12n
0165
(K12n
0165
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 12 11 4 5 7 6 10 9
Solving Sequence
6,11 3,7
4 10 12 5 2 1 9 8
c
6
c
3
c
10
c
11
c
5
c
2
c
1
c
9
c
8
c
4
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
41
+ u
40
+ ··· + u
2
+ b, u
41
− u
40
+ ··· + a + 2, u
42
− 2u
41
+ ··· + 2u − 1i
I
u
2
= hu
6
− 2u
4
− u
3
+ u
2
+ b + u + 1, u
7
+ u
6
− 2u
5
− 2u
4
+ u
3
+ u
2
+ a + u + 1,
u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1i
* 2 irreducible components of dim
C
= 0, with total 51 representations.
1
The image of knot diagram is generated by the software “Draw programme” developed by An-
drew Bartholomew(http://www.layer8.co.uk/maths/draw/index.htm#Running-draw), where we modi-
fied some parts for our purpose(https://github.com/CATsTAILs/LinksPainter).
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
1
I.
I
u
1
= h−u
41
+u
40
+· · ·+u
2
+b, u
41
−u
40
+· · ·+a+2, u
42
−2u
41
+· · ·+2u−1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
−u
41
+ u
40
+ ··· + 2u − 2
u
41
− u
40
+ ··· + u
3
− u
2
a
7
=
1
u
2
a
4
=
−u
40
+ u
39
+ ··· + u − 1
−u
40
+ u
39
+ ··· − 2u
2
+ u
a
10
=
u
u
a
12
=
−u
3
−u
3
+ u
a
5
=
u
6
− u
4
+ 1
u
6
− 2u
4
+ u
2
a
2
=
u
39
− u
38
+ ··· + u − 2
u
41
− u
40
+ ··· + u
3
− 2u
2
a
1
=
u
11
− 2u
9
+ 2u
7
+ u
3
u
13
− 3u
11
+ 5u
9
− 4u
7
+ 2u
5
+ u
3
− u
a
9
=
u
3
u
5
− u
3
+ u
a
8
=
u
17
− 4u
15
+ 7u
13
− 4u
11
− 3u
9
+ 6u
7
− 2u
5
+ u
u
17
− 5u
15
+ 11u
13
− 12u
11
+ 5u
9
+ 2u
7
− 2u
5
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6u
41
+ 9u
40
+ 58u
39
− 101u
38
− 263u
37
+ 545u
36
+ 691u
35
−
1830u
34
− 1021u
33
+ 4173u
32
+ 344u
31
− 6595u
30
+ 2038u
29
+ 6914u
28
− 5159u
27
−
3793u
26
+ 6460u
25
− 1008u
24
− 4436u
23
+ 3755u
22
+ 744u
21
− 2743u
20
+ 1497u
19
+
47u
18
− 1234u
17
+ 1369u
16
− 10u
15
− 898u
14
+ 560u
13
− 14u
12
− 332u
11
+ 300u
10
+
6u
9
− 87u
8
+ 54u
7
− 69u
6
− 18u
5
+ 67u
4
− 23u
3
+ 7u
2
+ 4u − 13
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
42
+ 6u
41
+ ··· + 14u + 1
c
2
, c
4
u
42
− 10u
41
+ ··· + 10u − 1
c
3
, c
7
u
42
− u
41
+ ··· + 512u + 512
c
5
, c
9
u
42
+ 6u
41
+ ··· − 134u − 17
c
6
, c
10
u
42
+ 2u
41
+ ··· − 2u − 1
c
8
u
42
+ 2u
41
+ ··· − 2773910u − 699025
c
11
u
42
+ 22u
41
+ ··· + 2u + 1
c
12
u
42
− 2u
41
+ ··· + 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
42
+ 70y
41
+ ··· − 202y + 1
c
2
, c
4
y
42
− 6y
41
+ ··· − 14y + 1
c
3
, c
7
y
42
− 57y
41
+ ··· − 4718592y + 262144
c
5
, c
9
y
42
+ 26y
41
+ ··· − 9490y + 289
c
6
, c
10
y
42
− 22y
41
+ ··· − 2y + 1
c
8
y
42
+ 90y
41
+ ··· − 6627902285450y + 488635950625
c
11
y
42
− 2y
41
+ ··· − 22y + 1
c
12
y
42
+ 54y
41
+ ··· − 2y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.758535 + 0.646533I
a = 0.556695 + 1.198170I
b = 0.884685 + 1.037480I
11.73740 + 1.53247I 0.138999 + 0.820904I
u = 0.758535 − 0.646533I
a = 0.556695 − 1.198170I
b = 0.884685 − 1.037480I
11.73740 − 1.53247I 0.138999 − 0.820904I
u = −0.850409 + 0.511409I
a = 0.510882 + 0.992064I
b = 0.224665 + 0.371706I
1.85125 + 3.14547I 0.50380 − 5.60471I
u = −0.850409 − 0.511409I
a = 0.510882 − 0.992064I
b = 0.224665 − 0.371706I
1.85125 − 3.14547I 0.50380 + 5.60471I
u = 0.799944 + 0.638230I
a = −0.94309 − 1.79340I
b = −0.338330 − 1.002640I
11.61710 − 6.48369I −0.21200 + 5.34025I
u = 0.799944 − 0.638230I
a = −0.94309 + 1.79340I
b = −0.338330 + 1.002640I
11.61710 + 6.48369I −0.21200 − 5.34025I
u = 0.894592
a = 1.03594
b = 0.336424
−1.45839 −6.12070
u = −0.674270 + 0.529870I
a = 0.671399 − 0.395254I
b = 0.627556 + 0.015747I
2.36371 + 1.10219I 2.10234 − 2.56596I
u = −0.674270 − 0.529870I
a = 0.671399 + 0.395254I
b = 0.627556 − 0.015747I
2.36371 − 1.10219I 2.10234 + 2.56596I
u = 0.229875 + 0.814047I
a = −0.318800 + 1.227780I
b = −2.11276 − 1.36196I
8.65210 + 8.20700I −1.50156 − 4.22989I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.229875 − 0.814047I
a = −0.318800 − 1.227780I
b = −2.11276 + 1.36196I
8.65210 − 8.20700I −1.50156 + 4.22989I
u = 0.265629 + 0.793421I
a = 0.226149 − 0.606330I
b = 2.08149 + 0.30188I
9.24719 + 0.37833I −0.624670 + 0.106792I
u = 0.265629 − 0.793421I
a = 0.226149 + 0.606330I
b = 2.08149 − 0.30188I
9.24719 − 0.37833I −0.624670 − 0.106792I
u = 1.104060 + 0.374333I
a = 0.75446 + 1.54813I
b = −0.75382 + 1.34478I
−2.74442 − 1.12803I −6.06477 − 0.07417I
u = 1.104060 − 0.374333I
a = 0.75446 − 1.54813I
b = −0.75382 − 1.34478I
−2.74442 + 1.12803I −6.06477 + 0.07417I
u = −1.129850 + 0.429601I
a = 0.69978 − 2.03550I
b = −0.45833 − 2.41418I
−5.34146 + 2.73778I −7.76940 − 4.35221I
u = −1.129850 − 0.429601I
a = 0.69978 + 2.03550I
b = −0.45833 + 2.41418I
−5.34146 − 2.73778I −7.76940 + 4.35221I
u = −1.178670 + 0.271644I
a = 2.19851 + 0.72892I
b = 1.52165 − 0.58407I
4.73543 + 2.88234I −5.97327 − 2.64287I
u = −1.178670 − 0.271644I
a = 2.19851 − 0.72892I
b = 1.52165 + 0.58407I
4.73543 − 2.88234I −5.97327 + 2.64287I
u = −0.117108 + 0.770698I
a = 0.612360 − 0.464938I
b = −1.219860 + 0.084521I
−1.15293 − 3.18904I −1.05779 + 4.41031I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.117108 − 0.770698I
a = 0.612360 + 0.464938I
b = −1.219860 − 0.084521I
−1.15293 + 3.18904I −1.05779 − 4.41031I
u = 1.135850 + 0.467533I
a = −0.617308 − 1.067990I
b = 1.12086 − 1.09303I
−5.06407 − 5.10089I −8.06417 + 3.90479I
u = 1.135850 − 0.467533I
a = −0.617308 + 1.067990I
b = 1.12086 + 1.09303I
−5.06407 + 5.10089I −8.06417 − 3.90479I
u = −1.203160 + 0.305558I
a = −2.23809 + 0.67343I
b = −0.75733 + 2.10032I
4.19530 − 4.64476I −6.50760 + 1.85624I
u = −1.203160 − 0.305558I
a = −2.23809 − 0.67343I
b = −0.75733 − 2.10032I
4.19530 + 4.64476I −6.50760 − 1.85624I
u = −1.132780 + 0.508544I
a = −0.90981 + 1.95883I
b = 0.74904 + 2.04170I
−1.76893 + 6.56054I −3.91539 − 6.62369I
u = −1.132780 − 0.508544I
a = −0.90981 − 1.95883I
b = 0.74904 − 2.04170I
−1.76893 − 6.56054I −3.91539 + 6.62369I
u = 1.192020 + 0.396256I
a = −1.338870 + 0.229283I
b = −1.18542 − 1.02127I
−4.97897 − 0.77808I −5.12685 − 0.91316I
u = 1.192020 − 0.396256I
a = −1.338870 − 0.229283I
b = −1.18542 + 1.02127I
−4.97897 + 0.77808I −5.12685 + 0.91316I
u = 0.680819 + 0.286111I
a = 0.32342 + 1.54495I
b = −0.624981 + 0.021795I
−1.21863 − 1.30834I −6.02556 + 4.39977I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.680819 − 0.286111I
a = 0.32342 − 1.54495I
b = −0.624981 − 0.021795I
−1.21863 + 1.30834I −6.02556 − 4.39977I
u = 1.154970 + 0.552374I
a = 1.32333 − 2.31600I
b = 2.34332 − 0.76541I
6.62233 − 5.39094I −3.64487 + 3.52784I
u = 1.154970 − 0.552374I
a = 1.32333 + 2.31600I
b = 2.34332 + 0.76541I
6.62233 + 5.39094I −3.64487 − 3.52784I
u = −1.186050 + 0.496366I
a = −1.24916 − 1.10553I
b = −1.95101 + 0.16201I
−4.27225 + 7.86289I −4.24616 − 7.85401I
u = −1.186050 − 0.496366I
a = −1.24916 + 1.10553I
b = −1.95101 − 0.16201I
−4.27225 − 7.86289I −4.24616 + 7.85401I
u = −0.229937 + 0.670128I
a = 0.500950 + 0.538105I
b = 0.526961 − 1.227640I
0.81664 − 2.02421I −0.01141 + 3.23458I
u = −0.229937 − 0.670128I
a = 0.500950 − 0.538105I
b = 0.526961 + 1.227640I
0.81664 + 2.02421I −0.01141 − 3.23458I
u = 1.173930 + 0.546675I
a = −0.61187 + 3.18910I
b = −2.91661 + 2.11720I
5.8554 − 13.2425I −4.63609 + 7.63130I
u = 1.173930 − 0.546675I
a = −0.61187 − 3.18910I
b = −2.91661 − 2.11720I
5.8554 + 13.2425I −4.63609 − 7.63130I
u = −0.657519
a = −2.62357
b = −1.65716
−2.37590 1.74590
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.088074 + 0.595763I
a = −0.85711 − 1.31323I
b = −0.101404 + 0.925885I
−2.22400 + 0.97106I −5.17623 + 0.64972I
u = 0.088074 − 0.595763I
a = −0.85711 + 1.31323I
b = −0.101404 − 0.925885I
−2.22400 − 0.97106I −5.17623 − 0.64972I
9
II. I
u
2
= hu
6
− 2u
4
− u
3
+ u
2
+ b + u + 1, u
7
+ u
6
− 2u
5
− 2u
4
+ u
3
+ u
2
+
a + u + 1, u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
−u
7
− u
6
+ 2u
5
+ 2u
4
− u
3
− u
2
− u − 1
−u
6
+ 2u
4
+ u
3
− u
2
− u − 1
a
7
=
1
u
2
a
4
=
−u
7
− u
6
+ 2u
5
+ 2u
4
− u
3
− u
2
− u − 1
−u
6
+ 2u
4
+ u
3
− u
2
− u − 1
a
10
=
u
u
a
12
=
−u
3
−u
3
+ u
a
5
=
u
6
− u
4
+ 1
u
6
− 2u
4
+ u
2
a
2
=
−u
7
− 2u
6
+ 2u
5
+ 3u
4
− u
3
− u
2
− u − 2
−2u
6
+ 4u
4
+ u
3
− 2u
2
− u − 1
a
1
=
−u
6
+ u
4
− 1
−u
6
+ 2u
4
− u
2
a
9
=
u
3
u
5
− u
3
+ u
a
8
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
8
− 6u
7
+ u
6
+ 12u
5
+ 5u
4
− 10u
3
− 7u
2
− 7u − 6
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
9
c
3
, c
7
u
9
c
4
(u + 1)
9
c
5
u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1
c
6
u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1
c
8
, c
12
u
9
− u
8
+ 2u
7
− u
6
+ 3u
5
− u
4
+ 2u
3
+ u + 1
c
9
u
9
− 3u
8
+ 8u
7
− 13u
6
+ 17u
5
− 17u
4
+ 12u
3
− 6u
2
+ u + 1
c
10
u
9
− u
8
− 2u
7
+ 3u
6
+ u
5
− 3u
4
+ 2u
3
− u + 1
c
11
u
9
+ 5u
8
+ 12u
7
+ 15u
6
+ 9u
5
− u
4
− 4u
3
− 2u
2
+ u + 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
9
c
3
, c
7
y
9
c
5
, c
9
y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1
c
6
, c
10
y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1
c
8
, c
12
y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1
c
11
y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y −1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.772920 + 0.510351I
a = −0.147032 − 1.012940I
b = −0.848670 − 0.225310I
0.13850 + 2.09337I −1.56547 − 4.18932I
u = −0.772920 − 0.510351I
a = −0.147032 + 1.012940I
b = −0.848670 + 0.225310I
0.13850 − 2.09337I −1.56547 + 4.18932I
u = 0.825933
a = −1.95176
b = −1.33142
−2.84338 −16.7240
u = 1.173910 + 0.391555I
a = 0.679689 + 0.626017I
b = 0.25695 + 1.39155I
−6.01628 − 1.33617I −11.45029 + 1.01794I
u = 1.173910 − 0.391555I
a = 0.679689 − 0.626017I
b = 0.25695 − 1.39155I
−6.01628 + 1.33617I −11.45029 − 1.01794I
u = −0.141484 + 0.739668I
a = −0.541407 + 0.753907I
b = 0.443165 − 0.284059I
−2.26187 − 2.45442I −5.68179 + 2.62939I
u = −0.141484 − 0.739668I
a = −0.541407 − 0.753907I
b = 0.443165 + 0.284059I
−2.26187 + 2.45442I −5.68179 − 2.62939I
u = −1.172470 + 0.500383I
a = 0.484630 + 0.655708I
b = 1.314260 + 0.168567I
−5.24306 + 7.08493I −8.94033 − 5.11095I
u = −1.172470 − 0.500383I
a = 0.484630 − 0.655708I
b = 1.314260 − 0.168567I
−5.24306 − 7.08493I −8.94033 + 5.11095I
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
42
+ 6u
41
+ ··· + 14u + 1)
c
2
((u − 1)
9
)(u
42
− 10u
41
+ ··· + 10u − 1)
c
3
, c
7
u
9
(u
42
− u
41
+ ··· + 512u + 512)
c
4
((u + 1)
9
)(u
42
− 10u
41
+ ··· + 10u − 1)
c
5
(u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1)
· (u
42
+ 6u
41
+ ··· − 134u − 17)
c
6
(u
9
+ u
8
+ ··· − u − 1)(u
42
+ 2u
41
+ ··· − 2u − 1)
c
8
(u
9
− u
8
+ 2u
7
− u
6
+ 3u
5
− u
4
+ 2u
3
+ u + 1)
· (u
42
+ 2u
41
+ ··· − 2773910u − 699025)
c
9
(u
9
− 3u
8
+ 8u
7
− 13u
6
+ 17u
5
− 17u
4
+ 12u
3
− 6u
2
+ u + 1)
· (u
42
+ 6u
41
+ ··· − 134u − 17)
c
10
(u
9
− u
8
+ ··· − u + 1)(u
42
+ 2u
41
+ ··· − 2u − 1)
c
11
(u
9
+ 5u
8
+ 12u
7
+ 15u
6
+ 9u
5
− u
4
− 4u
3
− 2u
2
+ u + 1)
· (u
42
+ 22u
41
+ ··· + 2u + 1)
c
12
(u
9
− u
8
+ ··· + u + 1)(u
42
− 2u
41
+ ··· + 2u + 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
9
)(y
42
+ 70y
41
+ ··· − 202y + 1)
c
2
, c
4
((y −1)
9
)(y
42
− 6y
41
+ ··· − 14y + 1)
c
3
, c
7
y
9
(y
42
− 57y
41
+ ··· − 4718592y + 262144)
c
5
, c
9
(y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1)
· (y
42
+ 26y
41
+ ··· − 9490y + 289)
c
6
, c
10
(y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1)
· (y
42
− 22y
41
+ ··· − 2y + 1)
c
8
(y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1)
· (y
42
+ 90y
41
+ ··· − 6627902285450y + 488635950625)
c
11
(y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y −1)
· (y
42
− 2y
41
+ ··· − 22y + 1)
c
12
(y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1)
· (y
42
+ 54y
41
+ ··· − 2y + 1)
15
