Topological Quantum Information Theory

Transcription

1 Proceedings of Symposi in Applied Mthemtics Topologicl Quntum Informtion Theory Louis H. Kuffmn nd Smuel J. Lomonco Jr. This pper is dedicted to new progress in the reltionship of topology nd quntum physics. Abstrct. This pper is n introduction to reltionships between quntum topology nd quntum computing. In this pper we discuss unitry solutions to the Yng-Bxter eqution tht re universl quntum gtes, quntum entnglement nd topologicl entnglement, nd we give n exposition of knottheoretic recoupling theory, its reltionship with topologicl quntum field theory nd pply these methods to produce unitry representtions of the brid groups tht re dense in the unitry groups. Our methods re rooted in the brcket stte sum model for the Jones polynomil. We give our results for lrge clss of representtions bsed on vlues for the brcket polynomil tht re roots of unity. We mke seprte nd self-contined study of the quntum universl Fiboncci model in this frmework. We pply our results to give quntum lgorithms for the computtion of the colored Jones polynomils for knots nd links, nd the Witten-Reshetikhin-Turev invrint of three mnifolds. 1. Introduction This pper describes reltionships between quntum topology nd quntum computing. It is modified version of Chpter 14 of our book [18] nd n expnded version of [58]. Quntum topology is, roughly speking, tht prt of lowdimensionl topology tht intercts with sttisticl nd quntum physics. Mny invrints of knots, links nd three dimensionl mnifolds hve been born of this interction, nd the form of the invrints is closely relted to the form of the computtion of mplitudes in quntum mechnics. Consequently, it is fruitful to move bck nd forth between quntum topologicl methods nd the techniques of quntum informtion theory. We sketch the bckground topology, discuss nlogies (such s topologicl entnglement nd quntum entnglement), show direct correspondences between certin topologicl opertors (solutions to the Yng-Bxter eqution) nd universl 2000 Mthemtics Subject Clssifiction. Primry 57M25; Secondry 81P10, 81P13. Key words nd phrses. Quntum computing, quntum topology, knots, links, stte sum, brcket stte sum, Jones polynomil, Yng-Bxter eqution, spin networks, Fiboncci model. 1 c 0000 (copyright holder)

2 2 LOUIS H. KAUFFMAN AND SAMUEL J. LOMONACO JR. quntum gtes. We then describe the bckground for topologicl quntum computing in terms of Temperley Lieb (we will sometimes bbrevite this to T L) recoupling theory. This is recoupling theory tht generlizes stndrd ngulr momentum recoupling theory, generlizes the Penrose theory of spin networks nd is inherently topologicl. Temperley Lieb recoupling Theory is bsed on the brcket polynomil model [37, 44] for the Jones polynomil. It is built in terms of digrmmtic combintoril topology. The sme structure cn be explined in terms of the SU(2) q quntum group, nd hs reltionships with functionl integrtion nd Witten s pproch to topologicl quntum field theory. Nevertheless, the pproch given here will be unrelentingly elementry. Elementry, does not necessrily men simple. In this cse n rchitecture is built from simple beginnings nd this rchictecture nd its recoupling lnguge cn be pplied to mny things including, e.g. colored Jones polynomils, Witten Reshetikhin Turev invrints of three mnifolds, topologicl quntum field theory nd quntum computing. In quntum computing, the ppliction of topology is most interesting becuse the simplest non-trivil exmple of the Temperley Lieb recoupling Theory gives the so-clled Fiboncci model. The recoupling theory yields representtions of the Artin brid group into unitry groups U(n) where n is Fiboncci number. These representtions re dense in the unitry group, nd cn be used to model quntum computtion universlly in terms of representtions of the brid group. Hence the term: topologicl quntum computtion. In this pper, we outline the bsics of the Temperely Lieb Recoupling Theory, nd show explicitly how the Fiboncci model rises from it. The digrmmtic computtions in the section 11 nd 12 re completely self-contined nd cn be used by reder who hs just lerned the brcket polynomil, nd wnts to see how these dense unitry brid group representtions rise from it. The outline of the prts of this pper is given below. (1) Knots nd Brids (2) Quntum Mechnics nd Quntum Computtion (3) Briding Opertors nd Univervsl Quntum Gtes (4) A Remrk bout EPR, Entnglement nd Bell s Inequlity (5) The Arvind Hypothesis (6) SU(2) Representtions of the Artin Brid Group (7) The Brcket Polynomil nd the Jones Polynomil (8) Quntum Topology, Cobordism Ctegories, Temperley-Lieb Algebr nd Topologicl Quntum Field Theory (9) Briding nd Topologicl Quntum Field Theory (10) Spin Networks nd Temperley-Lieb Recoupling Theory (11) Fiboncci Prticles (12) The Fiboncci Recoupling Model (13) Quntum Computtion of Colored Jones Polynomils nd the Witten- Reshetikhin-Turev Invrint We should point out tht while this pper ttempts to be self-contined, nd hence hs some expository mteril, most of the results re either new, or re new points of view on known results. The mteril on SU(2) representtions of the Artin brid group is new, nd the reltionship of this mteril to the recoupling theory is

3 TOPOLOGICAL QUANTUM INFORMATION THEORY 3 new. The tretment of elementry cobordism ctegories is well-known, but new in the context of quntum informtion theory. The reformultion of Temperley-Lieb recoupling theory for the purpose of producing unitry brid group representtions is new for quntum informtion theory, nd directly relted to much of the recent work of Freedmn nd his collbortors. The tretment of the Fiboncci model in terms of two-strnd recoupling theory is new nd t the sme time, the most elementry non-trivil exmple of the recoupling theory. The models in section 10 for quntum computtion of colored Jones polynomils nd for quntum computtion of the Witten-Reshetikhin-Turev invrint re new in this form of the recoupling theory. They tke prticulrly simple spect in this context. Here is very condensed presenttion of how unitry representtions of the brid group re constructed vi topologicl quntum field theoretic methods. One hs mthemticl prticle with lbel P tht cn interct with itself to produce either itself lbeled P or itself with the null lbel. We shll denote the interction of two prticles P nd Q by the expression PQ, but it is understood tht the vlue of PQ is the result of the interction, nd this my prtke of number of possibilities. Thus for our prticle P, we hve tht PP my be equl to P or to in given sitution. When intercts with P the result is lwys P. When intercts with the result is lwys. One considers process spces where row of prticles lbeled P cn successively interct, subject to the restriction tht the end result is P. For exmple the spce V [(b)c] denotes the spce of interctions of three prticles lbeled P. The prticles re plced in the positions, b, c. Thus we begin with (PP)P. In typicl sequence of interctions, the first two P s interct to produce, nd the intercts with P to produce P. (PP)P ( )P P. In nother possibility, the first two P s interct to produce P, nd the P intercts with P to produce P. (PP)P (P)P P. It follows from this nlysis tht the spce of liner combintions of processes V [(b)c] is two dimensionl. The two processes we hve just described cn be tken to be the qubit bsis for this spce. One obtins representtion of the three strnd Artin brid group on V [(b)c] by ssigning pproprite phse chnges to ech of the generting processes. One cn think of these phses s corresponding to the interchnge of the prticles lbeled nd b in the ssocition (b)c. The other opertor for this representtion corresponds to the interchnge of b nd c. This interchnge is ccomplished by unitry chnge of bsis mpping If F : V [(b)c] V [(bc)]. A : V [(b)c] V [(b)c] is the first briding opertor (corresponding to n interchnge of the first two prticles in the ssocition) then the second opertor B : V [(b)c] V [(c)b] is ccomplished vi the formul B F 1 RF where the R in this formul cts in the second vector spce V [(bc)] to pply the phses for the interchnge of b nd c. These issues re illustrted in Figure 1, where the prenthesiztion of the

4 4 LOUIS H. KAUFFMAN AND SAMUEL J. LOMONACO JR. R F Figure 1. Briding Anyons prticles is indicted by circles nd by lso by trees. The trees cn be tken to indicte ptterns of prticle interction, where two prticles interct t the brnch of binry tree to produce the prticle product t the root. See lso Figure 28 for n illustrtion of the briding B F 1 RF In this scheme, vector spces corresponding to ssocited strings of prticle interctions re interrelted by recoupling trnsformtions tht generlize the mpping F indicted bove. A full representtion of the Artin brid group on ech spce is defined in terms of the locl interchnge phse gtes nd the recoupling trnsformtions. These gtes nd trnsformtions hve to stisfy number of identities in order to produce well-defined representtion of the brid group. These identities were discovered originlly in reltion to topologicl quntum field theory. In our pproch the structure of phse gtes nd recoupling trnsformtions rise nturlly from the structure of the brcket model for the Jones polynomil. Thus we obtin knot-theoretic bsis for topologicl quntum computing. In modeling the quntum Hll effect [87, 26, 15, 16], the briding of qusiprticles (collective excittions) leds to non-trivl representtions of the Artin brid group. Such prticles re clled Anyons. The briding in these models is relted to topologicl quntum field theory. It is hoped tht the mthemtics we explin here will form bridge between theoreticl models of nyons nd their pplictions to quntum computing. Acknowledgement. The first uthor thnks the Ntionl Science Foundtion for support of this reserch under NSF Grnt DMS Much of this effort ws sponsored by the Defense Advnced Reserch Projects Agency (DARPA) nd Air Force Reserch Lbortory, Air Force Mteriel Commnd, USAF, under greement F The U.S. Government is uthorized to reproduce nd distribute reprints for Government purposes notwithstnding ny copyright nnottions thereon. The views nd conclusions contined herein re those of the uthors nd

5 TOPOLOGICAL QUANTUM INFORMATION THEORY 5 Figure 2. A knot digrm should not be interpreted s necessrily representing the officil policies or endorsements, either expressed or implied, of the Defense Advnced Reserch Projects Agency, the Air Force Reserch Lbortory, or the U.S. Government. (Copyright 2006.) It gives the uthors plesure to thnk the Newton Institute in Cmbridge Englnd nd ISI in Torino, Itly for their hospitlity during the inception of this reserch nd to thnk Hilry Crteret for useful converstions. 2. Knots nd Brids The purpose of this section is to give quick introduction to the digrmmtic theory of knots, links nd brids. A knot is n embedding of circle in threedimensionl spce, tken up to mbient isotopy. The problem of deciding whether two knots re isotopic is n exmple of plcement problem, problem of studying the topologicl forms tht cn be mde by plcing one spce inside nother. In the cse of knot theory we consider the plcements of circle inside three dimensionl spce. There re mny pplictions of the theory of knots. Topology is bckground for the physicl structure of rel knots mde from rope of cble. As result, the field of prcticl knot tying is field of pplied topology tht existed well before the mthemticl discipline of topology rose. Then gin long molecules such s rubber molecules nd DNA molecules cn be knotted nd linked. There hve been number of intense pplictions of knot theory to the study of DN A [82] nd to polymer physics [62]. Knot theory is closely relted to theoreticl physics s well with pplictions in quntum grvity [86, 79, 53] nd mny pplictions of ides in physics to the topologicl structure of knots themselves [44]. Quntum topology is the study nd invention of topologicl invrints vi the use of nlogies nd techniques from mthemticl physics. Mny invrints such s the Jones polynomil re constructed vi prtition functions nd generlized quntum mplitudes. As result, one expects to see reltionships between knot theory nd physics. In this pper we will study how knot theory cn be used to produce unitry representtions of the brid group. Such representtions cn ply fundmentl role in quntum computing.

6 6 LOUIS H. KAUFFMAN AND SAMUEL J. LOMONACO JR. I II III Figure 3. The Reidemeister Moves s 1 s 2 Brid Genertors s 3 s 1-1 s 1-1 s 1 1 s 1 s 2 s 1 s 2 s 1 s 2 s 1 s 3 s 3 s 1 Figure 4. Brid Genertors Tht is, two knots re regrded s equivlent if one embedding cn be obtined from the other through continuous fmily of embeddings of circles in three-spce. A link is n embedding of disjoint collection of circles, tken up to mbient isotopy. Figure 2 illustrtes digrm for knot. The digrm is regrded both s schemtic picture of the knot, nd s plne grph with extr structure t the nodes (indicting how the curve of the knot psses over or under itself by stndrd pictoril conventions). Ambient isotopy is mthemticlly the sme s the equivlence reltion generted on digrms by the Reidemeister moves. These moves re illustrted in Figure 3. Ech move is performed on locl prt of the digrm tht is topologiclly identicl to the prt of the digrm illustrted in this figure (these figures re representtive exmples of the types of Reidemeister moves) without chnging

7 TOPOLOGICAL QUANTUM INFORMATION THEORY 7 Hopf Link Trefoil Knot Figure Eight Knot Figure 5. Closing Brids to form knots nd links. b CL(b) Figure 6. Borromen Rings s Brid Closure the rest of the digrm. The Reidemeister moves re useful in doing combintoril topology with knots nd links, notbly in working out the behviour of knot invrints. A knot invrint is function defined from knots nd links to some other mthemticl object (such s groups or polynomils or numbers) such tht equivlent digrms re mpped to equivlent objects (isomorphic groups, identicl polynomils, identicl numbers). The Reidemeister moves re of gret use for nlyzing the structure of knot invrints nd they re closely relted to the Artin brid group, which we discuss below. A brid is n embedding of collection of strnds tht hve their ends in two rows of points tht re set one bove the other with respect to choice of verticl. The strnds re not individully knotted nd they re disjoint from one nother.

8 8 LOUIS H. KAUFFMAN AND SAMUEL J. LOMONACO JR. See Figures 4, 5 nd 6 for illustrtions of brids nd moves on brids. Brids cn be multiplied by ttching the bottom row of one brid to the top row of the other brid. Tken up to mbient isotopy, fixing the endpoints, the brids form group under this notion of multipliction. In Figure 4 we illustrte the form of the bsic genertors of the brid group, nd the form of the reltions mong these genertors. Figure 5 illustrtes how to close brid by ttching the top strnds to the bottom strnds by collection of prllel rcs. A key theorem of Alexnder sttes tht every knot or link cn be represented s closed brid. Thus the theory of brids is criticl to the theory of knots nd links. Figure 6 illustrtes the fmous Borromen Rings ( link of three unknotted loops such tht ny two of the loops re unlinked) s the closure of brid. Let B n denote the Artin brid group on n strnds. We recll here tht B n is generted by elementry brids {s 1,, s n 1 } with reltions (1) s i s j s j s i for i j > 1, (2) s i s i+1 s i s i+1 s i s i+1 for i 1, n 2. See Figure 4 for n illustrtion of the elementry brids nd their reltions. Note tht the brid group hs digrmmtic topologicl interprettion, where brid is n intertwining of strnds tht led from one set of n points to nother set of n points. The brid genertors s i re represented by digrms where the i-th nd (i + 1)-th strnds wind round one nother by single hlf-twist (the sense of this turn is shown in Figure 4) nd ll other strnds drop stright to the bottom. Brids re digrmmed verticlly s in Figure 4, nd the products re tken in order from top to bottom. The product of two brid digrms is ccomplished by djoining the top strnds of one brid to the bottom strnds of the other brid. In Figure 4 we hve restricted the illustrtion to the four-strnded brid group B 4. In tht figure the three brid genertors of B 4 re shown, nd then the inverse of the first genertor is drwn. Following this, one sees the identities s 1 s (where the identity element in B 4 consists in four verticl strnds), s 1 s 2 s 1 s 2 s 1 s 2, nd finlly s 1 s 3 s 3 s 1. Brids re key structure in mthemtics. It is not just tht they re collection of groups with vivid topologicl interprettion. From the lgebric point of view the brid groups B n re importnt extensions of the symmetric groups S n. Recll tht the symmetric group S n of ll permuttions of n distinct objects hs presenttion s shown below. (1) s 2 i 1 for i 1, n 1, (2) s i s j s j s i for i j > 1, (3) s i s i+1 s i s i+1 s i s i+1 for i 1, n 2. Thus S n is obtined from B n by setting the squre of ech briding genertor equl to one. We hve n exct sequence of groups 1 B n S n 1 exhibiting the Artin brid group s n extension of the symmetric group.

9 TOPOLOGICAL QUANTUM INFORMATION THEORY 9 In the next sections we shll show how representtions of the Artin brid group re rich enough to provide dense set of trnsformtions in the unitry groups. Thus the brid groups re in principle fundmentl to quntum computtion nd quntum informtion theory. 3. Quntum Mechnics nd Quntum Computtion We shll quickly indicte the bsic principles of quntum mechnics. The quntum informtion context encpsultes concise model of quntum theory: The initil stte of quntum process is vector v in complex vector spce H. Mesurement returns bsis elements β of H with probbility β v 2 / v v where v w v w with v the conjugte trnspose of v. A physicl process occurs in steps v U v U v where U is unitry liner trnsformtion. Note tht since Uv Uw v U U w v w when U is unitry, it follows tht probbility is preserved in the course of quntum process. One of the detils required for ny specific quntum problem is the nture of the unitry evolution. This is specified by knowing pproprite informtion bout the clssicl physics tht supports the phenomen. This informtion is used to choose n pproprite Hmiltonin through which the unitry opertor is constructed vi correspondence principle tht replces clssicl vribles with pproprite quntum opertors. (In the pth integrl pproch one needs Lngrngin to construct the ction on which the pth integrl is bsed.) One needs to know certin spects of clssicl physics to solve ny specific quntum problem. A key concept in the quntum informtion viewpoint is the notion of the superposition of sttes. If quntum system hs two distinct sttes v nd w, then it hs infinitely mny sttes of the form v +b w where nd b re complex numbers tken up to common multiple. Sttes re relly in the projective spce ssocited with H. There is only one superposition of single stte v with itself. On the other hnd, it is most convenient to regrd the sttes v nd w s vectors in vector spce. We thn tke it s prt of the procedure of deling with sttes to normlize them to unit length. Once gin, the superposition of stte with itself is gin itself. Dirc [23] introduced the br -(c)-ket nottion A B A B for the inner product of complex vectors A, B H. He lso seprted the prts of the brcket into the br < A nd the ket B. Thus A B A B In this interprettion, the ket B is identified with the vector B H, while the br < A is regrded s the element dul to A in the dul spce H. The dul element to A corresponds to the conjugte trnspose A of the vector A, nd the inner product is expressed in conventionl lnguge by the mtrix product A B (which is sclr since B is column vector). Hving seprted the br nd the ket, Dirc cn write the ket-br A B AB. In conventionl nottion, the

10 10 LOUIS H. KAUFFMAN AND SAMUEL J. LOMONACO JR. ket-br is mtrix, not sclr, nd we hve the following formul for the squre of P A B : P 2 A B A B A(B A)B (B A)AB B A P. The stndrd exmple is ket-br P A A where A A 1 so tht P 2 P. Then P is projection mtrix, projecting to the subspce of H tht is spnned by the vector A. In fct, for ny vector B we hve P B A A B A A B A B A. If { C 1, C 2, C n } is n orthonorml bsis for H, nd then for ny vector A we hve P i C i C i, Hence A C 1 A C C n A C n. B A B C 1 C 1 A + + B C n C n A One wnts the probbility of strting in stte A nd ending in stte B. The probbility for this event is equl to B A 2. This cn be refined if we hve more knowledge. If the intermedite sttes C i re complete set of orthonorml lterntives then we cn ssume tht C i C i 1 for ech i nd tht Σ i C i C i 1. This identity now corresponds to the fct tht 1 is the sum of the probbilities of n rbitrry stte being projected into one of these intermedite sttes. If there re intermedite sttes between the intermedite sttes this formultion cn be continued until one is summing over ll possible pths from A to B. This becomes the pth integrl expression for the mplitude B A Wht is Quntum Computer? A quntum computer is, bstrctly, composition U of unitry trnsformtions, together with n initil stte nd choice of mesurement bsis. One runs the computer by repetedly initilizing it, nd then mesuring the result of pplying the unitry trnsformtion U to the initil stte. The results of these mesurements re then nlyzed for the desired informtion tht the computer ws set to determine. The key to using the computer is the design of the initil stte nd the design of the composition of unitry trnsformtions. The reder should consult [72] for more specific exmples of quntum lgorithms. Let H be given finite dimensionl vector spce over the complex numbers C. Let {W 0, W 1,..., W n } be n orthonorml bsis for H so tht with i : W i denoting W i nd i denoting the conjugte trnspose of i, we hve i j δ ij where δ ij denotes the Kronecker delt (equl to one when its indices re equl to one nother, nd equl to zero otherwise). Given vector v in H let v 2 : v v. Note tht i v is the i-th coordinte of v.

11 TOPOLOGICAL QUANTUM INFORMATION THEORY 11 An mesurement of v returns one of the coordintes i of v with probbility i v 2. This model of mesurement is simple instnce of the sitution with quntum mechnicl system tht is in mixed stte until it is observed. The result of observtion is to put the system into one of the bsis sttes. When the dimension of the spce H is two (n 1), vector in the spce is clled qubit. A qubit represents one quntum of binry informtion. On mesurement, one obtins either the ket 0 or the ket 1. This constitutes the binry distinction tht is inherent in qubit. Note however tht the informtion obtined is probbilistic. If the qubit is ψ α 0 + β 1, then the ket 0 is observed with probbility α 2, nd the ket 1 is observed with probbility β 2. In speking of n idelized quntum computer, we do not specify the nture of mesurement process beyond these probbility postultes. In the cse of generl dimension n of the spce H, we will cll the vectors in H qunits. It is quite common to use spces H tht re tensor products of twodimensionl spces (so tht ll computtions re expressed in terms of qubits) but this is not necessry in principle. One cn strt with given spce, nd lter work out fctoriztions into qubit trnsformtions. A quntum computtion consists in the ppliction of unitry trnsformtion U to n initil qunit ψ n n with ψ 2 1, plus n mesurement of Uψ. A mesurement of Uψ returns the ket i with probbility i Uψ 2. In prticulr, if we strt the computer in the stte i, then the probbility tht it will return the stte j is j U i 2. It is the necessity for writing given computtion in terms of unitry trnsformtions, nd the probbilistic nture of the result tht chrcterizes quntum computtion. Such computtion could be crried out by n idelized quntum mechnicl system. It is hoped tht such systems cn be physiclly relized. 4. Briding Opertors nd Universl Quntum Gtes A clss of invrints of knots nd links clled quntum invrints cn be constructed by using representtions of the Artin brid group, nd more specificlly by using solutions to the Yng-Bxter eqution [10], first discovered in reltion to dimensionl quntum field theory, nd 2 dimensionl sttisticl mechnics. Briding opertors feture in constructing representtions of the Artin brid group, nd in the construction of invrints of knots nd links. A key concept in the construction of quntum link invrints is the ssocition of Yng-Bxter opertor R to ech elementry crossing in link digrm. The opertor R is liner mpping R: V V V V defined on the 2-fold tensor product of vector spce V, generlizing the permuttion of the fctors (i.e., generlizing swp gte when V represents one qubit). Such trnsformtions re not necessrily unitry in topologicl pplictions. It is

12 12 LOUIS H. KAUFFMAN AND SAMUEL J. LOMONACO JR. R I I R R I I R I R R I R I I R Figure 7. The Yng-Bxter eqution useful to understnd when they cn be replced by unitry trnsformtions for the purpose of quntum computing. Such unitry R-mtrices cn be used to mke unitry representtions of the Artin brid group. A solution to the Yng-Bxter eqution, s described in the lst prgrph is mtrix R, regrded s mpping of two-fold tensor product of vector spce V V to itself tht stisfies the eqution (R I)(I R)(R I) (I R)(R I)(I R). From the point of view of topology, the mtrix R is regrded s representing n elementry bit of briding represented by one string crossing over nother. In Figure 7 we hve illustrted the briding identity tht corresponds to the Yng-Bxter eqution. Ech briding picture with its three input lines (below) nd output lines (bove) corresponds to mpping of the three fold tensor product of the vector spce V to itself, s required by the lgebric eqution quoted bove. The pttern of plcement of the crossings in the digrm corresponds to the fctors R I nd I R. This crucil topologicl move hs n lgebric expression in terms of such mtrix R. Our pproch in this section to relte topology, quntum computing, nd quntum entnglement is through the use of the Yng-Bxter eqution. In order to ccomplish this im, we need to study solutions of the Yng-Bxter eqution tht re unitry. Then the R mtrix cn be seen either s briding mtrix or s quntum gte in quntum computer. The problem of finding solutions to the Yng-Bxter eqution tht re unitry turns out to be surprisingly difficult. Dye [25] hs clssified ll such mtrices of size 4 4. A rough summry of her clssifiction is tht ll 4 4 unitry solutions to the Yng-Bxter eqution re similr to one of the following types of mtrix: R 1/ / 2 0 1/ 2 1/ / 2 1/ 2 0 1/ / 2

13 TOPOLOGICAL QUANTUM INFORMATION THEORY 13 R R b 0 0 c d b c 0 d where,b,c,d re unit complex numbers. For the purpose of quntum computing, one should regrd ech mtrix s cting on the stmdrd bsis { 00, 01, 10, 11 } of H V V, where V is two-dimensionl complex vector spce. Then, for exmple we hve R 00 (1/ 2) 00 (1/ 2) 11, R 01 (1/ 2) 01 + (1/ 2) 10, R 10 (1/ 2) 01 + (1/ 2) 10, R 11 (1/ 2) 00 + (1/ 2) 11. The reder should note tht R is the fmilir chnge-of-bsis mtrix from the stndrd bsis to the Bell bsis of entngled sttes. In the cse of R, we hve R 00 00, R 01 c 10, R 10 b 01, R 11 d 11. Note tht R cn be regrded s digonl phse gte P, composed with swp gte S. P S b c d Compositions of solutions of the (Briding) Yng-Bxter eqution with the swp gte S re clled solutions to the lgebric Yng-Bxter eqution. Thus the digonl mtrix P is solution to the lgebric Yng-Bxter eqution. Remrk. Another venue relted to unitry solutions to the Yng-Bxter eqution s quntum gtes comes from using extr physicl prmeters in this eqution (the rpidity prmeter) tht re relted to sttisticl physics. In [91] we discovered tht solutions to the Yng-Bxter eqution with the rpidity prmeter llow mny new unitry solutions. The significnce of these gtes for qutnum computing is still under investigtion.

14 14 LOUIS H. KAUFFMAN AND SAMUEL J. LOMONACO JR Universl Gtes. A two-qubit gte G is unitry liner mpping G : V V V where V is two complex dimensionl vector spce. We sy tht the gte G is universl for quntum computtion (or just universl) if G together with locl unitry trnsformtions (unitry trnsformtions from V to V ) genertes ll unitry trnsformtions of the complex vector spce of dimension 2 n to itself. It is well-known [72] tht CNOT is universl gte. (On the stndrd bsis, CNOT is the identity when the first qubit is 0, nd it flips the second qbit, leving the first lone, when the first qubit is 1.) A gte G, s bove, is sid to be entngling if there is vector αβ α β V V such tht G αβ is not decomposble s tensor product of two qubits. Under these circumstnces, one sys tht G αβ is entngled. In [17], the Brylinskis give generl criterion of G to be universl. They prove tht two-qubit gte G is universl if nd only if it is entngling. Remrk. A two-qubit pure stte φ 00 + b 01 + c 10 + d 11 is entngled exctly when (d bc) 0. It is esy to use this fct to check when specific mtrix is, or is not, entngling. Remrk. There re mny gtes other thn CNOT tht cn be used s universl gtes in the presence of locl unitry trnsformtions. Some of these re themselves topologicl (unitry solutions to the Yng-Bxter eqution, see [56]) nd themselves generte representtions of the Artin brid group. Replcing CN OT by solution to the Yng-Bxter eqution does not plce the locl unitry trnsformtions s prt of the corresponding representtion of the brid group. Thus such substitutions give only prtil solution to creting topologicl quntum computtion. In this pper we re concerned with brid group representtions tht include ll spects of the unitry group. Accordingly, in the next section we shll first exmine how the brid group on three strnds cn be represented s locl unitry trnsformtions. Theorem. Let D denote the phse gte shown below. D is solution to the lgebric Yng-Bxter eqution (see the erlier discussion in this section). Then D is universl gte. D Proof. It follows t once from the Brylinski Theorem tht D is universl. For more specific proof, note tht CNOT QDQ 1, where Q H I, H is the 2 2 Hdmrd mtrix. The conclusion then follows t once from this identity nd the discussion bove. We illustrte the mtrices involved in this proof below:

15 TOPOLOGICAL QUANTUM INFORMATION THEORY 15 H (1/ ( 1 1 2) 1 1 Q (1/ 2) D QDQ 1 QDQ ) This completes the proof of the Theorem CNOT Remrk. We thnk Mrtin Roetteles [78] for pointing out the specific fctoriztion of CNOT used in this proof. Theorem. The mtrix solutions R nd R to the Yng-Bxter eqution, described bove, re universl gtes exctly when d bc 0 for their internl prmeters, b, c, d. In prticulr, let R 0 denote the solution R (bove) to the Yng-Bxter eqution with b c 1, d 1. Then R 0 is universl gte. R R b 0 0 c d Proof. The first prt follows t once from the Brylinski Theorem. In fct, letting H be the Hdmrd mtrix s before, nd ( ) ( ) 1/ 2 i/ 2 σ i/ 2 1/ 1/ 2 1/ 2, λ 2 i/ 2 i/ 2 ( ) (1 i)/2 (1 + i)/2 µ. (1 i)/2 ( 1 i)/2 Then CNOT (λ µ)(r 0 (I σ)r 0 )(H H). This gives n explicit expression for CNOT in terms of R 0 nd locl unitry trnsformtions (for which we thnk Ben Reichrdt).

16 16 LOUIS H. KAUFFMAN AND SAMUEL J. LOMONACO JR. Remrk. Let SWAP denote the Yng-Bxter Solution R with b c d SWAP SW AP is the stndrd swp gte. Note tht SW AP is not universl gte. This lso follows from the Brylinski Theorem, since SW AP is not entngling. Note lso tht R 0 is the composition of the phse gte D with this swp gte. Theorem. Let R 1/ / 2 0 1/ 2 1/ / 2 1/ 2 0 1/ / 2 be the unitry solution to the Yng-Bxter eqution discussed bove. Then R is universl gte. The proof below gives specific expression for CN OT in terms of R. Proof. This result follows t once from the Brylinksi Theorem, since R is highly entngling. For direct computtionl proof, it suffices to show tht CN OT cn be generted from R nd locl unitry trnsformtions. Let ( ) 1/ 2 1/ 2 α 1/ 2 1/ 2 ( ) 1/ 2 1/ 2 β i/ 2 i/ 2 ( ) 1/ 2 i/ 2 γ 1/ 2 i/ 2 ( ) 1 0 δ 0 i Let M α β nd N γ δ. Then it is strightforwrd to verify tht This completes the proof. CNOT MRN. Remrk. See [56] for more informtion bout these clcultions. 5. A Remrk bout EPR, Engtnglement nd Bell s Inequlity A stte ψ H n, where H is the qubit spce, is sid to be entngled if it cnnot be written s tensor product of vectors from non-trivil fctors of H n. Such sttes turn out to be relted to subtle nonloclity in quntum physics. It helps to plce this lgebric structure in the context of gednken experiment to see where the physics comes in. Thought experiments of the sort we re bout to describe were first devised by Einstein, Podolosky nd Rosen, referred henceforth s EPR.

17 TOPOLOGICAL QUANTUM INFORMATION THEORY 17 Consider the entngled stte S ( )/ 2. In n EPR thought experiment, we think of two prts of this stte tht re seprted in spce. We wnt nottion for these prts nd suggest the following: L ({ 0 } 1 + { 1 } 0 )/ 2, R ( 0 { 1 } + 1 { 0 })/ 2. In the left stte L, n observer cn only observe the left hnd fctor. In the right stte R, n observer cn only observe the right hnd fctor. These sttes L nd R together comprise the EPR stte S, but they re ccessible individully just s re the two photons in the usul thought experiement. One cn trnsport L nd R individully nd we shll write S L R to denote tht they re the prts (but not tensor fctors) of S. The curious thing bout this formlism is tht it includes little bit of mcroscopic physics implicitly, nd so it mkes it bit more pprent wht EPR were concerned bout. After ll, lots of things tht we cn do to L or R do not ffect S. For exmple, trnsporting L from one plce to nother, s in the originl experiment where the photons seprte. On the other hnd, if Alice hs L nd Bob hs R nd Alice performs locl unitry trnsformtion on her tensor fctor, this pplies to both L nd R since the trnsformtion is ctully being pplied to the stte S. This is lso spooky ction t distnce whose consequence does not pper until mesurement is mde. To go bit deeper it is worthwhile seeing wht entnglement, in the sense of tensor indecomposbility, hs to do with the structure of the EPR thought experiment. To this end, we look t the structure of the Bell inequlities using the Cluser, Horne, Shimony, Holt formlism (CHSH) s explined in the book by Nielsen nd Chung [72]. For this we use the following observbles with eigenvlues ±1. ) ( 1 0 Q 0 1 ( ) 0 1 R 1 0 ( ) 1 1 S 1 1 ( ) 1 1 T 1 1, 1, 1 / 2, 2 / 2. 2 The subscripts 1 nd 2 on these mtrices indicte tht they re to operte on the first nd second tensor fctors, repsectively, of quntum stte of the form φ 00 + b 01 + c 10 + d 11.

18 18 LOUIS H. KAUFFMAN AND SAMUEL J. LOMONACO JR. To simplify the results of this clcultion we shll here ssume tht the coefficients, b, c, d re rel numbers. We clculte the quntity finding tht φ QS φ + φ RS φ + φ RT φ φ QT φ, (2 4( + d) 2 + 4(d bc))/ 2. Clssicl probbility clcultion with rndom vribles of vlue ±1 gives the vlue of QS + RS + RT QT ±2 (with ech of Q, R, S nd T equl to ±1). Hence the clssicl expecttion stisfies the Bell inequlity E(QS) + E(RS) + E(RT) E(QT) 2. Tht quntum expecttion is not clssicl is embodied in the fct tht cn be greter thn 2. The clssic cse is tht of the Bell stte Here φ ( )/ 2. 6/ 2 > 2. In generl we see tht the following inequlity is needed in order to violte the Bell inequlity This is equivlent to (2 4( + d) 2 + 4(d bc))/ 2 > 2. ( 2 1)/2 < (d bc) ( + d) 2. Since we know tht φ is entngled exctly when d bc is non-zero, this shows tht n unentngled stte cnnot violte the Bell inequlity. This formul lso shows tht it is possible for stte to be entngled nd yet not violte the Bell inequlity. For exmple, if φ ( )/2, then (φ) stisfies Bell s inequlity, but φ is n entngled stte. We see from this clcultion tht entnglement in the sense of tensor indecomposbility, nd entnglement in the sense of Bell inequlity violtion for given choice of Bell opertors re not equivlent concepts. On the other hnd, Benjmin Schumcher hs pointed out [80] tht ny entngled two-qubit stte will violte Bell inequlities for n pproprite choice of opertors. This deepens the context for our question of the reltionship between topologicl entnglement nd quntum entnglement. The Bell inequlity violtion is n indiction of quntum mechnicl entnglement. One s intuition suggests tht it is this sort of entnglement tht should hve topologicl context. 6. The Arvind Hypothesis Link digrms cn be used s grphicl devices nd holders of informtion. In this vein Arvind [5] proposed tht the entnglement of link should correspond to the entnglement of stte. Mesurement of link would be modeled by deleting one component of the link. A key exmple is the Borromen rings. See Figure 8.

19 TOPOLOGICAL QUANTUM INFORMATION THEORY 19 Figure 8. Boromen Rings Deleting ny component of the Boromen rings yields remining pir of unlinked rings. The Borromen rings re entngled, but ny two of them re unentngled. In this sense the Borromen rings re nlogous to the GHZ stte GHZ (1/ 2)( ). Mesurement in ny fctor of the GHZ yields n unentngled stte. Arvind points out tht this property is bsis dependent. We point out tht there re sttes whose entnglement fter n mesurement is mtter of probbility (vi quntum mplitudes). Consider for exmple the stte ψ Mesurement in ny coordinte yields n entngled or n unentngled stte with equl probbility. For exmple ψ 0 ( ) so tht projecting to 1 in the first coordinte yields n unentngled stte, while projecting to 0 yields n entngled stte, ech with equl probbility. New wys to use link digrms must be invented to mp the properties of such sttes. One direction is to consider pproprite notions of quntum knots so tht one cn formlte superpositions of topologicl types s in [55]. But one needs to go deeper in this considertion. The reltionship of topology nd physics needs to be exmined crefully. We tke the stnce tht topologicl properties of systems re properties tht remin invrint under certin trnsformtions tht re identified s topologicl equivlences. In mking quntum physicl models, these equivlences should correspond to unitry trnsformtions of n pproprite Hilbert spce. Accordingly, we hve formulted model for quntum knots [60] tht meets these requirements. A quntum knot system represents the quntum embodiment of closed knotted physicl piece of rope. A quntum knot (i.e., n element K lying in n pproprite Hilbert spce H n, s stte of this system,

20 20 LOUIS H. KAUFFMAN AND SAMUEL J. LOMONACO JR. represents the stte of such knotted closed piece of rope, i.e., the prticulr sptil configurtion of the knot tied in the rope. Associted with quntum knot system is group of unitry trnsformtions A n, clled the mbient group, which represents ll possible wys of moving the rope round (without cutting the rope, nd without letting the rope pss through itself.) Of course, unlike clssicl closed piece of rope, quntum knot cn exhibit non-clssicl behvior, such s quntum superposition nd quntum entnglement. The knot type of quntum knot K is simply the orbit of the quntum knot under the ction of the mbient group A n. This leds to new questions connecting quntum computing nd knot theory. 7. SU(2) Representtions of the Artin Brid Group The purpose of this section is to determine ll the representtions of the three strnd Artin brid group B 3 to the specil unitry group SU(2) nd concomitntly to the unitry group U(2). One regrds the groups SU(2) nd U(2) s cting on single qubit, nd so U(2) is usully regrded s the group of locl unitry trnsformtions in quntum informtion setting. If one is looking for coherent wy to represent ll unitry trnsformtions by wy of brids, then U(2) is the plce to strt. Here we will show tht there re mny representtions of the threestrnd brid group tht generte dense subset of U(2). Thus it is fct tht locl unitry trnsformtions cn be generted by brids in mny wys. We begin with the structure of SU(2). A mtrix in SU(2) hs the form ( ) z w M, w z where z nd w re complex numbers, nd z denotes the complex conjugte of z. To be in SU(2) it is required tht Det(M) 1 nd tht M M 1 where Det denotes determinnt, nd M is the conjugte trnspose of M. Thus if z + bi nd w c + di where, b, c, d re rel numbers, nd i 2 1, then ( ) + bi c + di M c + di bi with 2 + b 2 + c 2 + d 2 1. It is convenient to write ( ) ( ) ( 1 0 i M + b + c i 1 0 nd to bbrevite this decomposition s where so tht nd 1 ( ) ( i 0, i 0 i M + bi + cj + dk ) ( 0 1, j, 1 0 i 2 j 2 k 2 ijk 1 ij k, jk i, ki j ji k, kj i, ik j. ) ( 0 i + d i 0 ) ( 0 i, k i 0 ), )

21 TOPOLOGICAL QUANTUM INFORMATION THEORY 21 The lgebr of 1, i, j, k is clled the quternions fter Willim Rown Hmilton who discovered this lgebr prior to the discovery of mtrix lgebr. Thus the unit quternions re identified with SU(2) in this wy. We shll use this identifiction, nd some fcts bout the quternions to find the SU(2) representtions of briding. First we recll some fcts bout the quternions. (1) Note tht if q + bi + cj + dk (s bove), then q bi cj dk so tht qq 2 + b 2 + c 2 + d 2 1. (2) A generl quternion hs the form q + bi + cj + dk where the vlue of qq 2 + b 2 + c 2 + d 2, is not fixed to unity. The length of q is by definition qq. (3) A quternion of the form ri+sj +tk for rel numbers r, s, t is sid to be pure quternion. We identify the set of pure quternions with the vector spce of triples (r, s, t) of rel numbers R 3. (4) Thus generl quternion hs the form q + bu where u is pure quternion of unit length nd nd b re rbitrry rel numbers. A unit quternion (element of SU(2)) hs the ddition property tht 2 +b 2 1. (5) If u is pure unit length quternion, then u 2 1. Note tht the set of pure unit quternions forms the two-dimensionl sphere S 2 {(r, s, t) r 2 + s 2 + t 2 1} in R 3. (6) If u, v re pure quternions, then uv u v + u v whre u v is the dot product of the vectors u nd v, nd u v is the vector cross product of u nd v. In fct, one cn tke the definition of quternion multipliction s ( + bu)(c + dv) c + bc(u) + d(v) + bd( u v + u v), nd ll the bove properties re consequences of this definition. Note tht quternion multipliction is ssocitive. (7) Let g + bu be unit length quternion so tht u 2 1 nd cos(θ/2), b sin(θ/2) for chosen ngle θ. Define φ g : R 3 R 3 by the eqution φ g (P) gpg, for P ny point in R 3, regrded s pure quternion. Then φ g is n orienttion preserving rottion of R 3 (hence n element of the rottion group SO(3)). Specificlly, φ g is rottion bout the xis u by the ngle θ. The mpping φ : SU(2) SO(3) is two-to-one surjective mp from the specil unitry group to the rottion group. In quternionic form, this result ws proved by Hmilton nd by Rodrigues in the middle of the nineteeth century. The specific formul for φ g (P) s shown below: φ g (P) gpg 1 ( 2 b 2 )P + 2b(P u) + 2(P u)b 2 u. We wnt representtion of the three-strnd brid group in SU(2). This mens tht we wnt homomorphism ρ : B 3 SU(2), nd hence we wnt elements g ρ(s 1 ) nd h ρ(s 2 ) in SU(2) representing the brid group genertors s 1 nd s 2. Since s 1 s 2 s 1 s 2 s 1 s 2 is the generting reltion for B 3, the only requirement on g nd h is tht ghg hgh. We rewrite this reltion s h 1 gh ghg 1, nd nlyze its mening in the unit quternions.

22 22 LOUIS H. KAUFFMAN AND SAMUEL J. LOMONACO JR. Suppose tht g +bu nd h c+dv where u nd v re unit pure quternions so tht 2 + b 2 1 nd c 2 + d 2 1. then ghg 1 c + dφ g (v) nd h 1 gh + bφ h 1(u). Thus it follows from the briding reltion tht c, b ±d, nd tht φ g (v) ±φ h 1(u). However, in the cse where there is minus sign we hve g + bu nd h bv + b( v). Thus we cn now prove the following Theorem. Theorem. If g + bu nd h c + dv re pure unit quternions,then, without loss of generlity, the brid reltion ghg hgh is true if nd only if h +bv, nd φ g (v) φ h 1(u). Furthermore, given tht g +bu nd h + bv, the condition φ g (v) φ h 1(u) is stisfied if nd only if u v 2 b 2 2b when u v. If u v then 2 then g h nd the brid reltion is trivilly stisfied. Proof. We hve proved the first sentence of the Theorem in the discussion prior to its sttement. Therefore ssume tht g + bu, h + bv, nd φ g (v) φ h 1(u). We hve lredy stted the formul for φ g (v) in the discussion bout quternions: φ g (v) gvg 1 ( 2 b 2 )v + 2b(v u) + 2(v u)b 2 u. By the sme token, we hve φ h 1(u) h 1 uh ( 2 b 2 )u + 2b(u v) + 2(u ( v))b 2 ( v) ( 2 b 2 )u + 2b(v u) + 2(v u)b 2 (v). Hence we require tht This eqution is equivlent to ( 2 b 2 )v + 2(v u)b 2 u ( 2 b 2 )u + 2(v u)b 2 (v). If u v, then this implies tht 2(u v)b 2 (u v) ( 2 b 2 )(u v). u v 2 b 2 2b 2. This completes the proof of the Theorem. An Exmple. Let g e iθ + bi where cos(θ) nd b sin(θ). Let h + b[(c 2 s 2 )i + 2csk] where c 2 + s 2 1 nd c 2 s 2 2 b 2 2b. Then we cn rewrite g nd h in mtrix 2 form s the mtrices G nd H. Insted of writing the explicit form of H, we write H FGF where F is n element of SU(2) s shown below. ( ) e iθ 0 G 0 e iθ ( ) ic is F is ic This representtion of briding where one genertor G is simple mtrix of phses, while the other genertor H FGF is derived from G by conjugtion by unitry mtrix, hs the possibility for generliztion to representtions of brid groups (on

23 TOPOLOGICAL QUANTUM INFORMATION THEORY 23 greter thn three strnds) to SU(n) or U(n) for n greter thn 2. In fct we shll see just such representtions constructed lter in this pper, by using version of topologicl quntum field theory. The simplest exmple is given by g e 7πi/10 f iτ + k τ h frf 1 where τ 2 +τ 1. Then g nd h stisfy ghg hgh nd generte representtion of the three-strnd brid group tht is dense in SU(2). We shll cll this the Fiboncci representtion of B 3 to SU(2). Density. Consider representtions of B 3 into SU(2) produced by the method of this section. Tht is consider the subgroup SU[G, H] of SU(2) generted by pir of elements {g, h} such tht ghg hgh. We wish to understnd when such representtion will be dense in SU(2). We need the following lemm. Lemm. e i e bj e ci cos(b)e i(+c) + sin(b)e i( c) j. Hence ny element of SU(2) cn be written in the form e i e bj e ci for pproprite choices of ngles, b, c. In fct, if u nd v re linerly independent unit vectors in R 3, then ny element of SU(2) cn be written in the form e u e bv e cu for pproprite choices of the rel numbers, b, c. Proof. It is esy to check tht e i e bj e ci cos(b)e i(+c) + sin(b)e i( c) j. This completes the verifiction of the identity in the sttement of the Lemm. Let v be ny unit direction in R 3 nd λ n rbitrry ngle. We hve nd where r 2 + s 2 + p 2 + q 2 1. So e vλ cos(λ) + sin(λ)v, v r + si + (p + qi)j e vλ cos(λ) + sin(λ)[r + si] + sin(λ)[p + qi]j [(cos(λ) + sin(λ)r) + sin(λ)si] + [sin(λ)p + sin(λ)qi]j. By the identity just proved, we cn choose ngles, b, c so tht Hence nd e vλ e i e jb e ic. cos(b)e i(+c) (cos(λ) + sin(λ)r) + sin(λ)si sin(b)e i( c) sin(λ)p + sin(λ)qi. Suppose we keep v fixed nd vry λ. Then the lst equtions show tht this will result in full vrition of b.

24 24 LOUIS H. KAUFFMAN AND SAMUEL J. LOMONACO JR. Now consider e i e vλ e ic e i e i e jb e ic e ib e i( +) e jb e i(c+c ). By the bsic identity, this shows tht ny element of SU(2) cn be written in the form e i e vλ e ic. Then, by pplying rottion, we finlly conclude tht if u nd v re linerly independent unit vectors in R 3, then ny element of SU(2) cn be written in the form e u e bv e cu for pproprite choices of the rel numbers, b, c. This Lemm cn be used to verify the density of representtion, by finding two elements A nd B in the representtion such tht the powers of A re dense in the rottions bout its xis, nd the powers of B re dense in the rottions bout its xis, nd such tht the xes of A nd B re linerly independent in R 3. Then by the Lemm the set of elements A +c B b A c re dense in SU(2). It follows for exmple, tht the Fiboncci representtion described bove is dense in SU(2), nd indeed the generic representtion of B 3 into SU(2) will be dense in SU(2). Our next tsk is to describe representtions of the higher brid groups tht will extend some of these unitry repressenttions of the three-strnd brid group. For this we need more topology. 8. The Brcket Polynomil nd the Jones Polynomil We now discuss the Jones polynomil. We shll construct the Jones polynomil by using the brcket stte summtion model [37]. The brcket polynomil, invrint under Reidmeister moves II nd III, cn be normlized to give n invrint of ll three Reidemeister moves. This normlized invrint, with chnge of vrible, is the Jones polynomil [35, 36]. The Jones polynomil ws originlly discovered by different method thn the one given here. The brcket polynomil, < K > < K > (A), ssigns to ech unoriented link digrm K Lurent polynomil in the vrible A, such tht (1) If K nd K re regulrly isotopic digrms, then < K > < K >. (2) If K O denotes the disjoint union of K with n extr unknotted nd unlinked component O (lso clled loop or simple closed curve or Jordn curve ), then where < K O > δ < K >, δ A 2 A 2. (3) < K > stisfies the following formuls < χ > A < > +A 1 <)(> < χ > A 1 < > +A <)(>,

25 TOPOLOGICAL QUANTUM INFORMATION THEORY 25 A -1 A -1 A A A -1 A < > A < > + A -1 < > -1 < > A < > + A < > Figure 9. Brcket Smoothings where the smll digrms represent prts of lrger digrms tht re identicl except t the site indicted in the brcket. We tke the convention tht the letter chi, χ, denotes crossing where the curved line is crossing over the stright segment. The brred letter denotes the switch of this crossing, where the curved line is undercrossing the stright segment. See Figure 9 for grphic illustrtion of this reltion, nd n indiction of the convention for choosing the lbels A nd A 1 t given crossing. It is esy to see tht Properties 2 nd 3 define the clcultion of the brcket on rbitrry link digrms. The choices of coefficients (A nd A 1 ) nd the vlue of δ mke the brcket invrint under the Reidemeister moves II nd III. Thus Property 1 is consequence of the other two properties. In computing the brcket, one finds the following behviour under Reidemeister move I: nd < γ > A 3 < > < γ > A 3 < > where γ denotes curl of positive type s indicted in Figure 10, nd γ indictes curl of negtive type, s lso seen in this figure. The type of curl is the sign of the crossing when we orient it loclly. Our convention of signs is lso given in Figure 10. Note tht the type of curl does not depend on the orienttion we choose. The smll rcs on the right hnd side of these formuls indicte the removl of the curl from the corresponding digrm. The brcket is invrint under regulr isotopy nd cn be normlized to n invrint of mbient isotopy by the definition f K (A) ( A 3 ) w(k) < K > (A),