1 Introdution Ever sine the seminl ppers [] nd [W], mthemtiins nd physiists hve een interested in the prolem of onstrution of topologil quntum eld the

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1 Stte-Sum Invrints of 4-Mnifolds, I Louis rne Deprtment of Mthemtis Knss Stte University Mnhttn, KS Louis H Kumn Deprtment of Mthemtis, Sttistis nd omputer Siene University of Illinois t higo 851 S Morgn Street higo, IL Dvid N etter Deprtment of Mthemtis Knss Stte University Mnhttn, KS strt: We provide, with proofs, omplete desription of the uthors' onstrution of stte-sum invrints nnouned in [], nd its generliztion to n ritrry (rtinin) semisimple tortile tegory We lso disuss the reltionship of these invrints to generliztions of rod's surgery invrints [r1,r2] using tehniques developed in the se of the semi-simple su-quotient ofrep(u q (sl 2 )) (q prinipl 4r th root of unity) y Roerts [Ro1] We riey disuss the generliztions to invrints of 4-mnifolds equipped with 2-dimensionl (o)homology lsses introdued y etter [6] nd Roerts [Ro2], whih re the sujet of the sequel HEP-TH

2 1 Introdution Ever sine the seminl ppers [] nd [W], mthemtiins nd physiists hve een interested in the prolem of onstrution of topologil quntum eld theories From the eginning of the sujet, the indition hs een tht the most importnt exmple would e in dimension four, nmely, Donldson- Floer theory The elief, whih hs never een rigorously sustntited in generl, is tht 4D-TQFTs n e onstruted, whih would generlize the line of development initited y Donldson [D] (f lso [DK]), whih hs led to the reent dvnes in our knowledge of the smooth strutures on 4 mnifolds If we exmine the urrent progress of the eorts t onstrution of TQFTs, we see tht there is very deep gp etween the sitution in dimensions 2 nd 3, nd tht in dimension 4 The piture in D2,3 is tht the TQFTs n e diretly onstruted in vrious wys y onneting strutures from strt lger (inluding tegoril lger) to vrious deompositions of mnifolds For exmple, in D2, we n onstrut TQFTs either y onneting hndleody deomposition to ommuttive Froenius lger, or y onneting tringultion to semisimple lger [FS] In D3, we n either produe theory y relting Heegrd splitting (or more generlly, hndleody deomposition) or tringultion to Hopf lger or modulr tensor tegory [TV,r1,Ku] More reently, it hs lso een shown how to get 3D theory y onneting surgery presenttion to Hopf lger [RT] (f lso [KLi]) In ontrst, the piture in D4 is muh less ler side from trivil exmples involving nite groups, the 4D theories under tive study hve not een onstruted in generl Speil ses re omputed, either y extremely diult methods involving nlysis on moduli spes of instntons [D,DK], or, on nonrigorous level, y mens of tehniques from quntum eld theory [W1] The purpose of this pper is to egin to ridge the gp etween the 3D nd 4D situtions In [], the uthors produed new 4D-TQFT y using methods nlogous to the lower dimensionl onstrutions More speilly, we showed how to produe 4D-TQFT from the tegory of representtions of the quntum group SU(2) q with q root of unity The present pper provides forml proofs nd generl setting for the onstrution We prove the theorem tht the nlog of the onstrution in [] for n rtinin semisimple tortile tegory gives rise to 4D-TQFT We lso present n pproh to proving tht stte sum is topologil, the lo property, whih myhve pplitions in other settings In our originl se, it hs een demonstrted in [K1] nd [Ro1], tht the invrint whih we otin for losed 4 mnifold is omintion of the Euler hrter nd the signture In prtiulr, our formul gives new solution to the lssil prolem, rst solved y Gelfnd, of nding omintoril formul for the signture of 4 mnifold There remins the question of how losely our theory is relted to Donldson-Floer theory One might nively think, tht sine the invrints we tth to losed mnifolds re topologil, ie, not dependent on smooth struture, tht there would e no hope of ny signint reltionship The tul sitution is somewht more omplex, nd turns on the issue of insertions It is quite generl phenomenon tht onstrutions of TQFTs n e extended to give invrints of mnifolds with lelled imedded sumnifolds For exmple, the SW 3D-TQFT n e esily extended to give invrints of frmed lelled grphs, whih re, in ft, generliztions of the ones polynomil [W2, r1, W RT] If we ould onstrut DF theory s TQFT,its topologil signine would depend, in n essentil mnner, on twotypes of insertions, one on surfes, nd one on points The insertion on points orresponds to twist in the undle, ie to hnging the seond hern lss of the undle in whih the DF theory tkes ple The insertion on surfes is represented in the piture of DF theory from moduli spes, y restriting to sumnifolds of moduli spe orresponding to onnetions whih hve nonvnishing index when restrited to the surfes In [W3], Witten formlly reprodues DF theory, in the speil se of Khler mnifolds, s TQFT with just suh insertions, lthough in 2

3 nonrigorous pproh It is then nturl to sk if we n nd topologilly invrint wys of modifying our stte sum to inlude insertions, nd if so, whether they produe results relted to DF theory Our investigtion of this question is so fr inomplete, ut the results re interesting In [2] nd [Ro2], two losely relted proedures hve een found for modifying our formul to inlude insertions on surfes These presriptions re invrint under homotopy of the surfes In [Ro2], it ws demonstrted tht the invrint of mnifold with emedded surfes ounts the intersetion numers of the surfes This is n intriguing result, sine there hs reently ppered some new informtion out DF theory, dedued rigorously in [KM] nd nonrigorously (ut very eutifully) in [W3] The formul these soures derive shows tht the generting funtion for the DF invrints n e expressed s n exponentil involving the Euler hrter nd signture, times qudrti exponentil involving the intersetion form, times sum of \sudominnt" exponentils whih re sensitive to the smooth struture of the mnifold It is not possile to see the eet of the sudominnt exponentils without looking t terms orresponding to twisted undles It follows tht the question of how muh informtion our theory n detet is losely onneted to the question of whether we n nd nturl wy to modify it to inlude twists in the undle, nd wht eet they hve on the sum We hve not solved this prolem s of this writing, ut we see severl nturl pprohes Unfortuntely, the dimension of the vetor spe whih our TQFT ssigns to S 3 is 1, so it is hrd to see how n \instnton" ould ontriute nything more thn multiplier Thus, it is still not ompletely ler whether nturl modition of our expression will mke it sensitive to smooth struture, ut it is most prole tht we hve reonstruted the titious ousin of DF theory disussed in [W3] Our formul is not the only possile pproh to 4D stte sum In [F], nother pproh is outlined, mking use of more sutle piee of lgeri struture, Hopf tegory There is reson to hope tht the entire piture in 4D n e rendered s lgeri s the lower dimensionl ses Physil pplitions Let us lso mention the possiility tht 4D topologil stte sums my ply role in the prolem of quntizing grvity The rst piee of evidene we n ite is the work of Regge nd Ponzno [RP] on spin networks They reinterpreted Penrose s spin network [P] pproh tn quntum grvity y using the tehniques of the grphil lulus to rewrite the evlution of spin network s sort of disretized pth integrl The form of their expression is identil to the TQFT of Turev nd Viro [TV], exept tht they use Lie group insted of quntum group Their stte sum gives n interesting pproh to quntum theory for 3D grvity It is then nturl to wonder if 4D stte sum model ould ply similr role This leds into gret omplexities of interprettion, ut see [r2] for possile pproh topologil stte sum hs mny ttrtive fetures s tool to desrie quntum theory of grvity It oupies position intermedite etween pth integrl for ontinuum theory nd lttie pproximtion to the theory, s sort of mgi lttie theory whih is invrint under ny hnge of the lttie This resontes niely with the old ide tht it is not possile to mesure the distne etween two physil points nd get vlue less thn the Plnk sle In summtion, topologil 4D stte sums re very new onstrution, whose possiilities hve not een explored fully, whih my hve mny pplitions Throughout, ll mnifolds re ssumed to e pieewise-liner (equivlently smooth) oriented, nd unless stted to the ontrry to hve empty oundry 3

4 2 Tortile tegories This setion hs two susetions In the rst, we review the properties of two versions of the reoupling theory ssoited with the quntum group U q (sl 2 ) These reoupling theories nd their properties n e used diretly to uild the simplest ses of the 4-mnifold invrints disussed in this pper This is the exmple whih hs een most fully understood The reder who is interested primrily in this se of the onstrution n red the rst susetion, then proeed diretly to Setion 3 The seond susetion desries the generl setting for our onstrution in terms of semi-simple tortile tegories This generl setting for the invrints is of gret potentil vlue sine it gives frmework in whih future pplitions to quntum groups or other tegories n e rdled 21 U q (sl 2 ) Reoupling Theory In this setion we give quik resumeoftwoversions of U q (sl 2 ) reoupling theory nd the reltionships etween them oth formultions re useful in studying our 4-mnifold invrints, nd in the following setions we shll express the invrints in terms of oth We egin with review of the knot-theoreti nd omintoril Temperley-Lie reoupling theory The prinipl referene for this version is [KLi] We shll refer to this s the TL theory The TL theory is sed on the rket polynomil model for the originl ones polynomil [] Rell tht the rket polynomil < K > is Lurent polynomil in stisfying the reltions in Figure K ( 2 2 ) K Figure 1: xioms for the rket Polynomil In the rst eqution, the smll digrms stnd for lrger link digrms diering only t the site shown In the seond, the irle stnds for n extr omponent disjoint from the rest of the digrm We let d 2 2 The rket evlution is n invrint of regulr isotopy the equivlene reltion generted y the Reidemeister moves II nd III, shown in Figure 2 Under the rst Reidemeister move, we hve the equtions of Figure 3 from whih it is redily seen tht the rket is, moreover, invrint under frmed isotopy the equivlene generted y Reidemeister Moves II nd III, nd the \frmed rst Reidemeister move" of Figure 4 n oservtion whih is importnt to the pplitions of the rket to surgery desriptions of mnifolds, nd to the tegoril formultion of the seond susetion The reoupling formultion sed on the rket polynomil involves pplying the rket polynomil to links with prllel les Prllel ling is indited y lelling link omponent with 4

5 II - ` `ZZ III - Figure 2: Regulr Isotopy Reidemeister Moves II nd III 3 3 Figure 3: The rket under Reidemeister I n integer 0 Tht omponent is then repled y prllel omponents 1 n exmple is given in Figure 5 We then dene q-symmetrizers (here p q) y the formul of Figure 6 in whih fg! P 2S n ( 4 ) t(), S n is the permuttion group on n letters, ^ is the usul lift of to positive rids, nd t() is the lest numer of trnspositions needed to express One then hs tht the q-symmetrizers re projetors in the sense given in Figure 7 in whih i rnges etween 1 nd 1, nd U i is the stndrd genertor for the Temperley-Lie lger shown in Figure 8 Equtions of the sort in Figure 7 men identities for rket evlutions of losed digrms Herefter, we omit the rkets when writing lelled link digrms in lgeri ontexts Thus, for exmple, we use the expression in Figure 9 insted of tht in Figure 10 Other useful formule out the symmetrizers re given in Figure 11 1 The reder must rememer tht we re deling with digrms, if one insists on thinking of the link in 3-dimensionl spe, we re using the \lkord frming" 5

6 Ẋ - Figure 4: The Frmed First Reidemeister Move T TT T T T T TT T" ""T T TT TT T T T TTT T" ""T T Figure 5: Exmples of ling The q-symmetrizers re used to uild 3-verties nd these re the ore of the reoupling theory They re dened in Figure 12 in whih i + j ; i + k ; j + k The (; ; )vertex exists extly when + + is even nd the sum of ny two elements of the set ; ; is greter thn or equl to the third (We will osionlly write verties where these onditions fil, the evlution of ny digrm with suh non-existent vertex is 0) One n hek tht rossings nd 3-verties re relted s in Figure 13 in whih x 0 x(x + 2) Other properties, suh s tht shown in Figure 14 follow diretly from the genesis of the 3-vertex in terms of rket evlution Two importnt network evlutions hve expliit formuls tht we omit here (see Kumn nd Lins [KLi]) These re the thet net, (; ; ), shown in Figure 15 nd the tetrhedrl net, Tet " e d f shown in Figure 16 typil instne of Shur s lemm (see the seond susetion) in the TL theory is the formul of Figure 17 in whih ;d is the Kroneker delt Reoupling of 3-verties is given in terms of q-6j symols, whih we write with susript TL to indite the Temperley-Lie lger formultion s in Figure 18 These q-6j symols hve n expliit expression in terms of net-evlutions given y ; 6

7 1 fg! ( 3 ) t() ^ 2S n Figure 6: q-symmetrizers ( i d j ) " i Tet i d j (; d; i)(; ; i) : The q-6j symols stisfy orthogonlity nd iedenhrn-elliot identities They n e used to onstrut 3-mnifold invrints, nd, s we shll see, 4-mnifold invrints The TL theory is rooted in the omintoris of link digrms, nd it is diret generliztion (q-deformtion) of Penrose spin networks (f Penrose [P]) Its dvntge to us here is tht there is no dependene in the digrmmtis of the TL theory on mxim or minim or on the orienttion of digrms with respet to diretion in the plne Thus TL networks n e freely emedded in mnifolds, feture tht we shll use in lter setions The Kirillov-Reshetikhin formultion of U q (sl 2 ) reoupling theory (f Kirillov nd Reshetikhin [KR]) is sed diretly on the representtion theory of the quntum group U q (sl 2 ) lthough it lks the geometri nturlness of the TL theory oserved in the lst prgrph, the KR theory hs good tegoril propeties: the verties (in two vors two-in-one-out nd one-in-two-out reding from top to ottom) re projetions nd inlusions from tensor produt of irreduile representtions to its irreduile diret summnds It is the KR theory whih is generlized to ritry semisimple tortile tegories in the next susetion The si informtion needed to trnsform KR nets into TL nets is the reltionship etween their 3-verties shown in Figure 19, in whih (; ; ) is the TL evlution of thet-net nd we indite the theory y lel on the vertex Note tht the KR vertex now hs distinguished diretion (sine the lel on the downwrd leg is distinguished from the others), while the TL vertex does not hve ny distinguished diretion dependent on leg plement The vlue of losed loop i is the sme in oth formultions 2 2 The reder should note tht KR nets re often lelled with hlf-integrl \spins" In tht onvention, the legs of the KR vertex would e lelled y ; ; Tovoid ompliting nottion, we shll lel oth KR nets nd TL nets with integers (twie spin, numer of les, or (non-quntum) dimension 1) 7

8 0 U i Figure 7: i i+1 Figure 8: The i th Temperley-Lie Genertor PP QQ Q Q P P Figure 9: rkets Hidden 8

9 Q Q P P PP QQ Figure 10: rkets Shown PP QQ Q Q P P ( 1) PP ( 1) (+2) 1 ( ) Figure 11: Useful Formule 9

10 j i k Figure 12: Denition of 3-verties!! ( 1) Figure 13: riding t 3-vertex d d ZZP P Figure 14: riding pst 3-vertex S S S S SS S S (; ; ) Figure 15: The Thet Net (; ; ) 10

11 PP f d Ṗ P e Tet " e d f Figure 16: The Tetrhedrl Net (;;) ;d d Figure 17: Shur s lemm in TL theory j d i ( i d j ) TL Q QQQ QQ i QQ Q Q d Q Q Figure 18: Reoupling vi q-6j symols 11

12 KR p p (;;) TL Figure 19: KR 3-verties in terms of TL 3-verties 12

13 The formul relting the KR vertex nd the TL vertex lets us derive the identities of KR theory involving ups nd ps diretly In the next susetion, we will see tht similr identities re generl phenomenon for projetion nd inlusion mps for diret summnds of tensor produts in semi-simple tortile tegories For exmple, we hve Proposition 21 KR p p KR Figure 20: Rottionl properties of KR verties s derived from TL theory Proof: Given in Figure 21 In this omprison, the ups nd ps of KR theory re the sme s those of TL theory, nd thus n lelled up (resp p) is the sme s TL (; ; 0) 3-vertex in the pproprite orienttion or p times KR 3-vertex with two upwrd legs lelled, nd downwrd one lelled 0 (resp times KR 3-vertex with two downwrd legs lelled nd n upwrd one lelled 0) In the next susetion, we develop the generl form of the tegoril dt needed for our onstrutions KR nd TL reoupling networks provide two formultions of the most fundmentl exmple: the representtion theory of U q (sl 2 ) 13

14 KR TL p p (;;) p p (;;) TL p p (;;) p (;;) p KR p p KR Figure 21: Proof of Proposition 21 14

15 22 Generl semi-simple tortile tegories The initil dt required for our onstrution is the sme s tht required for the onstrutions of etter [3]: semisimple tortile tegory over eld K Non-degenery onditions s in Turev [T] will e required only for the surgil versions given in Setion 4 We review the neessry xiomtis nd tegoril results The xioms fll into two types: those deling with the linerity struture over eld K, nd those deling with the monoidl nd dulity struture of the tegory We egin with the ltter We ssume fmilirity with the si notions of monoidl tegory theory nd elin tegory theory (f M Lne [WM]) nd with si notions ssoited with tegories of tngles (f Freyd/etter [F1,F2], oyl/street[s1,s2], Resetikhin/Turev [RT], Shum [S], etter [1]) Our tegories will ll e K-liner elin monoidl tegories with ext in oth vriles, ut will e equipped with dditionl struture One piee of struture we will require is the presene of dul ojets: Denition 22 right (resp left) dul to n ojet in monoidl tegory is n ojet (resp )equipped with mps :! I nd : I! (resp e :! I nd h : I! ) suh tht the omposites 1! I! ( ) 1! ( )! I! nd 1! I! ( )! ( )! I! (resp 1! I h! ( )! ( ) e! I! nd 1! I h! ( )! 1 ( ) e! I! ) re oth identity mps Oserve tht hoie of right (resp left) dul ojet for eh ojet of (smll) monoidl tegory extends to ontrvrint monoidl funtor from to its opposite tegory with reversed, nd tht there re nonil nturl isomorphisms k : ( )! nd :( )! The tegories we onsider will hve two-sided duls To mke sense of this in the non-symmetri setting, we need Denition 23 [Freyd/etter [F2]] monoidl tegory is sovereign if it is equipped with hoie of left nd right duls for ll ojets, nd (hosen) monoidl nturl isomorphism :! suh tht nd I 1 I k ( 1 ) 15

16 Denition 24 [Shum [S]] tortile (tensor) tegory is monoidl tegory (;;I;;;) in whih every ojet hs right dul, equipped moreover with nturl isomorphisms ; :! (the riding) nd :! (the twist) stisfying the hexgons ( ; ) ;; ( ; ) ;; ; ;; for 1 lne ; ; ( ) self-dul Of ourse, the presene of riding mkes right duls into left duls y e 1 nd h However, these re not two-sided duls in the sense of Denition 23 unless the tegory stises the lne xiom Indeed, we hve Proposition 25 [Deligne [D]] (f etter [1]) rided monoidl tegory with right duls for ll ojets is lned if nd only if the tegory is sovereign with the hosen right duls nd the left duls of the previous prgrph More preisely, hoie of twist is equivlent to hoie of the nturl isomorphism in Denition 23 Sine our tegories will e tortile, we will onsider hosen right duls s two-sided duls under the struture of the previous Proposition In wht follows, we will use digrmmti nottion, similr to Penrose s [P] nottion for tensors, for mps in our tegories (see ppendix on Digrmmti Nottion elow) Its use is justied y the following theorem of Shum [S] (f lso Freyd/etter [F2], oyl/street [S2], Reshetikhin/Turev [RT], etter[1]): Theorem 26 [Shum [S]] The tortile tegory freely generted y single ojet is monoidlly equivlent to the tegory of frmed tngles The seond sort of struture we require involves the liner nd elin struture on the tegory Denition 27 n ojet in K-liner tegory is simple if [; ] is 1-dimensionl Denition 28 K-liner elin tegory is ompletely reduile if every ojet is isomorphi to diret sum of simple ojets, nd ompletely reduile tegory is semisimple if there re only nitely mny isomorphism lsses of simple ojets If is equipped with n ext monoidl struture, we lso require tht the monoidl identity ojet I e simple In wht follows we will e onerned with K-liner semisimple elin tegories equipped with n ext tortile struture (for K eld) For revity we refer to these s semisimple tortile tegories over K Lemm 29 If S is simple ojet in ny tegory over K, then for ny ojet [; S] nd [S; ] re nonilly dul s vetor-spes proof: The omposition mp : [S; ] [; S] piring 2![S; S] K denes non-degenerte iliner 16

17 Lemm 210 Let e ny semisimple tegory over K with S fmily of representtives for the isomorphism lsses of simple ojets in Ifis ny ojet of, then hoie ofses 1;S ;:::; d S;S [; S] for eh S 2S (where d S dim K [; S]) determines diret sum deomposition M S2S in whih the i;s 's re the projetions onto the diret summnds, nd there re splittings i;s stisfying ds S2S i1 dsm i1 S: i;s i;s 1 nd i;s j;t is the zero mp from S to T, unless (i; S)(j; T ), in whih se it is 1 S proof: Now, y the denition of semisimpliity, dmits diret sum deomposition of the form given, though not priori hving the i;s s s projetions Let p i;s (resp p i;s ) denote the projetions (resp inlusions) of the summnds in this diret sum deomposition Now for eh S 2S,fp i;s g form sis for [; S], while fp i;s g form sis for [S; ] whih is the dul sis under the identition of the previous lemm ut if is the hnge of sis mtrix trnsforming the p i;s s to the i;s s then 1 trnsforms the p i;s s to the i;s s, nd thus the i;s s re the projetions for (generlly dierent) diret sum deomposition 2 We dopt the onvention tht if we hve sis of [; S] or of [S; ] for S ny simple ojet, then for ny sis element, x, x is the orresponding element of the dul sis, nd thus x x Denition 211 If f :! is ny endomorphism in tortile tegory over K then the tre of f, denoted tr(f), is the mp h(f 1 )", orequivlently, the orresponding element of K under the identition of [I;I] with K The dimension of n ojet is dim()tr(1 ) The nme tre follows from the following, originlly proved in the symmetri se y Kelly nd Lplz [KL]: Proposition 212 In ny tortile tegory over K, iff:! nd g :! then tr(fg)tr(gf): The following lemm is n immedite onsequene of the oherene theorem of Shum [S]: Lemm 213 If is ny ojet in tortile tegory over K, nd is its dul ojet, then dim()dim( ): On the other hnd in the presene of the dditive nd liner struture on semisimple tegory over K we hve Lemm 214 If nd re ojets in semisimple tortile tegory over K then dim( )dim()+dim( ): proof: It follows from the extness of () tht h I (h h ) nd " (" " )+ 2 17

18 Lemm 215 If nd re ojets in semisimple tortile tegory over K then dim( )dim()dim( ): proof: It is immedite from the denition of dimension nd the oherene theorem of Shum [S] tht dim( )dim() dim( ) when the dimensions re regrded s endomorphisms of I ut for endomorphisms of I,, omposition, nd multiplition of oeients of 1 I ll oinide 2 The following trivil lemm will e used throughout our onstrution: Lemm 216 (\Shur's Lemm") If is ny simple ojet in semisimple tortile tegory over tr(f ) K with dim() 6 0nd f :! is ny mp, then f is dim() 1 proof: Sine f is slr multiple of 1 y simpliity, it sues to oserve tht tr(f) must e the multiple of dim() y the sme slr 2 We shll ssume in wht follows tht ll of our semisimple tortile tegories re non-degenerte in the sense tht ll simple ojets hve dim() 6 0 n importnt, though esy onsequene of Lemm 216 onerns the ehviour of ses f 1 ; :::; n g nd dul ses f 1 ; :::; n g for [ ; ] nd [; ] under duliztion of one or more ojets Even without Lemm 216 it is ler tht ( ) rries to sis for [ ; ] nd to the dul sis for [ ; ] 3 Similrly, without resort to Lemm 216 we n see tht gives rise to ses for [ ; ] (resp [ ; ]; [; ]; nd [ ;])y pplying to the sis elements then preomposing with 1 nd postomposing with 1 (resp pplying to the splitting then preomposing with 1 h nd postomposing with e 1 ; pplying to the sis elements then preomposing with 1 h ; pplying to the splittings then postomposing with 1 ) (These re represented grphilly in Figure 22) [go through nd put in s] Wht is not immeditely ler is the reltionship etween the rst nd seond (resp third nd fourth) of the trnsformed ses in the previous prgrph In ft, lultion using Lemm 216 provides the following generliztion of the rules in the Kirillov-Reshetikhin [KR] formultion of U q (sl 2 )-reoupling theory: Lemm 217 The ses of [ ; ] nd [ ; ] (resp [ ;] nd [; ] ) otined y multiplying the trnsformed ses desried two prgrphs go y p dim() p dim() re splittings of eh other, moreover giving the projetion nd inlusions of diret sum deomposition for the given tensor produt proof: y pplition of Lemm 216 it sues to show tht the tre of omposition of n element in the seond sis of eh pir with one in the rst is dim( )dim() (resp dim()) if the elements re the multiple of the trnformtion of n element of nd its splitting, nd 0 otherwise ut this follows diretly from Lemm 210 We onlude with tegoril notion, introdued in [K2] whih will e importnt when we onsider the interprettion of the invrints onstruted: 3 Throughout this setion, we re using the oherene theorem of Freyd/etter [F2] for sovereign tegories to suppress mention of ertin nonil isomorphisms, for instne, etween ( ) nd 18

19 L L L "!, "! (( L D D "! l TT T "! ḣh Figure 22: Trnsformtion of ses under prtil duliztion Denition 218 The enter Z() of rided monoidl tegory is the full-sutegory of ll ojets with the property tht 8 2 O() ; ; 1 : rided monoidl tegory hs trivil enter if the enter is the full-sutegory of ojets isomorphi to nite (possily empty) diret sum of opies of I 19

20 3 oloring Tringultions nd Stte-Sum Invrints Throughout this setion we let e xed semisimple tortile tegory, let S e hosen set of representtives for the isomorphism lsses of simple ojets inluding, s representtive of its lss, the hosen monoidl identity ojet I, nd let e hoie for eh triple of elements of ; ; 2S of sis for the hom-spe [ ; ] nd y use of nottion the disjoint union of these ses ssume without loss of generlity tht the hoie of dul ojets hs een mde so tht () indues n involution on S For the reder who hs skipped Susetion 22, the spei exmple of Rep! (U q (sl 2 )), S f0; :::; r 2g nd given y hoosing the KR vertex with upwrd legs lelled nd nd downwrd leg lelled should e onsidered Notes direted to reders interested in this level of generlity will our from time to time The trnsltion from KR to TL reoupling theory is given in Susetion 51 If T is tringultion of 4-mnifold M, we let T (i) denote the set of (non-degenerte) i-simplies of the tringultion In wht follows we will e onerned with ordered tringultions, tht is tringultions equipped with totl orderings of their verties We re now in position to dene the olorings whih index our stte-sums Denition 31 S-oloring (or simply oloring if no onfusion is possile) of n ordered tringultion of 4-mnifold is triple of mps ( : T (2) [ T (3)!S; + :T (3)!; :T (3)!) suh tht + f; ; ; dg2 f;;dg;f;;dg f;;;dg nd f; ; ; dg 2 f;;dg;f;;g f;;;dg, where <<<din the ordering on the verties We denote the set of S-olorings of n ordered tringultion T y S (T) Reders interested in the U q (sl 2 ) se only should note tht in tht se, the ontent of oloring is given entirely y hoie of : T (2) [T (3)!Sfor whih the lels on the positive (resp negtive) prt of the oundry of eh 3-simplex ouple to the lel on the 3-simplex In wht follows, we let N 2S dim()2 nd n i jt (i) j Now, given n ordered tringultion T of 4-mnifold, we n ssign to eh oloring numer dened y N n 0 n 1 dim(()) dim(()) 1 k; k fes tetrhedr 4-simplies where k; k is given y the endomorphism of I grphilly in Figure 23 if the orienttion of indued y the ordering of the verties is the sme s the mient orienttion, nd y the endomorphism of I represented grphilly y the network otined from tht in Figure 23 y mirror-imging the network, pplying ( ) to ll ojet lels nd ( ) to ll lels y mps in For reders interested in the U q (sl 2 ) se, the piture is simpler: the nodes re KR verties with legs s shown in Figure 23 nd the dulizing of lels is unneessry (sine ll representtions of U q (sl 2 ) re self-dul) The min result of this setion is then Theorem 32 The stte-sum (M) 2 S (T) is independent of the hoie of ordered tringultion T, of representtive simple ojets S nd of ses Thus for ny semisimple tortile tegory, ( ) is n invrint of pieewise-liner 4-mnifolds In the se of Rep! (U q (sl 2 )), the invrint is the originl rne-etter invrint of [] 20

21 ^3 L L ^0 () % L % % ^1 () () L L S S L, S \ E E \ E!!! " LL "( (( L LL L LL L L LL LL ^2 LL ^4 L LL () \ \\ () Z ZZZ E \ \\ Z Z () () e ee!!!!! Figure 23: The generlized 15-j symol ssoited to orretly oriented vertex-ordered 4-simplex E EE E EE 21

22 To prove this we use n uxiliry notion Denition 33 d-lo is d-ell equipped with n ordered tringultion of its oundry lthough the initil verition of the invrine of the rne-etter stte-sum [] ws rried out using Phner s moves, the notion of los provides n lterntive method for verifying tht stte-sums on tringultions give rise to PL-mnifold invrints: in generl, one must show tht the stte-sum ssoited to ny lo with n ritrry extension of the tringultion to the interior nd xed initil dt on the oundry is independent of the extension of the tringultion to the interior Oserve tht Phner s moves in ny dimension re of this form t rst, it might pper tht the method suggested ove isworse thn verifying Phner s moves However, in our se (nd potentilly in the se of more rened stte-sums) the use of los llows uniform indutive proof Indeed, it follows from generl priniples enunited in etter [6] tht ny stte-summtion on ordered tringultions in whih weights re ssigned to simplies (or simpliil gs with suitle omptiilities imposed for shred simplies) must stisfy \lo lemm" of this sort if it is to e PL-invrint It will depend on the ext irumstnes whether it will e esier to verify this \lo lemm" indutively or to hek Phner s moves It should lso e noted, tht the use of los provides n immedite hek for fesiility of nding normliztion ftors on lower-dimensionl simplies (or simpliil gs) to mke proposed sttesum topologilly invrint: one must e le to nd presription for weight on lelled lo (for exmple s produt or rtio of reomintion digrms) whih restrits to the proposed weight on n ordered 4-simplex In the present se, we set up the indution (rried out in the proof of Lemm 35) s follows: rst, we desrie network nming n endomorphism of I, nd hene numer, for ny 4-lo in n oriented 4-mnifold, then show tht the stte-sum n e rewritten in terms of deomposition into los (s oserved ove, without regrd to the tringultion of their interiors) This will omplete the proof of invrine, nd independene from the tringultion, while independene from the ordering of the verties will follow from the simple expedient of oserving tht the str of vertex is 4-lo, rewriting with the vertex missing, then reversing the proess to insert the vertex somewhere else in the ordering To onstrut our networks ssoited to 4-los, rst notie tht vertex ordering on tetrhedron in the oundry of 4-lo (or 4-simplex s speil se) together with the orienttion gives rise to hosen side of the tetrhedron (Speilly, in lol oordinte system identied with ll in R 4 the orienttion gives wy to hoose fourth vetor orthogonl to ny given ordered triple of vetors here the vetors re the tngents to the edges from the lowest numered vertex to the others in order We only re out the side the fourth vetor lies on, inside the lo or outside, so the result doesn t depend on wht orienttion preserving mp we used to identify the hrt with the ll) Now, ple the highest numered vertex t 1, identity the rest of the ounding S 3 with R 3, nd hoose plne to projet on nd vertil nd horizontl diretions in the plne In eh 3-simplex ple vertil \dumell" (s in the network for the generlized 15j symol in Figure 23 with ends representing ples to e olored with mps, nd r representing ple to e olored with simple ojet) in plne prllel to the plne of projetion dd rs onneting the \dumells" so tht for tetrhedr with inwrd normls the ottom right (resp top right, ottom left, top left) is onneted through the fe otined y omitting the lowest (resp seond, third, highest) numered vertex of the tetrhedron, while for tetrhedr with outwrd normls, proeed s ove, ut reversing left nd right Finlly, in tetrhedr with inwrd norml vetors ple n overline in the lower end of the \dumell" (to indite tht the mp here will e the splitting of the olor from ), while in those with outwrd norml vetors, ple n overline nd in the upper end of the \dumell" nd 22

23 in the lower end (to indite tht the mp here will e the dul of the splitting, respetively the dul, of the olor from ) For revity it will e onvenient to refer to the \dumells" s \inwrd" or \outwrd dumells" ording to the struture of onnetions nd lellings Oserve Lemm 34 The generlized 15j symol of Figure 23 is preisely the network ssoited to 4-simplex whose order-orienttion grees with the indued orienttion y the proedure just outlined proof: It sues to oserve tht the norml vetor indued on ^0; ^2; ^4 re opposite to those indued on ^1; ^3, nd tht the greement of orienttions implies tht the norml vetor on ^0 is on the sme side of ^0 s 4, nd thus inwrd Finlly we re in position to stte the key lemm in the proof In its proof, we will regrd the stte-sum s n evlution of liner omintions of olored emedded trivlent (rion) grphs interpreted in the now stndrd wy (f Reshetikhin/Turev [RT]) Lemm 35 \The lo Lemm" If M is 4-mnifold equipped with n ordered tringultion T, nd D is 4-lo formed y the union of 4-simplies in T, then the stte-sum N n 0 n 1 dim(()) dim(()) 1 k; k 2 S (T) fes tetrhedr 4-simplies deomposes s oloring of Tj D oloring of Tj D extending N jt (0) [int(d)j jt (0) [int(d)j fes int(d) N jt nint(d)j jt nint(d)j (0) (0) fes oloring of Tj M n Mnint(D) int(d) extending dim(()) dim(()) tetrhedr int(d) tetrhedr M n int(d) dim(()) 1 dim(() 1 4-simplies D 4-simplies M n int(d) k; k k; k nd for eh the rst ftor (the sum on ) isequl to the evlution of the network ssoited to the oundry of the 4-lo D proof: We proeed y indution on the numer n of 4-simplies in D If n 1 there is nothing to show, y the preeding lemm Now suppose n>1, nd we hve shown for the lemm for ll 4-los with n 1 4-simplies Selet 4-simplex in D whih interset the oundry in ell of dimension 3 Then D 0 D n is 4-lo Now oserve tht D 0 \ is union of losed 3-simplies of T, nd tht those ssigned \inwrd dumells" in one of D 0 or re ssigned \outwrd dumells" in the other nd vie-vers Similrly, the pttern of onnetions etween the \dumells" in shred tetrhedr (nd from shred tetrhedr out to unspeied \dumells") will e mirror imges To omplete the proof, it sues to show tht lol evlution of the prt of the digrm desriing the stte-sum whih inludes the ontriutions of, nd ll simplies in D 0 \ is equl to the lol evluttion of the remining fes of series of digrmmti lultions veries this 23

24 k 2 ij k 2 lm k i j "! j "! k k dim 1 (k) "! "! l m m l i i j "! j "! i k k 2 lm k!! 2 ij k l "! m!! M "! l i!! j j S i S!!!!!!!! S S l m m l Figure 24: oining los whih shre one tetrhedron 24

25 k; l; m ; k; m SS \ \ SS k SS!! \ m \ \ S m SS H!! H \\ S m k l l k m \\ \\ S S k S S dim(m)dim 1 (l)dim 1 (k) dim(m)dim 1 (k) k SS SS H!! k ( (( ( H \\ h hhh Q S k SS SS H hhhh hhh ( (( (!! (((( " (((" " H h hhh Q S Figure 25: One se of joining los whih shre two tetrhedr 25

26 smple of the lultions re given in Figures 24, 25, 26, 27, 28 nd 29 The others (in whih other portions of the 15-j symol re involved) re ompletely nlogous, nd re left to the reder Only for the rst (Figure 24) do we give detiled desription of where lels re drwn from Independene of the hoie of representtive simple ojets will follow y inserting isomorphisms etween the hosen representtive ojets round eh of the nodes in the generlized 15-j symol, nd the following lemm, whih shows independene from the hoie of ses Lemm 36 The stte-sum is invrint under the hnge of ses from whih the mp omponents of the lels re hosen on ny one tetrehedron (nd hene onll) proof: It sues to show tht lol evlution of the digrm desriing the stte-sum inluding ll ourenes of the sis elements hosen on prtiulr tetrhedron is independent of the hoie of sis Oserve tht this follows immeditely from the rst of the lultions in the proof of Lemm 35 (Figure 24) This ompletes the proof of Theorem 32 26

27 i; j; k l; m; n dim(l)dim(m)dim(n) N dim(i)dim(j)dim(k) m (( k F TT j E Q QQQ n ( ((( l H H TT E j F k hhh h l n!hh! m ; ; D; E; F i D T T i D m; n; k ; F dim(m)dim(n) Ndim(k) m F l, l k Q QQQ n hhhhhh Z Z P P (( (((( F k PP `` P TT Figure 26: One se of joining los tht shre three tetrhedr (eginning) 27

28 m; k; dim(m) N Z Q Z QQQ Z hhhhhh hhhhhh PP (( ((((" l, m l k k (( (((( " PP `` P TT PP dim(m) N m, l Q l Ẋ, Z T Z TT Z T hhhhhh " (( ((((" PP `` P TT l Z Z Z PP hhhhhh " (( ((((" PP `` P TT Figure 27: One se of joining los whih shre three tetrhedr (onlusion) 28

29 h; i; j; k l; m; n; p; q; r ; ; ; G D; E; F; H dim(l)dim(m)dim(n)dim(p) dim(q)dim(r) N 3 dim(h)dim(i)dim(j)dim(k) (( k r F i D m T T TT E EE j q E Q QQQ n ((( ( l H H G E E q D DD G h H H hhh l n h h p % % D \ D!hh! TT m E p j D r i F k,, dim(p)dim(q)dim(r) N 2 dim(h) p; q; r;h G; H S r S p q % e D G DD G h H H h e e D D DD D DD D DD l ll D DD D D l ll Figure 28: One se of joining los whih shre four tetrhedr (eginning) 29

30 ,, D D DD l ll p; r;h;h dim(p)dim(r) N 2 r!! P P p h H H \ \ h D DD D DD D DD D DD D D l ll,, p; r dim(p)dim(r) N 2 r!! P P p " " \ \ l ll l ll,, l L ll L l LL ll L LL L L Figure 29: One se of joining los whih shre four tetrhedr (onlusion) 30

31 4 Surgil Versions nd n side out 3-Mnifold Invrints t out the sme time s the nnounement of [] ppered, rod [] nnouned the onstrution of 4-mnifold invrint lulted from surgery desription of the 4-mnifold (f Kiry [Ki]) y frmed link with distinguished unlink In this setion, we desrie nlogs of rod s invrint for ritrry semisimple tortile tegories nd of the generlized Reshetikhin/Turev 3-mnifold invrints of Turev [T] (without the \modulrity" ssumption on the tegory) The detour through 3-mnifold invrints is neessry, s the generliztion of Roerts results relting the surgil nd stte-sum invrints, nd interpreting the former in terms of signture requires the use of the 3-mnifold invrints The key here is the ide tht in ddition to eing le to lel omponents of frmed link digrm with ojets of k-liner tortile tegory, nd therey (vi the freeness theorem of Shum [S]) interpret the digrm s giving n endomorphism of I (tht is numer, when I is simple), we n lso lel them with liner omintions of ojets (gin otining numer) 4 If the liner omintion used to lel the omponents is refully hosen, the resulting frmed link invrint will e invrint under hndle-sliding, nd thus (upon suitle normliztion) n e turned into n invrint of the 3- or 4-mnifold desried y surgery on the link ( little re must e tken in the 4-mnifold se to orretly del with the distingushed unlink whose urves represent ples to \hollow out 2-hndle", equivlent to tthing 1-hndle, ut we will del with tht when the time omes) The key here is generliztion of the elegnt demonstrtion given in Likorish [L] (sed on ides of Roerts nd Viro, f lso Kumn/Lins [KLi]) tht the liner omintion of the simple ojets in the semisimple suquotient tegory of Rep(U q (sl 2 )) t root of unity with their (internl or quntum) dimensions s oeents gives rise to frmed link invrint whih is invrint under hndle-sliding In ft the phenomenon is quite generl: Proposition 41 If is ny semisimple k-liner tortile tegory with S nd s in the previous setion, then d 2S d2s d where 2Snd is ny ojet, nd the ox represents ny mp uilt out of the struturl mps for the tortile struture (thus representle y frmed tngle) proof: The proof is the digrmmti lultion given in Figure 30 The rst step is n pplition of Lemm 210, the seond uses the nturlity (nd dinturlity) properties of the struture mps, nd the third is n pplition of Lemm 217 Thus, the frmed link invrint rising y lelling eh strnd of the link with the liner omintion 4 The very tegorilly minded will reognize tht we re relly lelling eh strnd with diret sum of ojets, nd pling on eh strnd node with the mp whih multiplies eh diret summnd y the oeient This view of the onstrution will doutless seem it strethed nd onfusing to most reders, ut my e essentil to generliztions to more non-ommuttive \Hopf tegories"{f rne/frenkel [F] 31

32 \ 2S 2S d2s 2 l `` ( ( d d 2S d2s 2 d d2s d Figure 30: Proof of Proposition 41! s2s dim(s)s is invrint under hndle-sliding, regrdless of wht semisimple tortile tegory we use (It is trivil to see tht the invrint does not depend on the hoie of S) Now, if we let + (resp ) e the vlue of the +1- (resp 1-)frmed! -lelled unknot, nd ssume tht stises Denition 42 semisimple tortile tegory over eld k is 3-onformed if the vlues + nd re non-zero then letting x ( + ) 1 2 nd y ( + ) 1 2, it follows immeditely from the sme sort of rgument given in [RT] tht if M is the 3-mnifold otined s the oundry of the 4-dimensionl hndle-ody with one 0-hndle, nd 2-hndles tthed using L, then I (M)! (L)x jlj y (L) depends only on the dieomorphism type of M, where jlj is the numer of omponents of L nd (L) is the signture of the linking mtrix We shll ll I (M) generlized Reshetikhin/Turev invrint of M Note tht we hve used dierent non-degenery ondition thn tht used y Turev [T] In similr wy,ifwe let e the evlution of n! -lelled Hopf link, then if W is the 4-mnifold otined y tthing 2-hndles to undotted omponents of L, nd \hollowing out 2-hndles" long dotted omponents of L s in Kiry [Ki], then 32

33 (W )! (L) (L) jlj 2 N (L) depends only on the dieomorphism type of W, where jlj is s ove nd (L) is the nullity of the linking mtrix We will ll (W ) generlized rod invrint Note rst tht for (W )toe dened, we need dierent non-degenery ondition: Denition 43 semi-simple tortile tegory is 4-onformed if nd N re oth non-zero We will see in Setion 6 tht under hypotheses stised y the TL nd KR tegories t prinipl 4r th -roots of unity, N x 2 nd + nd re oth non-zero provided N is In the next setion, we present Roerts nlysis [Ro1] (f lso [K]) of the reltion etween the originl rne-etter invrint [], the originl rod invrint [], nd the orresponding 3- mnifold invrint, the Reshetikhin/Turev invrint [RT], ll in the TL formultion 33

34 5 TL trnsltion of (W) nd Roerts' hinmil Method 51 Trnsltion The purpose of this setion is to give sketh of ustin Roerts eutiful method of understnding the rne-etter 4-mnifold invrint in the se of U q (sl 2 ) In order to omplish this onnetion we need to trnslte the originl formultion of the rne-etter invrint in terms of Kirillov-Reshetikhin reoupling into Temperley-Lie reoupling, in tht Roerts method is lenest in the TL theory This trnsltion hs lredy een done in [K] For ompleteness, we repet this onstrution here First rell the generl denition for rne-etter invrint: where N n 0 n 1 fes (M) 2 S (T) dim(()) tetrhedr dim(()) 1 4-simplies k; k nd k; k is the 15j-network pproprite to the 4-simplex nd the oloring Rell n 0 (resp n 1 ) is the numer of verties (resp edges) in the tringultion nd N is the sum of the squres of the quntum dimensions In the se of where our tegory is the truntion of p Rep(U q (sl 2 )) t root of unity, wehoose s S, the irreduile representtions of U q (sl 2 )t (q)4r th -root of unity, lelling them f0; 1; 2;:::;r 2g; The ses for the hom-spes onsist of the projetions nd inlusions given y the KR 3-verties; nd N, the sum of the squres of the quntum dimensions, hs the spei vlue 2r N (q q 1 ) 2 (q exp(ir)) euse the lels t the node of the 15j-symol re uniquely determined y the lels of the rs inident, we n regrd the lelling s oloring of fes nd tetrhedr y integers (twie spins) Tht is, the 15j-symol eomes in this se simply prtiulr KR network Now, we hve remrked in the rst prt of Setion 2 tht there is simple trnsltion from KR theory to TL theory eeted y the formul of Figure 19 Rell tht in Figure 19 (; ; ) is the evlution of the TL thet net with lles ; ; nd The result of pplying this trnsltion to the 15j-symols results in formul for the (U q (sl 2 )) rne-etter invrint (W) in the TL theory: (M) 2(T) where the sum runs over ll lellings of fes nd tetrhedr y elements of f0; 1; :::; r N n 0 n 1 fes k; k T L 4-simplies dim(()) tetrhedr 2g nd dim(())(();( 0 );( 2 ))(();( 1 );( 3 )) Here k; k T L denotes the TL 15j-symol ssoited to nd the oloring, tht is the network given y the digrm of Figure 23, with TL 3-verties in ple of the ends of every \dumell", nd 0 ; 1 ; 2 ; 3 re the fes of the tetrhedron otined y omitting the lowest numered,,highest numered vertex of in the ordering on T This ompletes the trnsltion of (W)into the TL reoupling theory 34

35 52 Roerts' hin Mil Roerts onsiders tringulted 4-mnifold W nd its dul hndleody deomposition D The 0- hndles of D orrespond to the 4-simplies of W ; the 1-hndles of D orrespond to the tetrhedr of W ; the 2-hndles of D hve frmed tthing urves on the oundry of M N, where N is the union of the 0- nd 1-hndles of D these 2-hndles orrespond to the fes of the tringultion of W Letting N 0 denote N with the 2-hndles tthed nd M 0 N 0, note tht oth M nd M 0 re onneted sums of S 1 S 2 s y dding more S 1 S 2 s (y dding 1-hndles whih nel ll ut one of the 0-hndles of D ), Roerts produes surgery desription of N 00 y lelled link in S 3 (f Kiry [Ki]), where losing up N 00 y 3- nd 4-hndles give W 0 W d(s 1 S 3 ) for d n 4 1(n 4 is the numer of 4-simplexes in the originl tringultion on W ) The surgery urves of the presenttion of N 00 then tke the lol form shown in Figure 31 or its mirror imge, with one suh region ouring for eh 4-simplex of W S SS t S t Z hhh, e ZZZ, E ee, Z, E e ZZZ, E E e, E Q E QQQQ, \ \\ % % hhhhh````` \ %% l \ ```` Q S ( (( D S t QQ DD t S S %%% % t Figure 31: The Portion of hinmil orresponding to 4-simplex In Figure 31 the dotted urves re meridins orresponding to the 1-hndles (rell from Kiry [Ki] tht \hollowing out" 2-hndle long n unknot is equivlent to tthing 1-hndle), the other urves re (prts of) tthing urves for the 2-hndles 35

36 One n then dene the (originl) rod invrint [] of W 0 y lelling every urve with! P r 2 i0 i i Here i denotes the prllel ling of i rs (in lkord frming) with q-symmetrizer tthed s in Setion 2 One then normliztions of the rket evlution of the resulting sum y multiplying y N jlj+(l) 2, where jlj is the numer of omponents of the \hinmil link" L, nd (L) is the nullity of its linking mtrix The resulting invrint is then seen to e I(W ) (W0) where exp( i( 3 r2 ) 2r i 4 ) nd (W ) is the signture of the 4-mnifold W, ut sine W 0 is onneted sum of W with mnifold with trivil signture, (W )(W 0 ) This would e uninteresting if it were not for the formul of Figure 32 whih follows diretly from Likorish s enirlement lemm [L], d! N i (;;i)(;d;i) S S P P i d S S i 2f0; :::; r 2g S S i PP S S d Figure 32: Using Enirlement to ut 4 Strnds (Of ourse, in Figure 32 (; ; i) nd (; d; i)must e dmissile triples for 3-vertex This inludes the ondition (imposed y eing 4r th root of unity) tht + + i 2r 4) N is s usul the sum of the squres of the quntum dimensions pplition of this formul to I(W ) rewrites it s sum of produts involving i s orresponding to lelled tetrhedr; reiprols of -net evlutions orresponding to the \even" nd \odd" fes of the tetrhedr, i s on fes, nd evlutions of networks shown in Figure 33 The lels on the network in Figure 33 re given in terms of the ordered 4-simplex < > nd will llow the reder to ompre the omintoris of its struture to tht of the network in Figure 23 to see tht it is indeed just the quntum 15j-symol in the TL formultion (s usul ^ mens the fe otined y omitting ) Putting ll this together with the normliztion ftors for the 4-mnifold invrint I(W ), we nd tht I(W )N (W) 2 (W); where (W ) is the Euler hrteristi of W Thus, equivlently (W)N (W) 2 (W) : 36

37 ^3 s ^0 " " " s, < 134 > 02 " " " s " S S s s H `` "" HHHH " " " " " " " s " s ((,, PPPP Z E ELL Z s Z s D Z Z " D ^4 ^1 TT s ^2 Figure 33: 15j-networks otined y utting hinmil 37

38 6 The enter nd Roerts' hinmil Method The keys to Roerts pproh [Ro1] to interpreting the originl rne-etter invrint [] (f lso [K1]), ws the Likorish Enirlement Lemm (see [L]) The notion of the enter of rided monoidl tegory introdued in [K2] ws motivted y the desire to understnd this result in more generlity The nlog of Likorish s lemm t the pproprite level of generlity is Lemm 61 n j N Z(l) (n)id n j 2S j Figure 34: where, s usul N is the sum of the squres of the dimensions of the ojets in S nd Z() (n) 1if n is n ojet in the enter, nd 0 otherwise The proof of this Lemm is nlogous to tht given in [KLi] or [L]: If n is in the enter, we n unrid it from the loops lelled j, nd otin NId n Otherwise, selet n ojet with whih n does not rid trivilly, hndle-slide this over the! -lled loop, nd oserve tht the non-trivility of the riding implies tht the endomorphism of n depited on the right must e 0, sine otherwise it nnot ompose with two dierent mps to give the sme result (y simpliity of n) Oserve tht in the se where the tegory hs trivil enter, this lemm llows us to simply erse (!) omponents pssing through unknotted loops lelled! just s Likorish s originl formultion did in the se of TL digrms Moreover, in the nottion of Setion 4, we hve Proposition 62 If is semi-simple tortile tegory with trivil enter, then is 3- nd 4-onformed, with N x 2 Moreover, the generlized rod invrint ssoited tostises (W )y (W) nd the generlized rne-etter invrint stises (W)N (W) 2 y (W) : proof: One we estlish the rst sttement, the seond follows y n nlysis essentilly identil to tht in the previous setion The rst sttement follows y pplying the generlized Likorish enirlement lemm nd trivility of the enter to the! -lelled 0-frmed Hopf link, nd Lemm 41 followed y the generlized Likorish enirlement lemm nd trivility of the enter to the! lelled disjoint union of +1-frmed unknot nd 1-frmed unknot Proposition 62 is generliztion of the result of [K1] (f lso [Ro1]) expressing the originl rne-etter invrint in terms of the signture nd Euler hrter of the mnifold s noted in 38

39 [K1] nd in the introdution, this result should e regrded s giving purely omintoril expression for the signture of 4-mnifold in terms of tringultion t rst it my not e ler tht this result n e red in this wy sine y is (for known exmples) root of unity However, in the se of the TL formultion of Rep! (U q (sl 2 )) with hosen to e the prinipl 4r th root of unity, y expliit lultions of Roerts [Ro1], we hve tht y e i(3+r2 ) 2r i 4 It is not hrd to show tht if r is hosen to e multiple of 4 nd reltively prime to 3, then y will e primitive 2r th root of unity Thus hoosing suh nrgreter thn the rnk of the seond homology of the mnifold (or for simpliity, greter thn the numer of 2-simplies in the tringultion used), it is then possile to extrt the signture of the mnifold from the originl rne-etter invrint [] Expliitly, ifwe let r 4n((n; 3) 1) for n suiently lrge, then the signture of W is the unique solution etween r nd r to the eqution i(3 + 2n) 8n log( (W)N (W) 2 ) (mod 2) where log is the prinipl rnh of the omplex nturl logrithm urious (though not prtiulrly importnt) question is whether there re ny rtinin semisimple tortile tegories whih re suiently \non-unitry" tht y is not root of unity If so, the generlized rne-etter invrint for those tegories would llow us to ompute the signture diretly s logrithm of stte-sum n importnt point for further reserh is omprison of this omintoril expression for the signture with tht given y Gelfnd nd Mpherson [GM] nother spet of this formultion of the signture of 4-mnifold ers onsidertion: The generlized rne-etter invrints re ll invrints ssoited to TQFT This ft follows from generl priniples set down in [6] The trnsition mplitudes of n generlized rne-etter TQFT, n thus e regrded s giving \reltive signtures" for oordisms Put nother wy, rne-etter theory llows us to \ftor" 4-mnifold signtures long ny 3-mnifold Still, from nother point of view, Proposition 62 is rther disppointing: the rne-etter nd rod onstrutions for ny semi-simple tortile tegory with trivil enter merely give rise to vrious enodings of the signture nd Euler hrter of the 4-mnifold The question of whether this onstrution n give yield other informtion, sy out homotopy type or smooth struture, turns ruilly on the properties of the enter of the tegory used Two prtiulr points in the onstrution suggest lines of further reserh: The rst is the demonstrtion in the se of trivil enter tht x 2 N In generl, the hndlesliding followed y the enirlement lemm shows tht x 2 Nz + Nz P where z is the vlue of 1-frmed unknot lelled with i2z() dim(i)i Thus, our tegory will fil to e 3-onformed if z 0 In this se, the redution of the generlized rod invrint to signture will rek down Similrly there is property whih semi-simple tortile tegories ould possess whih would destroy the redution of the generlized rne-etter invrint to generlized rod invrint: oserve tht in Roerts hin-mil rgument, the enirlement lemm is used to ut n edge lelled with summnd of some i i (for i n ojet in our set of simple ojets) The redution of the generlized rne-etter invrint to generlized rod invrint isthus depends on the tegory stifying: 39