We prove that the dimension of the space of primitive Vassiliev invariants of
degree n grows - as n tends to infinity - faster than Exp(c Sqrt(n)) for any c
< Pi Sqrt (2/3).
  The proof relies on the use of the weight systems coming from the Lie algebra
gl(N). In fact, we show that our bound is - up to multiplication with a
rational function in n - the best possible that one can get with gl(N)-weight
systems.Comment: 11 pages, 12 figure

Dasbach, Oliver T.

English

arXiv

AbstractBy the theory of Vassiliev invariants, the knowledge of a certain algebra B, defined as generated by graphs, is essentially as good as the knowledge of the space of Vassiliev invariants. We will prove that the subspace Bc2,uof B generated by all connected diagrams with 4+2uvertices, including u univalent ones, has dimension dimBc2,u=⌊(u2+12u)/48⌋+1 for u even. Bc2,uis trivial foruodd

Dasbach, Oliver T

Elsevier - Publisher Connector 

On the Combinatorial Structure of Primitive Vassiliev Invariants, II 

By the theory of Vassiliev invariants, the knowledge of a certain algebra B, defined as generated by graphs, is essentially as good as the knowledge of the space of Vassiliev invariants. We will prove that the subspace Bc2,uof B generated by all connected diagrams with 4+2uvertices, including u univalent ones, has dimension dimBc2,u=⌊(u2+12u)/48⌋+1 for u even. Bc2,uis trivial foruodd. © 1998 Academic Press

Louisiana State University

On the Combinatorial Structure of Primitive Vassiliev Invariants, II

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Journal of Combinatorial Theory, Series ATA2821
Journal of Combinatorial Theory, Series A 81, 127139 (1998)
On the Combinatorial Structure of Primitive
Vassiliev Invariants, II
Oliver T. Dasbach*
Mathematisches Institut, Abt. M.+M.D., Universita tsstrasse 1,
D-40225 Du sseldorf, Germany
Communicated by the Managing Editors
Received April 1, 1997
By the theory of Vassiliev invariants, the knowledge of a certain algebra B,
defined as generated by graphs, is essentially as good as the knowledge of the space
of Vassiliev invariants. We will prove that the subspace Bc2, u of B generated by all
connected diagrams with 4+2u vertices, including u univalent ones, has dimension
dim Bc2, u=w(u2+12u)48x+1 for u even. Bc2, u is trivial for u odd.  1998 Academic
Press
1. INTRODUCTION
Since the theory of Vassiliev knot invariants started (see e.g. [4, 2]),
many people have been working for a good combinatorial understanding
of it. Vassliev invariants form an algebra with an underlying filtered vector
space [0]=V1/ } } } /Vn/ } } } /V with finite dimensional subspaces Vn .
By some deep results (see e.g. [2]) the knowledge of a certain algebra
defined by graphs is as good as the knowledge of the spaces VnVn&1. One
description of this algebra can be given as a space B generated by diagrams
(graphs) having only univalent and trivalent vertices modulo relations,
which are defined in terms of subdiagrams.
Because all information on B is contained in connected diagrams and
because the relations are homogeneous in the number of univalent vertices
in a diagram we define the subspace Bcj, u to be the space generated by all
connected diagams in B with 2(u+ j) vertices including exactly u univalent
vertices.
In [6] we have shown that for every odd u the spaces Bcj, u are trivial for
j5. Whether this is always the case is an important open problem.
Article No. TA972821
127
0097-316598 25.00
Copyright  1998 by Academic Press
All rights of reproduction in any form reserved.
* E-mail: kastenmath.uni-duesseldorf.de; http:www.math.uni-duesseldorf.detkasten.
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For j=0 and j=1 and u even the dimensions of the spaces Bcj, u can be
easily computed to be dim Bc0, u=1 and dim B
c
1, u=wu6x+1, where wxx is
the lower Gauss bracket, i.e., the greatest integer less than or equal to x.
By the results in [6] we know that for fixed j the dimensions dim Bcj, 2u
grow at most as O((2u) j) and dim Bcj, 2u+1=O((2u+1)
j&1). Furthermore,
by results in [5], dim Bcj, 2u{O((2u)
j&1).
In the main part of this paper we will compute the dimension of Bc2, u for
arbitrary u. We will construct a filtration
Du, 0Du, 1 } } } Du, u$Bc2, u
and we will show that Bc1, u$Du, 0 .
We will compute the dimensions of Du, k Du, k&1 . An important role in
our computations is played by a polynomial-valued map on the Bcj, u that
was recently defined in a nice paper by Chmutov and Duzhin [5]. The
summation of the dimensions yields:
dim Bc2, u=dim B
c
1, u+dim B
c
2, u&4.
Finally, for u even we will see:
dim Bc2, u=\u
2+12u
48 +1.
We think that some of our ideas can be extended to computations for the
spaces Bcj, u for some higher j. We will outline this in a forthcoming paper.
I am grateful to Stefan Ku hnlein for his comments on an earlier version
of this paper and to Sergei Chmutov and Sergei Duzhin for various
stimulating discussions.
2. SETTING THE SCENE
We briefly recall some theory from [2]. We assume all spaces to be over
a field F of characteristic 0. A central role in the theory of Vassiliev
invariants is played by a Hopf algebra A, that isas a vector space
a (infinite) direct sum of finite-dimensional subspaces An .
Vassiliev invariants can be seen as linear functionals in the graded dual
A* :=n An*. The grading [0]=V1/ } } } /V on Vassiliev invariants,
where Vn is the vector space of all Vassiliev invariants of order at most n,
corresponds to the grading on A, that is VnVn&1 is isomorphic to a sub-
space of An*.
A is isomorphic as a vector space to an algebra B generated by
diagrams (graphs) that have only vertices of valency one or three and at
128 OLIVER T. DASBACH
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least one univalent vertex occurs. At each of the trivalent vertices one of the
two possible cyclic orderings of the adjacent three edges is assigned.
Furthermore, the following two relations hold:
(i) The IHX-relation:
(ii) The antisymmetry relation: if in a diagram D the orientation at
one trivalent vertex is changed then D changes to &D.
(We will draw all diagrams so that the counterclockwise ordering at each
trivalent vertex is chosen.)
Because of the isomorphism between A and B we can translate ques-
tions about Vassiliev invariants to questions about B and vice versa.
It can be shown that A is a primitively generated Hopf algebra and the
primitive elements of A correspond to connected diagrams in B.
The IHX-relation and the antisymmetry relation are homogeneous in the
number of external edges and we set the following:
Definition 2.1. Bcj, u is the subspace of B generated by all connected
diagrams having 2( j+u) vertices including u univalent vertices.
We have proved:
Theorem 2.2 [6]. For j5 and all odd u the spaces Bcj, u are trivial.
It is easy to see that in a nontrivial diagram a vertex cannot be adjacent
to two different univalent vertices, so we ‘‘abbreviate’’ the univalent vertices
and adjacent edges by a number (see [6]), for example,
Therefore we see a diagram as a cubic graphi.e., only trivalent vertices
occurtogether with a labeling of the edges. The side of the edges at which
the labeling is posted gives the cyclic ordering at the trivalent vertices. As
a basic formula we have in this notation:
Lemma 2.3. Let D(a, b, c) be a diagram with the following subdiagram:
129PRIMITIVE VASSILIEV INVARIANTS
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We see by the IHX-relation that
D(a, b, c)= :
a
j=0 \
a
j+ D(0, b+a&j, c+ j ).
3. THE USE OF LIE ALGEBRAS
For a more detailed account of the following see [5]. For every Lie
algebra G equipped with an ad-invariant nondegenerate bilinear form t we
have a well-defined map W G, t of the spaces B
c
j, u into a naturally defined
quotient of the center ZU(G) of the universal enveloping algebra of G. In
the case of G=glN and the trace as the invariant form t(x, y) :=Tr(xy) we
have:
Proposition 3.1 [5]. For Nu we get a homomorphism
W glN : B
c
j, u  F[x0 , ..., xN]
defined by the following process:
We resolve every trivalent vertex in a diagram in Bcj, u according to the
rule:
(1)
We will call the two resolutions of a trivalent vertex positive and negative.
The resulting object is a sum of disjoint unions of circles with some points
missing. To every term in this sum having l disjoint circles with j1 , ..., jl
missing points we assign the monomial xj1 } } } xjl to get a polynomial in
F[x0 , ..., xN].
Note 3.2. For the proof of the proposition a reference to Lie algebras
is not needed. It is easy to see directly that the rule (1) yields a well-defined
polynomial invariant on the Bcj, u .
Definition 3.3. For a diagram D in Bcj, u the polynomial CD(D) is
defined as the highest degree homogeneous part of the polynomial
W glN (D).
130 OLIVER T. DASBACH
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Example 3.4 [5].
So
CD(m)=2(x0x2&x21).
4. THE GROWTH OF Bcj, u FOR FIXED j
In [5] the polynomial CD is used to prove the following theorem:
Theorem 4.1 [5]. Let u be even and j be fixed. For every n1 , ..., nj+1 , all
even, with n1+ } } } +nj+1=u and nl>l&1k=1 nk for every l j and
 jk=1 nk<u3, there is a diagram Dn1 , ..., nj+1 # B
c
j, u such that:
The family Dn1 , ..., nj+1 is linearly independent in B
c
j, u .
Thus, we can state:
Corollary 4.2. For a fixed j : dim Bcj, u{O(u
j&1) for even u.
Together with [6] we have the following interesting result:
Theorem 4.3. dim Bcj, 2u=O((2u)
j){O((2u) j&1) and dim Bcj, 2u+1=
O((2u+1) j&1) as u ranges to infinity.
5. THE SUBSPACE Bc2, u
Definition 5.1. Let D2(a, b, c, d, e) be the diagram
Lemma 5.2. Bc2, u is generated by diagrams of the form D(a, 0, c, d, 0)
with a+c+d=u.
131PRIMITIVE VASSILIEV INVARIANTS
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Proof. By application of the IHX-relation it is easy to see that every
diagram in Bc2, u is the sum of diagrams that have an underlying cubic
graph with no circle of length one or two. But D2(0, 0, 0, 0, 0) is the only
cubic graph with this property.
Moreover, by Lemma 2.3 we see that every diagram with underlying
cubic graph D2(0, 0, 0, 0, 0) can be written as a sum of diagrams of the
form that we claimed. K
Lemma 5.3. For a+c+d even,
CD(D2(a, 0, c, d, 0))
=2 :
a
j1=0
:
c
j2=0
:
d
j3=0
(&1) j1+ j2+ j3 \ aj1+\
c
j2+\
d
j3+
_xj1 xj2 xj3 xa+c+d&j1&j2&j3 .
Proof. The number of circles that we get after a total resolution of the
trivalent vertices of a diagram by some positive or negative resolutions
does not depend on the resolutions at the vertices that are adjacent to
univalent vertices.
For the computation of the polynomial CD we need the highest degree
homogeneous part of the polynomial W glN . So we have to look for resolu-
tions that lead to a maximal number of disjoint circles. It is easy to see that
for the diagrams D2(a, 0, c, d, 0) we get a maximal number of disjoint
circles in two cases: Resolve all four trivalent vertices that are not adjacent
to a univalent vertex by a positive resolution or resolve all by a negative
resolution. (A more elegant way to see this can be extracted from [1].)
For the resolution at the trivalent vertices that are adjacent to univalent
vertices we choose the following abbreviation:
where a& j and j stand for the missing points.
Hence we have the two pictorial representations of all resolutions that
lead to terms in CD:
132 OLIVER T. DASBACH
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For the first diagram the corresponding part of CD is
:
a
j1=0
:
c
j2=0
:
d
j3=0
(&1) j1+ j2+ j3 \ aj1+\
c
j2+\
d
j3+ xj1 xj2 xj3xa+c+d& j1& j2& j3 ;
for the second diagram the corresponding part is
:
a
j1=0
:
c
j2=0
:
d
j3=0
(&1)a+c+d& j1& j2& j3 \ aj1+\
c
j2+\
d
j3+ xj1 xj2 xj3 xa+c+d& j1& j2& j3 .
a+c+d is even and so the two polynomials are equal, hence CD is two
times one of the two polynomials, as claimed. K
6. A FILTRATION OF Bc2, u
It is useful to define a natural extension of the space Bc1, u :
Definition 6.1. B 1, u is the space of all triples
[(a, b, c) | a+b+c=u; a, b, c # N]
modulo the relations
(a, b, c)=(&1)u (a, c, b)=(&1)u (b, a, c), (2)
and for a1,
(a, b, c)=&(a&1, b+1, c)&(a&1, b, c+1). (3)
Lemma 6.2 ([6], see also [3]). The space Bc1, u is trivial for u odd. For
u even we have: Bc1, u is isomorphic to the space B 1, u and has dimension
dim Bc1, u=\u6+1.
133PRIMITIVE VASSILIEV INVARIANTS
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Lemma 6.3. For u odd we have:
dim B 1, u=\u+36  .
Proof. We proceed as in [6] for Bc1, u : By relation (2) we can change
every triple (a, b, c), such that abc. We order these triples (a, b, c),
abc, by the lexicographical order.
We rewrite the terms in (3) so that the greatest element is on the left-
hand side. It is easy to see that every element occurs as the greatest element
in at most one equation. So we have to count the non-trivial elements that
do not occur as greatest elements. Because (a, a, c)=0 for every a and c,
these are precisely the elements
[(a+1, a, b) | a>b0; 2a+1+b=u].
The cardinality of this set is w(u+3)6x. K
Definition 6.4. Let Du, k be the subspace of Bc2, u generated by all diagrams
D2(a, b, c, d, 0) with a+b+c+d=u and dk, where D2(a, b, c, d, e) is the
diagram of Definition 5.1.
Note 6.5. By definition we have
Du, 0Du, 1 } } } Du, u$Bc2, u .
The last isomorphism follows from Lemma 5.2. In the course of this text we
will see that Du, 0 is naturally isomorphic to B
c
1, u .
Lemma 6.6. For u even, we have well-defined surjective homomorphisms
u, k : B 1, u&k  Du, kDu, k&1
given by
u, k((a, b, c)) :=(&1)c D2(a, b, c, k, 0)
Proof. By construction the map is surjective. Because of the IHX-
relation in Bc2, u we have for a1 and a+b+c=u&k:
u, k((a, b, c))=&u, k((a&1, b+1, c))&u, k((a&1, b, c+1)).
So it remains to show that the equations
u, k((a, b, c))=(&1)k u, k((a, c, b))=(&1)k u, k(b, a, c))
are fulfilled.
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We have
D2(a, b, c, k, 0)=D2(b, a, c, 0, k)
= :
k
j=0
(&1) j \kj+ D2(b, a, c+k& j, j, 0)
=(&1)k D2(b, a, c, k, 0) mod Du, k&1.
By reflection of D2(a, b, c, k, 0) we get the remaining equation. K
Lemma 6.7. We have a well-defined homomorphism
Du, kDu, k&1  F[xk , ..., xu]
given by the map
CD : Bc2, u  F[xk , ..., xu]
with the setting x0= } } } =xk&1=0.
Proof. Every element D2(a, b, c, k&1, 0) # Du, k&1 can be writtenby
Lemma 2.3as a sum of some D2(a~ , 0, c~ , k&1, 0).
Hence, by Lemma 5.3, the polynomial CD vanishes on Du, k&1 after
setting x0= } } } =xk&1=0. K
Lemma 6.8. Let
P(a, b) := :
a
j1=0
:
b
j2=0
(&1) j1+ j2 \ aj1+\
b
j2+ xj1 xj2 xa+b& j1& j2 .
The space Pl generated by all P(a, b) with a+b=l has dimension at least:
dim Pl{\
l
6+1
\l&36 +1
for l even
for l odd.
The polynomials P(a, b) # Pl with b even, awl3x, are linearly independent.
If we set for a fixed k : x0= } } } =xk&1 :=0 then we get:
P(a, b) |x0= } } } =xk&1 :=0{\
l
6+1&\
k+1
2 
\l&36 +1&\
k
2
for l even
for l odd.
135PRIMITIVE VASSILIEV INVARIANTS
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The polynomials P(a, b) |x0= } } } =xk&1 :=0 with b even, kawl3x, are
linearly independent.
Proof. Obviously P(a, b) |x0= } } } =xa&1 :=0 is trivial for a&1wb2x or
equivalently a>w(a+b)3x.
Furthermore,
P(a, b) |x0= } } } =xa&1 :=0=(&1)
a xa :
b
j2=0
(&1) j2 \ bj2+ xj2 xb& j2 }x0= } } } =xa&1 :=0
is trivial for b odd (map j2 to b& j2) and has for b even and ab2 a non-
trivial term (&1)a xa(&1)b2 ( bb2) x
2
b2 .
Hence, to get a lower bound, we just have to count all pairs (a, b) with
a+b=l, b even, and aw(a+b)3x.
The rest of the lemma follows as well. K
Lemma 6.9. As a basis for B 1, u we can choose:
{(a, 0, b) | b even; a+b=u; a\u3= . (4)
Proof. The number of elements fulfilling (4) is precisely the dimension
of the spaces B 1, u as stated in Lemma 6.2 and Lemma 6.3.
So it remains to show that the elements are linearly independent.
For u even we look at the homomorphisms of Lemma 6.6 and of
Lemma 6.7:
B 1, u w

Du, 0 w
CD F[x0 , ..., xu].
Hence, for b even,
CD((a, 0, b))=CD(D2(a, 0, b, 0, 0))
=2x0 :
a
j1
:
b
j2
(&1) j1+ j2 \ aj1+\
b
j2+ xj1 xj2 xa+b& j1& j2 .
Therefore, we get 2x0 times the polynomials of Lemma 6.8. For our com-
binations of a and b these are independent.
For u odd we have
B 1, u w

Du+1, 1 Du+1, 0 www
CD |x0 :=0 F[x1 , ..., xu+1].
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Hence,
CD((a, 0, b)) |x0 :=0
=CD(D2(a, 0, b, 1, 0)) |x0 :=0
=&2x1 :
a
j1
:
b
j2
(&1) j1+ j2 \ aj1+\
b
j2+ xj1 xj2 xa+b& j1& j2 } x0 :=0 .
Again, we get &2x1 times the polynomials of Lemma 6.8 and can apply it
with the restriction x0 :=0. K
7. THE DIMENSION OF Bc2, u
Theorem 7.1. For the filtration Du, k we have for ku4
dim Du, k Du, k&1={\
u&k
6 +1&
k
2
\u&k&36 +1&
k&1
2
for k even
for k odd.
For k>u4 we have Du, k&1$Du, k .
Proof. By Lemma 6.6 we have a surjective homomorphism
u, k : B 1, u&k  Du, kDu, k&1.
By Lemma 6.9 we know that B 1, u&k is generated by all
{(a, 0, b) | b even, a\u&k3  and a+b=u&k= .
If ak&1 then a triple (a, 0, b) lies in the kernel. Therefore
dim Du, k Du, k&1{\
u&k
6 +1&
k
2
\u&k&36 +1&
k&1
2
for k even
for k odd.
Again we regard the homomorphism CD | x0= } } } =xk&1 :=0 on the image of
u, k on the set
{(a, 0, b) | b even, ka\u&k3  and a+b=u&k= .
137PRIMITIVE VASSILIEV INVARIANTS
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Hence,
CD(u, k(a, 0, b))=2(&1)k xk :
a
j1=0
:
b
j2=0
(&1) j1+ j2
_\ aj1+\
b
j2+ xj1 xj2 xa+b& j1& j2 }x0= } } } =xk&1 :=0 .
Therefore, CD(u, k(a, 0, b))=2(&1)k xkP(a, b) under the restriction
x0= } } } =xk&1 :=0, where P(a, b) is the polynomial of Lemma 6.8 and so
the CD(u, k(a, 0, b)) are independent on the above set. K
Directly from the above formula we can see:
Lemma 7.2. dim Du, kDu, k&1=dim Du&4, k&1 Du&4, k&2.
A very suggestive formula is:
Proposition 7.3. dim Bc2, u=dim B
c
2, u&4+dim B
c
1, u .
Proof. By Theorem 7.1 and Lemma 7.2 we know that
dim Bc2, u=dim Du, 0+ :
1ku4
dim Du, kDu, k&1
=dim Bc1, u+ :
0ku4&1
dim Du&4, k Du&4, k&1
=dim Bc1, u+dim B
c
2, u&4. K
Now we have all the tools to prove:
Main Theorem 7.4. dim Bc2, u=w(u
2+12u)48x+1 for u even and
dim Bc2, u=0 for u odd.
Proof. For u odd the theorem is clear by Theorem 2.2. So let u be even.
For u=2 and u=4 we have as desired
dim Bc2, 2=dim B 1, 2=1
dim Bc2, 4=dim B 1, 4+dim B 1, 3=2.
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We will show that the formula satisfies the recursion of Proposition 7.3.
\u
2+12u
48 &\
(u&4)2+12(u&4)
48 
=\u
2+12u
48 &\
u2&8u+16+12u
48 +1. (5)
Write u=6u1+u2 with u2=0, 2 or 4. So dim Bc1, u=u1+1.
For u2=2 we get that Eq. (5)=u1+1 as claimed.
For u2=0 we have that
Eq. (5)=\3u
2
1+6u1
4 &\
3u21+6u1
4
+
1
3+u1+1.
A simple check shows that this yields u1+1 for u1 even as well as for u1
odd. Finally, the case u2=4 can seen in essentially the same way as u2=0,
again with a case-by-case check for u1 even, as well as for u1 odd. K
Note 7.5. This formula verifies the dimensions of the Bc2, u for low u
given in [3] that were achieved by computer computations.
REFERENCES
1. D. Bar-Natan, Lie algebras and the four color theorem, Combinatorica, to appear.
2. D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995), 423472.
3. D. Bar-Natan, ‘‘Some Computations Related to Vassiliev Invariants,’’ Revised version,
May 1996.
4. J. S. Birman and X.-S. Lin, Knot polynomials and Vassiliev’s invariants, Invent. Math. 111
(1993), 225270.
5. S. V. Chmutov and S. V. Duzhin, ‘‘A Lower Bound for the Number of Vassiliev Knot
Invariants,’’ preprint, November 1996.
6. O. T. Dasbach, ‘‘On Subspaces of the Space of Vassiliev Invariants,’’ Doctoral thesis,
University of Du sseldorf, 1997.
139PRIMITIVE VASSILIEV INVARIANTS


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On the Combinatorial Structure of Primitive Vassiliev Invariants, III - A Lower Bound

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