SOME COMPUTATIONS RELATED TO VASSILIEV INVARIANTS

DROR BAR-NATAN

Last updated: May 5, 1996

Abstract. This is a regularly updated collection of results of computations related to Vassiliev invariants. It is distributed using photocopiers, laser printers, and the internet. If you have data to add, please let me know!

Contents

1. Knots 2 2. String Links 4

2.1. Chinese Characters 2.2. The representation E_{n} 2.3. Invariants

3. Braids 16 4. The data file 17 5. Acknowledgement 18 References 18

As a warmup, let as start with a reproduction of the table in section 6.1 of [1]:

m  0 1 2 3 4 5  6  7  8  9
dim G_{m}A 1 1 2 3 6 10 19  33  60  104
dim G_{m}W 1 0 1 1 3 4  9  14  27  44
dim G_{m}P 1 1 1 1 2 3  5  8  12  18
dim G_{m}Lie 1 1 2 3 6 10 19  33  60  104
diagrams 1 0 1 2 7 36 300 3, 218 42, 335 644, 808
relations 0 0 0 2 15 144 1, 645 21, 930 334, 908 5, 056, 798

The latest edition of this collection of results is available at http://www.ma.huji.ac.il/ ∼drorbn and at file://ftp.ma.huji.ac.il/drorbn.

Remark 1. • G_{m} obj denotes the degree m piece of a graded object obj. A is the space of chord diagrams modulo the 4T relation (see e.g. [1]). W is the space of weight systems, that is, functionals A → Q that vanish on chord diagrams having an isolated chord. P is the space of primitive elements of A (recall that A is a commutative and co-commutative Hopf algebra). G_{m}Lie is the subspace of (G_{m}A)^{∗} spanned by the Lie algebraic weight systems of [1]. In fact, up to degree 9, it is spanned by the weight systems corresponding to classical groups.

• Vogel [10] proved that dim G_{m}SSLie < dim G_{m}A for large enough values of m, where SSLie denotes the class of semi-simple Lie and super-Lie algebras.

• The numbers in the last two rows indicate the size of the matrices that had to be row reduced in order to compute dim G_{m}W. These numbers have no real significance — they somewhat depend on the details of the algorithm chosen — and are displayed only so as to give an impression of the complexity of the problem.

• The problem is highly exponential and it is unlikely that it will be possible to use the same techniques to compute dim G_{10}A.

• It took about 2, 106 lines of C++ code and about 10 days of CP U time to compute the above numbers. The programs are available at my ftp/http sites.

1. Knots

The following is a table of the dimensions β_{m,u} and φ_{m,u} of G_{m}B_{u} and of im Φ|_{G_{m}B_{u}} where G_{m}B_{u} is the space of connected Chinese Characters of degree m having exactly u univalent vertices, divided by the AS and IHX relations. This space as well as the map Φ : B_{u} → {marked surfaces} are defined in [1].

u = 0 u = 2 u = 4 u = 6 u = 8 u = 10 total (u > 0)
m = 1 β  1  1  1
φ  1  1  1
m = 2 β  1  1  1
φ  1  1  1
m = 3 β  1  1  1
φ  1  1  1
m = 4 β  2  1  1  2
φ  2  1  1  2
m = 5 β  2  2  1  3
φ  2  2  1  3
m = 6 β  3  2  2  1  5
φ  3  2  2  1  5
m = 7 β  4  3  3  2  8
φ 4 (3)  3  3  2  8
m = 8 β  5  4  4  3  1  12
φ 5 (4) 4 (3)  4  3  1  12 (11)
m = 9 β  ?  5  6  5  2  18
φ ≥ 6 (4) 5 (4)  6 (5)  5  2  18 (16)
m = 10 β  ?  ?  ?  ?  4  1  ?
φ ≥ 8 (5) ≥ 6 (4) ≥ 8 (6) ≥ 8 (7)  4  1  ≥ 27 (22)
m = 11 β  ?  ?  ?  ?  8  2  ?
φ  ?  ≥ 8 (5) ≥ 10 (7) ≥ 11 (9) 8 (7)  2  ≥ 39 (30)

Remark 2. • Up to and including m = 9, β_{m,u} = 0 if u is odd. For m = 10 this

is not known yet. φ_{m,u} = 0 if u is odd for all m.

• When different than φ_{m,u}, the dimension of im Φ projected to orientable surfaces appears in parenthesis after φ_{m,u}.

• As was noticed by M. Kontsevich [6], for every m one has β_{m,0} = β_{m+1,2}, β_{m,1} = 0, and similarly for φ.

• Vogel [10] proved that φ_{m,u} < β_{m,u} for large enough values of m and for u = 0, 2. The same is expected for u > 2 as well.

• The symbol ‘≥’ appearing in some of the entries above can most likely be replaced by ‘=’. φ_{m,u} was computed by computing Φ on a large number of random Chinese characters, but not on all of them. So there is a (small) probability that something in the image of Φ was missed.

• The programs used in the computation of the above numbers are available

from my ftp/http sites. Similarly, here are some values of φ_{m,u} for higher m and u. Notice that most of

these numbers are just lower bounds, for the same reason as in remark 2. Due to time constrains, the lowest entry in each column was computed by computing Φ on a relatively small number of random Chinese characters, and the probability of mistake in these cases is higher.

u = 6  u = 8  u = 10 u = 12 u = 14 u = 16 u = 18
m = 12 ≥ 14 (10) ≥ 12 (10)  5  1
m = 13  ?  ≥ 18 (13) ≥ 10 (9)  3
m = 14  ?  ≥ 22 (14) ≥ 17 (14)  7  1
m = 15  ?  ?  ?  ≥ 15 (13)  3
m = 16  ?  ?  ?  ≥ 27 (21)  8
m = 17  ?  ?  ?  ?  ≥ 19 (16)  3
m = 18  ?  ?  ?  ?  ?  10 (?)  1
m = 19  ?  ?  ?  ?  ?  ?  4

Remark 3. The dimensions β_{m,u} discussed above are the same as the dimensions β_{m}(u′) discussed below, when u′ = (u) is a partition of length 1. Thus some more β_{m,u}’s, for odd values of u, can be read from the tables in sections 2.1.10–2.1.20. These additional dimensions all vanish. See also remarks 10 and 11.

2. String Links

Next, let us turn to Vassiliev invariants of string links.

2.1. Chinese Characters. By [2], the associated graded space to the filtered space of Vassiliev invariants of string links is the dual of A^{sl}, which is isomorphic to B^{sl}, where B^{sl} is the space of Chinese characters with ‘colored’ univalent vertices (modulo IHX and AS, of course). Thus B^{sl} has many different gradings — by the total degree m (= half the number of vertices), and by the number of univalent vertices of each color. For any partition u = (u_{1} ≥ u_{2} ≥ ⋯ ≥ u_{l}), let β_{m}(u) be the dimension of the space B_{m}(u) of connected Chinese characters of total degree m and of u_{i} univalent vertices of color υ_{i} for each 1 ≤ i ≤ l (modulo IHX and AS), where Υ = {υ_{i}} is some set of colors. For simple counting reasons, it is clear that β_{m}(u) = 0 unless |u| ≤ m + 1, where |u| = ∑ u_{i}. The following are lists of the dimensions β_{m}(u) presently known to me:

2.1.1. Degree 1.

β_{1}() = 1 β_{1}(1) = 0 β_{1}(2) = 1 β_{1}(11) = 1

2.1.2. Degree 2.

β_{2}() = 1 β_{2}(2) = 1 β_{2}(3) = 0 β_{2}(111) = 1
β_{2}(1) = 0 β_{2}(11) = 1 β_{2}(21) = 0

SOME COMPUTATIONS RELATED TO VASSILIEV INVARIANTS 5

2.1.3. Degree 3.

β_{3}() = 1 β_{3}(11) = 1 β_{3}(111) = 1 β_{3}(22) = 1
β_{3}(1) = 0 β_{3}(3) = 0 β_{3}(4) = 0 β_{3}(211) = 1
β_{3}(2) = 1 β_{3}(21) = 0 β_{3}(31) = 0 β_{3}(1111) = 2

2.1.4. Degree 4.

β_{4}() = 2 β_{4}(21) = 0 β_{4}(211) = 2 β_{4}(311) = 1
β_{4}(1) = 0 β_{4}(111) = 1 β_{4}(1111) = 3 β_{4}(221) = 1
β_{4}(2) = 1 β_{4}(4) = 1 β_{4}(5) = 0 β_{4}(2111) = 3
β_{4}(11) = 1 β_{4}(31) = 1 β_{4}(41) = 0 β_{4}(11111) = 6
β_{4}(3) = 0 β_{4}(22) = 2 β_{4}(32) = 0

2.1.5. Degree 5.

β_{5}() = 2 β_{5}(31) = 1 β_{5}(221) = 2 β_{5}(321)  = 2
β_{5}(1) = 0 β_{5}(22) = 3 β_{5}(2111) = 6 β_{5}(3111) = 4
β_{5}(2) = 2 β_{5}(211) = 3 β_{5}(11111) = 12 β_{5}(222)  = 4
β_{5}(11) = 2 β_{5}(1111) = 5 β_{5}(6)  = 0 β_{5}(2211) = 6
β_{5}(3) = 0 β_{5}(5) = 0 β_{5}(51) = 0 β_{5}(21111) = 12
β_{5}(21) = 0 β_{5}(41) = 0 β_{5}(42) = 1 β_{5}(111111) = 24
β_{5}(111) = 2 β_{5}(32) = 0 β_{5}(411) = 1
β_{5}(4) = 1 β_{5}(311) = 2 β_{5}(33) = 1

2.1.6. Degree 6.

β_{6}()  = 3 β_{6}(5)  = 0 β_{6}(321)  = 6 β_{6}(4111)  = 5
β_{6}(1) = 0 β_{6}(41)  = 0 β_{6}(3111) = 10 β_{6}(331)  = 3
β_{6}(2) = 2 β_{6}(32)  = 0 β_{6}(222)  = 11 β_{6}(322)  = 4
β_{6}(11) = 2 β_{6}(311) = 3 β_{6}(2211) = 16 β_{6}(3211)  = 10
β_{6}(3) = 0 β_{6}(221) = 3 β_{6}(21111) = 30 β_{6}(31111) = 20
β_{6}(21) = 0 β_{6}(2111) = 10 β_{6}(111111) = 60 β_{6}(2221)  = 14
β_{6}(111) = 2 β_{6}(11111) = 22 β_{6}(7)  = 0 β_{6}(22111) = 30
β_{6}(4) = 2 β_{6}(6)  = 1 β_{6}(61)  = 0 β_{6}(211111) = 60
β_{6}(31) = 2 β_{6}(51)  = 1 β_{6}(52)  = 0 β_{6}(1111111) = 120
β_{6}(22) = 5 β_{6}(42)  = 3 β_{6}(511)  = 1
β_{6}(211) = 5 β_{6}(411) = 3 β_{6}(43)  = 0
β_{6}(1111) = 8 β_{6}(33)  = 3 β_{6}(421)  = 2

6 DROR BAR-NATAN

2.1.7. Degree 7.

β_{7}()  = 4 β_{7}(2111) = 16 β_{7}(43)  = 1 β_{7}(5111)  = 6
β_{7}(1) = 0 β_{7}(11111) = 34 β_{7}(421)  = 6 β_{7}(44)  = 2
β_{7}(2) = 3 β_{7}(6)  = 2 β_{7}(4111) = 15 β_{7}(431)  = 5
β_{7}(11) = 3 β_{7}(51)  = 2 β_{7}(331)  = 10 β_{7}(422)  = 9
β_{7}(3) = 0 β_{7}(42)  = 6 β_{7}(322)  = 12 β_{7}(4211)  = 15
β_{7}(21) = 0 β_{7}(411) = 6 β_{7}(3211) = 30 β_{7}(41111) = 30
β_{7}(111) = 3 β_{7}(33) = 6 β_{7}(31111) = 60 β_{7}(332) = 10
β_{7}(4) = 3 β_{7}(321) = 12 β_{7}(2221) = 42 β_{7}(3311)  = 20
β_{7}(31) = 3 β_{7}(3111) = 20 β_{7}(22111) = 90 β_{7}(3221)  = 30
β_{7}(22) = 7 β_{7}(222) = 22 β_{7}(211111) = 180 β_{7}(32111) = 60
β_{7}(211) = 7 β_{7}(2211) = 32 β_{7}(1111111) = 360 β_{7}(311111) = 120
β_{7}(1111) = 11 β_{7}(21111) = 60 β_{7}(8) = 0 β_{7}(2222) = 48
β_{7}(5) = 0 β_{7}(111111) = 120 β_{7}(71)  = 0 β_{7}(22211) = 90
β_{7}(41) = 0 β_{7}(7)  = 0 β_{7}(62)  = 1 β_{7}(221111) = 180
β_{7}(32) = 0 β_{7}(61)  = 0 β_{7}(611)  = 1 β_{7}(2111111) = 360
β_{7}(311) = 5 β_{7}(52)  = 0 β_{7}(53)  = 1 β_{7}(11111111)= 720
β_{7}(221) = 5 β_{7}(511) = 3 β_{7}(521)  = 3

SOME COMPUTATIONS RELATED TO VASSILIEV INVARIANTS 7

2.1.8. Degree 8. The symbol ? below means that on May 5, 1996 I did not know the relevant dimension.

β_{8}(221111) = 630
β_{8}()  = 5 β_{8}(52)  = 0 β_{8}(2111111) = ?
β_{8}(1)  = 0 β_{8}(511)  = 6 β_{8}(11111111) = ?
β_{8}(2)  = 4 β_{8}(43)  = 1 β_{8}(9)  = 0
β_{8}(11)  = 4 β_{8}(421)  = 12 β_{8}(81)  = 0
β_{8}(3)  = 0 β_{8}(4111) = 32 β_{8}(72)  = 0
β_{8}(21)  = ? β_{8}(331)  = 19 β_{8}(711)  = 1
β_{8}(111)  = ? β_{8}(322)  = 24 β_{8}(63)  = 1
β_{8}(4)  = 4 β_{8}(3211) = 63 β_{8}(621)  = 3
β_{8}(31)  = ? β_{8}(31111) = 131 β_{8}(6111)  = 7
β_{8}(22)  = ? β_{8}(2221) = 88 β_{8}(54)  = 1
β_{8}(211)  = ? β_{8}(22111) = 193 β_{8}(531)  = 7
β_{8}(1111) = ? β_{8}(211111) = ? β_{8}(522)  = 9
β_{8}(5)  = 0 β_{8}(1111111) = ? β_{8}(5211)  = 21
β_{8}(41)  = 0 β_{8}(8)  = 1 β_{8}(51111)  = 42
β_{8}(32)  = 0 β_{8}(71)  = 1 β_{8}(441)  = 8
β_{8}(311)  = 7 β_{8}(62)  = 4 β_{8}(432)  = 16
β_{8}(221)  = ? β_{8}(611)  = 4 β_{8}(4311)  = 35
β_{8}(2111) = ? β_{8}(53)  = 5 β_{8}(4221)  = 51
β_{8}(11111) = ? β_{8}(521)  = 12 β_{8}(42111)  = 105
β_{8}(6)  = 3 β_{8}(5111) = 21 β_{8}(411111) = 210
β_{8}(51)  = 4 β_{8}(44)  = 8 β_{8}(333)  = 24
β_{8}(42)  = 11 β_{8}(431)  = 19 β_{8}(3321)  = 70
β_{8}(411)  = 12 β_{8}(422)  = 33 β_{8}(33111)  = 140
β_{8}(33)  = 11 β_{8}(4211) = 54 β_{8}(3222)  = 102
β_{8}(321)  = 22 β_{8}(41111) = 105 β_{8}(32211)  = 210
β_{8}(3111) = 36 β_{8}(332)  = 38 β_{8}(321111) = 420
β_{8}(222)  = 39 β_{8}(3311) = 70 β_{8}(3111111) = 840
β_{8}(2211) = ? β_{8}(3221) = 108 β_{8}(22221)  = 312
β_{8}(21111) = ? β_{8}(32111) = 210 β_{8}(222111) = 630
β_{8}(111111) = ? β_{8}(311111) = 420 β_{8}(2211111) = 1260
β_{8}(7)  = 0 β_{8}(2222) = 171 β_{8}(21111111) = 2520
β_{8}(61)  = 0 β_{8}(22211) = 318 β_{8}(111111111) = 5040

8 DROR BAR-NATAN

2.1.9. Degree 9. For degrees 9 and up, only the dimensions known as of May 5, 1996 are displayed, and two digit numbers in the partition u are underlined.

β_{9}(1) = 0 β_{9}(44) = 20 β_{9}(91) = 0 β_{9}(5221) = 84
β_{9}(2) = 5 β_{9}(9)  = 0 β_{9}(82) = 1 β_{9}(52111) = 168
β_{9}(11) = 5 β_{9}(81) = 0 β_{9}(811) = 1 β_{9}(511111) = 336
β_{9}(3) = 0 β_{9}(72) = 0 β_{9}(73) = 1 β_{9}(442)  = 38
β_{9}(4) = 6 β_{9}(711) = 4 β_{9}(721) = 4 β_{9}(4411) = 70
β_{9}(5) = 0 β_{9}(63) = 3 β_{9}(7111) = 8 β_{9}(433)  = 46
β_{9}(6) = 5 β_{9}(621) = 12 β_{9}(64) = 3 β_{9}(4321) = 140
β_{9}(7) = 0 β_{9}(6111) = 28 β_{9}(631) = 9 β_{9}(43111) = 280
β_{9}(61) = 0 β_{9}(54) = 4 β_{9}(622) = 16 β_{9}(4222) = 216
β_{9}(8) = 2 β_{9}(531) = 28 β_{9}(6211) = 28 β_{9}(42211) = 420
β_{9}(71) = 3 β_{9}(522) = 36 β_{9}(61111) = 56 β_{9}(3331) = 186
β_{9}(62) = 10 β_{9}(5211) = 84 β_{9}(55) = 3 β_{9}(3322) = 280
β_{9}(611) = 11 β_{9}(441) = 32 β_{9}(541) = 14
β_{9}(53) = 13 β_{9}(432) = 64 β_{9}(532) = 28
β_{9}(521) = 30 β_{9}(10) = 0 β_{9}(5311) = 56

2.1.10. Degree 10.

β_{10}(1)  = 0 β_{10}(73) = 8 β_{10}(911) = 1 β_{10}(632) = 40
β_{10}(7)  = 0 β_{10}(721) = 20 β_{10}(83) = 1 β_{10}(6311) = 84
β_{10}(8)  = 4 β_{10}(7111) = 36 β_{10}(821) = 4 β_{10}(6221) = 124
β_{10}(71) = 6 β_{10}(64) = 16 β_{10}(8111) = 9 β_{10}(62111) = 252
β_{10}(81) = 0 β_{10}(631) = 44 β_{10}(74) = 2 β_{10}(611111)= 504
β_{10}(72) = 0 β_{10}(622) = 74 β_{10}(731) = 12 β_{10}(551) = 25
β_{10}(711) = 10 β_{10}(55) = 16 β_{10}(722) = 16 β_{10}(542) = 60
β_{10}(10) = 1 β_{10}(541) = 66 β_{10}(7211) = 36 β_{10}(5411) = 126
β_{10}(91) = 1 β_{10}(11) = 0 β_{10}(71111) = 72 β_{10}(533) = 84
β_{10}(82) = 5 β_{10}(101) = 0 β_{10}(65) = 3 β_{10}(5321) = 252
β_{10}(811) = 5 β_{10}(92) = 0 β_{10}(641) = 20 β_{10}(443) = 102

SOME COMPUTATIONS RELATED TO VASSILIEV INVARIANTS 9

2.1.11. Degree 11.

β_{11}(8)  = 8 β_{11}(83) = 5 β_{11}(921) = 5 β_{11}(7221) = 180
β_{11}(9)  = 0 β_{11}(821) = 20 β_{11}(9111) = 10 β_{11}(72111) = 360
β_{11}(81) = 0 β_{11}(8111) = 45 β_{11}(84) = 5 β_{11}(66) = 9
β_{11}(10) = 2 β_{11}(74) = 10 β_{11}(831) = 15 β_{11}(651) = 42
β_{11}(91) = 3 β_{11}(731) = 60 β_{11}(822) = 25 β_{11}(642) = 110
β_{11}(82) = 15 β_{11}(65) = 16 β_{11}(8211) = 45 β_{11}(6411) = 210
β_{11}(811) = 16 β_{11}(12) = 0 β_{11}(81111) = 90 β_{11}(633) = 141
β_{11}(11) = 0 β_{11}(111) = 0 β_{11}(75) = 6 β_{11}(552) = 126
β_{11}(101) = 0 β_{11}(102) = 1 β_{11}(741) = 30
β_{11}(92) = 0 β_{11}(1011) = 1 β_{11}(732) = 60
β_{11}(911) = 5 β_{11}(93) = 2 β_{11}(7311) = 120

2.1.12. Degree 12.

β_{12}(10) = 5 β_{12}(921) = 30 β_{12}(103) = 1 β_{12}(85) = 7
β_{12}(101) = 0 β_{12}(9111) = 55 β_{12}(1021) = 5 β_{12}(841) = 40
β_{12}(92) = 0 β_{12}(84) = 29 β_{12}(10111) = 11 β_{12}(832) = 80
β_{12}(12) = 1 β_{12}(75) = 38 β_{12}(94) = 3 β_{12}(8311) = 165
β_{12}(111) = 1 β_{12}(13) = 0 β_{12}(931) = 18 β_{12}(76) = 9
β_{12}(102) = 6 β_{12}(121) = 0 β_{12}(922) = 25 β_{12}(751) = 66
β_{12}(1011) = 6 β_{12}(112) = 0 β_{12}(9211) = 55 β_{12}(661) = 75
β_{12}(93) = 12 β_{12}(1111) = 1 β_{12}(91111) = 110

2.1.13. Degree 13.

β_{13}(11) = 0 β_{13}(103) = 8 β_{13}(113) = 2 β_{13}(101111)= 132
β_{13}(12) = 3 β_{13}(1021) = 30 β_{13}(1121) = 6 β_{13}(95) = 11
β_{13}(111) = 4 β_{13}(94) = 20 β_{13}(11111) = 12 β_{13}(941) = 55
β_{13}(13) = 0 β_{13}(14) = 0 β_{13}(104) = 7 β_{13}(932) = 110
β_{13}(121) = 0 β_{13}(131) = 0 β_{13}(1031) = 22 β_{13}(86) = 19
β_{13}(112) = 0 β_{13}(122) = 1 β_{13}(1022) = 36 β_{13}(851) = 99
β_{13}(1111) = 6 β_{13}(1211) = 1 β_{13}(10211) = 66 β_{13}(77) = 19

10 DROR BAR-NATAN

2.1.14. Degree 14.

β_{14}(12) = 7 β_{14}(113) = 16 β_{14}(123) = 2 β_{14}(11211) = 78
β_{14}(121) = 0 β_{14}(1121) = 42 β_{14}(1221) = 6 β_{14}(111111)= 156
β_{14}(14) = 1 β_{14}(15) = 0 β_{14}(12111) = 13 β_{14}(105) = 13
β_{14}(131) = 1 β_{14}(141) = 0 β_{14}(114) = 5 β_{14}(1041) = 70
β_{14}(122) = 7 β_{14}(132) = 0 β_{14}(1131) = 26 β_{14}(96) = 22
β_{14}(1211) = 7 β_{14}(1311) = 1 β_{14}(1122) = 36 β_{14}(87) = 28

2.1.15. Degree 15.

β_{15}(13) = 0 β_{15}(123) = 12 β_{15}(1321) = 7 β_{15}(115) = 18
β_{15}(14) = 3 β_{15}(1221) = 42 β_{15}(13111) = 14 β_{15}(1141) = 91
β_{15}(131) = 5 β_{15}(16) = 0 β_{15}(124) = 9 β_{15}(106) = 36
β_{15}(15) = 0 β_{15}(151) = 0 β_{15}(1231) = 30 β_{15}(97) = 47
β_{15}(141) = 0 β_{15}(142) = 1 β_{15}(1222) = 49 β_{15}(88) = 58
β_{15}(132) = 0 β_{15}(1411) = 1 β_{15}(12211) = 91
β_{15}(1311) = 7 β_{15}(133) = 2 β_{15}(121111)= 182

2.1.16. Degree 16.

β_{16}(14) = 8 β_{16}(133) = 21 β_{16}(1421) = 7 β_{16}(131111)= 210
β_{16}(141) = 0 β_{16}(17) = 0 β_{16}(14111) = 15 β_{16}(125) = 21
β_{16}(16) = 1 β_{16}(161) = 0 β_{16}(134) = 7 β_{16}(116) = 42
β_{16}(151) = 1 β_{16}(152) = 0 β_{16}(1331) = 35
β_{16}(142) = 8 β_{16}(1511) = 1 β_{16}(1322) = 49
β_{16}(1411) = 8 β_{16}(143) = 2 β_{16}(13211) = 105

2.1.17. Degree 17.

β_{17}(15) = 0 β_{17}(1511) = 8 β_{17}(153) = 3 β_{17}(14211) = 120
β_{17}(16) = 3 β_{17}(143) = 16 β_{17}(1521) = 8 β_{17}(141111)= 240
β_{17}(151) = 5 β_{17}(18) = 0 β_{17}(15111) = 16 β_{17}(135) = 28
β_{17}(17) = 0 β_{17}(171) = 0 β_{17}(144) = 12
β_{17}(161) = 0 β_{17}(162) = 1 β_{17}(1431) = 40
β_{17}(152) = 0 β_{17}(1611) = 1 β_{17}(1422) = 64

SOME COMPUTATIONS RELATED TO VASSILIEV INVARIANTS 11

2.1.18. Degree 18.

β_{18}(16) = 10 β_{18}(1611) = 9 β_{18}(1711) = 1 β_{18}(1531) = 45
β_{18}(161) = 0 β_{18}(153) = 27 β_{18}(163) = 2 β_{18}(1522) = 64
β_{18}(18) = 1 β_{18}(19) = 0 β_{18}(1621) = 8 β_{18}(15211) = 136
β_{18}(171) = 1 β_{18}(181) = 0 β_{18}(16111) = 17 β_{18}(151111)= 272
β_{18}(162) = 9 β_{18}(172) = 0 β_{18}(154) = 9 β_{18}(145) = 32

2.1.19. Degree 19.

β_{19}(17) = 0 β_{19}(172) = 0 β_{19}(1811) = 1 β_{19}(1631) = 51
β_{19}(18) = 4 β_{19}(1711) = 9 β_{19}(173) = 3 β_{19}(1622) = 81
β_{19}(171) = 6 β_{19}(20) = 0 β_{19}(1721) = 9 β_{19}(16211) = 153
β_{19}(19) = 0 β_{19}(191) = 0 β_{19}(17111) = 18 β_{19}(155) = 41
β_{19}(181) = 0 β_{19}(182) = 1 β_{19}(164) = 15

2.1.20. Degree 20.

β_{20}(181) = 0 β_{20}(21) = 0 β_{20}(1821) = 9 β_{20}(17211) = 171
β_{20}(20) = 1 β_{20}(201) = 0 β_{20}(18111) = 19 β_{20}(165) = 46
β_{20}(191) = 1 β_{20}(192) = 0 β_{20}(174) = 12
β_{20}(182) = 10 β_{20}(1911) = 1 β_{20}(1731) = 57
β_{20}(1811) = 10 β_{20}(183) = 3 β_{20}(1722) = 81

2.1.21. Remarks.

Remark 4. The β_{m}(u)’s were computed by a rather complicated mathematica [11] program. Mathematica is a very expressive language, but it is rather slow. So some of the harder parts of the computation of some of the harder to compute β_{m}(u)’s were farmed out (through MathLink [12]) to programs written in C and C++. In particular, the problem of deciding whether two graphs are the same or not was farmed out to B. D. McKay’s graph isomorphism and automorphism program nauty [8] through a custom-made MathLink interface. I wish to thank Prof. McKay for his assistance.

Remark 5. The program used to compute β_{m}(u) is not yet available on my ftp site. Sorry, it is way too messy at this stage. The numbers β_{m}(u) them self are available, in the file table.m.gz in my ftp site. See section 4.

Remark 6. There is one known bug in the that program, but it is unlikely that it caused a mistake in the end result. Roughly speaking, the problem is as follows. Let G be the graph whose vertices are all Chinese characters (corresponding to some fixed m and u), and whose edges correspond to I → H and I → X moves. Let G′ be the graph obtained from G by removing all Chinese characters that have an automorphism which reverses the orientation of an odd number of trivalent vertices (that is, the Chinese characters that vanish already mod AS alone), and all edges connected to them. If G′ is not connected, my program may miss a part of the space and give a too low dimension. The problem is technical and easy to fix, and, in fact, I also have a corrected version of the program in which the problem doesn’t exist. But for some technical reason, the fixed version is much slower, and so, except for a small number of test cases (on which the two versions of the program happily agreed), most results were obtained with the wrong program. Generally speaking, G is highly connected and G′ is obtained from it by removing a small number of ‘randomly placed’ vertices, and so I believe the results reported here are correct.

Remark 7. In degree 7 and up, some of the computations were performed modulo a large prime rather than over the rationals, to prevent overflow problems. It is unlikely that this affected the results in any way.

Remark 8. Just for the record, it took about 8 months of CPU time on a 1992 workstation to compute the above 614 dimensions, and they add up to a total of 30743.

Remark 9. In [2] it is proven that β_{m}(1^{m+1}) = (m − 1)!.

Remark 10. It is easy to check that β_{m}(m) = 1 for even m, and β_{m}(m) = 0 for odd m.

Remark 11. With some effort one can prove that β_{m}(m − 1) = ⌈ m ; 6⌉ for odd m, and β_{m}(m − 1) = 0 for even m, where ⌈x⌉ denotes the smallest integer not smaller than x.

2.2. The representation E_{n}. The symmetric group S_{n} acts in a natural way on the space of Chinese characters having exactly n univalent vertices colored by n different colors (modulo IHX and AS). Knowing the dimensions β_{m}(u) for all m and u is equivalent to knowing the decomposition of the resulting graded representation E_{n} of S_{n} into irreducibles, and the β_{m}(u)’s shown above are sufficient to determine the structure of E_{n} for some small values of n and in some small degrees. With R_{u} denoting the irreducible representation of S_{n} corresponding to the partition u of n (so that R_{n} and R_{1^{n}} are the trivial and the alternating representations respectively), here’s what we can do:

\begin{aligned}G_{1}E_{2} & = & R_{2} \\ G_{2}E_{2} & = & R_{2} \\ G_{3}E_{2} & = & R_{2} \\ G_{4}E_{2} & = & R_{2} \\ G_{5}E_{2} & = & 2 R_{2} \\ G_{6}E_{2} & = & 2 R_{2} \\ G_{7}E_{2} & = & 3 R_{2} \\ G_{8}E_{2} & = & 4 R_{2} \\ G_{9}E_{2} & = & 5 R_{2} \\ G_{2}E_{3} & = & R_{111} \\ G_{3}E_{3} & = & R_{111} \\ G_{4}E_{3} & = & R_{111} \\ G_{5}E_{3} & = & 2 R_{111} \\ G_{6}E_{3} & = & 2 R_{111} \\ G_{7}E_{3} & = & 3 R_{111} \\ G_{3}E_{4} & = & R_{22} \\ G_{4}E_{4} & = & R_{4} + R_{22} \\ G_{5}E_{4} & = & R_{4} + 2 R_{22} \end{aligned} \begin{aligned}G_{6}E_{4} & = & 2 R_{4} + 3 R_{22} \\ G_{7}E_{4} & = & 3 R_{4} + 4 R_{22} \\ G_{4}E_{5} & = & R_{311} \\ G_{5}E_{5} & = & 2 R_{311} \\ G_{6}E_{5} & = & 3 R_{311} + R_{2111} \\ G_{7}E_{5} & = & 5 R_{311} + R_{2111} \\ G_{5}E_{6} & = & R_{42} + R_{222} + R_{3111} \\ G_{6}E_{6} & = & R_{6} + 2 R_{42} + 2 R_{222} + R_{321} + R_{3111} + R_{21111} \\ G_{7}E_{6} & = & 2 R_{6} + 4 R_{42} + 4 R_{222} + 2 R_{321} + 2 R_{3111} + 2 R_{21111} \\ G_{6}E_{7} & = & R_{331} + R_{421} + R_{511} + R_{3211} + R_{22111} \\ G_{7}E_{7} & = & R_{43} + 3 R_{331} + 2 R_{421} + 3 R_{511} + R_{2221} + 3 R_{3211} \\ & & + R_{4111} + 2 R_{22111} + R_{1111111} \\ G_{7}E_{8} & = & R_{44} + R_{62} + 2 R_{422} + R_{431} + R_{521} + R_{2222} + R_{3221} \\ & & + 2 R_{3311} + R_{4211} + R_{5111} + R_{32111} + R_{41111} + R_{221111} \\ G_{8}E_{9} & = & R_{63} + 2 R_{333} + 2 R_{432} + R_{441} + R_{522} + 3 R_{531} \\ & & + R_{621} + R_{711} + R_{3222} + 2 R_{3321} + 3 R_{4221} + 3 R_{4311} \\ & & + 3 R_{5211} + R_{6111} + 3 R_{32211} + R_{33111} + 3 R_{42111} \\ & & + R_{222111} + R_{321111} + R_{411111} + R_{3111111} \end{aligned}

Remark 12. As we are restricting our attention to connected Chinese Characters, it is clear that G_{n}E_{m} = 0 if n < m − 1.

Remark 13. Notice that many of the possible irreducible representations of S_{1}–S_{9} are missing in the above decompositions. I do not know why this is so.

Remark 14. Kontsevich’s observation (see remark 2) implies that E_{1} = 0 and that E_{2} is a multiple of R_{2}. Vogel [10] proved that R_{111} appears in E_{3} with the same graded multiplicity as R_{2} in E_{2} (but it is not known whether E_{3} is a multiple of R_{111}). The degree m multiplicity in both cases is β_{m,2} = β_{m−1,0}.

Remark 15. The decompositions presented here were computed from the β_{m}(u)’s using Stembridge’s maple package SF [9]. I wish to thank N. Bergeron for his help with both the mathematical and the computational aspects of this computation.

2.3. Invariants.

2.3.1. Primitive invariants. The information in the lists 2.1.1–2.1.7 can be used to determine the dimension of the space of type m primitive Vassiliev invariants of n-component string links, for sufficiently small values of m:

\begin{matrix}m\n & 1 & 2 & 3 & 4 & 5 & 6 & general n \\ 1 & 1 & 3 & 6 & 10 & 15 & 21 & n (1+n) ; 2 \\ 2 & 1 & 3 & 7 & 14 & 25 & 41 & n(5+n^{2}) ; 6 \\ 3 & 1 & 4 & 13 & 34 & 75 & 146 & n(10−n+2 n^{2}+n^{3}) ; 12 \\ 4 & 2 & 9 & 34 & 105 & 271 & 608 & n(154+45 n+20 n^{2}+15 n^{3}+6 n^{4}) ; 120 \\ 5 & 3 & 16 & 81 & 321 & 1012 & 2679 & n(258+31 n+30 n^{2}+25 n^{3}+12 n^{4}+4 n^{5}) ; 120 \\ 6 & 5 & 32 & 208 & 1040 & 3987 & 12452 & n(1208+490 n+175 n^{2}+105 n^{3}+77 n^{4}+35 n^{5}+10 n^{6}) ; 420 \\ 7 & 8 & 62 & 547 & 3515 & 16469 & 60767 & n(4042+1175 n+700 n^{2}+350 n^{3}+238 n^{4}+140 n^{5}+60 n^{6}+15 n^{7}) ; 840 \end{matrix}

Remark 16. The dimensions presented above (and in sections 2.3.2, 2.3.3 and 3) are of the appropriate degree subspace of the associated graded space. Remember that a Vassiliev invariant of type 2 is also of type 4, and thus the total dimension of the space of type 4 (say) primitive Vassiliev invariants of 3-component (say) string links is 1 + 6 + 7 + 13 + 34 = 61, where the first summand (1) comes from the type 0 (constant) invariants, not listed in the above table.

2.3.2. Framed invariants. The information about primitive invariants can be assembled to give the dimension of the space of all Vassiliev invariants of type m of ncomponent framed string links (excluding invariants of lower type):

n = 1 n = 2 n = 3 n = 4 n = 5  n = 6
m = 1  1  3  6  10  15  21
m = 2  2  9  28  69  145  272
m = 3  3  23 111  394  1130  2778
m = 4  6  60 413 2035  7781  24632
m = 5  10 148 1461 9849 49455 198981
m = 6  19 366 5027 45680 297622 1506218
m = 7  33 884 16924 205612 1722724 10875542

The formulas for general n are (the formula for m = 7 is too messy to print):

m type m invariants (excluding lower types)
1 n (1+n) ; 2
2 n(26+9 n+10 n^{2}+3 n^{3}) ; 24
3 n(48+30 n+41 n^{2}+17 n^{3}+7 n^{4}+n^{5}) ; 48
4 n(10512+9140 n+6900 n^{2}+4415 n^{3}+2568 n^{4}+830 n^{5}+180 n^{6}+15 n^{7}) ; 5760
5 n (2+n) (3+n)(4320+920 n+2682 n^{2}+805 n^{3}+638 n^{4}+192 n^{5}+40 n^{6}+3 n^{7}) ; 11520
6 n(10607616+13964832 n+11387656 n^{2}+8527428 n^{3}+5541270 n^{4}+3139353 n^{5}+1376706 n^{6}+477729 n^{7}+115514 n^{8}+17955 n^{9}+1638 n^{10}+63 n^{11}) ; 2903040

2.3.3. Unframed invariants. Similarly, here are some dimensions of spaces of Vassiliev invariants of type m of n-component unframed string links (excluding invariants of lower type):

n = 1 n = 2 n = 3 n = 4 n = 5 n = 6
m = 1  0  1  3  6  10  15
m = 2  1  4  13  35  80  161
m = 3  1  8  44 174  545  1441
m = 4  3  23 158 834 3436 11639
m = 5  4  51 527 3807 20474 87728
m = 6  9 130 1772 16987 117567 630207
m = 7  14 300 5813 74240 656859 4370368

The formulas for general n are (the formula for m = 7 is too messy to print):

m type m invariants (excluding lower types)
1 (−1+n) n ; 2
2 n (1+n)(14−5 n+3 n^{2}) ; 24
3 n(32−10 n+15 n^{2}+9 n^{3}+n^{4}+n^{5}) ; 48
4 n(9072+1940 n+4380 n^{2}+575 n^{3}+888 n^{4}+350 n^{5}+60 n^{6}+15 n^{7}) ; 5760
5 n(23616−464 n+11532 n^{2}+4680 n^{3}+4489 n^{4}+1371 n^{5}+658 n^{6}+170 n^{7}+25 n^{8}+3 n^{9}) ; 11520
6 n(10123776+3844512 n+6322456 n^{2}+2320164 n^{3}+1893570 n^{4}+931833 n^{5}+485382 n^{6}+153153 n^{7}+43694 n^{8}+7875 n^{9}+882 n^{10}+63 n^{11}) ; 2903040

2.3.4. Factoring knots out. Divide B^{sl} further by the ideal generated by ‘boring’ connected Chinese character — Chinese characters all of whose univalent vertices are colored by the same color. The resulting dimensions (“invariants of string links that do not have a factor that depends on only a single component”, or, “genuine string link invariants”) are:

n = 1 n = 2 n = 3 n = 4 n = 5 n = 6
m = 1  0  1  3  6  10  15
m = 2  0  2  10  31  75  155
m = 3  0  4  32 146  490  1345
m = 4  0  10 107 668 2986 10586
m = 5  0  22 347 2963 17359 78179
m = 6  0  53 1132 12923 97787 552517
m = 7  0 121 3653 55576 538394 3781984

3. Braids

Just for the sake of comparison, here are some dimensions of spaces of Vassiliev invariants of braids. Notice that even the dimensions in section 2.3.4 are bigger than the dimensions below.

m\n 1 2 3  4  5  6 general n
1 0 1 3  6  10  15 (−1+n) n ; 2
2 0 1 7 25  65  140 (−1+n) n (1+n) (−2+3 n) ; 24
3 0 1 15 90  350  1050 \begin{aligned}(−1+n)^{2} & n^{2} & (1+n) & (2+n) \\ & 48 \end{aligned}
4 0 1 31 301 1701  6951 (−1+n) n (1+n) (2+n) (3+n)(8−10 n−15 n^{2}+15 n^{3}) ; 5760
5 0 1 63 966 7770 42525 \begin{aligned}(−1+n)^{2} & n^{2} & (1+n) & (2+n) & (3+n) & (4+n) & (−6+n+3 n^{2}) \\ & & & 11520 \end{aligned}
6 0 1 127 3025 34105 246730 (−1+n) n (1+n) (2+n) (3+n) (4+n) (5+n)(−96+140 n+224 n^{2}−315 n^{3}+63 n^{5}) ; 2903040
7 0 1 255 9330 145750 1379400 too messy to print

Remark 17. These dimensions d(m, n) satisfy d(m, n) = (n−1)d(m−1, n)+d(m, n− 1) for m ≥ 1 and n ≥ 2, d(0, n) = 1 for n ≥ 1 and d(m, 1) = 0 for m ≥ 1. This fact, which makes their computation very easy, follows from Drinfel’d’s presentation of (his notation) 𝔞_{n} as a semidirect product of free algebras [4, page 847]. See also [3, 5, 7].

4. The data file

The file table.m.gz in my ftp site contains, in a mathematica readable format, some of the data presented in this article as well as explicit basis for some of the spaces whose dimensions were given here. To read this file, get it from my ftp site, uncompress it by typing gunzip table.m at the UNIX prompt, start up mathematica and load it in:

Mathematica 2.2 for SPARC
Copyright 1988-93 Wolfram Research, Inc.
 -- Open Look graphics initialized --
In[1]:= << table.m

The space B_{5}(22) is of dimension 3. Indeed,

In[2]:= DimB[5,{2,2}]

Out[2]= 3

A basis of B_{3}(22) (which is 1-dimensional) is

In[3]:= BBasis[3,{2,2}] // InputForm

Out[3]=

  {Graph[C1[6], C1[7], C2[6], C2[7], Dashed[6,7], OV[1,3,5], OV[2,4,5]]}

Let us explain the notation: A Graph is an object of the form Graph[v_{1}, v_{2}, …], where each v_{j} is a vertex, that is, an object of the form vname_{j}[p_{j1}, p_{j2}, …]. Such a Graph object represents a graph with numbered and named vertices, so that the jth vertex is named vname_{j}, and is connected by an arc to the vertices numbered p_{j1}, p_{j2}, . . .. Chinese Characters are represented by such Graph object containing three types of vertices — oriented trivalent vertices named OV, colored univalent vertices named Ci for some integer i, and artificial bivalent vertices named Dashed placed in the middle of each dashed line connecting two trivalent vertices.

Therefore, the Chinese Character in Out[3] is:

[figure: OV; Dashed; 5; 7; 6; OV; C1; 1; C2; 3; 2; C1; C2; 4]

5. Acknowledgement

I wish to thank A. Stoimenow for his comments.

References

1. D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 423–472 (1995).

2. , Vassiliev homotopy string link invariants, Jour. of Knot Theory and its Ramifications 4 (1995) 13–32.

3. , Vassiliev and quantum invariants of braids, in Proc. of Symp. in Appl. Math. 51 (1996) 129–144 The interface of knots and physics, (L. H. Kauffman, ed.), Amer. Math. Soc., Providence.

4. V. G. Drinfel’d, On quasitriangular Quasi-Hopf algebras and a group closely connected with Gal(ℚ̄/ℚ), Leningrad Math. J. 2 (1991) 829–860.

5. M. Hutchings, Integration of singular braid invariants and graph cohomology, Harvard University preprint, May 1995.

6. M. Kontsevich, Private communication.

7. X-S. Lin, Braid algebras, trace modules and Vassiliev invariants, Columbia Univ. preprint, 1994.

8. B. D. McKay, Nauty users guide (Version 1.5), Technical Report TR-CS-90-02, Computer Science Department, Australian National University, 1990.

9. J. R. Stembridge, A maple package for symmetric functions, available via anonymous file transfer

  from ftp.math.lsa.umich.edu, user name ftp, subdirectory pub/jrs/software. Read the file
  READ_ME and get SF.tar.Z.

10. P. Vogel, Algebraic structures on modules of diagrams, Université Paris VII preprint, July 1995.

11. S. Wolfram, Mathematica — a system for doing mathematics by computer, Addison-Wesley, 1989.

12. Wolfram Research, MathLink reference guide, Wolfram Research technical report, 1993.

Department of Mathematics, Harvard University, Cambridge, MA 02138 Current address: Institute of Mathematics, The Hebrew University, Giv’at-Ram, Jerusalem

91904, Israel E-mail address: drorbn@math.huji.ac.il