Knots, Skein Theory and q-series

Transcription

1 Lousaa State Uversty LSU Dgtal Coos LSU Doctoral Dssertatos Graduate School 205 Kots, Se Theory ad q-seres Mustafa Hajj Lousaa State Uversty ad Agrcultural ad Mechacal College, Follow ths ad addtoal wors at: Part of the Appled Matheatcs Coos Recoeded Ctato Hajj, Mustafa, "Kots, Se Theory ad q-seres" (205) LSU Doctoral Dssertatos Ths Dssertato s brought to you for free ad ope access by the Graduate School at LSU Dgtal Coos It has bee accepted for cluso LSU Doctoral Dssertatos by a authorzed graduate school edtor of LSU Dgtal Coos For ore forato, please cotact gcoste@lsuedu

2 KNOTS, SKEIN THEORY AND Q-SERIES A Dssertato Subtted to the Graduate Faculty of the Lousaa State Uversty ad Agrcultural ad Mechacal College partal fulfllet of the requreets for the degree of Doctor of Phlosophy The Departet of Matheatcs by Mustafa Hajj BS, Daascus Uversty, 2004 MS, Jorda Uversty for Scece ad Techology, 2008 August 205

3 Acowledgets I would le to express y grattude to Olver Dasbach for hs advce ad patece I a grateful to Pat Gler for ay ay dscussos, deas ad sght I a also thaful to y topology teachers at Lousaa State Uversty Da Cohe ad Neal Stoltfus I would le to tha the atheatcs departet at Lousaa State Uversty for provdg excellet study evroet Fally, I would le to tha y faly ad freds for ther ecourageet ad support

4 Table of Cotets Acowledgets Lst of Fgures Abstract v v Chapter : Itroducto Chapter 2: Bacgroud 4 2 Prelares 4 2 Alteratg ad Adequate Ls 5 22 Se Theory 6 22 Se Maps Quatu Sp Networs 23 The Colored Joes Polyoal 4 Chapter 3: The Colored Kauffa Se Relato ad the Head ad the Tal of the Colored Joes Polyoal 6 3 Itroducto 6 32 The Head ad the Tal of the Colored Joes Polyoal for Alteratg Ls 7 33 The Colored Kauffa Se Relato 8 34 The Ma Theore 23 Chapter 4: The Bubble Se Eleet ad the Tal of the Kot Itroducto A Recursve Forula for the Bubble Se Eleet 3 43 The Bubble Expaso Forula Applcatos The Theta Graph The Head ad the Tal of Alteratg Kots 44 Chapter 5: The Tal of a Quatu Sp Networ ad Roger-Raauja Type Idettes 55 5 Itroducto Bacgroud Roger Raauja type dettes Exstece of the Tal of a Adequate Se Eleet Coputg the Tal of a Quatu Sp Networ Va Local Se Relatos Tal Multplcato Structures o Quatu Sp Networs 80

5 56 Applcatos The Tal of the Colored Joes Polyoal The Tal of the Colored Joes Polyoal ad Adrews- Gordo Idettes 88 Refereces 92 Vta 95 v

6 Lst of Fgures 2 The three Redeester oves 4 22 A ad B soothgs 5 23 The ot 6 2, ts A-graph o the left ad ts B-graph o the rght 6 24 A eleet the odule of the ds ad ts appg to T 3,2,2, 2 25 The se eleet τ a,b,c the space T a,b,c 2 26 The eleet τ a,b,c 3 27 The evaluato of a quatu sp etwor as a eleet S(S 2 ) 3 28 The relatve se odule T a,b,c,d 4 29 A bass of the odule T a,b,c,d 4 3 The -colored A ad B soothgs colored B-state 2 33 Adequate ad adequate se eleets All arcs are colored A local pcture a adequate se eleet Local vew of the se eleet Υ (+) (s ) 26 4 The bubble se eleet B,, (, l) Correspodece betwee two bases the odule T,,, The se eleet Λ(,, ) The ot 4, ts A-graph (left) Its reduced A-graph (rght) Obtag S () B (D) fro a ot dagra D The ot The ot Γ The reduced B-graph for Γ The se eleet S () B (Γ) 46 5 A local pcture for S (+) B (D) 59 v

7 52 Expadg f (2) A crossgless atchg that appears the expaso of f (2) The graph Γ wth a trvalet graph τ 2,2, The product [Γ, Γ 2 ] 8 56 The graph Υ (left) ad the graph Ξ (rght) The product [Υ, Υ 2 ] 2 (left) ad the product [Ξ, Ξ 2 ] 3 (rght) The graph G The graph G,l The (2, f) torus ot 89 v

8 Abstract The tal of a sequece {P (q)} N of foral power seres Z[q ][[q]], f t exsts, s the foral power seres whose frst coeffcets agree up to a coo sg wth the frst coeffcets of P The colored Joes polyoal s l varat that assocates to every l S 3 a sequece of Lauret polyoals I the frst part of ths wor we study the tal of the ureduced colored Joes polyoal of alteratg ls usg the colored Kauffa se relato Ths gves a atural exteso of a result by Kauffa, Murasug, ad Thstlethwate regardg the hghest ad lowest coeffcets of Joes polyoal of alteratg ls Furtherore, we show that our approach gves a ew ad atural proof for the exstece of the tal of the colored Joes polyoal of alteratg ls I the secod part of ths wor, we study the tal of a sequece of adssble trvalet graphs wth edges colored or 2 Ths ca be cosdered as a geeralzato of the study of the tal of the colored Joes polyoal We use local se relatos to uderstad ad copute the tal of these graphs Furtherore, we cosder certa se eleets the Kauffa bracet se odule of the ds wth ared pots o the boudary ad we use these eleets to copute the tal quatu sp etwors We also gve product structures for the tal of such trvalet graphs As a applcato of our wor, we show that our se theoretc techques aturally lead to a proof for the Adrews-Gordo dettes for the two varable Raauja theta fucto as well to correspodg ew dettes for the false theta fucto v

9 Chapter Itroducto I [7] Dasbach ad L cojectured that, up to a coo sg chage, the hghest 4( + ) (the lowest resp) coeffcets of the th ureduced colored Joes polyoal J,L (A) of a alteratg l L agree wth the frst 4( + ) coeffcets of the polyoal J +,L (A) for all Ths gves rse to two power seres wth teger coeffcets assocated wth the alteratg l L called the head ad the tal of the colored Joes polyoal The exstece of the head ad tal of the colored Joes polyoal of adequate ls was prove by Arod [4] usg se theory Idepedetly, ths was show by Garoufalds ad Le for alteratg ls usg R-atrces [8] ad geeralzed to hgher order stablty of the coeffcets of the colored Joes polyoal Let L be a alteratg l ad let D be a reduced l dagra of L Wrte S A (D) to deote the A-soothg state of D, the state obtaed by replacg each crossg by a A-soothg The state S B (D) of D s defed slarly Usg the Kauffa se relato Kauffa [7], Murasug [28], ad Thstlethwate [33] showed that the A state (respectvely the B state) realzes the hghest (respectvely the lowest) coeffcet of the Joes polyoal of a alteratg l I chapter 2, we exted ths result to the colored Joes polyoal by usg the colored Kauffa se relato 42 We show that the -colored A state ad the -colored B state realze the hghest ad the lowest 4 coeffcets of the th ureduced colored Joes polyoal of a alteratg l Furtherore we show that ths gves a atural layout to prove the stablty of the hghest ad lowest coeffcets of the

10 colored Joes polyoal of alteratg ls I other words we prove that the head ad the tal of the colored Joes polyoal exst Let S () B (D) be the se eleet obtaed fro S B(D) by decoratg each crcle ths state wth the th Joes-Wezl depotet ad replacg each place where we had a crossg D wth the (2) th projector It was prove [3] that for a adequate l L the frst 4( + ) coeffcets of th ureduced colored Joes polyoal cocde wth the frst 4(+) coeffcets of the se eleet S () B (D) I chapter 3, we study a certa se eleet, called the bubble se eleet, the relatve Kauffa bracet se odule of the ds wth soe ared pots, ad expad ths eleet ters of learly depedet eleets of ths odule The we use the bubble se eleet the study of the tal of 8 5 cosderg the se eleet S () B (D) obtaed fro a alteratg dagra of 8 5 The ot 8 5 s the frst ot o the ot table whose tal could ot detered drectly by the techques developed [3] I chapter 4, we exted the study of the tal of the colored Joes polyoal that we started the prevous chapter to study the tal of quatu sp etwors A quatu sp etwor s a baded trvalet graph wth edges labeled by oegatve tegers, also called the colors of the edges, ad the three edges eetg at a vertex satsfy soe adssblty codtos Se theoretc techques have bee used [3] ad [4] to uderstad the head ad tal of a adequate l It was prove [3] that for a adequate l L the frst 4( + ) coeffcets of th ureduced colored Joes polyoal cocde wth the frst 4( + ) coeffcets of the evaluato S(S 2 ) of a certa se eleet S 2 We deostrate here that ths se eleet ca be realzed as quatu sp etwor obtaed fro the 2

11 l dagra D Hece, studyg the tal of the colored Joes polyoal ca be reduced to studyg the tal of these quatu sp etwors Our ethod to study the tal of such graphs reles aly o adaptg varous se theoretc dettes to ew oes that ca tur be used to copute ad uderstad the tal of such graphs We use ths structure to gve atural product structures o the tal of quatu sp etwors The q-seres obtaed fro ots ths way appear to be coected to classcal uber theoretc dettes Ha [4] realzed that that Rogers-Raauja dettes appear the study of the colored Joes polyoal of torus ots I [4] Arod ad Dasbach calculate the head ad the tal of the colored Joes polyoal va ultple ethods ad use these coputatos to prove uber theoretc dettes I chapter 4, we show that the se theoretc techques we developed here ca be also used to prove classcal dettes uber theory I partcular we use se theory to prove the Adrews-Gordo dettes for the two varable Raauja theta fucto, as well as correspodg ew dettes for the false theta fucto 3

12 Chapter 2 Bacgroud 2 Prelares A ot the 3-sphere S 3 s a pecewse-lear oe-to-oe appg f : S S 3 A l S 3 s a fte ordered collecto of ots, called the copoets of the l, that do ot tersect each other Two ls are cosdered to be equvalet f they are abet sotopc A l dagra of a l L s a projecto of L o S 2 such that ths projecto has a fte uber of otagetoal tersecto pots (called crossgs), each of whch s cog fro exactly two pots of the l L We usually draw a sall brea the projecto of the lower strad to dcate that t crosses uder the other strad Redeester s theore asserts that two ls are abet sotopc f ad oly f ther dagras ca be trasfored to the other by a fte sequece of Redeester oves See Fgure 2 FIGURE 2 The three Redeester oves A fraed l s a l together wth a sooth secto of the oral budle over the l called a frag Fraed ls are also cosdered up to abet sotopy I ths thess we wll deal ostly wth fraed ls We wll assue that l dagras are equpped wth blacboard frag Ay dagra of a l L gves rse to a frag of L by tag a ozero vector feld that s everywhere parallel to the projecto plae of the dagra Ths frag s called the blacboard 4

13 frag Ay arbtrary fraed l ca be represeted by a l dagra wth the blacboard frag Approprate serto of curls the dagra adjusts the blacboard frag of the dagra so that oe ca realze ay frag 2 Alteratg ad Adequate Ls Let L be a l S 3 ad let D be a alteratg ot dagra of L For ay crossg D there are two ways to sooth ths crossg, the A-soothg ad the B-soothg See Fgure 22 A B FIGURE 22 A ad B soothgs We replace a crossg wth a soothg together wth a dashed le jog the two arcs After applyg a soothg to each crossg D we obta a plaar dagra cosstg of a collecto of dsjot crcles the plae We call ths dagra a state for the dagra D The A-soothg state, obtaed fro D by replacg every crossg by a A soothg, ad the B-soothg state for D are of partcular portace for us Wrte S A (D) ad S B (D) to deote the A soothg ad B soothg states of D respectvely For each state S of a l dagra D oe ca assocate a graph obtaed by replacg each crcle of S by a vertex ad each dashed l by a edges I partcular we are terested the graphs obtaed by the A-soothg state ad the B-soothg state of the dagra D We wll deote these two graphs by A(D) ad B(D) respectvely 5

14 FIGURE 23 The ot 6 2, ts A-graph o the left ad ts B-graph o the rght Recall that a l dagra s alteratg f as we travel alog the l, fro ay startg pot, the crossgs alteratg betwee uder ad over crossgs A l s called alteratg f t possesses such a dagra The followg defto ca be cosdered as a geeralzato of the alteratg ls Defto 2 A l dagra D s called A-adequate (B-adequate, respectvely) f there are o loops the graph A(D) (the graph B(D), respectvely) A l dagra D s called adequate f t s both A-adequate ad B-adequate [22] It s ow that a reduced alteratg l dagra s adequate See for exaple 22 Se Theory I ths secto we revew the fudaetals of the Kauffa Bracet Se Modules ad troduce the se odules that wll be used for our purpose Furtherore, we dscuss the recursve defto of Joes-Wezl depotet ad recall soe of ts basc propertes For ore detals about lear se theory assocated wth the Kauffa Bracet, see [9], [22], ad [29] 6

15 Defto 22 (J Przytyc [29] ad V Turaev [34]) Let M be a oreted 3- afold Let R be a coutatve rg wth detty ad a fxed vertable eleet A Let L M be the set of sotopy classes of fraed ls M cludg the epty l Let RL M be the free R-odule geerated by the set L M Let K(M) be the sallest subodule of RL M that s geerated by all expressos of the for () A A, (2) L + (A 2 + A 2 )L where L cossts of a fraed l L M ad the trval fraed ot The Kauffa bracet se odule, S(M; R, A), s defed to be the quotet odule S(M; R, A) = RL M /K(M) A relatve verso of the Kauffa bracet se odule ca be defed whe M has a boudary The defto s exteded as follows We specfy a fte (possbly epty) set of fraed pots x, x 2,, x 2 o the boudary of M A bad s a surface that s hoeoorphc to I I A eleet the set L M s a sotopy class of a oreted surface ebedded to M ad decoposed to a uo of fte uber of fraed ls ad bads jog the desgated boudary pots Let K(M) be the sallest subodule of RL M that s geerated by Kauffa relatos specfed above The Kauffa bracet se odule s defed to be the quotet odule S(M; R, A, {x } 2 =) = RL M /K(M) The relatve Kauffa bracet se odule depeds oly o the dstrbuto of the pots {x } 2 = aog the dfferet coected copoets of M ad t does ot deped o the exact posto of the pots {x } 2 = I partcular, f M s coected the the defto of the relatve Kauffa bracet se odule s depedet of the choce of the exact posto of the pots {x } 2 = For ore detals see [29] ad [30] 7

16 Whe the afold M s hoeoorphc to F [0, ], where F s a surface wth a fte set of pots (possbly epty) ts boudary F, oe could defe the (relatve) Kauffa bracet se odule of F I ths case oe cosders a approprate verso of l dagras F stead of fraed ls M The soorphs betwee S(M; R, A) ad S(F ; R, A) that seds a fraed l to ts l dagra wll be used to detfy these two se odules We wll wor wth the se odule of the sphere S(S 2 ; R, A) Ths se odule s freely geerated a oe geerator Let D be ay dagra S 2 Usg the defto of the oralzed Kauffa bracet [7], we ca wrte D =< D > φ S(S 2 ; R, A), where φ deotes the epty l Ths provdes a soorphs <>: S(S 2 ; R, A) R, duced by sedg D to < D > I partcular ths soorphs seds the epty l φ S(S 2 ; R, A) to the detty R We wll also wor wth the relatve se odule S([0, ] [0, ]; R, A, {x } 2 =), where the rectagular ds [0, ] [0, ] has desgated pots {x } = o the top edge, where x = (, + ) for, ad desgated pots o the botto edge {x } 2 =+, where x = (0, ) for + 2 As we etoed above, the relatve se + odule S([0, ] [0, ]; R, A, {x } 2 =) does ot deped o the exact posto of the pots {x } 2 = However, we are ag a choce for the posto of the pots here because we wll defe a algebra structure o S([0, ] [0, ]; R, A, {x } 2 =) where the posto of these pots s requred Ths relatve se odule ca be thought of as the R-odule geerated by all (, )-tagle dagras [0, ] [0, ] odulo the Kauffa relatos I fact the odule S([0, ] [0, ]; R, A, {x } 2 =) s a free R-odule o +( 2 ) free geerators For a proof of ths fact see [29] The relatve se odule S([0, ] [0, ]; R, A, {x } 2 =) adts a ultplcato gve by juxtaposto of two dagras [0, ] [0, ] More precsely, let D 8

17 ad D 2 be two dagras [0, ] [0, ] such that D j, where j =, 2, cossts of the pots {x } 2 = specfed above Defe D D 2 to be the dagra [0, ] [0, ] obtaed by attachg D o the top of D 2 ad the copress the result to [0, ] [0, ] Ths ultplcato o dagras exteds to a well-defed ultplcato o sotopy classes of dagras [0, ] [0, ] Fally t exteds by learty to a ultplcato o S([0, ] [0, ]; R, A, {x } 2 =) Wth ths ultplcato S([0, ] [0, ]; R, A, {x } 2 =) s a assocatve algebra over R ow as the th Teperley-Leb algebra T L For each there exsts a depotet f () T L that plays a cetral role the Wtte-Resheth-Turaev Ivarats for SU(2) See [9], [2], ad [32] The depotet f (), ow as the th Joes-Wezl depotet, was dscovered by Joes [6] ad t has a recursve forula due to Wezl [35] We wll use ths recursve forula to defe f () Further, we wll adapt a graphcal otato for f () whch s due Lcorsh [20] I ths graphcal otato oe ths of f () as a epty box wth strads eterg ad strads leavg the opposte sde The Joes-Wezl depotet s defed by: = ( 2 ) 2, = (2) where = ( ) A2(+) A 2(+) A 2 A 2 The polyoal s related to [ + ], the ( + ) th quatu teger, by = ( ) [ + ] It s coo to use the substtuto a = A 2 whe oe wors wth quatu tegers 9

18 We assue that f (0) s the epty dagra The Joes-Wezl depotet satsfes =, =, = 0, (22) () =, (2) = (23) + + ad j () = +, (2) + j = A j + j (24) I ths thess R wll be Q(A), the feld geerated by the deterate A over the ratoal ubers Before we troduce ay other se odules we tal brefly about lear aps betwee se odules 22 Se Maps We ca relate varous se odules by lear aps duced fro aps betwee surfaces Let F ad F be two oreted surfaces wth ared pots o ther boudares A wrg s a oretato preservg ebeddg betwee F ad F alog wth a fxed wrg dagra of arcs ad curves F F such that the boudary pots of the arcs cossts of all the ared pots of F ad F Ay dagra D F duces a dagra W(D) F by extedg D by wrg dagra A wrg 0

19 W of F to F duces the odule hooorphs S(W ) : S(F ) S(F ) defed by D W (D) for ay D dagra F For ore detals see [?] Exaple 23 Cosder the square I I wth ared pots o the top edge ad ared pots o the botto edge Ebed I I S 2 ad jo the pots o the top edge to the pots o the botto edge by parallel arcs as follows: For each, ths wrg duces a odule hooorphs: tr : T L S(S 2 ) Ths ap s usually called the Marov trace o T L 222 Quatu Sp Networs Before gvg the defto of quatu sp etwor we wll eed to troduce a few relatve se odules Cosder the relatve se odule of the ds wth a + + a ared pots o the boudary We are terested a subodule of ths odule costructed as follows Partto the set of the a + + a pots to sets of a,, a ad a pots respectvely At each cluster of these pots we place a approprate depotet, e the oe whose color atches the cardalty of ths cluster We wll deote ths relatve se odule by T a,,a Hece a eleet T a,,a s obtaed by tag a eleet the odule of the ds wth a + + a ared pots ad the addg the depotets f (a ),, f (a)

20 o the outsde of the ds Fgure 24 shows a eleet the Kauffa bracet se odule of the ds wth ared pots o the boudary ad the correspodg eleet the space T 3,2,2, FIGURE 24 A eleet the odule of the ds ad ts appg to T 3,2,2, Of partcular terest are the spaces T a,b, T a,b,c Joes-Wezl projector ply that the space T a,b ad T a,b,c,d The propertes of the s zero desoal whe a b ad oe desoal whe a = b, spaed by f (a) Slarly, T a,b,c s ether zero desoal or oe desoal The space T a,b,c s oe desoal f ad oly f the eleet τ a,b,c show Fgure 25 exsts Ths occurs whe oe has o-egatve tegers x, y ad z such that the followg three equatos are satsfed a = x + y, b = x + z, c = y + z (25) a x y z b c FIGURE 25 The se eleet τ a,b,c the space T a,b,c Whe τ a,b,c exsts we wll refer to the outsde colors of τ a,b,c by the colors a, b ad c ad to the sde colors of τ a,b,c by the colors x, y ad z The followg defto characterzes the exstece of the eleet τ a,b,c ters of the outsde colors 2

21 Defto 24 A trple of colors (a, b, c) s adssble f a + b + c 0( od 2) ad a + b c a b Note that f the trple (a, b, c) s adssble, the wrtg x = (a + b c)/2, y = (a + c b)/2, ad z = (b + c a)/2 we have that x,y ad z satsfy the equatos 25 If the trple (a, b, c) s ot adssble the the space T a,b,c s zero desoal The fact that the sde colors are detered by the outsde colors allows us to replace τ a,b,c by a trvalet graph as follows: a a x y z b c b c FIGURE 26 The eleet τ a,b,c Ths otvates the followg defto Defto 25 A quatu sp etwor S 2 s a ebedded trvalet graph S 2 I wth edges labeled by o-egatve tegers ad the labels of the edges eetg at a vertex satsfy the adssblty codto Let Γ be a quatu sp etwor S 2 The Kauffa bracet evaluato of Γ s defed to be bracet of the expaso of Γ after replacg every edge labeled by the projector f () ad every vertex labeled (a, b, c) by the se eleet τ a,b,c See Fgure 27 a a x y, z b c b c FIGURE 27 The evaluato of a quatu sp etwor as a eleet S(S 2 ) 3

22 We assue that a dagra represet the zero eleet the odule f t has a strad labeled by a egatve uber The last relatve Kauffa bracet se odule that we eed here s the odule T a,b,c,d, see Fgure 28 a b d c FIGURE 28 The relatve se odule T a,b,c,d Ths odule s free o the set of geerators gve Fgure 29 Here rus over all possble postve tegers such that (a, b, ) ad (c, d, ) are adssble a b d c FIGURE 29 A bass of the odule T a,b,c,d 23 The Colored Joes Polyoal I 980s a vast faly of ew ot varats, called quatu varats were dscovered ad later exteded to varats of 3-afolds The frst quatu varat was dscovered by Joes [5] The Joes polyoal s a Lauret polyoal ot varat the varable q wth teger coeffcets The Joes polyoal geeralzes to a varat J g K,V (q) Z[q± ] of a zero-fraed ot K colored by a represetato V of a sple Le algebra g, ad oralzed so that J g O,V (q) =, where O deotes the zero-fraed uot The varat J g K,V (q) s called the quatu varat of the ot K assocated wth the sple Le algebra g ad the represetato V The Joes polyoal correspods to the 2- desoal rreducble represetato of sl(2, C) ad the th colored Joes polyoal, deoted by 4

23 J,K (q), s the quatu varat assocated wth the +-desoal rreducble represetato of sl(2, C) Aother verso of the colored Joes polyoal s the ureduced colored Joes polyoal Let L be zero-fraed fraed l S 3 The th ureduced colored Joes polyoal of a l L s obtaed by decoratg every copoet of L, accordg to ts frag, by the th Joes-Wezl depotet ad cosder ths decorated fraed l as a eleet of S(S 3 ) The two versos of the colored Joes polyoal are related by a chage of varable ad a shft the dex by I what follows we assue A 4 = q I the ext chapter we start studyg the hghest ad the lowest coeffcets of the colored Joes polyoal usg aly the se theoretc defto of the ureduced colored Joes polyoal 5

24 Chapter 3 The Colored Kauffa Se Relato ad the Head ad the Tal of the Colored Joes Polyoal 3 Itroducto I [7] Dasbach ad L cojectured that, up to a coo sg chage, the hghest 4( + ) (the lowest resp) coeffcets of the th ureduced polyoal J,L (A) of a alteratg l L agree wth the frst 4( + ) coeffcets of the polyoal J +,L (A) for all Ths gves rse to two power seres wth teger coeffcets assocated wth the alteratg l L called the head ad the tal of the colored Joes polyoal The exstece of the head ad tal of the colored Joes polyoal of adequate ls was prove by Arod [4] usg se theory Idepedetly, ths was show by Garoufalds ad Le for alteratg ls usg R-atrces [8] ad geeralzed to hgher order tals Let L be a alteratg l ad let D be a reduced l dagra of L Wrte S A (D) to deote the A-soothg state of D, the state obtaed by replacg each crossg by a A-soothg The state S B (D) of D s defed slarly Usg the Kauffa se relato Kauffa [7] showed that the A state (respectvely the B state) realzes the hghest (respectvely the lowest) coeffcet of the Joes polyoal of a alteratg l I ths chapter we exted ths result to the colored Joes polyoal by usg the colored Kauffa se relato 42 We show that the -colored A state ad the -colored B state realze the hghest ad the lowest 4 coeffcets of the th ureduced colored Joes polyoal of a alteratg l Furtherore we show that ths gves a atural layout to prove the 6

25 stablty of the hghest ad lowest coeffcets of the colored Joes polyoal of alteratg ls I other words we prove that the head ad the tal of the colored Joes polyoal exst 32 The Head ad the Tal of the Colored Joes Polyoal for Alteratg Ls Let P (q) ad P 2 (q) be eleets Z[q ][[q]], we wrte P (q) = P 2 (q) f ther frst coeffcets agree up to a sg Defto 3 Let P = {P (q)} N be a sequece of foral power seres Z[q ][[q]] The tal of the sequece P- f t exsts - s the foral power seres T P (q) Z[[q]] that satsfes T P (q) = P (q), for all N Observe that the tal of the sequece P = {P (q)} N exsts f ad f oly f P (q) = P + (q) for all We wll eed the defto of the al degree of such foral power seres If p Z[q ][[q]] the we wll deote by (p) to the al degree of p Rear 32 Cosder the sequece {f (q)} N where f (q) s a ratoal fucto Q(q) Every ratoal fucto Q(q) ca be represeted uquely as a eleet Z[q ][[q]] Usg ths coveto oe ca study the tal of the sequece {f (q)} N Furtherore, ths ca be used to defe the al degree of a ratoal fucto Let D = {D (q)} N be a sequece of se eleets S(S 2 ) The evaluato of D (q) gves geeral a ratoal fucto Usg the observato rear 32, oe could study the tal of the sequece D 7

26 33 The Colored Kauffa Se Relato We start ths secto by provg the colored Kauffa se relato 45 Ths relato s plct the wor of Yaada [36] The colored Kauffa se relato wll be used the ext secto to uderstad the hghest ad the lowest coeffcets of the colored Joes polyoal The followg two Leas are bascally due to Yaada [36] We clude the proof here wth odfcato for copleteess Lea 33 (The colored Kauffa se relato) Let 0 The we have = A 2+ + A (2+) Proof Applyg the Kauffa relato we obta = A + A = A + A Usg property (2) 24 we obta the result Note that f we specalze to the prevous Lea we obta the Kauffa se relato Recall that the quatu boal coeffcets are defed by: [ ] []! = []![ ]! A 8

27 where [] = ( ) ad []! = [][] Lea 34 Let 0 The we have = C, =0 Where C, = A ( 2) [ ] A Proof Lea 45 ples that = C, (3) =0 where C, s a polyoal wth tegral coeffcets A Let us prove by ducto o that we have C, = C, (32) For = relato (32) holds sce ths s just the Kauffa se relato Applyg the detty (3) o each ter of the colored Kauffa se relato, 9

28 we obta : C, = A 2 C, =0 =0 + A 2+ =0 C, + + = A 2 C, =0 + A 2+ = C, The se eleets, where 0, are learly depedet ( see rear 40 ) ad hece C, = A 2 C, + A 2+ C, (33) However Hece [ ] [ ] C, = A ( 2) (A 2 + A 2 2 ) A A [ ] [ ] = A A A A C, = A 2 C, + A 2+ C, (34) Relatos (33) ad (34) ad the ducto hypothess yeld the result Motvated by Lea 45 we defe the -colored states for a l dagra D for every postve teger Suppose that the l dagra has crossgs Label the 20

29 crossgs of the l dagra D by,, A -colored state s () for a l dagra D s a fucto s () : {,, } {, +} If the color s clear fro the cotext, we wll drop fro the otato of a colored state A B FIGURE 3 The -colored A ad B soothgs Gve a l dagra D ad a colored state s for the dagra D We costruct a se eleet Υ () D (s) obtaed fro D by replacg each crossg labeled + by a -colored A-soothg ad each crossg labeled by a -colored B-soothg, see Fgure 3 Two partcular se eleets obtaed ths way are portat to us The se eleet obtaed by replacg each crossg by the -colored A- soothg wll be called the -colored A-state ad deoted by Υ () D (s +), where s + deotes the colored state for whch s + () = + for all {,, } The -colored B-state s defed slarly See Fgure 32 for a exaple FIGURE 32 -colored B-state Cosder a crossgless se eleet D S(S 2 ) cosstg of arcs coectg Joes-Wezl depotets of varous colors Let D be the dagra obtaed fro 2

30 D by replacg each depotet f () wth the detty d T L The dagra D thus cossts of o-tersectg crcles We say that D s adequate f each crcle D passes at ost oce through ay gve rego where we replaced the depotets D See Fgure 34 for a local pcture of a adequate se eleet ad ote that the crcle dcated the fgure bouds a ds Fgure 33 shows a exaple of adequate se eleet o the left ad a adequate se eleet o the rght FIGURE 33 Adequate ad adequate se eleets All arcs are colored FIGURE 34 A local pcture a adequate se eleet Usg the coveto 32, we wll deote by (f) to the u degree of f expressed as a Lauret seres q or A Furtherore, deote D(S) := ( S) The followg lea s due to Arod [4] Lea 35 If S S(S 2 ) s expressed as a sgle dagra cotag the Joes- Wezl depotet, the (S) D(S) If the dagra for S s a adequate se dagra, the (S) = D(S) 22

31 34 The Ma Theore The colored Kauffa se relato provdes a atural fraewor to uderstad the hghest ad the lowest coeffcets of the colored Joes polyoal I ths secto we wll use ths relato to prove that the the hghest (the lowest respectvely) 4 coeffcets of the th ureduced colored Joes polyoal agree up to a sg wth the -colored A-state (the -colored B-state respectvely) We use ths result to prove exstece of the the tal of the colored Joes polyoal Theore 36 Let L be a alteratg l dagra The J,L =4 Υ () L (s ) Proof Assue that the l dagra L has crossgs ad label the crossg of the l dagra by,, The colored Kauffa se relato ples that J,L = s α L (, s)υ () L (s) where α L (, s) = A (2 ) = s() ad the suato rus over all fuctos s : {, 2,, } {, +} Now for ay colored state s of the l dagra L there s a sequece of states s 0, s, s r such that s 0 = s, s r = s ad s j () = s j () for all {,, } except for oe teger l for whch s j ( l ) = ad s j ( l ) = It s eough to show that the lowest 4 ters of α(, s )Υ () L (s ) are ever caceled by ay ter fro α(, s) L Υ () L (s) for ay s For ay colored state s of the l dagra L oe could wrte Υ () L (s) =,, =0 j= C,j Λ s,(,, ) where Λ s,(,, ) s the se eleet that we obta by applyg 48 to every crossg Υ () (s) The theore follows fro the followg three leas L 23

32 Lea 37 (α L (, s )) = (α L (, s )) (α L (, s r )) (α L (, s r+ )) (C, ) (C, 2 ) = 2 (C, ) (C, ) Proof It s clear that α L (, s ) = A 2 ad α L (, s ) = A Furtherore, α L (, s r ) = A (2 ) = sr() = A (2 )( +2r) = A 2 2r+4r hece (α L (, s r )) (α L (, s r+ )) = 2 4 Fally, t s clear that (C, ) = Hece, the result follows Lea 38 (Λ s,(,, )) = D(Λ s,(,, )) 2 D(Λ s,(, j, j, j+, )) = D(Λ s,(, j, j, j+,, )) ± 2 24

33 Proof Whe we replace the depotet by the detty the se eleet Υ () (s ) we obta the dagra L C where C s a l dagra coposed of a dsjot uo of ut crcles each oe of the bouds a ds ad L s the ( )-parallel of L Note that state Λ s,(,, ) s exactly the all B-state of the l dagra L ad t s also the all B-state of L C Sce L s alteratg the the the uber of crcles Λ s,(,, ) s oe less tha the uber of crcles Λ s,(,, ) I other words, the uber of crcles the all B-state of Υ () (s ) s oe less tha the uber of crcles the all B-state of the dagra L C Thus, D(Λ s,(,, )) = D(Λ s,(,, )) 2 Moreover the se eleet Λ s,(,, ) s adequate sce L s alteratg Hece by Lea 35 we have (Λ s,(,, )) = D(Λ s,(,, )) 2 For the secod part, ote that the uber of crcles the dagras Λ s,(, j, j, j+, ) ad Λ s,(, j, j, j+,, ) dffers by Hece by 35 we obta D(Λ s,(, j, j, j+, )) = D(Λ s,(, j, j, j+,, )) ± 2 Lea 39 (Υ () (s )) = ((C, ) Λ s,(,, )) Proof The prevous two leas ply drectly that the lowest ter Υ () (s ) s cog fro the se eleet (C, ) Λ s,(,, ) ad ths ter s ever caceled by ay other ter the suato 25

34 Theore 30 Let L be a alteratg l dagra ad let Υ (+) (s ) be ts correspodg ( + )-colored B-state se eleet The Υ (+) L (s ) = 4 J,L Proof Sce L s a alteratg l dagra the the se eleet Υ (+) (s ) ust loo locally as Fgure 35 FIGURE 35 Local vew of the se eleet Υ (+) (s ) It follows fro Theore 9 ad Lea 0 [4] that = 4 The detals of the prevous equato ca be foud [4] ad we wll ot repeat the here The prevous step ca be appled aroud the crcle utl we reach the fal depotet : = 4 = 4 26

35 Equato 24 ples = 4 + = 4 Applyg ths procedure o every crcle Υ (+) (s ), we evetually obta Υ (+) (s ) = 4 J,L Theores 56 ad 57 ply edately the followg result Corollary 3 Let L be a alteratg l dagra The J +,L =4 J,L 27

36 Chapter 4 The Bubble Se Eleet ad the Tal of the Kot Itroducto Let L be a alteratg l ad let D be a reduced l dagra of L Recall that S B (D) deotes the all-b soothg state of D, the state obtaed by replacg each crossg by a B soothg Let S () B (D) be the se eleet obtaed fro S B (D) by decoratg each crcle ths state wth the th Joes-Wezl depotet ad replacg each place where we had a crossg D wth the (2) th projector It was prove [3] that for a adequate l L the frst 4( + ) coeffcets of th ureduced colored Joes polyoal cocde wth the frst 4( + ) coeffcets of the se eleet S () (D) Our wor here tally aed to uderstad S() (D) for B B a alteratg l dagra D Ital exaatos of varous exaples of S () B (D) showed that a certa se eleet the relatve Kauffa bracet se odule of the ds wth soe ared pots ostly shows up as sub-se eleet of S () B (D) We wll call ths se eleet the bubble se eleet Ths chapter s based o our wor [] I ths chapter we study a certa se eleet the relatve Kauffa bracet se odule of the ds wth soe ared pots, ad expad ths eleet ters learly depedet eleets of ths odule The we use ths se eleet the study of the tal of 8 5 The ot 8 5 s the frst ot o the ot table that ot coputable drectly by the techques developed [3] I chapter 4 we use ths se eleet varous other applcatos 28

37 l FIGURE 4 The bubble se eleet B,, (, l) As we etoed earler, we are terested a partcular se eleet the odule T,,, Ths eleet s show Fgure 4 ad t s deoted by B,, (, l), where, l We wll call such a eleet T,,, a bubble se eleet For every bubble se eleet B,, (, l), the tegers,,,,, l 0 ad they satsfy + = + l ad + = + l The a wor of ths chapter gves a expaso the bubble se eleet B,, (, l), defed the prevous secto, ters of a set of Q(A)-learly depedet se eleets the odule T,,, ad gves a explct deterato of the coeffcets obtaed fro ths expaso Recall that the quatu boal coeffcets are defed by [ ] l = q (q; q) l (q; q) (q; q) l where (a; q) s q-pochhaer sybol whch s defed as (a; q) = ( aq j ) j=0 Rear 4 Note that our choce for the quatu boal coeffcet s slghtly dfferet fro the choce we ade the prevous chapter We wll use the coveto we have here the cosequet chapters as well 29

38 Theore 42 (The bubble expaso forula) Let,,, 0, ad l;, l The = (,,l) =0 l l + where l l := ( A 2 ) ( l) l j=0 j s s s=0 l + t + t t=0 [ ] l l A 4 j=0 ++ j We gve two applcatos of the prevous theore The frst oe gves a relato betwee the coeffcet ad the theta graph Λ(,, ) (see Fgure 43) 0 S(S 2 ) Proposto 43 = + 0 The secod applcato that we gve to the bubble expaso forula s showg that ths expaso ca be used to study ad copute the tal of alteratg ls I partcular, we gve a sple forula for the tal of the ot 8 5 : Proposto 44 T 85 (q) = (q; q) 2 =0 q +2 (q; q) ( =0 q ( 2( )) 2 q ) 30

39 42 A Recursve Forula for the Bubble Se Eleet I ths secto we use the recursve defto of the Joes-Wezl depotet to obta a recursve forula for the bubble se eleet B,, (, l) the odule T,,, To obta a recursve forula for the eleet B,, (, l) we start by expadg the sple bubble eleet B,, (, ) Lea 45 ad the use ths expaso to obta a recurso equato for a arbtrary bubble se eleet Lea 48 The recursve forula of B,, (, l) obtaed ths secto wll be used Theore 4 to wrte a bubble se eleet as a Q(A)-lear su of learly depedet eleets T,,, We deote the ratoal fuctos ad β,, respectvely ad + + by α, Lea 45 Let,,, 0; The we have = α, + β, Proof Frst we cosder the trval cases whe oe of the tegers,,, s zero Observe that + = + ad + = + Sce, we ow that ad If = = 0, the ths ples that ad ust be zero ad s Hece B0,0(, ) = α0,0 = ad we are doe Here we used our coveto that a dagra s zero f t has a strad colored by a egatve uber If (, ) = 0 or (, ) = 0, the the result follows fro (24) Whe (, ) 0, we use ducto o For = we apply the recursve defto of the Joes-Wezel depotet (2) o the projector f (+) that 3

40 appears B,, (, ) to obta = (4) Usg detty (24) the frst ter ad expadg the projector f (+) the secod ter (5) ples = + + (42) For 2, we use the recursve defto (2) o the projector f (+) that appears B,, (, ) Hece = (43) Usg detty (24) ad expadg the projector f (+) (43) ples = (44) We apply the ducto hypothess o the bubble se eleet B,, (, ) that appears the secod ter of the last equato: 32

41 = ( ) Collectg slar ters together, we obta = Rear 46 Note that, whle the syetry wth respect to the varables ad the fucto β, s clear, the ratoal fucto α, appears to be asyetrc 33

42 wth respect to these varables It s route to verfy that + + = + + ad hece α, s also syetrc wth respect to the varables ad Ths also could be see to follow fro the proof of Lea 45 To see ths ote that Lea 45 we obta (44) by expadg f (+) frst ad f (+) secod Further, the coeffcets α, ad β, are a result of a teratve applcato of (44) We could have doe the expaso of the bubble the opposte order, by expadg f (+) frst ad f (+) secod, ad obta a verso of the detty (44) except ad are swapped A terato of ths detty would yeld the sae coeffcet β, ad forces α, to be the sae as α, We wll eed the followg detty A proof ca be foud [9] ad [25] Lea = ++ Note that the prevous detty ca be used to obta a ore aageable forula for α, Lea 48 Let,,, 0;, l ad l The we have = α, + β, (45) l l l Proof We start aga by cosderg the trval cases whe oe of the tegers,,, s zero Note that, as Lea 45, ad If = = 0, the B 0,0 0,0(, ) = α 0,0 = If (, ) = 0 or (, ) = 0, the the result follows fro (24) 34

43 Now suppose that (, ) 0 ad apply the recursve defto of the Joes- Wezel depotet o the projector f (+) that appears B,, (, l) l = l l (46) Reovg the loop that appears the frst ter ad expadg the projector f (+) the secod ter of (46) we obta = (47) l l l If l = the the result follows fro Lea 45 Otherwse we apply Lea 45 to the se eleet B,, (, ) appearg the secod ter of (47) to obta = l l 2 + ( l ) l 35

44 The last equato ples: = l l l Rear 49 Let,,, 0 Wthout loss of geeralty assue that + + I the space T,,,, there s a oe-to-oe correspodece betwee the two set of eleet show Fgure 42 The correspodece s show also the sae Fgure ad t ca be paraeterzed by a teger Note that the dagoal le the Fgure represets 2 ( + ) parallel les The se eleets o the rght-had sde of Fgure 42 for a bass for the space T,,, For a proof of ths fact see Lea 49 [22] O the other had the se eleets of the left-had sde spas the space T,,, See also the proof of Lea 49 [22] Oe cocludes that, by the correspodece show Fgure 42 the se eleets o the left-had sde of Fgure 42 ust be learly depedet ad hece they for a bass for the space T,,, FIGURE 42 Correspodece betwee two bases the odule T,,, Rear 40 We are terested the coeffcets the bubble se eleet B,, (, l) ters of the bass show o the left-had sde Fgure 42 Sce + = +l ad + = +l the we ust have + = + ad hece the 36

45 dagoal le the bass show o the left-had sde Fgure 42 wll ot appear whe we expad the eleet B,, (, l) ters of ths bass 43 The Bubble Expaso Forula I ths secto we wll use the recursve forula we obtaed Lea 48 for B,, (, l) to expad ths eleet as a Q(A)-lear su of certa learly depedet se eleets T,,, The we wll use Theore 4 together wth Lea 48 to detere a recursve forula for the coeffcets of B,, (, l) ters of these learly depedet eleets Fally, the recursve forula wll be used to detere a closed for of these coeffcets Theore 4 Let,,, 0;, l The () For l: (2) For l : l l where := l ratoal fuctos = = (,,l) =0 (,,) l =0 l (A) ad l l + l + (48) (49) := (A) are l l Proof () The trval cases whe (,, l) were dscussed Lea 48 Suppose that (,, l) 2 ad cosder the detty (45) We apply ths detty to the bubble se eleets appearg the frst ad the secod ters of 37

46 (45) We obta that B,, (, l) s equal to a Q(A)-lear su of the three se eleets 2 2,, l 2 l 2 l 2 Note that a bubble se eleet appears each of the three eleets above Moreover, each bubble has a dex o at least oe of ts strads that s less tha the dex of the correspodg strad (45) Oe could apply the detty (45) teratvely o each bubble that appears the su Ths teratve process wll evetually terate all bubbles each eleet the suato Ths yelds that B,, (, l) s equal to a Q(A)-lear su of the T,,, se eleets l, l +,, s l + s where s = (,, l) Now the result follows by otcg that the varables,,,, ad l are related by the equatos + = + l ad + = + l ad hece t s suffcet to dex the coeffcets the expaso of B,, (, l) ters of the prevous se eleets by fve dces (2) Follows fro () Now we detere the coeffcets l Proposto 42 Let,,, 0,, l Let B,, (, l) be a bubble se eleet T,,, such that l The, for 0 (,, l), the ratoal fucto satsfes the followg recursve detty: l 38

47 l = α, l + β, l (40) Proof Substtute (48) both sdes of (45): (,,l) =0 l l + = α, (,,l ) =0 l l + + β, (,,l ) =0 l l + + Hece (,,l) =0 l l + = (,,l ) =0 α, l l + + (,,l ) =0 β, l + l

48 The latter ca be wrtte as (,,l) =0 l l + = (,,l ) =0 α, l l + + (,,l) = β, l l + Now ote that the eleets l +, where 0 (,, l), are learly depedet the odule T,,, by Rears 49 ad 40 Hece we coclude that l = α, l + β, l ( Rear 43 The coeffcets behave le the boal coeffcets l ) l the sese that = 0 whe < 0 or > l Note also that the recurso l forula for, whe oe focuses the varables l ad, s aalogues to the l recurso forula of the boal coeffcets ( l ) The followg theore gves a closed forula for the ratoal fucto l 40

49 Theore 44 Let,,, l 0, ad l ad 0 (,, l) The l = ( A 2 ) ( l) l j=0 j s s s=0 l + t + t t=0 [ ] l l A 4 j=0 ++ j (4) Proof Ths equato agrees wth the recurso detty (40) The syetry of the bubble se eleet, or the prevous forula for the coeffcets, ples edately the followg l Proposto 45 Let,,, l 0 Let l The l = l (42) It s preferred soetes to wrte the the coeffcets ters of l quatu tegers rather tha deltas Recall that = ( ) [ + ] ad hece the l sg of the ter [ j] ca be easly calculated to be ( ) ( 2 )( l)( ++2 l) j=0 l Slarly, the sg of [ s] s ( ) ( 2 )(+ 2), the sg of [ + t] s s=0 ( ) ( 2 )l( 2+l 2), ad the sg of l t=0 [++ j+] s ( ) ( 2 )( l)(++2 l+2+2) j=0 Thus the expoet of of the whole ter s 2 l+2l+4l 2l 2 +2l+2l = + l (od 2) Hece the prevous theore ca be rewrtte as follows: 4

50 Corollary 46 Let,,, l 0 Let l ad 0 (,, l) The l = ( ) +l a ( l) [ j] [ s][ s] l j=0 s=0 l [ + t][ + t] t=0 [ ] l l a 2 j=0 [++ j+] 44 Applcatos I ths secto we relate the coeffcet to the theta graph evaluato 0 S 2 ad use the bubble forula to copute the tal of the ot The Theta Graph A theta graph s a sp etwor S 2 that plays a portat role coputg arbtrary sp etwor S 2 The evaluato of a theta graph S(S 2 ) s equvalet to fd the evaluato of the se eleet Fgure 43 Explct deterato FIGURE 43 The se eleet Λ(,, ) of ths se eleet s doe [9] ad [25] We wll deote ths se eleet by Λ(,, ) Our frst edate applcato of the bubble expaso forula s coputg ths se eleet The followg lea shows that s 0 alost equal to the evaluato of ths se eleet 42

51 Lea 47 = 0 + (43) Proof We apply the bubble se forula (48) o that bubble B,,(, ) appears Fgure 43: = (,,) =0 - - The prevous suato s zero except whe = 0 ad hece t reduces to 0 = 0 + Rear 48 I the prevous lea oe could apply the bubble se forula o the other bubbles the se eleet 43 ad obta = 0 + = 0 + = 0 + Ths reflects the syetry of the theta graph 43

52 442 The Head ad the Tal of Alteratg Kots We recall here soe basc facts about the tal of the colored Joes polyoal Recall that B(D) deotes the B-graph assocated wth the the state S B (D) The reduced B-graph of D, deoted by B (D), s defed to be the graph that s obtaed fro B (D) by eepg the sae set of vertces of B(D) ad replacg parallel edges by a sgle edge See for 55 a exaple We defe the A graph slarly FIGURE 44 The ot 4, ts A-graph (left) Its reduced A-graph (rght) Let D be a l dagra ad cosder the se eleet obtaed fro S B (D) by decoratg each crcle S B (D) wth the th Joes-Wezl depotet ad replacg each dashed le S B (D) wth the (2) th Joes-Wezl depotet See Fgure 45 for a exaple Wrte S () (D) to deote ths se eleet B FIGURE 45 Obtag S () (D) fro a ot dagra D B 44

53 We wll eed the followg theore Theore 49 (C Arod [4]) Let L be a l S 3 ad D be a reduced alteratg ot dagra of L The J,L (A) = 4(+) S () B (D) Theore 49 has a portat cosequece, whch bascally tells us that the tal (the head) of a alteratg l oly depeds o the reduced A-graph (reduced B-graph): Theore 420 (C Arod, O Dasbach [3]) Let L ad L 2 be two alteratg ls wth alteratg dagras D ad D 2 If the graph A (D ) cocdes wth A (D 2 ), the T K = T K2 Slarly, f B (D ) cocdes wth B (D 2 ), the H K = H K2 Fdg a exact for for the head ad tal seres s a terestg tas Explct calculatos were doe o the ot table to detere these two power seres [3] Usg ultple techques Arod ad Dasbach detered the head ad tal for a fte faly of ots ad ls The ot 8 5, Fgure 46, s the frst ot o the ot table whose tal could ot be detered by a drect applcato of techques [3] FIGURE 46 The ot

54 The techques we developed here appear to be helpful uderstadg the head ad the tal for soe ots We deostrate ths by studyg the faly of ots Fgure 47 Note that we obta the ot 8 5 fro ths faly by replacg each crossg rego by a sgle crossg FIGURE 47 The ot Γ The reduced B-graph for each ot ths faly ca be easly see to cocde wth the graph Fgure 54 FIGURE 48 The reduced B-graph for Γ Hece the se eleet S () B for ay ot fro the faly of the ots show 47 s gve Fgure 49 Note that the bubble se eleet appears ultple places ths se eleet FIGURE 49 The se eleet S () B (Γ) 46

55 Let Γ be ay ot fro the faly Fgure 47 Theore 49 ples J,Γ (A) = 4(+) Rear 42 The algorth of Masbau ad Vogel [25] ca be used to copute the evaluato of ay quatu sp etwor S(S 2 ) I partcular, t ca be used to gve a forula for the se eleet S () B (Γ) However, t s dffcult to copute the tal of Γ usg the forula obtaed fro ths algorth For ths reaso, we wll use the techques we developed here to copute the evaluato of the se eleet S () B (Γ) Now we copute the tal of the ot Γ Lea 422 J,Γ (A) = 4(+) =0 j=0 j j j 0 j j 0 + +j Proof We use the bubble expaso forula o the top left bubble the se eleet S () (Γ) we obta: B J,Γ (A) = 4(+) =0 - - Usg propertes of the Joes-Wezl projector J,Γ (A) = 4(+) =

56 Usg the detty (24) J,Γ (A) = 4(+) = Now apply the bubble expaso forula to the lower ostrght bubble J,Γ (A) = 4(+) =0 j=0 2 j + - -j -j j j = =0 j=0 2 2 j + +j - -j j j 48

57 Slarly, we apply the bubble expaso forula o the top bubble that appears the prevous equato J,Γ (A) = 4(+) =0 j=0 (j, ) =0 j j j j + - -j j j = =0 j=0 (j, ) =0 j j j j - -j j Note that the prevous su s zero uless = 0 Hece j J,Γ (A) = 4(+) =0 j=0 j j j -j j 49

58 Slarly j j J,Γ (A) = 4(+) =0 j=0 j j j j j Applyg the bubble expaso oe last te o the ddle bubble prevous suato, we obta J,Γ (A) = 4(+) =0 j=0 (,j) =0 j j j j 0 j 0 j j j The prevous suato s zero uless = 0 Hece J,Γ (A) = 4(+) j =0 j=0 j 0 j 0 j 0 j j j whch ples that J,Γ (A) = 4(+) =0 j=0 j j j 0 j j 0 + +j (44) 50

59 I the ext proposto we wor wth the varable q Recall that A 4 = q Proposto 423 T Γ (q) = (q; q) 2 =0 q +2 (q; q) ( =0 q ( 2( )) 2 q ) Proof Usg Corollary 46 ad the fact that j [ ] = q (2+3j+j2 2 2j)/4 ( q) j (q; q) (45) (q; q) j =0 Oe obtas: = ( ) + q (2+42 2)/4 (q; q)6 (q; q) 3 + (q; q) 2 2(q; q) 2+ (q; q) 2 (q;, (46) q)3 ad j = ( ) q ( )/2 (q; q) +j(q; q) (q; q) +(q; q) 2+j+ (47) (q; q) (q; q) 2 (q; q) j+ (q; q) +j++ 0 Fally Lea 47 ples j +j = Λ(,, j), (48) 0 ad oe could use (56) ad the forula for the theta graph gve [9] or [25] to wrte j +j = ( ) +j+ q (+j+)/2 (q; q) (q; q) j (q; q) (q; q) +j++ (49) ( q)(q; q) + (q; q) j+ (q; q) j+ 0 Puttg (47), (48), ad (49) (44) we obta 5

60 where J,Γ (q) = =0 P (,, j), (420) P (,, j) = ( )+j+ q j 2 +j2 5 2 (q; q) +j (q; q) 5 (q; q) ++2 (q; q) +j+2 (q; q) +3 (q; q) j+3 ( q)(q; q) 2 (q; q)2 j (q; q)6 2 (q; q)3 (q; q) +(q; q) 3 j (q; q) j+(q; q) ++j+ (q; q) 2 +2 j= j Now Slarly, (q; q) (q; q) 2 = = = ( q + ) =0 2 ( q + ) =0 ( q + ) 2 = =0 ( q ++ ) = (q; q) (q; q) 2+ = Moreover, ad Hece, =0 (q; q) 3 + (q; q) 2+ = q O(2 + 3) = P (,, j) = j=0 (q; q) 2++ (q; q) + = =0 = j=0 3+ ( q + ) =0 + =0 3+ ( q + ) ( q + ) = =+ q +2 +j+j 2 (q; q) +j (q; q) 8 ( q)(q; q) 2 (q; q)2 j (q; q)3 (q; q)3 j 52

61 Further splfcato yelds: =0 q +2 +j+j 2 (q; q) +j (q; q) 8 ( q)(q; q) 2 (q; q)2 j (q; q)3 (q; q)3 j j=0 = (q; q) 2 = (q; q)2 ( q) q +2 +j+j 2 (q; q) +j ( q)(q; q) 2 j=0 (q; q)2 j 2 q +2 2( ) (q; q) =0 = =0 q The defto of the quatu boal coeffcets allows us to wrte (q; q) 2 ( q) =0 = q +2 2( ) (q; q) 2 q = (q; q)2 q =0 q +2 (q; q) ( =0 q ( 2( )) 2 q ) Hece T Γ (q) = J,Γ (q) (q) = (q; q) 2 =0 q +2 (q; q) ( =0 q ( 2( )) 2 q ) Usg Matheatca we coputed the frst 20 ters of T Γ (q): T 85 =20 2q + q 2 2q 4 + 3q 5 3q 8 + q 9 + 4q 0 q 2q 2 2q 3 3q 4 + 3q 5 + 7q 6 + 2q 7 4q 8 4q 9 4q 20 5q 2 + 3q q q 24 4q 26 9q 27 8q 28 5q 29 q q 3 + 3q q q 34 0q 35 3q 36 5q 37 2q 38 7q q q q q 44 3q 45 6q 46 28q 47 3q 48 2q 49 2q q 5 + 6q q q q q 56 6q 57 34q 58 48q 59 5q 60 49q 6 32q 62 8q q q q q q q q 70 36q 7 74q 72 99q 73 0q 74 79q 75 52q 76 7q q q q q q q q 84 0q 85 82q 86 45q 87 74q 88 82q 89 60q 90 5q 9 37q q

62 9q q q q q q q 00 7q 0 26q q 03 33q q q q 07 57q 08 28q q 0 +24q + 343q q q q q 6 +22q 7 +4q 8 50q 9 324q 20 54

63 Chapter 5 The Tal of a Quatu Sp Networ ad Roger-Raauja Type Idettes 5 Itroducto I ths chapter we geeralze the study of the tal of the colored Joes polyoal to study the tal of certa trvalet graphs S 2 I partcular, we study the tal of a sequece of adssble trvalet graphs wth edges colored or 2 We use local se relatos to uderstad ad copute the tal of these graphs We also gve product forulas for the tal of such trvalet graphs Fally, we show that our se theoretc techques aturally lead to a proof for the Adrews-Gordo dettes for the two varable Raauja theta fucto as well to correspodg ew dettes for the false theta fucto Ths chapter s based o our wor [0] Se theoretc techques have bee used [3] ad [4] to uderstad the head ad tal of a adequate l It was prove [3] that for a adequate l L the frst (+) coeffcets of th ureduced colored Joes polyoal, cosderg the q varable, cocde wth the frst ( + ) coeffcets of the evaluato S(S 2 ) of a certa se eleet S 2 We deostrate here that ths se eleet ca be realzed as quatu sp etwor obtaed fro the l dagra D Hece, studyg the tal of the colored Joes polyoal ca be reduced to studyg the tal of these quatu sp etwors A quatu sp etwor s a baded trvalet graph wth edges labeled by oegatve tegers, also called the colors of the edges, ad the three edges eetg at a vertex satsfy soe adssblty codtos The a purpose of ths chapter 55

64 s to uderstad the tal of a sequece of plaer quatu sp etwor wth edges colored or 2 Our ethod to study the tal of such graphs reles aly o adaptg varous se theoretc dettes to ew oes that ca tur be used to copute ad uderstad the tal of such graphs Studyg the tal of these graphs va local se relatos does ot oly gve a tutve ethod to copute the tals but also deostrates certa equvalece betwee the tals of dfferet quatu sp etwors as well as the exstece of the tal of graphs that are ot ecessarly derved fro alteratg ls The q-seres obtaed fro ots ths way appear to be coected to classcal uber theoretc dettes Ha [4] realzed that that Rogers-Raauja dettes appear the study of the colored Joes polyoal of torus ots I [4] Arod ad Dasbach calculate the head ad the tal of the colored Joes polyoal va ultple ethods ad use these coputatos to prove uber theoretc dettes I ths chapter we show that the se theoretc techques we developed here ca be also used to prove classcal dettes uber theory I partcular we use se theory to prove the Adrews-Gordo dettes for the two varable Raauja theta fucto, as well as correspodg dettes for the false theta fucto 52 Bacgroud I ths secto we gve the deftos of the geeral Raauja theta fucto ad false theta fuctos ad we lst soe of ther propertes 52 Roger Raauja type dettes The geeral two varable Raauja theta fucto, see [2], s defed by : f(a, b) = a (+)/2 b ( )/2 + =0 a ( )/2 b (+)/2 (5) = 56

65 The defto of f(a, b) ples f(a, b) = f(b, a) The Jacob trple product detty of f(a, b) s gve by f(a, b) = ( a; ab) ( b, ab) (ab, ab), It follows edately fro the Jacob trple product detty that f( q 2, q) = (q; q) The fucto f(a, b) specalzes to (5) f( q 2, q) = ( ) q (2 +) q ( )/2 + =0 ( ) q (2 ) q (+)/2 (52) = The Adrews-Gordo detty for the Raauja theta fucto s gve by where j f( q 2, q) = (q, q) = s=j Raauja detty l =0 l 2 =0 l =0 ( j ( j +)) j= q (53) (q, q) lj l s Ths detty s a geeralzato of the secod Rogers- j= f( q 4, q) = (q, q) =0 q 2 + (q, q) (54) 2 The geeral two varable Raauja false theta fucto s gve by (eg [23]): I partcular Ψ(a, b) = a (+)/2 b ( )/2 =0 a ( )/2 b (+)/2 (55) = Ψ(q 2, q) = q 2 +( ) =0 = q (2 )+ (56) 57

66 We wll show that the Adrews-Gordo dettes (53) have correspodg dettes for the false Raauja theta fucto: where j Ψ(q 2, q) = (q, q) = s=j l =0 l 2 =0 l =0 ( j ( j +)) j= q (57) (q, q) 2 2 l (q, q) lj l s The latter detty s a geeralzato of the followg detty (Raauja s oteboo, Part III, Etry 9 [6]) j= Ψ(q 3, q) = (q, q) =0 q 2 + (q; q) 2 (58) Usg se theory, we recover ad prove the dettes (53) ad (57) Theore Exstece of the Tal of a Adequate Se Eleet I [3] C Arod proves that the tal of the colored Joes polyoal of alteratg ls exst Ths was doe by provg that the tal of the colored Joes polyoal of a alteratg l L s equal to the tal a sequece of certa se eleets S(S 2 ) obtaed fro a alteratg l dagra of L I fact, Arod proved ths for a larger class of ls, called Adequate ls Followg [3], we brefly recall the proof of exstece of the tal the colored Joes polyoal ad we llustrate how ths ca be appled to our study Every alteratg l L duces a faly of adequate se eleets S 2 Let D be a alteratg dagra of L These adequate se eleets are the se eleets S () (D) that we troduced the prevous chapter Arod proved B that the tal of faly {S () B (D)} N exsts by showg that S (+) B (D)(q) = + S () B (D)(q) usg three basc steps: 58

67 Sce the l dagra D s alteratg oe ca observe that S (+) B (D) s a adequate se eleet ad t actually loos locally le Fgure 5 + FIGURE 5 A local pcture for S (+) B (D) Furtherore, we have the followg equalty + = + Ths s doe by usg the recursve defto of the Joes-Wezl depotet ad showg that all of the other ters resultg fro applyg the recursve defto of the depotet do ot cotrbute to the frst + coeffcets of S (+) B (D) 2 Step oe ca be appled aroud the crcle utl we reach the fal depotet: = + = + ad fally oe ca show that reovg the crcle colored do ot affect the frst + coeffcets of S (+) B (D) 3 Step oe ad two ca be appled o every crcle S (+) B (D) ad evetually we reduce S (+) B (D) to S () B (D) 59

68 Now let D = {D (q)} N be a sequece of se eleets S(S 2 ) The prevous dscusso ples that the tal of the sequece D exsts wheever the se eleets D (q) are adequate O the other had, we ow that the tal of the sequece D exsts f ad oly f D + (q) = D (q) Ths codto ad the prevous proof suggest that adequateess of every dagra the faly D ay be a ecessary codto for the tal of D to exst Ths s ot true however, ad the ext secto we gve fte faly of sequeces of adequate se eleets whose tal exst See Exaple Coputg the Tal of a Quatu Sp Networ Va Local Se Relatos Let D be a plaer trvalet graph Recall that a adssble colorg of D s a assget of colors to the edges of D so that at each vertex, the three colors eetg there for a adssble trple Cosder a sequece of adssble quatu sp etwors {D } N obtaed fro D by labelg each edge by or 2 Recall that the evaluato of the quatu sp etwor D the se odule S(S 2 ) gves geeral a ratoal fucto Usg defto 3 ad rear 32 oe could study the tal of the sequece {D } N I ths secto we wll study the tal of such se eleets We start ths secto wth a sple calculato for a certa coeffcet of a crossgless atchg dagra the expaso of the Joes-Wezl projector ad we use ths coeffcet to derve our frst local se relato We the use the bubble se relato to copute ore coplcated local se relatos Rear 5 Sce we wll be worg closely wth dettes such as the bubble expaso equato t wll be easer to wor wth Joes-Wezl projectors tha to wor wth trvalet graphs For ths reaso we wll ot state our results ters of trvalet graphs otato 60

69 Oe ca regard ay se eleet Γ the lear se space T a,a 2,,a as a eleet of the dual space T a,a 2,,a Ths s doe by ebeddg the space T a,a 2,,a S 2 ad wrg the outsde soeway to obta a se eleet S(S 2 ) Let Γ be a eleet of the se space T a,a 2,,a ad let x be a wrg the ds S 2 that s copleetary to T a,a 2,,a wth the sae specfed boudary pots Deote by Γ to the eleet T a,a 2,,a duced by the se eleet Γ We call the se eleet Γ (x) S(S 2 ) a closure of Γ I the followg defto we assue that α ad β are adssble trvalet graphs wth edges labeled or 2 the se space T a,a 2,,a, where a {, 2}, wth α ad β are the correspodg dual eleets Defto 52 Let α,β,α ad β be as above Let S be a subset of T a,a 2,,a We say that α = β o S f α (x) = β (x) for all x S Rear 53 The set S etoed the defto ca be chose to be the set of all wrg x such that the se eleets α(x) ad β(x) are adequate However, adequateess sees to be uecessary soe cases ad oe could loose ths codto o the set S further We wll gve exaples of such cases ths chapter Ideally, the set S s supposed to be the set of all wrg x such that the tal of the se eleets α(x) ad β(x) exst It s ot ow to the author what s the largest set for whch ths codto holds 6

70 Rear 54 If we are worg wth T a,b,c, where (a, b, c) {(2, 2, 2), (,, 2)}, the for ay se eleet α T a,b,c we ca wrte α = P (q)τ a,b,c (59) for soe ratoal fucto P (q) Hece f x s a eleet T a,b,c the the tal of the sequece {α (x)} N exsts f ad oly f the tals of the sequeces {τ a,b,c (x)} N ad {P (q)} N exst I partcular, 59 also ples that f the tal of {P (q)} N exsts ad x s a wrg T a,b,c such that τa,b,c (x) s a adequate se eleet, the the tal of the sequece {α (x)} N exsts Note that for every such x oe has P (q) = α (x)/τ a,b,c (x) Followg Morrso [27], wrte coeff f () (D) to deote the coeffcet of the crossgless atchg dagra D appearg the th Joes-Wezl projector We wll use Morrso s recursve forula to calculate certa coeffcets of the Joes-Wezl depotet The recursve forula s explaed very well [27], see Proposto 4 ad the exaples wth, ad we shall ot repeat t here Lea 55 coeff f (2) = ([]!)2 [2]! (50) 62

71 Proof Applyg Morrso s ducto forula, Proposto 4 [27], o the left had sde of (50), we obta coeff = [] coeff f (2) [2] f (2 ) 2 [][ ] = coeff [2][2 ] f (2 2) = []! 2 =+ [] = ([]!)2 [2]! 2 2 Proposto 56 For all adequate closures of the eleet τ,,2 ad for all 0: = (q; q) (5) 2 Proof Wrte Γ to deote the se eleet that appears o the left had sde of FIGURE 52 Expadg f (2) Cosder the depotet f (2) that appears Γ sde the square Fgure 52 ad expad ths eleet as a Q(A)-lear suato of crossgless atchg dagras Every crossgless atchg dagra ths expaso, except for the dagra that appears Fgure 53, s gog to produce a hoo to the botto 63