Louis H. Kauffman- From Knots to Quantum Groups (and Back)

8/3/2019 Louis H. Kauffman- From Knots to Quantum Groups (and Back)

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d y n a m i c s ,

g r o u p s , J.

FROM KNOTS TO QUANTUM GROUPS (AND BACK)

LOUIS 1-1. l (AUFFMAN

D e p a r t m e n t of M at h em at i c s , S t a t i s t i c s an d C o m p u t e r Sc i en ce ,

The Universi ty of I l l inois at Chicago,Chicago, I11 60680, USA

1. INTRODUCTION

Th is pap er t races how the Jon es polyi loiri ial leads natura l ly to th e not ion of ~ U A I I ~ , I I I I I

grou p, and how q uan tum group s g ive r ise to invarianbs of l inks v i as o l u t io ~~ s.o t11c Ya11g-

Baxter equat ion. Sect ioi~ is al l origirial t reatment of Lhe construct ion of the universal

R-matrix. All the other m aterial Ims, or will appe ar elsewhere in s imilar form.

2. KNOT TXIEORY

L et ' s b eg i n b y r eca l l i ~ ~ g. l ~ et e i t l e~ i ~ e i s t e r oves:

I /--------c3

11 3 : c

These m o v es can b e p e r ro rn ~ ed 11 a l i n k d i ag rn 1 1 1 . A l ink Oiagrn~n s n locally

four-valen t p lane gra l ) l~wit11 exl.ra sl. ru cl. ~lr e t llle vcrl.iccs i l l t11e for111o r c r o s s i i ~ g s

Th ese cross ings are taken to int l icate the projecl ioi l of arcs elnbed(led i n a Lllrce-s[)ace,

and pro jected t o the p lane. Ti le broken ar c pai r a t a c ro s s i n g i n d i ca t e s t h e a r c t h a t

passes u n d ern ea t1 1 t h e o t l ~ e r r c it1 space. Any l i n k (a link is a collection of circles

embedded in a t h r ee -s p h e re o r E r~ c l i r l ca~ ihree s$n.ce.) llas a p o i n t o f p r o j c c t i o ~ ~.0 t h e

surface of a t w o d i ~ n e l ~ s i o n a lp h e re o r t o a ~)In l le , o t l ~ a t he pro jccl io l~ wit11 1111tlcr

and over-cross ing in t l i ca t io l~s ) cco ~ i te s d ia gra r l~Cor t l ~ a t ink.



L.11. I < ~ I I I ~ I I I ~ I I

T w o l inks are sa id to be n l l lb ie l i t i s o t op ic i f there is a cont ir iuous t ime-pararr ie ter

f am il y of e m be d di n gs s t a r t i ~ ~ git11 one l inlc and ending wit11 the ot he r one. T h e t l~ eo ry

of kn ots an d l inks is the t l~ eo ry f l ink enibedd ings under t l ie equivalence reiat ion of

ambie nt i so topy. (A k n o t is a l ink rv it li one cornpol le l it . T h a t i s , a knot i s an en~ l ledd ing

of a s ingle circle into t l~ree-space.)

I t i s as s um ed t ha t all t lre e inbeddings are represented (u p to amb ient i so topy) by an

embedding that i s a d iKere~ i t i ab le urve(s ) in the three-space. Links that t lo not admi t osuch a representat ion are cal led wi ld , a n d r r~ us t e t reated separate ly .

T h e l t e idemeis ter moves genera te the theory of knots and l inks in t l rree-d imens ional

s p a c e in t h e s e ns e o f t l ~ eol lowing t l~eorem:

THEOREMR E I D E M E I S T E Rla]). L et Ii' a r ~ d (' be trvo l i r ~k s mbeddecl ' in three-

d imens ional space (e it lrer the three- t l i~ r lens ional l~ he re r the Eucl idean s pace n3).T h e n , I( a n d I{' a r e a ~ ~ ~ h i e r ~ tso top ic if n11d o111yif diagrams for Ir' and I(' a re r e l a t ed !1by a f in i te sequence of t he m oves I , 11, 111.

R E M A I < : 111 Iteiden ~eist er 'sday t he no t iou o f an h i e n t i s o topy was a l so com b i ~ ~ a t o r i a l .

Let R den ote Iteidelneis(.er. For R , ambie nt i so topy was generated by a s ing l e move type

c alle d a n e l e ~ i l e n t n r y o l i ~ l > i l l n t o r i i ~ ls o t o py (o r e l en ien tn ry i s o topy for s lro r t ) . T l ~ e

knots and l inks for R a re p i e ce w i s e l i l ~ e n l - m ea ll ing t l ra t t hey co~ i s i s t f in t ercon -

nec t ions o f s t r a i g l ~ t iue s e g ~ r ~ e ~ ~ t sm bedded in E ~ ~ c l i d e a npace. Ver t ices are regarded i fas t h e endpo i n t s of t hese s eg ~ nc n t s , nd any s t xa i g l ~ t eg ~ ne n t an be r ega rt led a s tile

connec t i on o f t wo s eg~ r i e~ i t s ,y ad t l ing a ver tex a t an in ter ior poin t . Tl le e le lnentary

co l nbi na to r i al i s o topy has two d ir ec ti ons: ex pn l l s i o l ~ , nd co l l t rn c t i o l i . I n an expan-

s ion , one ta kes two ver ti ces on the l ink , and a new ver tex i l l t h e c o m p l e ~ n e n t ~ l c ll h a tthe ( tw o d imens ional ) t r i a~ lg le panned by t l lese ver t ices in tersect s the l i r~k n ly a t one

o f i t s t h ree edges . Ex l ~ a~ l s i o l lonsists in replacing this edge in tlie link by tlie two

remaining edges in t l i e t r i angle . Cor~ t ract ion s t il e o p l ~ o s i t e f e x p a n s i o ~ i t l ~ r e e o in ts

on t he l i nk s pan a tr ia ng le i ~ ~ t e r s e c t i ~ ~ ghe l i ~ l k nly a l o ~ ~ gwo edges ; t l ~ e s e dges a re

replaced by the th i rd e t lge of t l ~ er iangle .

cu

T h e b i de m ei s t er rnoves co li le a l )out v ia exan l inat io ~ l f Llie forn ls of p lanar pro jec-

t ions of t l ie elementary isotopies. For eexa~r~ple,he diagram below sl lows how a t ype IReiderneister move is t l ~ e l ~ a do w f an e l em cn l a ry i s ot opy . ...\

I l e iderneis ter' s ap pro nc l~ o l~is l ~ e o r e ~ r ~s a good wily to get a ge o~n et~ r iccel for the



o no t a dm i t

n t l ~ re e -

R3) .

a re r e l a t ed

imove type 1

s l ~ o rt ) . l l e

r t led as the

lementary

h a t

n l y a t one

y t l ~ e wo

three poin t s

ow a t y p e I

Fro111Knots to Quanturn Groups (and Back) 163

s i t ua t i on . F o r a m odern t r ea t m e n t o f t he Theo rem , u si ng the con t i n r ~ o r~ sor difreren-

tiable) notion of isotopy, see [5].

While Reidemeister 's T heo rem is an excel lent s ta rt i ng point for a comhinatorial theory

of knots a nd l inks, i t doe s not m ake l ife easy. T h e easiest way t o i llustra.t ,e this is toexhib i t a demon such as the one shown below. (This demon - s h o w n t o t h e a u t h o r b y

Ken Mil let - mproves over previous culpri ts , a nd is the smallest possible for project ions

on a sphere.)

T h i s d e m o n D is unkn o t t ed , bu t does no t a d ~ n i t ny s i n~ l) l ify i ng t e i de~ ne i s t e r noves ,

no r does i t ad m i t a lly l ype b l ~ reemoves. ( A I l e i de~ ne i s t e rm o ve i s s ai d t o b e s i l r ~ p I i f y i n g

if i t reduces the number of cr os s i ~~ gsn t l ~ e iag ram .) 111 orde r to unk not D it is necessary

t o f i rs t m ake t h e d i ag ra ~ n l o re co ~ np l ex ~ e fo re t ca ll becom e s i n~ p l e r .E x a ~ n p l e s f t l ~ i s

s o r t s ho w t h a t l l ~ e q u i v a l e ~ ~ c ee l a l i o ~ ~enerat ,ed by the Itei t lemeister nloves is subtle,

and t h a t t he m a t t e r o f const r r~ cL ing nva r i ant s is non-trivial .

The re a re m any accoun t s of t he c la s si ca l co ~ ~ s t ru c t i o nf k no t a n d link i ~ ~ v a r i a ~ ~ l s[ I ] ,

[B], [4], [ l l . ] , [23.]). 111 t l ~ e ex t sect ion , 1 s l ~ a l l o t li rec tl y t o a rnot lc l fo r t he J o ~ ~ e s

po l ynom i al and d i s cuss i t s p l ~ ys i ca l n t e rp re t a t i o~ ~ s .or these purposes i t t loes make

s ens e t o m ake one r en~ ar l ca b o ~ ~ the process o l abs t ract io l l l eading to rnnt l~ernal icalknots . I f we were to make a kno t or l ink fro111 rope or o t l ~ e rm a te r ia l , t l ~ e n h e a ~ r ~ o u ~ ~

of twis t ing on the rope would ~nalce d i l rerence in the behaviour or t l ~ ee s r~ ll in g ~ ~ o t t e d

form. Such twis t ing has bee11 abs t ra cted when we go to t l ~ e i ag ra m o r I ,l~ e ~ ~ a l l ~ e ~ r ~ a t i c a

curve embedd ed in space. We can recover some of th is s t ruct r r re by cons i t l er i~~ g~ i ~ l r l c t l

l i n k s . A framed link is a l i ~ ~ ku ch t l ~ a b a c l ~ o m l ) o ~ ~ e n t~ a s c o ~ ~ t i ~ ~ ~ ~ o r r s~orrn al ector

f ie ld . T h i s i s eq u i v a l e ~ ~ to t l ~ i ~ ~ l t i ~ ~ gl ) o ~ ~ tm b e d t l i ~ ~ g sl ' I)nl l t ls rather ( , I I ~ I Icircles.

T hu s th e f igure below indicates a f ra ~n ed refo i l, wit11 s tn l lt l ard f raming i r l l~er i te t l ro ~n

i t s p l ana r em b edd i ng .

If we keep trac k oT I.l ~e ra111i11g l ~ e n I I C no lollget I~ils r ~ v a r i a ~ ~ c e~ ~ t l e rl ~ e ,yl)e Imove:

F or t h i s r eason i t i s us eful l o have t l ~ e o t ~ cep t f r e g u l a r i s o t o I )y . ' l'wo l i r~ ks r e

said to be regularly isotopic i f one can be ob t n i r~ ed rom t he o t he r by a sequence of typ e

eel lor t l ~ e 11 an d ty pe 111 moves only. Regular i so topy i s lhe ecl ~~ iv i l l e ~~ cee la ti 011 ge~ ~ er a t e t ly t he



type I1 and type 111 Inoves. Note: I

f

?&.- Opposile curls cancel. 'l'l~is eg~~larsotopy is l l ~ e not tl~eoretic ersior~or the Whit-s- ]ley trick 1241. Act~~al ly ,cgrtlar isolopy is a bit subller Illan sirliple fr a ~ ~ ~ i n g .The bands

3;EE3z*p= are isotopic, I ~ u the corresl)ot~cling tring diagrams are 11ot regularly isolopic (tlley l~avez- (limerent Wllitney degree (241.).

,

['A useful invariant or rcg~rlar sol.opy is tlie writhe, i u ( 1 i ) . The wri t l~es the sun1 of

i: the crossing signs

in a given diagram. 'l'l~~rs

It is easy to see that the writl~es a regular isot.ol)y it~v ar ia t~ tor diagrar r~s . t is very

rlseful for riorlnalizing ot l~ er r~varinllts f regr~lar solopy. It t r ~ r ~ ~ sut l l~ a l 11os1f the

invariants we sl~ all iscuss neecl sucl~ ~lornlalizalion.

111. LINK 1NVAn.IANTS AS VACUUM-VACUUM AMPL ITUD ES

First, a quick description of t l ~ e rackel moclel (121 or l l~ e or~cs oly~~ornialg]: Me

give a me tl~o d f associal il~g well-defined polyl~omial ll three variables, ( I i ) ( A , B, d ) ,

to an unoriented link I< . 'l'llis polyl~otnial s tlefinetl recursively by the for tl~ulas:



tI Fro1111C11ots o Q I I ~ I ~ ~ I ~ I I IT O I I J ) ~ailid Dnck) 165

! Th e first forinr ~la sserts tha t the polynomial for a given diagram is obtained as a n

Whit-additive combination of tl ~ e olynomials for t l ~ e iagrams obtained by splicing away t l ~ eI

I given crossing in two possible ways. T l ~ u she small diagrams indicate larger diagrams

that diKer only as i~~tlicated.'11e second formula says tha t t l ~ e alue of a loop (simpleI closed curve in the p l a ~ ~ e )s d , and ll~atf a loop occurs (isolated) inside a larger d ingra rr~,

then the value of the poly~~otnialccluires a factor of (1 frorn this loop. In particrrlar,

the value of a disjoint union of N sirnple closcd curves is d N .

i Ibgetller, the two forr~lrrlasco~nplelcly lctermine (A'), a114 (I<)s well-tlefined just

so long as A , B ant1 d c o ~ n ~ n u l eith olle a~ ~a tl ~e l. .~ ' I I I I S tl ~i s olynoniial takes values in

the ring Z [ A ,B,n ] of polyr~or~lialsn 1,llree variables will1 integer coefIiciel~ts.y have As it stantls, ( K ) is not all i~lvaria~~tf any of the Reiden~cistermoves. IIowever, t l ~ e

following forrnula is a11 easy collsequence of 1. and 2 . above.1

As a result we see t l ~ a t It-) is invariant untler tile type I1 move i f we select U = A-'

and d = - A 2 - A-' . I~rlrll~errnore,t now follows tlireclly that (I<) s i~~vnri ; \~ i t~ntler

the move 111:1I

t f tlie

Thus, with B = A - I and d = --/I2 - we have that (I<)s an invariant of regular

isotopy. To obtain an invariant of ambient isotopy for oriented links, we form

fK (A) = ( - A ~ ) - " ( ~ ) ( K )(

where Ii' is oriented, w(Ii) is the writhe of K as defined in the previous section, and

(Ii) s the bracket evaluated on the unoriented link underlying I i . The reason for this



L.11. I c ~ l l f r 1 1 l ~ ~ l l

factor of - A ~s t ha t

a rn<3-> A( 0 > +A=-'&->

- A(-A'-A')<-> +A?->

=CR)<'").

On e t hen has the

TIIEOREM12]. For al ly o rie i~ le d i ilh. I ( , VIc(t) = j f i (l - '1 ') ~vl lereVfi del lotcs t l ~ e

original one-variab le Jone s j )olynorr~ial .

l ' l ~ u s , he bra cket , suilill,ly norrnalizetl, gives a d i rec t ~ no t l e l o r t l ~ e o r ~ e s o l y l ~ o ~ ~ ~ i n l .

THE A C U U M - V A C U U MM P I , I ' ~ U D E .

In the rest of this sec t io ~l s l~ al l t ick to the bra.cket. , an d show how i t c an I:)(: s t :c ~~s

a "vacu um-vac uum ampli lude" in a combina torial version of topological qu x n tu ~ r ~icld

theory [ 2 5 ] . More generally, we ca n co ~ls ide r n a.m plitude associa ted to a, given diagrn.111

by regard ing the p lane a5 1+ 1 spacet ime. By convention, let t ime run vert ical ly 1 1 1 )co

the pag e, a.nd space proceed from lel t to right . (This is the convention of th e rcadcr of

English.) Posi t ion the l ink diagra m so th at i t is transversa.1 to the spa ce levels csccpt ,

a t cr it i ca l poin t s c orrespo~ld ing o maxima, minima and cross ings .as

Each m i n i m um can be regart let l as a cr~ 'at io11r ~ W O)itl.Li~les r0111 I I C V ; I C I I I I I I I , each

m ax im um a n a n n i l ~ i l ; ~ l i o ~ ~ ,nt1 each crossi llg is an inl.crac1io11 t l ~ o l ~ g l ~ lf as i ~ l vo l v i r~ g

b r ai di ng i n t h e e x t r a s p a li a l t l i n ~ e n s i o ~ ~r l l ~ o g o ~ l n l.o Ll~e ag e). 'KO e a c l ~ f t he s e e v e ~ l t s

we associa te a mat r ix wl~ose nt li ces go over (say) t l ~ c p il ls of t . l ~ e a r ti c le s , a ~ ~ t lv l~ose

values a re th e amp li t r~cle s or each of these processes.

Th e ampl i tudes used here arc a generalization of a .mpl itudes in quant .um m c c l~ a ~ ~ ic s ,

sui tably general ized for the purposes of topology ( I n the process we ta.ke leave or t.lre

usual in terpreta t ions of observat ion in qu an tum mechanics. In th e topology t , l ~e .nq)li-

tud e i tself is a real pro perty of the system . Th er e is no "col lapse of the wave fi lnct. ionn).

Therefo re the a mpli t udes wil l take values in a cornrnutat ive ring (e.g. in Z [ A ,A- ' I ) ,



II Fro111 ICl~ots o Qua1itt1111 roups (and Dnck) 167I1 and l l~epins will run over an arl)itrary finite index set (e.g. {-1,+1)). Arnplitudcsi are calculaled accordi~rg o the pl .i ~~ ci pl csf c l u al ~ t um ~ ~ e c l l n l ~ i c s7]:I

sc:c~ias

icltl

1113

dcr of

s csct pt,

(1) If an event occrirs in a way th at call be decomposed iiito a se t of intlivitlual steps(e.g. creations, a~i~iiliilatio ns,nteractions), tlieri llie alnplitride of tlie given event

is the product of tlie ai~ipli tude s l tlie i~~d iv id ua lteps.

(2) If an event rnay occrir in several disjoint alkernative ways, tlien tlre aml)lit.ude of

this event is the slim of tlie amplitudes of t l ~ e ays.

Given a diagram I(, and a set of matrices as above, we can calc r~la te lie ainplilude

for particles t o be created from tlie vacuum, il~ ter actn llie patlerri of l l ~ eil~k iagrarr~,

and return to tlie vactiiim. Tliis a~npli trlde eco~nposesas a sum of llre arnplit.udcs

for c on fi gu ra ti or ~s f llre clingram. (1 lalie corifiguralioi~as a iieulral term hcre. For

an~ pli tud es ne may prcfer tlie terin Iristory. For a 1~11relypalial interpretation o ~ i c ay

preler the terrn s t a t e . ) l<ncli co~ifigurat.ioii a is all assiglimerit of spi r~ s o llie r lodcs

of llie diagram. ('Tlie ~iotlesare tlie input and oulp ut nodcs of tlie s~nall l ia g r a ~ ~ ~ scorresporidirig to tlie matrices) . Civc~i configr~~.at.iol~ ,ac l~ na lr ix ias a wcll-tlcfiried

value, and tlie ar~ ipl ilu de f this configuralio~i s tlie product of these valucs. Thus

the vacuum-vacuorri a~iil)lilude, (I(), for a given diagra~nK is llie sum (over the

configurations) of tlie pr od i~ct f the matrix values for eacli corifiguralion.

Symbolically, lhis works out in accord with llie usr~ alEir~stein onveiltion for repeated

indices: Write down a prot lr~ct f all tlie matrices for the give11diagrarii, in i~idices,witli

one index for eacli node. Th e amplitude is t l i e ~ ~lie value of tliis expression iirteroretetl

as a sum over all cases of repet,itioiis of an index i ll lower and upper posilions.

a

ach

i r~volvi~~g

s IIaving defined tlie vacr~u~~r-vacoumml)liLutle T'(I<), we rn~ rs l ee wlicli il will I)c an

invariant of regular isotopy, and wlien il \ \ r i l l motlel llic bracket. 111 order lor T (I<)

to be an invariant of rc g~ ~l arsotopy, we lieetl l l ~ eollowi~rg estrictions on llie matrices

[15]:

a.

ve of t.lie

"). c

A-'I) ,



( I n 3. t h er e is a c o r r c s l l o ~ ~ t l i ~ ~ gc l ' l - l r ;~ ~ ~ t lwist , , i \ ~ ~ ( ll l 4 . t . l~ cre s irlso t lre snnle

e q ~ ~ a t i o r ~o r a l l c r o s s i ~ ~ g sc v c r s c t l . ) E q ~ r a l . i o ~ ~. is c illl etl I .lrc Y ~ ~ i~ g - l l i ~ s l , c rS ( l ~ r a l i o ~ ~3]

( h e r e g iv e1 1 w it ,l1 01 11 n 1) it li ly ~ ~ a l . a ~ ~ ~ c l . c r ) .

I ~ E M A R K : t i s i n( .c rc s li l lg Lo s p e c ~ ~ l i r t c~ l ~ o ~ ~ l ., l r e ~ ) l~ys icn l11ea1li11gr l l ~ c s e es t r ic -

t io ns . T h e tw is l c o ~ ~ t l i l i o ~ ~. i s the 111os1 ~ys l . e r io ~ isi l ~ c c t r c l at e s 12. n l ~ t l I - ' v i a

c r e a t i o n s a n t1 a n i~ i l~ i l a t i o n s . s i l n p l e r ~ ) l r y s i c i~ lil,~l:ll.ion111;ly lei~ tl oln e irlsi gl~ the re .

111 h e s i m p li li et l s c c ~ ~ i l r i o ,I re a ssl rrne t l ~ a l . I ro lt ls , n~ rd~.II;IL )ilrallel itle111il.y i ~ ~ e sr e

t

II

STATISTICS'L%

T h i s I ra s b e e n i n t e r p r e l e tl a s a d e p i c t i o ~ r r t h e e q ~ i i~ ln l er r c c/spilr n11d stni is t ic s (see

e .g. [22] a n d r e r e r e ~ ~ c e sh e r e i n ) i i ~ l ~ e r ep i l l i s ~ . egar t le t l a s ca la logr ~cd by llre twist

o f fr a m in g (b ec on ie c ur l o r d i d g r n ~ ~ ~ )lrd s t a t is t i c s c o r ~ . c s p o ~ ~ t l so l l ~ e r a i c l i~ ~ gr t l ~ e

two l ines. Tl~is l ~ o w s l ~ a l a r t o f orlr t l i a g r a ~ n ~ ~ ~ n t i c so ~ . r e s l ) o ~ ~ do o r t l i ~ ~ a r y,llysical

i n te rp r et a ti o ns , a n d t lr at w l i c r c t l ~ c o p o l o g y G c g ii is t l i c c t l r ~ i v n lc l r c cof s p i l l

a n d s t a t i s t i c s l ct lv cs off. 11 this sen se , l l re topolo gy is ; I I I i ~ ~ t l c xf I . l ~ e I O I I - s t a r ~ t l a r t l. . . ..



1 MODELLINGI I E RRACI<E,I'.

IIn order to model t lre I ) rnclie t wit11 a v a c ~ ~ u ~ r l - r ~ a c ~ ~ ~ ~ ~ i ~m p l i t r ~ t l ewe ~ ~ e e do f ind

c r ea t io r i a n d a t i ~ i i l i i l e t i o ~ ii la t ri c es t l ra t i ll ve rs e t o o n e a ~ io t l ~ e r ,nt1 t l i a t give a loop

I va lue of -A2 - A - * . Ilere is arr answer 1.0 t l l ; r t pi~zzle :

31 1I1

v ia

N o t e t h a t t h e ~ l l ; ~ l . r i . u l I i i is sc l t lnre t l ie i t le~lt . i l .~ ,~ ~ t l. l int I,lie loop v alue is t l~ er ef or e

tli e s u m of t l ~ e cl\rc lrcs of l . l~c r ~ l . ~es of A! ( S e e [13],[14],[15]or nlol . ival . iol~s or this

constructio!i.)

Wit l i a given c l~o ice or b l~e r en t . io r~s111tl n ~ i ~ ~ i l ~ i l n t i o ~ ~ s ,he re i s one c l~ oic e or t l re R

ma t r ix t o g i v e t l ~ e r nc li et .:

I

i REMARK :n fac t i t i s in te re s t ing to no te t ha t if t .he c rea t io~ i nd annih i la t ion a rc inve r sc

ma t r i c e s , a n d R is dcfinetl a5 a b o v e , t h e n 2. follows easily, while 3. g o e s as b e l o ~ v ,

ar e

a n d 4. is proved by Grs t c l~ec l< ing

(see t b & L a

ul y4wK:!$e=M-b eJ , .~ gf t h e

tspill/

a n d t h e n p e r ro r mi ri g t li c f ol lo w ir ig ~ a r i a t ~ i o ~ i11 our bracke t t le r ivn t io~ io f t li e i~ ~ v c i r i a n c e

u n d e r t h e I11 move

Witl i t l i is choice , T ( I i ) w ill s a t is f y t l ~ cl e f in i l ~ g c l~ ~ ; l t . i o ~ l sf t l ~ e )racket, ant1 tlierc fore

(I( ) = T'(I( ) (s ince wc I ~ i ~ v eo ~ ~ e c l . l yt l j~ ~ s ( . c t l. l ~ e0 0 1 ) v i ~ l t ~ e ) .



Ii1

l?lle upsllot or tl ~i s I ~ S ( . I I S S ~ O ~ I s I.II:II. )y si1111)ly I ~ I , ~ I I S I . ~ I I ~l lc (: l .c!~ltio~~I I I ~

nrrriilrilntio~r rlir~t.l.ic:csc:orrc?c:tly, we: n t~ tn ~r ~r tt ic :r ~l ly)r.otl~tc:c?l ~ r~ o t l ( ! lf th o

brkcket n ~ l d so l~ l t io rr o t l ~ o 1111g-1311xt.arclrlr~t.iorl.I'llis is 1 . 1 1 ~ irl~l)lcsl,~ ~ s l . ; ~ ~ l c e

of a solutiorl to tllc Ynllg-llnsl.cr ecl~lnl.io~~plwnl.i~~~;I : I I . I I I . ; I I I ~ ~ I . O I I I,11(: k11oI. 1.I1coty.

Tllis is the well-~IIOIVIIID ] It .--~~~alrixo r r e s o o ~ ~ t l i ~ ~ g,o l.l~c 1,(2) cl11r7111.11111gro111). I I

fact, the structure tllnt we llnve crcnletl so Tar \ \ t i l l 11ow c ~ ~ a l ~ l es 1.0 see olie 111ol.ivn1io11

for the constructior~ r l l ~ c I I I ~ ~ I I ~ I I I ~ I1.0111).

Note that we call wriI.e

ca

Ite

A

the

cor

T

diag

I

for

grou

is to

Ltt

with

I to t

E=IOI.. i

Th e matrix E is sigllificnllt i l l lillenr i~lgcl>ra)cci~rrse t. expresses Llle tl cI .c r~ ~~ ir ~n ~l tra 2 x 2 matri x. 1'l1i11 s, lel I' Lc n 2 x 2 ~ l l n l . l . i s \ ~ i l . l l O I I I I I ~ I I ~ . ~ I I ~'111.rics.I ' I I ( ? I I

J ' E ~ J " ' = l le l (I7)&.

(T denbles transpose.)



Now SL(2) (over R c o ~ n ~ n r l l n l i v ei l ~ g ) s Llle s e l o r n ln lr ic c s o r d e l c r l ~ l i l ~ n ~ ~ l ,l lc, and

I can t i lere lore b e c l~orncler izct l s 1 . 1 1 ~ c:t o r 111nI.l.iccs c n v i ~ ~ gl ~ c l,silor~ l1vnrinllI.::

IIere T denotes mat , r ix t ranspose.

A t A = 1 the bracket do cs not discriminate bet.ween undc r ant1 ov cr cr os si~ ~g s,nd

the ident i ty

11 i1 T h u s a t A = 1 (and a l so a t A = -1) the diag ram s becom e interprctc:d as (.cllsorI diagrams for SL (2) invar ian t express ions.i1 I t is t h e n n a t u r a l t o a s k w l lc t l lc r t h e r e i s a generalization o f t h i s s y l l l n l c t l .y

wf o r t h e t o p o l o g y a n d I i n k d i a g r a m s . S pe cif ic ally , we as k w h e th e r E ha s n S ~ I I I I I I C ~ ~ ~

g

I group arialogous to SL(2 ). Som e expe rimen tat ion shows th at t he way to ask t.11isquesl.ion

is to

with

consider

associat ive, possibly non-commu tat ive entries , and ask for the invariances:

t I t is then an exercise in elem entary alg ebra to see th at thcse condit ions a re cclrlivalc~lt~

to the equat ions : (q =a)

ca=

qac db=

qbdba = qa b dc = qcd

bc = cb

a d - da = (g-' - q)bc

a d - q - ' b ~ = 1

These are the defining relat>ionsfor the a lgebra U* = S L ( 2 ) q ( [G I , [17]) , so11let.imcs

ca ll ed t h e S L(2 ) q r~ a l i t um roup . It is not a gro up, but r ather a IIopf algebra. ?'lie

co-algebra s t r uctu re i s g iven by the ma p



Thus

In this case the 1lol)f nlgc1,)l.a l~ns1.11 nl~lipode I I I ~111is is tlireclly rel;rtctl to llte fact

that the rnatrix P 11as~ I Il~vei.se (1"):

Recall tllat the aulil,oclc is n Illap y : /I ' - / * s11cl1 . l l n l l,lle rollowi~ig lii~graln

commu tes

where E and rl are t l ~ c o-1111it11c1 l l ~ i t , csl)ccl.ivcly ; 1 11 t l 111 tle~~ol.cs.l1e~null,il~lic:~tion

in the algebra. Here

& ( I ? ) = 6; and v(6 j ) = 6 j .

7 ~ 1 1 ~ slie conditio~ ~.lt;rt y I)e all al~I.ipotle s j~ l s l .l~aI.

And this is the same as saying tha t P and y ( P ) are inverse matrices.

We could now discuss a number of things about the relationship of this quantutn

group to solutions to the Yang-Baxter equation, and to its dual form as a deformation

of the Lie Algebra for S L ( 2 ) (see [6],19]). But here there is not space for this. The

purpose of this section has been to show how the quantum group arises naturally from

a combination of the topology and a desire to extend the algebraic symmetry inllerent

in a significant special case of the vacuum-vacuum expectat ion model.

in

co

ris

fol

D

sati



V. AND BACK

In order to indicate how the trail looks going back from quantum groups to link

invariants I sllall make a leap to the formalism behind the so-called quantum double

construction of Drinfeld [GI. We shall then see how a TIopf algebra structure can give

rise, through i ts matrix representations, to invariants of links.

We begin with an algebra with generators eo,el, ..., n and eD, el, n, ..,en and Ihe

gfollowing relations describing m~lltiplication n the algebra

( A ) e, el = 7 1 i l t ei

(13) eS e' = 111' ei

where i t is u~ltlerslootl I I ~ I I I.l~c mses cli,~lolr I( : ~)l.otl~tctspn~~siorloclliciel~Ls,altd

hcnce tile I)oxcs c o ~ t t ~ ~ r ~ r l cv i l . l ~ Ll~ee-~~otlcsI , t ) i ~ r ~ t lv i l . l ~ each oI,l~el.. \'e fu~t,lrer

p assume llle follo!ving rc l; ~l io ~r sl ri ~~~c(.wcclr 1111lli01yi11gII)I)( : I . i111~lO I VC T CIS:F

f (C )

Dsatisfies t h e a lgebra ic I'orm of Ll~el'i~rrg-lli~stcr l~rit{iorl:

[/)I? = e, @ e 1 , 1 ~ 1 3 1 , @ 1 &, e" ...IJ J *

this. The

PROOF :See [15]).



In a matrix rcprcce~lt.aLiorltile e - ~ ~ o d c sp r o~ l t ~ltl iccs, litl all a1gcl)rnic soll llio~ i o

tlie Yang-Baxter Eq~liiLio~ l ecol~ lcs ll I;llot t.l~eoiisL's111at.rix oll~t.io~lin all atldctl

permutation. Tllus i f

denotes tlie algebraic solr~tiol~,.llcli

denotes t l~e orrespolltling I i110 l~ l.l~corc:(.ic l -~i lal .ri s l l sollle rel)reserlt;~.t.iorlI.lle i~ltli ces

of this representntio~l orrcsl)o~~tl.0 1 . 1 1 ~ I C ~i l~es).

As \ve know rro~llLllc 1)reviorls sccl.io~l, .lie I;r~ol, llcory t l c n~ ; ~~ i t l srclnl.iollsl~ip e-

tween the crea t io~~I I I ~ L I I I I ~ I I ~ I : I L ~ ~ I I11i11.riccs111tI I.11e li, ... i~~ :~l .ri x.I'l~is L~vist. cIaLi011"

is given diagran~nlalically s rollo\vs:

We conclude l l ~ a l llcrc sllo111tl )e a11 ;~ ii l. ol ll orl )l ~i s~ ~~

3: -A7 t!)

or the abstract algel~ril . I i i ~ t .corresoontls 1.01.llc I I I ~ I O l l t . 1 1 ~ ~ ~ ) r e s ~ ~ i t . i ~ I . i o ~ i

and

and

a IIotion

inver

cond

th e q

Iie

Recal

find

We are

Drinfel

is give1

in deno

Here

that; if

e, @ e

the vac

constru

abstrac

gives a

Acknow

October



-- a n d we ~ ~ e e dl ~ a l (e , ) @ e ' i s l l ~ enverse of e, 8 e"(wit.11 sr~in~nalionI I repealed lower-- and upper indices ) . I t is 1lle1111o1 1:1rtI LO s ee t l ~ a t , f \ lrc were to ~ l la l rc l~ e lgebra i l lto-?c a I Iop f a lge l )r a s r ~ c l ~l ~ a l.11e co - ~ n r ~ l l i l ) l i c n l i o ~ ~or ( I I C lo lvcr i ndex e ' s is l l ~ e l ~ r~ l l i l ) l i ca -

Lion for the upper ~ I I ( ~ L ' S' s a l ~ d ice ve r sa ( I .l ~i s s I .l~ e l o t~ l )l e o~ ~ s l r ~ ~ c I . i o n ) ,l1e11 l ~ i s-= in ve rse r e q u i r e n ~ e ~ ~ ts eq r~ i va l en t .0 y being all a111il)odc (Sce t . 1 1 ~ ppendix . ) ' l ' l~e wis t- c o nd it io n s of t h e k l ~ o t l ~e o l- y r e i ~ ~ t i ~ n a l e l yiecl \vi tl~ .lle IIopC algcl> rn st rl ~c t. r~ reor-

t he qua n t u m g roup . 'I 'l~ is con~ l ) l e l e s l ~ eor1111cy ,acl(, a ll ~e it l l all absl.rnct illode.

A P P E N D I X

IIere we verify that 7, as tlelir~etl ll sectio n 6 , is a l l a ~ ~ t i l ~ o t l el l the I lopf a lgebra .

R ec all t h a t we d e n ~ a ~ ~ d e t ll ~ a t ( e ,) @ eJ be illverse to e, @ e S , h ~ l r i l l ip l y i ~ ~ gl ~ i s u t , w e

h find

c

=

L

.g We a re s r ~ n ~ m i ~ ~ gr1 rcl)ealetl illtliccs, ant1 ill 1 . 1 1 ~Iilst. sl cp t~s etl .lle S:icl. (.hat ill th eBg

D rin fe ld d or ll) le c o ~ ~ s l . r r ~ c t , i o ~ ~ ,. l~el iagor~al l l I Ile ;~lgc:I)~.af 1 . 1 1 ~ ' s wil l1 lowel. s l ~b sc ri pl sg i s g iven by the mul l i l ) l i ca t io~~ocllicic:~ll,s ll IIlc al&cl)ra of 1.lle 1ll)l)er e's . Nol c al so I.llal

c in d en otes ~ ~ l ~ ~ l t i ] ) l i c a ~ . i o ~ ~ ,i ld ( I le fo r r n~ ~ l aor y to I>e a11 an lip ot le isE--

Here we take eo t o b e 1, he identi ty elemenl in the IIopf algeb ra. T hu s we Ilave shown

that i f y i s an an t ipode for the Drinfeld double cons t ruct ion , then y (e , ) @ eJ and

e, @ e ' a r e i nve r ses . Th i s , in t u rn , shows t ha t t he d i ag ram m at i c t w is t cond it ion for

the vacuum-vacuum expecta t ion corresponds , through representat ion of the Drinfeld

cons t ruct ion , to the ex i s tence of an an t ipo de in th e I Iopf a lgebra . I n l l ~ i s e n sc , t h e

a b s t r a ct q u a n t u m g r o u p d e fi ne d b y t l ~ e ormalism of the Drinfeld double construct ion

gives a universal link invariant.

. ...A c k n o w l c d g e ~ n c ~ i t s .I.'l~e ar lll ~o rs ~)le :~sct l.0 I.)e ;lble 1.0 . l~a.~i l<l ~ e r gi ll ~i zc r s f l l ~ e IOct ober 1989 M o r ~ l r e a lC o ~ ~ f e r e n c e I I1nmilt.011ia.11 ysl.e;'i~s,I'larisfor~~lnt.ionG r o u p s

an d Sp ec tra l Trailsforrli Mcl.l~ocls or llreir I1osl)it.ali1.y, rico r~ra gern elit nd st.ii1l1112'on .



I~~:I 'EIII~NC:I:;S

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351-399.

1

2

th

w

co

fu

ge

th

A

a i ja i j

We

rep

(1.4

Louis H. Kauffman- From Knots to Quantum Groups (and Back)

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