6 The esult ow follows usig the defiitio ad simplifyig the epessio See, fo eample, the case fo = The figue o the ight epeset the li of the educed fom of Δ, 6 + vetices i the smoothig cube which is 4 489

10 Table 6 Khovaov bacets ad -dimesios fo smoothigs of + Level Khovaov Bacet -dimesio 0 V ( + ) V 9 { } ( 6 + ) ( + ) { } V V 8 8 { } ( ( { } 4 V V 6 { } + ( ( ( + ) V + ( 4 + ) V ( + )( + + ( 4 + )( V { 6 + } ( ) Table 7 Khovaov bacet ad -dimesios fo smoothigs of Δ Level Khovaov bacet -dimesio 0 V ( + ) V 9 { } 9( + ) V { } V 8 8 { } 8 ( ( + 4 { } 4 V V 6 { } 6 ( ( { } { } V 4 V 4 V { 4} 4 ( ( ( { } 4 { } V 5 V 5 V { 5} 60 ( ( ( { } { } 5 { } V 6 V 6 V 6 V 8 54 { 6} 8 ( ( + + ( + + ( { } 4 { } V 7 V 7 V { 7} ( + + ( + + ( { } V 8 V 6 { 7} 6 ( + ) + ( V 9 { 9} ( + ) 4 Table 8 Homology of Δ Homological gadig, Kh Quatum gadig

11 Fo Khovaov bacets ad -dimesios fo smoothigs of (see Table 7) We ultimately eceive χ ( ) = The homology of is peseted i Table 8 The poofs of othe pats ae simila to the poof of Pat 4 Refeeces [] Khovaov, M (000) A Categoificatio of the Joes Polyomial Due Mathematical Joual,, [] Ba-Nata, D (00) O Khovaov s Categoificatio of the Joes Polyomial Algebaic ad Geometic Topology,, [] Reidemeiste, K (96) Elemetae begudug de otetheoie Abhadluge aus dem Mathematische Semia de Uivesität Hambug, 5, 4- [4] Matuov, V (004) Kot Theoy Chapma ad Hall/CRC, Boca Rato [5] Ati, E (947) Theoy of Baids Aals of Mathematics, 48,0-6 [6] Aleade, J (9) Topological Ivaiats of Kots ad Lis Tasactios of the Ameica Mathematical Society, 0, [7] Joes, V (985) A Polyomial Ivaiat fo Kots via Vo Neuma Algebas Bulleti of the Ameica Mathematical Society,, 0- [8] Kauffma, LH (987) State Models ad the Joes Polyomial Topology, 6, [9] Reidemeiste, K (948) Kot Theoy Chelsea Publ ad Co, New Yo 49