T U Q UOC T HANG L E J UN M URAKAMI
The universal Vassiliev-Kontsevich invariant for framed oriented links
Compositio Mathematica, tome 102, n
o1 (1996), p. 41-64
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The universal Vassiliev-Kontsevich invariant for framed oriented links
LE TU
QUOC
THANG and JUN MURAKAMI© 1996 Kluwer Academic Publishers. Printed Netherlands.
Max-Planck Institut für Mathematik, Gottfried-Clarenstr., 26, 53225 Bonn, Germany email:letu@newton.math.buffalo.edu, jun@math.sci.osaka-u.ac.jp
Received 21 February 1995; accepted in final form 12 March 1995
Abstract. We give a generalization of the Reshetikhin-Turaev functor for tangles to get a combinato- rial formula for the Kontsevich integral for framed oriented links. The rationality of the Kontsevich
integral is established. Many properties of the universal Vassiliev-Kontsevich invariant are dis- cussed. Connections to invariants coming from quantum groups and to multiple zeta functions are explained.
Introduction
The aim of the paper is to
give
apresentation
of the universal Vassiliev-Kontsevich invariant of framed oriented links and to establish some of itsproperties, including
the
rationality.
We define amapping
from the set of all framed orientedtangles
to rather
complicated
sets. Thismapping,
when restricted to the set of all framed oriented links in the3-sphere S3,
is anisotopy
invariant called the universal Vassiliev-Kontsevich invariant of framed oriented links. It is aspowerful
as theset of all invariants of finite
type (or
Vassilievinvariants)
of framed oriented links.Hence it dominates all the invariants
coming
fromquantum
groups in which the R-matrix is a deformation ofidentity,
as in[Re-Tu; Tu1].
Similar constructions of the universal Vassiliev-Kontsevich invariants appear in[Car; Piul].
The values of the universal Vassiliev-Kontsevich invariant of framed knots lie in an
algebra,
and if weproject
to anappropriate quotient algebra,
weget
the Kontsevichintegral
of knots(Theorem 6).
Actually
the universal Vassiliev-Kontsevich invariant is constructedusing
anobject
called the Drinfeld associator. This is a solution of asystem
ofequations.
Every
solution of thissystem gives
rise to a universal Vassiliev-Kontsevich invari- ant which we prove(Theorem 8)
isindependent
of the solution used. As acorollary
we
get
therationality
of the universal Vassiliev-Kontsevich invariant and Kontse- vichintegral.
The
rationality
of the Kontsevichintegral
was claimed in[Kontl ],
withoutproof, citing only
Drinfeld’s paper[Drin2].
The result of[Drin2]
cannot beapplied directly
to this case because the spaces
involved, though related,
are in fact different. Herewe
modify
Drinfeld’sproof
to oursituation, using
asuggestion
of Kontsevich.Fig. 1.
Many properties
of the universal Vassiliev-Kontsevich invariant are established.By
a suitable substitution we canget
all the link invariantscoming
from quan- tum groups. Weexplain
how tocompute
the Vassiliev-Kontsevich invariantusing
braids. Connections to
quasi-Hopf algebras
and tomultiple
zeta values are dis-cussed.
For a detailed
exposition
of thetheory
of the Kontsevichintegral
and theuniversal Vassiliev-Kontsevich invariant for
(unframed)
knots see[Barl]. Many arguments
in[Bar1]
aregeneralized
here.For technical convenience we use
q-tangles
instead oftangles.
Thisconcept
is similar to that of ac-graph
introduced in[Al-Co]. Actually
thecategory
ofq-tangles
and the
category
oftangles
are the same,by
Maclane’s coherence theorem.1. Chord
diagrams
Suppose
X is a one-dimensionalcompact
oriented smooth manifold whose com-ponents
are numbered. A chorddiagram
with support X is a setconsisting
of afinite number of unordered
pairs
of distinctnon-boundary points
onX, regarded
up to orientation and
component preserving homeomorphisms.
We view eachpair
of
points
as a chord on X andrepresent
it as a dashed lineconnecting
the twopoints.
Thepoints
are called the vertices of chords.Let
A(X)
be the vector spaceover Q (rational numbers) spanned by
all chorddiagrams
withsupport X, subject
to the 4-term relation:DI - D2
+D3 - D4
=0,
where
Dj
are four chorddiagrams
identical outside a ball in whichthey
differ asindicated in
Figure
1.The space
4(X)
isgraded by
the number of chords. We denote thecompletion
with
respect
to thisgrading
alsoby A4(X). Every homeomorphism f: X -
Yinduces an
isomorphism
betweenA(X)
andA(Y).
On the
plane R2
with coordinates(x, t)
consider the setX n consisting
of nvertical lines x = al, x = a2,..., x = an, ai ... an,
lying
between twohorizontal lines t =
bi and t
=b2, b1
1b2.
All the lines are orienteddownwards;
they
are numbered from left toright,
i.e. the number of the line x = ai is i. The spaceA(Xn)
will be denotedby Pn.
Acomponent
ofXn
is called astring.
Thevector space
Pn
is analgebra
with thefollowing multiplication.
IfD and D2
aretwo chord
diagrams
inPn,
thenD
1 xD2
is the chorddiagram gotten by putting D above D2.
The unit is the chorddiagram
without any chord. LetP0 = Q.
PROPOSITION 1.
[Kontl].
Thealgebra P1 is
commutative.Suppose
thatX, X’
havedistinguished components 1, £1,
and that X consists ofloop components only.
Let D EA(X)
and D’ eA(X’)
be two chorddiagrams.
From each of
Ê, £’
we remove a small arc which does not contain any vertices.The
remaining part
of ~ is an arc which weglue to £’
in theplace
of the removedarc such that the orientations are
compatible.
The new chorddiagram
is called the connected sumof D, D’ along
thedistinguished
components. It does notdepend
on the locations of the removed arcs, which follows from the 4-term relation and the fact that all
components
of X areloops.
Theproof
is the same as in caseX
= X’ = SI as in [Barl].
In case when X
= X’
=S1,
the connected sum defines amultiplication
whichtums
A(S1)
into analgebra
which we will denotesimply by A.
If D is a chorddiagram
inA, by removing
a small arc of thesupport S
1 which does not contain anyvertices,
weget
a chorddiagram
inP1.
This defines anisomorphism
betweenthe
algebras A and Pi (see [Kontl; Barl]).
Suppose again
that X has adistinguished component
Ê. LetX’
be the manifoldgotten
from Xby reversing
the orientation of 1. We define a linearmapping
S~:A(X) ~ A(X’)
as follows. If D eA(X) represents by
adiagram
with nvertices of chords on Ê.
Reversing
the orientationof ~,
thenmultiplying by (-1)n,
from D we
get
the chorddiagram S~(D)
EA(X’).
Now let us define
Ai: Pn - Pn+1,
for1
i n.Suppose
D is a chorddiagram
inP n
with m vertices on the ithstring. Replace
the ithstring by
twostrings,
the left and theright,
very close to the old one. Mark thepoints
on this newset of n + 1
strings just
as inD ;
if apoint
of D is on the ithstring
then ityields
two
possibilities, marking
on the left or on theright string. Summing
up all 2"2possible
chorddiagrams
of thistype,
weget Di ( D )
EPn+1.
Define sj by sj(D)
= 0 if thediagram
D has a vertex of chords on the ithstring.
Otherwise let03B5i(D)
be thediagram
inPn- gotten by throwing
away the ithstring.
Wecontinue 03B5i
to a linearmapping
fromPn
toPn-1.
Notation: we will write A for
03941: P1 ~ P2, id
~ ···0 A
~ ··· ~ id(the A
isat the ith
position)
forAj ; s for 03B51: P1 ~ P0
=Q, id ~ ··· ~ 03B5
~ ··· ~ id(the -
isat the ith
position)
for 03B5i.REMARK. The reader should not confuse A with the
co-multiplication
introducedin
[Bar1]
forA.
PROPOSITION 2. We have
(0394 ~ id)A
=(id
~A)A.
This follows
immediately
from the definitions.THEOREM 1. The
image of 03B4n: P1 ~ P n+
1 lies in the centerofP n+
1.The
proof
is similar as in case n = 0 which isProposition
1 and isproved
in[Bar1].
2. Non-associative words
A non-associative word on some
symbols
is an element of the free non-associativealgebra generated by
thosesymbols.
Consider the set of all non-associative wordson two
symbols
+ and -. If w is such aword,
different from + and - and theunit,
then w can bepresented
in aunique
way w = w1w2, where W 1, W2 are non-associative non-unit words. Define
inductively
thelength l(w)
=l(w1)
+l(w2)
ifw = w1w2 and
/(+)
=l(-)
= 1. A non-associative word can berepresented
as asequence of
symbols
andparentheses
which indicate the order ofmultiplication.
There is a map which transfers each non-associative word into an associative word
by simply forgetting
the non-associative structure, thatis, forgetting
theparentheses.
An associative wordis just
a finite sequence ofsymbols.
If we have a finite sequence of
symbols +, -,
then we can form a non-associative wordby performing
themultiplication step by step
from the left. It will be called the standard non-associative word of the sequence.Suppose
w1, w2 are non-associative words.Replacing
asymbol
in the word W2by
w, onegets
another word w. In such a case we will call wl a subword of w, and write wi w.3.
q-tangles
We fix an oriented 3-dimensional Euclidean
space R 3with
coordinates(x,
y,t).
A
tangle
is a smooth one-dimensionalcompact
oriented manifold L~ R3 lying
between two horizontal
planes
=al, tt - b},
a b such that all theboundary points
arelying
on two linestt
= a, y =0}, tt - b, y
=0},
and at everyboundary point
L isorthogonal
to these twoplanes.
These lines are called thetop
and the bottom lines of thetangle.
A normal vector
field
on atangle
L is a smooth vector field on L which isnowhere
tangent
to L(and,
inparticular,
is nowherezero)
and which isgiven by
the vector
(0, - 1, 0)
at everyboundary point.
Aframed tangle
is atangle
enhancedwith a normal vector field. Two framed
tangles
areisotopic if they
can be deformedby
a1-parameter family
ofdiffeomorphisms
into one another within the class of framedtangles.
We will consider a
tangle diagram
as theprojection
ontoJae 2( x, t)
oftangle
ingeneric position. Every
doublepoint
isprovided
with asign
+ or -indicating
anover or under
crossing.
Two
tangle diagrams
areequivalent
if one can be deformed into anotherby using: (a) isotopy
ofR2(x, t) preserving
every horizontal linef t
=constl, (b) rescaling
ofr2(x,t), (c)
translationalong
the t-axis. We will considertangle diagrams
up to thisequivalent
relation.Two
isotopic
framedtangles
mayproject
into twonon-equivalent tangle
dia-grams. But if T is a
tangle diagram,
then T defines aunique
class ofisotopic framed tangles L(T) :
letL(T)
be atangle
whichprojects
into T and is coincident with Texcept
for a smallneighborhood
of the doublepoints,
the normal vector at everypoint
ofL(T)
is(0, -1, 0).
One can
assign
asymbol
+ or - to all theboundary points
of atangle diagram according
to whether thetangent
vector at thispoint
directs downwards orupwards.
Then on the
top boundary
line of atangle diagram
we have a sequence ofsymbols consisting
of + and -.Similarly
on the bottomboundary
line there is also asequence of
symbols
+ and -.A
q-tangle diagram
T is atangle diagram
enhanced with two non-associative wordswb(T)
andwt(T )
such that whenforgetting
about the non-associative struc- ture frommt(T) (resp. wb(T))
weget
the sequence ofsymbols
on thetop (resp.
bottom) boundary
line.Similarly, a framed q-tangle
L is a framedtangle
enhancedwith two non-associative words
wb(L)
andwt(L)
such that whenforgetting
aboutthe non-associative structure from
wt(L) (resp. wb(L))
weget
the sequence ofsymbols
on thetop (resp. bottom) boundary
line.If
Tl,T2
areq-tangle diagrams
such thatwb(T1)
=wt (T2)
we can define theproduct
T =Ti
xT2 by putting Tl
aboveT2.
Theproduct
is aq-tangle diagram with wt(T)
=wt(T1), wb(T) = wb(T2).
Every q-tangle diagram
can bedecomposed (non-uniquely)
as theproduct
ofthe
following elementary q-tangle diagrams:
(la) X+w,v,
where v w are two non-associative words on one onesymbol
+,v = ++, the
underlying tangle diagram
is inFigure 2a,
with wt = ivb = w,the two
strings
of thecrossing correspond
to twosymbols
of the word v.(1 b) X -@v:
the same asX+w,v, only
theovercrossing
isreplaced by
theundercrossing (Figure 2b).
(2a) Yw,v
with v =(+-)
w, all thesymbols
in w outside v are +. Theunderlying tangle
is inFigure
2c. Here wt = w, wb is obtained from wby removing
v.(2b) Yjl
with v =(-+)
w, all thesymbols
in w outside v are +. Theunderlying tangle
is inFigure
2d. Here wb = w, wt is obtained from wby removing
v.(3a) T;%lW2W3,W
where w1, w2, w3, w are non-associative words on onesymbol
+, and
((w1w2)w3)
is a subword of w. Theunderlying tangle diagram
istrivial, consisting
ofL(w) parallel lines,
all are directeddownwards,
andFig. 2.
wb(T+w1w2w3,w) = w
whilewt(T+w1w2w3,w) is
obtained from wby substituting ((w1w2)w3) by (w1(w2w3)).
(3b) T-w1w2w3,w
where wl, w2, w3, w are non-associative words on onesymbol
+, and
((w1w2)w3)
is a subword of w. Theunderlying tangle diagram
istrivial, consisting
ofl ( w ) parallel lines,
all are directeddownwards,
andWt(T;;lW2W3,W)
= w whilewb(T;;lW2W3,W)
is obtained from wby substituting ((WIW2)W3) by (W 1 (W2W3» -
(4)
All the above listedq-tangle diagrams
with reversed orientations on somestrings
and thecorresponding change of signs
of theboundary points.
4. The Drinfeld associator
Let M be the
algebra
over C of all formal series on twonon-commutative,
asso-ciative
symbols A,
B. Consider a function G:(0, 1) ~
Msatisfying
thefollowing
Knizhnik-Zamolodchikov
equation
Let
GA (A, B)
be the value at t = 1 - À of the solution to thisequation
which takes the value 1 at t = À. Thefollowing
limit existsand can be written in the form
where the summation is over all the associative words on two
symbols
A and B.The coefficients
fW
arehighly
transcendent and arecomputed
in[Le-Mu2].
EachfW
is the sum of a finite number of numbersof type
with natural numbers
il, ... , ik,ik
2. Thesenumbers,
calledmultiple
zetavalues,
have
recently gained
much attention among number theorists(see [Za]).
More
precisely,
let bold letters p, q, r, s stand fornon-negative
multi-indices.For a multi-index p =
(pl, ..., pk), k
is thelength
of p. Letlk
be the multi-indexconsisting
of k letters 1.Let 1 Pl = E
pi .For two multi-indices p, q of the same
length k,
define:q(p; q)
= 0 if one ofpi, qi
is 0,
otherwiseThen
Here the sum is over all multi-indices p, q, r, s of the same
length k, k
=1, 2, 3,....
This formula follows from the result of
[Le-Mu2].
Denote
by flij
the chorddiagram
inP3
with one chordconnecting
the ith andthe jth strings.
The element03A6KZ
=~(1 203C0-103A912, 1 203C0-103A923)
EP3 0 C
is calledthe KZ Drinfeld associator. It is a solution of the
following equations (for
aproof
see
[Drin 1 ; Drin2]):
Here
03A6ijk
is the elementof P3
0 C obtained from (bby permuting
thestrings:
thefirst to the
ith,
the second to thejth,
the third to the kth andR’3*
=exp(flij /2).
Equation (A1)
holds inP4 ~ C, equations (A2a, A2b, A3)
inP3
0C,
andequation (A4)
inP2 ~
C.REMARK.
(A2b)
follows from the other identities in(Al-A4).
The first follows from
(1)
and theorem 1 for any03A6,
the second istrivial,
the third follows from theorem 1. Theequations (A1-A4, B1-B3)
mimic the axioms of aquasi-triangular quasi-Hopf algebra
in[Drin1].
Every
solution $ of(A1-A4)
is called an associator. TheoremA"
of[Drin2]
asserts that there is an associator
03A6Q
with rationalcoefficients,
i.e.03A6Q
EP3.
5. The
représentation
of framedq-tangles
Every tangle diagram
T defines a framedtangle L(T),
and every framedtangle
1(is
L(T)
for sometangle diagram.
Suppose
T is aq-tangle diagram.
ThenL(T)
is a framedq-tangle. Regarding L(T)
as a 1-dimensionalmanifold,
we can define the vector spaceA( L(T)),
whichwe will abbreviate as
A(T).
This vector spacedepends only
on theunderlying tangle diagram
of T but not on the words wb and wt.If Di
EA(Ti), i = 1, 2 and T = Tl
xT2,
then we can defineDi
xD2
EA(T)
in the obvious way,
just putting D1
aboveD2.
We will define a
mapping
T ~Zf(T)
EA(T) ~
C for anyq-tangle diagram
such that if T =
Tl
xT2
thenZf(T) = Zf(T1)
xZf (T2).
Such amapping
isuniquely
definedby
the values of theelementary q-tangle diagrams
listed in theprevious
section. Letwhere nn is the chord
diagram
in4(X+,)
with nchords,
each isparallel
to thehorizontal line and connects the two
strings
that form the doublepoint
ofXw,w
i.e.
Zf(Yw,v)
is the chorddiagram
inA(Yw,v)
without any chord.i.e.
Zf(Y-1w,v)
is the chorddiagram
inA(Y-1w,v)
without any chord.For a
q-tangle diagram Te. w2w3,w
let #1 and #r berespectively
the number ofstrings (in
theunderlying tangle diagram)
left andright
of the block ofstrings
which form the word
((w1w2)w3).
DefineHere
1~n
1 0 D 01~n2,
for D EP n,
is the element ofPn1+n+n2
which has nochords on the first ni
strings
and on the last n2strings,
and on the middle nstrings
it looks like D.
Finally,
ifT’
is aq-tangle diagram
obtained from Tby reversing
the orientation of somecomponents £1,... ék,
thenZf(T’)
is obtained fromZf(T) by
succes-sively applying
themappings S~1,..., S~k.
The result does notdepend
on the orderof these
mappings.
THEOREM 2. 1. The
mapping
T -Z f (T)
iswell-defined:
it does notdepend
onthe
decomposition of a q-tangle diagram
intoelementary q-tangle diagrams.
2.
Let ~ = Zf(U) ~ A ~ C,
where U istangle diagram
inFigure
3. Thenwhere the RHS is the connected sum
of 0
andZ f along
the indicated component, andT,
T’are twoq-tangles
identical outside a ball in whichthey differas
indicatedin
Figure
4.3.
Suppose
thecoordinate function
t on the ith componentof
T has si maximalpoints. Define
Fig. 4.
where the
right
hand side is the element obtainedby successively taking
the con-nected sum
of ~-Si
andZf(T) along
the ith component.If two q-tangle diagrams Tl , T2 define
the sameisotopic
classof framed q-tangles,
thenZ¡(T1)
=Z¡(T2),
hence
f
is anisotopy
invariantof framed q-tangles.
In
particular, Z f
is anisotopy
invariant offramed
oriented linksregarded
asframed
q-tangles
withoutboundary points.
There are at least two ways to prove Theorem 2. In the first which is more
algebraic,
we use MacLane’s coherence theorem to reduce thecategory of q-tangles
to the
category of tangles
and thenverify
thatZ f
does notchange
under certain localmoves
(see
the definition of the local moves in[Re-Tu; Al-Co]). Similar proofs
aresketched in
[Car; Piul].
In the second which is moreanalytical (see [Le-Mu3]),
wefirst define the
regularized
Kontsevichintegral
for framed orientedtangles,
then weprove that the value
Z f
of aq-tangle
is the limit(in
somesense)
of theregularized
Kontsevich
integrals.
In thisapproach
we can avoidusing
MacLane’s coherence theorem andverifying
the invariance under local moves. The secondproof
alsoshows the relation between
Z f
and theoriginal
Kontsevichintegral (see
Theorem6
below).
REMARK. We have chosen the normalization in which
Z f
of the unframed trivial knot is~-1,
of theempty
knot is 1.THEOREM 3. Let 03C9 be the
unique
elementof 4
with one chord. Then achange
inframing
results onf by multiplying by exp(03C9/2):
where
T+, T_,
T are identical outside a ball in whichthey differ
as indicated inFigure
4.This can be
proved easily by moving
the minimumpoint
to the left thenusing
the
representations
ofq-tangles.
The invariant
f
is called a universal Vassiliev-Kontsevich invariant of framed oriented links. As in[Barl],
it is easy to prove thatZ f (K1)
=f(K2)
if andonly
if all the invariants of finite
type
are the same for framed linksIi7j
1 andK2.
Thismeans
Zy
is aspowerful
as the set of all invariants of finitetype,
inparticular
it dominates all invariants
coming
from R-matrices which are deformations ofidentity,
as in[Tu1; Re-Tu].
Let be a
component
of a one-dimensionalcompact
manifold X and X’ be obtained from Xby replacing î by
twocopies
of ~. In a similar manner as inSection 1 one can define the
operator 0394~:A(X) ~ A(X’).
THEOREM 4.
Suppose
L isa framed
oriented link with adistinguished
component~, L’ is obtained from
Lby replacing
~by
two itsparallels,
close to~, L"
is obtainedfrom
Lby reversing
the orientationof ~.
ThenThe second
identity
followstrivially
from the definition ofÎf .
The first is moredifficult and can be
proved by analyzing
theparallel
of theelementary q-tangles.
Note that the chosen normalization of
Z f plays important
role in theidentity.
Theformula for the
parallel
version ofZf (not Éj)
wouldrequire
an additional factor.Applying
thisidentity
to the unknot weget
a beautiful formulaTHEOREM 5.
Suppose LI, L2
areframed
links withdistinguished
components and L is the connected linkalong
thedistinguished
components. ThenThe
right
hand side is the connected sum of0, f(L1)
andf(L2) along
thedistinguished components.
Theorem 5 followseasily
from the construction off.
6. The Kontsevich
integral
Let
,Ao
=A/(03C9A),
where w is theonly
chorddiagram
in A with one chord.With
respect
to connected sum,A0
is a commutativealgebra.
There is a naturalprojection
p:A - A0.
Let 1( be the
image
of anembedding
of the oriented circle intoR3 lying
betweentwo horizontal
planes {t
=tmin}, {t
=tmax}.
We will consider the 2-dimensionalplane (x, y)
as thecomplex plane
with coordinate z = x +y 1 .
The Kontsevichintegral
of Il is defined as an element of.Ao by
where for fixed
tmin tj
1t2
...tm tmax the object
P is a choice ofunordered
pairs
of distinctpoints zi, zi lying
in 1( n{t
=ti}
for i =1 ... ,
m; the summation in(7)
is over all suchchoices; D p
is the chorddiagram
inA0
obtainedby connecting pairs zi, z’i by
dashedlines; # P~
is the number ofzi , z’i
at which theorientation of Il is downwards. Here we
regard zi, z’i
as functionsof ti (for
moredetails on the Kontsevich
integral
see[Bar1]).
The
integral Z(K)
is not anisotopy
invariant.Let y
=p(~).
Kontsevichproved [Kont1]
thatÎ(K)
:=03B3-s. Z(K),
where s is the number of maximumpoints
ofthe coordinate function t on
K,
is anisotopy
invariant of(unframed)
orientedknots. Note that in
[Bar1]
instead ofZ
another normalizationZ = y . Z
is used.This invariant is as
powerful
as the set of all invariants of finitetype.
The relation betweenZ f
and the Kontsevichintegral
isexplained
in thefollowing
THEOREM 6. For
a framed
oriented knot 1(This theorem and the trivial
generalization
for links areproved
in[Le-Mu3].
Knowing (K)
eAo
one can alsocompute f(K) (see [Le-Mu3]).
7.
Symmetric twisting
An element D E
P2 ~
C is calledsymmetric
ifD21
=D,
whereD 21
is obtained from Dby permuting
the twostrings
of thesupport. Suppose
that F eP2 ~
Csatisfies
(T2)
F issymmetric.
Then there exist
F-1 in P2
0 Csatisfying (T1, T2).
If 4l is an element ofP3
0C,
thenis said to be obtained from (1)
by twisting
via F.Note that the first two terms in the
right
hand side of(8)
commute with eachother,
so do the last two terms. If 4l eP3
0 C is anassociator,
i.e. a solution of(Al-A4),
then it is not difficult to checkthat 4l
is also an associator.For a non-associative word w on one
symbol
+ defineFw
ePl(w) by
inductionas follows.
Let F~ =
1 EQ,F+ =
1EPI, 0++ =
F EP2
0 C. For w = w1w2let
Then
(8) implies
that $ =F+(++)03A6(F(++)+)-1.
Fix F E
P2 ~
Csatisfying (Tl, T2).
Consider a newmapping
T -3Z f (T)
defined for
q-tangle diagrams by
the same rules as forZf, only replacing
the valueslisted in Section 3 for
elementary q-tangle diagrams by:
Here
S1
and,S’2
arerespectively
theoperators
which actby reversing
the orientation of the first and the secondcomponents
of thesupport
of chorddiagrams
inP2
as described in Section 1. The values ofZFf(T±w1w2w3,w)
are definedby
the sameformulas
(D3a,b), replacing 03A6KZ by
obtainedfrom 03A6KZ by twisting
via F.THEOREM 7. The map
Z f is well-defined and for
everyq-tangle diagram
TIn
particular, ZFf(L)
=Zf(L) for
anytangle diagram
L withoutboundary points.
Proof.
We needonly
to checkidentity (10).
It’s sufficient to check for the cases when T areelementary q-tangle diagrams.
These cases followstrivially
from thedefinition. ~
Note that if à has rational
coefficients,
i.e. if 4) EP3,
then from the defini-tion,
the invariantZ f
of a framed link(not
framedq-tangle)
has rational coeffi-cients, f
EA(K). Although
the coefficients of F may be irrational and in(D2a’, D2b’)
the elementsF, F-1 are involved, they
appear inpairs
which annihi-late each other in every link
diagram.
REMARK. In
[Drinl ; Drin2]
Drinfeld defined twists forquasi-triangular quasi- Hopf algebras.
Here weadapt
a similar definition for the series ofalgebras P n
which
play
the role of asingle quasi-triangular quasi-Hopf algebra.
If we use therepresentation
of section 10 below then weget
aquasi-triangular quasi-Hopf alge- bra,
and the construction of twists herecorresponds only
to Drinfeld’s twist which does notchange
theco-multiplication.
If F is notsymmetric,
then A isreplaced
by = F210394F-1 = (F21F-1)0394.
THEOREM 8.
Ifip, ip’
E(P3 0 C)
areassociators,
then 4) isobtained from 03A6’ by twisting
via some F EP2 ~
Csatisfying (Tl, 12).
As a
corollary,
from every solution 03A6 of(Al -A4)
we can construct an invariantof framed
q-tangles.
All suchinvariants,
when restrict to the sets of framed orientedlinks,
are the same and contain all invariants of framed oriented links of finitetype.
By
theorem A" of[Drin2]
there is a solution03A6Q
with rationalcoefficients,
thus weget
COROLLARY. The universal Vassiliev-Kontsevich invariant
of framed
links hasrational
coefficients
in the sense thatÎf (L) belongs
toA(L) for
everyframed
link L. The Kontsevich
integral of a
knot has rationalcoefficient
in the sense thatZ(1()
E,Ao.
Proof of
Theorem 8. LetHere
4ln, 4)’ n
are thehomogeneous part
ofgrading
n.Suppose
wealready
haveComparing
thek -grading parts
of(A1-A4)
for(D,
03A6’ weget:
where d:
Pn ~ Pn+1
is themapping:
PROPOSITION 3.
If 1/;
EP3 ~
Cof grading
k andsatisfying (CI-C4)
thenthere is a
symmetric
elementf
EP2 (D
Cof grading
k such thatd( f )
=0;
03B51(f) = E2(1)
= 0.Suppose
for the moment that this is true. Pickf
as in thisproposition.
Then onecan check
immediately
that the twistby
F = 1 +f
transfers 4)to 03A6 with 03A6i
=03A6’i
for 0 i k.
Fig. 5.
Continue the process we can find a element F E
P2 0
Csatisfying (T1, T2)
which transfers 03A6 into
03A6’.
There remains
Proposition
3 to prove.9. Proof of
Proposition
39.1. OTHER REALIZATIONS OF
P n
A Chinese character
([Barl])
is agraph
whose vertices are either trivalent and oriented or univalent. Here an orientation of a trivalent vertex isjust
acyclic
order of the three
edges
incident to this vertex. The trivalent vertices are calledinternal,
the univalent vertices are called external. Theedges
of thegraph
willbe
represented by
dashed lines on theplane. By
convention all the orientations infigures
are counterclockwise for Chinese characters.An n-marked Chinese
character 03B6
is a Chinese character with at least one exter- nal vertex in each connectedcomponent,
where in addition the extemal verticesare
partitioned
into n labeled sets03981(03B6),..., 0398n(03B6).
Let
03B5n
be the vector spaceover Q spanned by
all n-marked Chinese characterssubject
to thefollowing
identities(see
also[Barl]):
(1)
theantisymmetry
of internal vertices:03B61
+03B62
=0,
for every two Chinese characters03B61
and03B62
identical identicaleverywhere except
for the orientation at one internal vertex.(2)
the Jacobiidentity: 03B61 = 03B62
+03B63,
for every three Chinese characters identical outside a ball in whichthey
differ as inFigure
5.Let us define linear
mappings Aç : 03B5n ~ 03B5n+1 and 03B5i:03B5n ~ En-1. Suppose
theset
0398i(03B6)
of an n-marked Chinesecharacter 03B6 E En
containsexactly
m vertices.There are 2"L ways of
partition 0398i(03B6)
into an orderedpair
of subsets. For each suchpartition q
let03B6q
be the(n + 1)-marked
Chinese character with the sameunderlying
Chinese character
as 03B6, 0398j(03B6q) = 0398j(03B6) if j i, 8j(çq)
=8j-1(Ç) if j
i +2,
while
0398i(03B6q), 0398i+1(03B6q)
are two subsets of8i(ç) corresponding
to thepartition q.
Define
0394i(03B6)
as the sum of all 2m(n
+1 )-marked
Chinese charactersçq.
If
0398i(03B6) ~ 0
define03B5i(03B6)
= 0. Otherwise defineéi(Ç)
as the(n - l)-marked
Chinese character with the same
underlying
Chinese character and0398j(03B5i(03B6)) =
0398j(03B6) if j 2, 0398j(03B5i(03B6)) = 0398j+1(03B6) if j
i.Fig. 6.
The
Zn-grading
of an n-marked Chinesecharacter 03B6
is thetuple (k 1,
...,kn)
ofintegers, where ki
is the number of elements of0i( ç).
The number03A3ni=1 ki
is calledthe
Z-grading of 03B6.
Note that all themappings 03B4i, 03B5i respect
theZ-grading.
We define the linear
mapping
d:03B5n ~ 03B5n+1 by
Here
1 ~ 03B6 and 03B6
0 1 are the(n
+1 )-marked
Chinese charactersgotten
frommodifying
themarking on 03B6 by setting 03981(1
003B6) = ~, 0398j(1
003B6)
=03B8j-1(03B6)
for2 j n + 1, 0398n+1(1 ~ 03B6) = ~, 0398j(1 ~ 03B6) = 0398j(03B6) for 1 j n.
Now we define a linear
mapping
x:în Pn
as follows. First we define~’(03B6)
for an n-marked Chinese
character 03B6 of Zn-grading (k1,..., kn).
There areki!
waysto
put
vertices from0398i(03B6)
on the ithstring
and each of thek1!
...kn! possibilities gives
us an element ofP n. Summing
up all suchelements,
weget ~’(03B6).
Now we use the STU relation indicated in
Figure
6 to convert everydiagram appearing
in~’(03B6)
into a chorddiagram, obtaining
in total~(03B6)
fromX’(03B6).
THEOREM 9. The
linear mapping x is well-defined
and is anisomorphism
betweenvector
spa ces En and Pn commuting
with all the operators0394i,
Et.REMARK. x,
however,
does not preservegradings.
The
proof
for the case n = 1 ispresented
in[Barl,
Theorems 6 &8].
Thisproof
does not concern thesupport
of chorddiagrams except
for the firststep
of the induction which is trivial in our case(see
also[Bar2]).
Consider the
following subspaces Yn
ofEn 0 C, Yn
=~ni=
1ker(si).
It can bechecked that
d(Gn)
CGn+1.
We will nowstudy
thehomology
of thefollowing
chain
complex:
Note that d preserves the
Z-grading,
hence it suffices to consider thepart
of Z-grading
m of thecomplex.
where
9;;:
is thehomogeneous part
ofZ-grading
m of9n.
We will find ageometric
interpretation
of thiscomplex.
9.2. A SIMPLICIAL COMPLEX OF THE CUBE Let lm be the m-dimensional
cube,
where vl , ... , vm form a base of Jaem. We
partition Im
into m!m-simplexes:
apermutation ( i 1, ... , im )
of(1,..., m) gives
rise to them-simplex
which is theconvex hull of m
+ 1 points 0,
vi1, Vil+
Vi2’ ... , vi+ ···
+ vim . This tums I "2 intoa
simplicial complex,
denotedby C(Im).
The spaceCk(lm)
is the vector spaceover C
spanned by
all the k-facets of all m! abovem-simplexes.
Theboundary
~(Im)
is asimplicial subcomplex.
The spacee k( 8( lm))
isspanned by
all k-facetswhich lie
entirely
in8( lm).
Let
Ck
be the vector space over Cspanned by
alltuples (03B81,..., 03B8k)
which arepartitions
of theset {1, 2,..., m}, each 0, non-empty.
Define 8:Ck ~ Ck- 1 by
Then the chain
complex (C,,, 0)
isisomorphic
to thequotient complex
C(Im)/C(~(Im)).
Infact,
themapping
which sends(03B81,..., 0k )
to thek-simplex
with vertices
0,
v03B81, v03B81 + V02’ ... , VOl + ··· + VO, is anisomorphism
between these twocomplexes, where v03B8 = ¿jEO
Vj.Let
Em
be the dual chaincomplex
of(C*, 0), Em
=(C*, d). Using
the abovebase of
Ck,
we canidentify eZ
withCk
with the same base. Then theco-boundary
d can be written
explicitly
aswhere for a
non-empty
subset 0of {1, 2,..., m}
we setd(03B8) = 03A3(03B8’, 03B8"),
thesummation is over all
possible partitions
of 0 into an orderedpair
ofnon-empty
subsets.PROPOSITION 4. The
homology of the
chaincomplex Em is given by Hm(Em) =
This follows from the fact that the
homology
ofEm
is the reducedcohomology of Im/~(Im).
Since every