The file below shows the image in the Temperley-Lieb algebra
of the following 6-string 27-crossing braid:
EdCCBaaEEEdcBBBabCDebceecDe
This braid is algebraically right 1-split which means that the image in
TL_{6,1} looks like the last string of the braid is split.
More precisely: this means that the coefficients of (nine) Kauffman diagrams
with the last (rightmost) string n o t going straight down are non-zero only
for those with at least two cups (and for the unit). In fact, the braid
is not right split which is easily seen from the fact that the image has
many non-zero coefficients for non-split generators with more than 1 cup.
When this braid is suitably manipulated one obtains
FgHHIjjFFFghIIIjiHGfihffhGf
which is an 11-string braid that is trivial on the first 5 strings
and the image in TL_{11,1} looks like it is trivial on the
first s i x strings. The image of the commutator ot the two is shown
in TL11_176-2kernel.txt.
The image in TL_{6} is given in the following manner:
generator number/number of cups/first term degree/coefficients with step 4.
The generators of TL_{6}, the Kauffman diagrams, are listed as
described in Piwocki, Traczyk
"Representations of small braid groups in Temperley-Lieb algebra"
Lists of generators are given as TLgensxx.txt
1 0 1 1
2 1 3
3 1 1
4 1 1
5 1 -9 1 -2 3 -2 1
6 1 3
7 1 -11 1 -3 5 -5 4 -2 1
8 2 -23 -1 5 -12 18 -17 7 7 -17 18 -12 5 -1
9 2 -29 1 -6 18 -34 45 -40 16 16 -40 45 -34 18 -6 1
10 1 3
11 1 -15 1 -2 4 -5 5 -3 1
12 1 -17 1 -3 6 -9 11 -9 6 -3 1
13 2 -25 1 -6 18 -34 45 -40 16 16 -40 45 -34 18 -6 1
14 2 -31 -1 7 -24 55 -90 109 -92 36 36 -92 109 -90 55 -24 7 -1
15 1 1
16 1 -5 -1 3 -4 4 -2 1
17 2 -23 -1 7 -21 40 -53 46 -17 -21 48 -52 38 -19 6 -1
18 1 -11 -1 3 -5 8 -8 6 -3 1
19 2 -25 1 -7 25 -58 97 -119 101 -40 -40 101 -119 97 -58 25 -7 1
20 2 -19 1 -4 9 -13 12 -5 -5 12 -13 9 -4 1
21 2 -17 2 -10 25 -40 42 -24 -7 34 -43 34 -18 6 -1
22 2 -17 2 -8 18 -27 27 -15 -5 22 -28 23 -13 5 -1
23 2 -19 -1 8 -27 58 -87 91 -57 -5 64 -92 83 -53 24 -7 1
24 2 -27 1 -4 9 -13 12 -5 -5 12 -13 9 -4 1
25 2 -33 -1 5 -13 23 -28 22 -5 -15 27 -27 18 -8 2
26 2 -27 1 -5 13 -23 28 -21 3 17 -28 26 -16 6 -1
27 2 -19 1 -3 5 -5 2 2 -5 5 -3 1
28 2 -21 -1 5 -12 19 -18 7 10 -22 23 -15 6 -1
29 1 1
30 1 -17 1 -2 4 -4 3 -1
31 1 -19 1 -3 6 -8 8 -5 3 -1
32 2 -31 -1 6 -19 38 -52 48 -21 -17 46 -53 40 -21 7 -1
33 2 -37 1 -7 25 -58 97 -119 101 -40 -40 101 -119 97 -58 25 -7 1
34 1 -13 -1 3 -5 7 -5 3 -1
35 2 -31 -1 7 -24 55 -92 113 -96 38 38 -96 113 -92 55 -24 7 -1
36 2 -23 -1 6 -16 26 -28 17 3 -21 28 -23 13 -5 1
37 2 -25 1 -7 24 -53 81 -86 55 3 -59 86 -78 50 -23 7 -1
38 2 -33 -1 6 -18 34 -43 34 -7 -24 42 -40 25 -10 2
39 2 -39 1 -7 24 -53 83 -92 64 -5 -57 91 -87 58 -27 8 -1
40 2 -33 -1 7 -23 50 -78 86 -59 3 55 -86 81 -53 24 -7 1
41 2 -25 -1 6 -15 23 -22 10 7 -18 19 -12 5 -1
42 2 -27 1 -7 23 -48 68 -62 26 26 -62 68 -48 23 -7 1
43 1 3
44 1 1 1 -2 3 -2 1
45 2 -21 1 -3 3 3 -15 25 -24 10 9 -21 21 -13 5 -1
46 1 -5 1 -2 4 -5 5 -3 1
47 2 -19 1 -2 -2 16 -37 52 -47 19 18 -44 48 -35 18 -6 1
48 2 -19 1 -4 7 -5 -5 18 -24 17 -1 -13 17 -12 5 -1
49 2 -17 2 -7 11 -6 -12 35 -47 36 -7 -22 35 -30 17 -6 1
50 2 -17 1 -5 11 -14 10 0 -10 14 -11 5 -1
51 2 -19 1 -2 1 4 -9 10 -4 -5 11 -10 5 -1
52 1 -7 1 -2 4 -4 3 -1
53 2 -21 -1 7 -22 42 -54 45 -14 -24 49 -51 36 -18 6 -1
54 2 -23 -1 4 -6 0 17 -37 45 -31 1 27 -38 31 -17 6 -1
55 2 -23 -1 7 -21 37 -42 27 3 -31 42 -34 18 -6 1
56 2 -25 -1 4 -5 -2 19 -34 35 -15 -12 30 -29 17 -6 1
57 2 -15 -1 4 -8 10 -7 0 7 -10 8 -4 1
58 2 -21 1 -4 8 -9 4 5 -12 12 -6 -1 4 -3 1
59 2 -13 -2 8 -16 21 -17 4 11 -20 19 -12 5 -1
60 2 -19 2 -8 16 -19 11 6 -22 26 -17 3 6 -7 4 -1
61 2 -15 -1 3 -4 3 0 -3 4 -3 1
62 2 -21 1 -3 4 -2 -2 5 -5 2 1 -2 1
63 2 -19 1 -5 11 -15 13 -4 -7 14 -14 9 -4 1
64 2 -25 -1 5 -11 13 -6 -8 20 -21 11 2 -9 8 -4 1
65 2 -21 1 -5 10 -12 8 1 -9 12 -9 4 -1
66 2 -27 -1 5 -10 10 -1 -12 20 -16 5 5 -7 4 -1
67 3 3
68 3 -23 -1 4 -8 10 -7 0 7 -10 8 -4 1
69 3 -29 1 -4 8 -9 4 5 -12 12 -6 -1 4 -3 1
70 3 -29 1 -5 12 -19 20 -11 -4 17 -21 16 -8 2
71 3 -35 -1 5 -12 18 -16 3 15 -26 23 -10 -3 8 -6 2
72 3 -19 1 -4 8 -10 7 0 -7 10 -8 4 -1
73 3 -29 1 -5 12 -17 12 5 -25 34 -25 5 12 -17 12 -5 1
74 3 -35 -1 5 -12 16 -8 -13 34 -37 17 12 -29 25 -9 -4 7 -4 1
75 3 -35 -1 6 -17 31 -38 26 5 -39 55 -44 17 7 -16 12 -5 1
76 3 -41 1 -6 17 -29 29 -4 -41 78 -77 34 22 -55 51 -25 2 6 -4 1
77 2 -27 1 -4 8 -10 7 0 -7 10 -8 4 -1
78 2 -33 -1 5 -12 19 -20 11 4 -17 21 -16 8 -2
79 2 -27 1 -4 9 -14 14 -7 -4 13 -15 11 -5 1
80 2 -19 1 -3 4 -3 0 3 -4 3 -1
81 2 -21 -1 4 -9 12 -9 1 8 -12 10 -5 1
82 2 -17 1 -4 7 -7 3 3 -7 7 -4 1
83 2 -19 1 -1 -1 6 -9 8 -2 -4 6 -4 1
84 2 -15 -1 2 -2 1 1 -2 2 -1
85 2 -21 1 -2 1 3 -7 9 -6 2 1 -1
86 3 -17 2 -8 16 -21 17 -4 -11 20 -19 12 -5 1
87 3 -27 1 -5 12 -16 7 17 -44 55 -39 5 26 -38 31 -17 6 -1
88 3 -33 -1 5 -12 15 -3 -25 52 -54 23 23 -53 50 -24 -2 13 -11 5 -1
89 3 -33 -1 6 -17 31 -37 21 17 -58 76 -58 17 21 -37 31 -17 6 -1
90 3 -39 1 -6 17 -28 24 11 -71 119 -115 50 39 -98 100 -60 16 7 -10 5 -1
91 1 3
92 1 -19 1 -2 3 -2 1
93 1 -21 1 -3 5 -5 4 -2 1
94 2 -33 -1 5 -13 21 -21 9 10 -24 25 -15 3 3 -3 1
95 2 -39 1 -6 18 -35 48 -44 18 19 -47 52 -37 16 -2 -2 1
96 1 -15 -1 3 -4 4 -2 1
97 2 -33 -1 6 -18 36 -51 49 -24 -14 45 -54 42 -22 7 -1
98 2 -25 -1 5 -11 14 -10 0 10 -14 11 -5 1
99 2 -27 1 -6 18 -34 42 -31 3 27 -42 37 -21 7 -1
100 2 -35 -1 5 -12 17 -13 -1 17 -24 18 -5 -5 7 -4 1
101 2 -41 1 -6 17 -30 35 -22 -7 36 -47 35 -12 -6 11 -7 2
102 2 -35 -1 6 -17 31 -38 27 1 -31 45 -37 17 0 -6 4 -1
103 2 -27 -1 5 -10 11 -5 -4 10 -9 4 1 -2 1
104 2 -29 1 -6 17 -29 30 -12 -15 35 -34 19 -2 -5 4 -1
105 1 -9 1 -2 3 -2 1
106 2 -23 -1 5 -12 18 -17 7 7 -17 18 -12 5 -1
107 2 -25 -1 3 -3 -2 11 -17 14 -3 -9 14 -11 5 -1
108 2 -25 -1 5 -11 14 -9 -3 14 -17 11 -2 -3 3 -1
109 2 -27 -1 3 -2 -4 13 -15 7 7 -15 13 -4 -2 3 -1
110 2 -21 1 -4 7 -7 3 3 -7 7 -4 1
111 2 -27 -1 4 -7 6 0 -7 10 -7 1 3 -3 1
112 2 -23 1 -4 6 -4 -2 8 -9 6 -1 -1 1
113 2 -29 -1 4 -6 3 5 -11 10 -2 -5 7 -3 0 1
114 3 -21 1 -3 4 -1 -6 12 -12 5 4 -9 8 -4 1
115 3 -31 1 -4 7 -4 -9 25 -29 12 17 -37 34 -13 -8 16 -12 5 -1
116 3 -37 -1 4 -7 4 9 -24 25 -4 -26 40 -26 -4 25 -24 9 4 -7 4 -1
117 3 -37 -1 5 -11 13 -2 -24 50 -53 23 23 -54 52 -25 -3 15 -12 5 -1
118 3 -43 1 -5 11 -12 -1 29 -53 46 0 -57 81 -55 1 39 -43 22 -1 -6 4 -1
119 2 -29 1 -3 4 -1 -6 12 -12 5 4 -9 8 -4 1
120 2 -35 -1 4 -7 6 3 -17 26 -22 6 11 -19 16 -8 2
121 2 -29 1 -4 8 -9 2 11 -21 20 -8 -6 13 -11 5 -1
122 2 -21 1 -2 1 2 -5 5 -2 -2 4 -3 1
123 2 -23 -1 4 -7 5 5 -16 20 -12 -1 10 -10 5 -1
124 2 -19 1 -3 3 1 -7 10 -7 0 6 -7 4 -1
125 2 -21 1 0 -3 7 -5 -2 10 -11 5 3 -6 4 -1
126 2 -17 -1 1 2 -6 9 -7 3 1 -2 1
127 2 -23 1 -1 -2 7 -8 4 4 -8 7 -2 -1 1
128 3 -19 2 -6 8 -3 -10 23 -26 15 3 -16 18 -12 5 -1
129 3 -29 1 -4 6 2 -25 51 -55 22 34 -77 78 -41 -4 29 -29 17 -6 1
130 3 -35 -1 4 -6 -1 22 -43 39 1 -55 81 -57 0 46 -53 29 -1 -12 11 -5 1
131 3 -35 -1 5 -10 7 16 -60 100 -98 39 50 -115 119 -71 11 24 -28 17 -6 1
132 3 -41 1 -5 10 -6 -18 61 -92 69 17 -119 164 -119 17 69 -92 61 -18 -6 10 -5 1
1517