9 2 Quantum Invariants
From Knot Atlas
Revision as of 18:07, 4 December 2005 by
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A1 Invariants.
Weight
Invariant
1
−
q
21
+
q
15
+
q
5
+
q
{\displaystyle -q^{21}+q^{15}+q^{5}+q}
2
q
60
−
q
56
−
q
50
+
q
46
−
q
30
−
q
28
+
q
18
+
q
16
+
q
14
+
q
8
+
q
2
{\displaystyle q^{60}-q^{56}-q^{50}+q^{46}-q^{30}-q^{28}+q^{18}+q^{16}+q^{14}+q^{8}+q^{2}}
3
−
q
117
+
q
113
+
q
111
−
q
107
+
q
103
−
q
99
−
q
97
+
q
93
+
q
73
+
q
71
−
q
67
−
q
61
−
q
59
−
q
53
+
q
49
+
q
47
−
q
43
−
q
41
−
q
39
+
q
37
−
q
33
−
q
31
+
q
29
+
2
q
27
−
q
23
+
q
21
+
2
q
19
+
q
17
+
q
11
+
q
3
{\displaystyle -q^{117}+q^{113}+q^{111}-q^{107}+q^{103}-q^{99}-q^{97}+q^{93}+q^{73}+q^{71}-q^{67}-q^{61}-q^{59}-q^{53}+q^{49}+q^{47}-q^{43}-q^{41}-q^{39}+q^{37}-q^{33}-q^{31}+q^{29}+2q^{27}-q^{23}+q^{21}+2q^{19}+q^{17}+q^{11}+q^{3}}
4
q
192
−
q
188
−
q
186
−
q
184
+
q
182
+
q
180
+
q
178
−
2
q
174
+
q
170
+
q
168
+
q
166
−
q
164
−
q
162
−
q
160
+
q
156
−
q
134
−
q
132
+
q
128
+
2
q
126
−
q
122
+
q
118
+
2
q
116
−
2
q
112
−
q
110
+
q
106
−
2
q
102
−
q
100
+
q
96
+
q
94
+
q
90
+
q
88
+
q
86
−
q
82
−
q
80
+
q
76
−
q
72
−
q
70
−
q
68
−
q
66
+
q
62
−
q
60
−
2
q
58
−
2
q
56
+
2
q
52
+
q
50
−
2
q
46
+
q
44
+
2
q
42
+
q
40
−
q
38
−
2
q
36
+
q
34
+
2
q
32
+
q
30
−
q
26
+
q
24
+
q
22
+
q
20
+
q
18
+
q
14
+
q
4
{\displaystyle q^{192}-q^{188}-q^{186}-q^{184}+q^{182}+q^{180}+q^{178}-2q^{174}+q^{170}+q^{168}+q^{166}-q^{164}-q^{162}-q^{160}+q^{156}-q^{134}-q^{132}+q^{128}+2q^{126}-q^{122}+q^{118}+2q^{116}-2q^{112}-q^{110}+q^{106}-2q^{102}-q^{100}+q^{96}+q^{94}+q^{90}+q^{88}+q^{86}-q^{82}-q^{80}+q^{76}-q^{72}-q^{70}-q^{68}-q^{66}+q^{62}-q^{60}-2q^{58}-2q^{56}+2q^{52}+q^{50}-2q^{46}+q^{44}+2q^{42}+q^{40}-q^{38}-2q^{36}+q^{34}+2q^{32}+q^{30}-q^{26}+q^{24}+q^{22}+q^{20}+q^{18}+q^{14}+q^{4}}
5
−
q
285
+
q
281
+
q
279
+
q
277
−
q
273
−
2
q
271
−
q
269
+
q
265
+
2
q
263
+
q
261
−
q
259
−
2
q
257
−
q
255
+
q
251
+
2
q
249
+
q
247
−
q
243
−
q
241
−
q
239
+
q
235
+
q
213
+
q
211
−
q
207
−
2
q
205
−
2
q
203
+
2
q
199
+
2
q
197
−
2
q
193
−
2
q
191
−
q
189
+
2
q
187
+
4
q
185
+
2
q
183
−
q
181
−
2
q
179
−
2
q
177
+
2
q
173
+
2
q
171
−
2
q
167
−
2
q
165
−
q
163
+
q
159
+
q
157
−
q
155
−
q
153
−
q
151
+
q
147
+
2
q
145
+
q
143
−
q
139
−
q
137
+
q
135
+
2
q
133
+
q
131
−
q
127
+
q
123
−
q
119
−
2
q
117
−
q
115
+
q
113
+
2
q
111
+
q
109
−
q
105
−
q
103
−
2
q
101
+
q
97
+
2
q
95
+
2
q
93
+
q
91
−
2
q
89
−
3
q
87
−
2
q
85
+
q
81
+
q
79
−
q
77
−
2
q
75
−
3
q
73
−
q
71
+
q
69
+
q
67
−
q
63
+
q
57
−
q
53
+
2
q
49
+
3
q
47
+
q
45
−
q
43
−
q
41
−
q
39
+
q
37
+
2
q
35
+
q
33
+
q
23
+
q
21
+
q
19
+
q
17
+
q
5
{\displaystyle -q^{285}+q^{281}+q^{279}+q^{277}-q^{273}-2q^{271}-q^{269}+q^{265}+2q^{263}+q^{261}-q^{259}-2q^{257}-q^{255}+q^{251}+2q^{249}+q^{247}-q^{243}-q^{241}-q^{239}+q^{235}+q^{213}+q^{211}-q^{207}-2q^{205}-2q^{203}+2q^{199}+2q^{197}-2q^{193}-2q^{191}-q^{189}+2q^{187}+4q^{185}+2q^{183}-q^{181}-2q^{179}-2q^{177}+2q^{173}+2q^{171}-2q^{167}-2q^{165}-q^{163}+q^{159}+q^{157}-q^{155}-q^{153}-q^{151}+q^{147}+2q^{145}+q^{143}-q^{139}-q^{137}+q^{135}+2q^{133}+q^{131}-q^{127}+q^{123}-q^{119}-2q^{117}-q^{115}+q^{113}+2q^{111}+q^{109}-q^{105}-q^{103}-2q^{101}+q^{97}+2q^{95}+2q^{93}+q^{91}-2q^{89}-3q^{87}-2q^{85}+q^{81}+q^{79}-q^{77}-2q^{75}-3q^{73}-q^{71}+q^{69}+q^{67}-q^{63}+q^{57}-q^{53}+2q^{49}+3q^{47}+q^{45}-q^{43}-q^{41}-q^{39}+q^{37}+2q^{35}+q^{33}+q^{23}+q^{21}+q^{19}+q^{17}+q^{5}}
A2 Invariants.
Weight
Invariant
1,0
−
q
32
−
q
30
+
q
24
+
q
22
+
q
8
+
q
6
+
q
2
{\displaystyle -q^{32}-q^{30}+q^{24}+q^{22}+q^{8}+q^{6}+q^{2}}
1,1
q
84
+
2
q
80
−
2
q
78
+
2
q
76
−
4
q
74
+
2
q
72
−
4
q
70
+
4
q
62
−
q
60
+
4
q
58
−
4
q
56
+
2
q
54
−
4
q
52
+
2
q
50
−
2
q
48
+
2
q
46
−
2
q
42
−
2
q
40
−
2
q
38
−
2
q
36
+
2
q
30
+
2
q
28
+
2
q
26
+
2
q
24
+
q
20
+
2
q
16
+
2
q
12
+
2
q
8
+
q
4
{\displaystyle q^{84}+2q^{80}-2q^{78}+2q^{76}-4q^{74}+2q^{72}-4q^{70}+4q^{62}-q^{60}+4q^{58}-4q^{56}+2q^{54}-4q^{52}+2q^{50}-2q^{48}+2q^{46}-2q^{42}-2q^{40}-2q^{38}-2q^{36}+2q^{30}+2q^{28}+2q^{26}+2q^{24}+q^{20}+2q^{16}+2q^{12}+2q^{8}+q^{4}}
2,0
q
82
+
q
80
+
q
78
−
q
76
−
q
74
−
q
72
−
q
70
−
q
68
−
q
66
+
q
64
+
q
62
+
q
60
−
q
44
−
2
q
42
−
2
q
40
−
q
38
+
q
32
+
q
30
+
q
28
+
2
q
26
+
q
24
+
q
20
+
q
18
+
q
16
+
q
12
+
q
10
+
q
4
{\displaystyle q^{82}+q^{80}+q^{78}-q^{76}-q^{74}-q^{72}-q^{70}-q^{68}-q^{66}+q^{64}+q^{62}+q^{60}-q^{44}-2q^{42}-2q^{40}-q^{38}+q^{32}+q^{30}+q^{28}+2q^{26}+q^{24}+q^{20}+q^{18}+q^{16}+q^{12}+q^{10}+q^{4}}
A3 Invariants.
Weight
Invariant
0,1,0
q
66
+
q
62
−
q
58
−
q
56
−
q
54
−
q
52
−
q
50
−
q
46
−
q
42
+
q
38
+
q
36
+
q
34
+
q
32
+
q
30
+
q
16
+
q
12
+
2
q
10
+
q
8
+
q
4
{\displaystyle q^{66}+q^{62}-q^{58}-q^{56}-q^{54}-q^{52}-q^{50}-q^{46}-q^{42}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}+q^{16}+q^{12}+2q^{10}+q^{8}+q^{4}}
1,0,0
−
q
43
−
q
41
−
q
39
+
q
33
+
q
31
+
q
29
+
q
11
+
q
9
+
q
7
+
q
3
{\displaystyle -q^{43}-q^{41}-q^{39}+q^{33}+q^{31}+q^{29}+q^{11}+q^{9}+q^{7}+q^{3}}
B2 Invariants.
Weight
Invariant
0,1
−
q
66
−
q
62
−
q
58
+
q
56
−
q
54
+
q
52
+
q
50
+
q
46
+
q
42
−
2
q
40
+
q
38
−
q
36
+
q
34
−
q
32
+
q
30
+
q
16
+
q
12
+
q
8
+
q
4
{\displaystyle -q^{66}-q^{62}-q^{58}+q^{56}-q^{54}+q^{52}+q^{50}+q^{46}+q^{42}-2q^{40}+q^{38}-q^{36}+q^{34}-q^{32}+q^{30}+q^{16}+q^{12}+q^{8}+q^{4}}
1,0
q
108
+
q
100
−
q
96
−
q
94
−
q
88
−
q
86
+
q
82
−
q
76
−
q
68
−
q
66
+
q
56
+
q
54
+
q
48
+
q
46
+
q
26
+
q
18
+
q
16
+
q
14
+
q
6
{\displaystyle q^{108}+q^{100}-q^{96}-q^{94}-q^{88}-q^{86}+q^{82}-q^{76}-q^{68}-q^{66}+q^{56}+q^{54}+q^{48}+q^{46}+q^{26}+q^{18}+q^{16}+q^{14}+q^{6}}
G2 Invariants.
Weight
Invariant
1,0
q
156
+
q
152
−
q
150
+
q
142
−
2
q
140
+
q
138
−
q
136
−
q
134
−
2
q
130
−
q
128
−
q
126
−
q
124
−
q
118
+
q
112
−
q
108
+
q
106
+
q
104
+
2
q
102
+
q
98
+
q
94
+
q
92
−
2
q
90
+
q
88
+
q
86
+
q
76
−
q
72
+
q
66
−
q
62
−
q
52
+
q
48
+
q
38
+
q
34
+
q
28
+
q
24
+
q
20
+
q
14
+
q
10
{\displaystyle q^{156}+q^{152}-q^{150}+q^{142}-2q^{140}+q^{138}-q^{136}-q^{134}-2q^{130}-q^{128}-q^{126}-q^{124}-q^{118}+q^{112}-q^{108}+q^{106}+q^{104}+2q^{102}+q^{98}+q^{94}+q^{92}-2q^{90}+q^{88}+q^{86}+q^{76}-q^{72}+q^{66}-q^{62}-q^{52}+q^{48}+q^{38}+q^{34}+q^{28}+q^{24}+q^{20}+q^{14}+q^{10}}
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