lipsoid dynamics, lodular groups, J. FROM KNOTS TO QUANTUM GROUPS (AND BACK) LOUIS 1-1. l(AUFFMAN Department o f Mathematics, Statistics and Computer Science, The University of Illinois at Chicago, Chicago, I11 60680, U SA 1. INTRODUCTION This paper traces how the Jones polyiloiriial leads naturally to the notion of ~UAII~,IIIII group, and how quantum groups give rise to invarianbs of links viasolutio~~ s .o t11c Ya11g- Baxter equation. Sectioi~ is all origirial treatment of Lhe construction of the universal R-matrix. All the other material Ims, or will appear elsewhere in similar form. 2. KNOT TXIEORY Let's begin by recalli~~g .l~e teitle~i~eister oves: I /---- ----c 3 1 1 3 : c These moves can be perrorn~ed 11 a link diagrn111. A link Oiagrn~n s n locally four-valent plane gral)l~ wit11 exl.ra sl. rucl.~lre t llle vcrl.iccs i l l t11e for111 or cros sii~gs These crossings are taken to intlicate the projeclioil of arcs elnbed(led i n a Lllrce-s[)ace, and projected to the plane. Tile broken arc pair a t a crossing indicates the arc that passes underneat11 the otl~er rc it1 space. Any link ( a link is a collection of circles embedded in a three-sphere or Er~clirlca~i hree s$n.ce.) llas a point of projcctio~~ .0 t h e surface of a two di~nel~sional phere or to a ~)Inlle, o tl~at he projccliol~ wit11 1111tlcr and over-crossing intlicatiol~s) cco~ites diagrarl~ Cor t l ~ a t ink.
17
Embed
Louis H. Kauffman- From Knots to Quantum Groups (and Back)
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
8/3/2019 Louis H. Kauffman- From Knots to Quantum Groups (and Back)
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
8/3/2019 Louis H. Kauffman- From Knots to Quantum Groups (and Back)
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
8/3/2019 Louis H. Kauffman- From Knots to Quantum Groups (and Back)
?&.- 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:
8/3/2019 Louis H. Kauffman- From Knots to Quantum Groups (and Back)
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
8/3/2019 Louis H. Kauffman- From Knots to Quantum Groups (and Back)
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) ,
8/3/2019 Louis H. Kauffman- From Knots to Quantum Groups (and Back)
( 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. . . ..
8/3/2019 Louis H. Kauffman- From Knots to Quantum Groups (and Back)
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.)
8/3/2019 Louis H. Kauffman- From Knots to Quantum Groups (and Back)
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
8/3/2019 Louis H. Kauffman- From Knots to Quantum Groups (and Back)
-- 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 .
8/3/2019 Louis H. Kauffman- From Knots to Quantum Groups (and Back)