12a
0440
(K12a
0440
)
A knot diagram
1
Linearized knot diagam
3 6 10 9 2 5 11 12 1 7 8 4
Solving Sequence
7,11
8 12
4,9
5 1 6 10 3 2
c
7
c
11
c
8
c
4
c
12
c
6
c
10
c
3
c
2
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h8.17973 × 10
78
u
78
− 3.32534 × 10
79
u
77
+ ··· + 1.08264 × 10
78
b + 1.17493 × 10
79
,
7.38204 × 10
76
u
78
+ 1.20719 × 10
79
u
77
+ ··· + 2.16527 × 10
78
a − 2.53946 × 10
79
, u
79
− 5u
78
+ ··· + 9u − 1i
I
u
2
= hb, a
3
+ a
2
+ 2a + 1, u − 1i
I
u
3
= hb + u, a + u + 1, u
2
+ u − 1i
* 3 irreducible components of dim
C
= 0, with total 84 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
= h8.18 × 10
78
u
78
− 3.33 × 10
79
u
77
+ · · · + 1.08 × 10
78
b + 1.17 × 10
79
, 7.38 ×
10
76
u
78
+1.21×10
79
u
77
+· · ·+2.17×10
78
a−2.54×10
79
, u
79
−5u
78
+· · ·+9u−1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
u
2
a
12
=
−u
−u
3
+ u
a
4
=
−0.0340929u
78
− 5.57525u
77
+ ··· − 79.3261u + 11.7282
−7.55538u
78
+ 30.7152u
77
+ ··· + 80.5660u − 10.8525
a
9
=
−u
2
+ 1
−u
4
+ 2u
2
a
5
=
8.72790u
78
− 40.8993u
77
+ ··· − 174.062u + 24.6495
−7.44447u
78
+ 31.6128u
77
+ ··· + 76.5696u − 9.71049
a
1
=
−10.5701u
78
+ 37.8812u
77
+ ··· + 73.4125u − 11.9931
−7.43744u
78
+ 25.5627u
77
+ ··· + 41.0173u − 4.39937
a
6
=
4.78292u
78
− 22.0655u
77
+ ··· − 38.3048u + 4.25699
−1.06013u
78
+ 0.567165u
77
+ ··· − 17.7043u + 2.68876
a
10
=
u
u
a
3
=
−0.513025u
78
− 2.90158u
77
+ ··· − 83.6483u + 13.0441
−8.03431u
78
+ 33.3889u
77
+ ··· + 76.2438u − 9.53651
a
2
=
−11.7439u
78
+ 49.5318u
77
+ ··· + 111.423u − 15.7495
−2.05455u
78
+ 9.78315u
77
+ ··· + 32.7790u − 3.84880
(ii) Obstruction class = −1
(iii) Cusp Shapes = 80.7160u
78
− 362.186u
77
+ ··· − 1054.12u + 134.511
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
79
+ 26u
78
+ ··· + 39u + 1
c
2
, c
5
u
79
+ 4u
78
+ ··· − 13u − 1
c
3
u
79
+ u
78
+ ··· − 1072u + 71
c
4
u
79
+ 3u
78
+ ··· + 108u + 43
c
7
, c
8
, c
10
c
11
u
79
+ 5u
78
+ ··· + 9u + 1
c
9
u
79
− 4u
78
+ ··· − 16u + 4
c
12
u
79
+ 7u
78
+ ··· + 20u − 8
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
79
+ 58y
78
+ ··· − 337y −1
c
2
, c
5
y
79
− 26y
78
+ ··· + 39y −1
c
3
y
79
− 75y
78
+ ··· + 614980y −5041
c
4
y
79
− 87y
78
+ ··· + 1126052y −1849
c
7
, c
8
, c
10
c
11
y
79
− 93y
78
+ ··· + 81y −1
c
9
y
79
− 18y
78
+ ··· + 488y −16
c
12
y
79
+ 19y
78
+ ··· + 720y −64
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.866803 + 0.533665I
a = −0.0297098 − 0.0022778I
b = −0.441116 − 0.602987I
−4.07309 − 0.36413I 0
u = 0.866803 − 0.533665I
a = −0.0297098 + 0.0022778I
b = −0.441116 + 0.602987I
−4.07309 + 0.36413I 0
u = −0.788989 + 0.582780I
a = 0.187892 − 0.000787I
b = −0.17742 − 1.57226I
2.77854 + 7.24987I 0
u = −0.788989 − 0.582780I
a = 0.187892 + 0.000787I
b = −0.17742 + 1.57226I
2.77854 − 7.24987I 0
u = −0.826444 + 0.598529I
a = −0.161790 − 0.057817I
b = 0.10767 + 1.55410I
1.83680 + 13.25190I 0
u = −0.826444 − 0.598529I
a = −0.161790 + 0.057817I
b = 0.10767 − 1.55410I
1.83680 − 13.25190I 0
u = −0.853346 + 0.449925I
a = 0.0578767 + 0.0709751I
b = 0.263441 + 1.345450I
−4.45438 + 7.50984I 0
u = −0.853346 − 0.449925I
a = 0.0578767 − 0.0709751I
b = 0.263441 − 1.345450I
−4.45438 − 7.50984I 0
u = 0.636803 + 0.681178I
a = −0.205565 − 0.221493I
b = −0.155312 − 0.659709I
−0.43638 − 4.87894I 0
u = 0.636803 − 0.681178I
a = −0.205565 + 0.221493I
b = −0.155312 + 0.659709I
−0.43638 + 4.87894I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.17006
a = −0.220626
b = 0.350149
−2.29502 0
u = 0.514546 + 0.651061I
a = 0.226947 + 0.377376I
b = 0.077277 + 0.523891I
−0.067693 + 0.265761I 0
u = 0.514546 − 0.651061I
a = 0.226947 − 0.377376I
b = 0.077277 − 0.523891I
−0.067693 − 0.265761I 0
u = −0.114852 + 0.812432I
a = 1.029070 + 0.541721I
b = −0.410388 − 0.393922I
3.99517 − 8.58385I 0
u = −0.114852 − 0.812432I
a = 1.029070 − 0.541721I
b = −0.410388 + 0.393922I
3.99517 + 8.58385I 0
u = −0.718534 + 0.395646I
a = 0.034558 − 0.356407I
b = −0.42088 − 1.46065I
−0.60334 + 4.40814I 0
u = −0.718534 − 0.395646I
a = 0.034558 + 0.356407I
b = −0.42088 + 1.46065I
−0.60334 − 4.40814I 0
u = −0.151437 + 0.772666I
a = −1.080720 − 0.571184I
b = 0.325154 + 0.460637I
4.71159 − 2.74858I 0
u = −0.151437 − 0.772666I
a = −1.080720 + 0.571184I
b = 0.325154 − 0.460637I
4.71159 + 2.74858I 0
u = −0.747171 + 0.215526I
a = 0.399822 + 0.577326I
b = 0.62169 + 1.40665I
−3.74765 + 0.59468I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.747171 − 0.215526I
a = 0.399822 − 0.577326I
b = 0.62169 − 1.40665I
−3.74765 − 0.59468I 0
u = 1.120250 + 0.498783I
a = −0.004779 + 0.192620I
b = −0.730479 − 0.456211I
0.26087 + 4.05570I 0
u = 1.120250 − 0.498783I
a = −0.004779 − 0.192620I
b = −0.730479 + 0.456211I
0.26087 − 4.05570I 0
u = 1.157870 + 0.428882I
a = −0.048722 − 0.212728I
b = 0.715991 + 0.330451I
0.72655 − 1.44588I 0
u = 1.157870 − 0.428882I
a = −0.048722 + 0.212728I
b = 0.715991 − 0.330451I
0.72655 + 1.44588I 0
u = −0.620939 + 0.363470I
a = 1.64470 + 0.21723I
b = 0.740983 − 1.001040I
2.97369 + 6.23589I −6.00000 − 9.23175I
u = −0.620939 − 0.363470I
a = 1.64470 − 0.21723I
b = 0.740983 + 1.001040I
2.97369 − 6.23589I −6.00000 + 9.23175I
u = 0.052448 + 0.681727I
a = 0.825797 + 0.735148I
b = −0.171743 − 0.089114I
−1.71576 − 3.77103I −9.94460 + 7.06655I
u = 0.052448 − 0.681727I
a = 0.825797 − 0.735148I
b = −0.171743 + 0.089114I
−1.71576 + 3.77103I −9.94460 − 7.06655I
u = 0.654189 + 0.184431I
a = −0.538618 − 0.070989I
b = 0.391553 + 0.549679I
−1.260360 − 0.423717I −7.99674 + 0.91931I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.654189 − 0.184431I
a = −0.538618 + 0.070989I
b = 0.391553 − 0.549679I
−1.260360 + 0.423717I −7.99674 − 0.91931I
u = −0.529989 + 0.397570I
a = −1.67202 − 0.34888I
b = −0.596201 + 0.881402I
3.95414 + 0.38627I −1.85947 − 3.38777I
u = −0.529989 − 0.397570I
a = −1.67202 + 0.34888I
b = −0.596201 − 0.881402I
3.95414 − 0.38627I −1.85947 + 3.38777I
u = 0.639852 + 0.144046I
a = 3.08474 + 0.58209I
b = −1.381130 + 0.226724I
1.84996 − 3.15841I 9.47520 − 9.12090I
u = 0.639852 − 0.144046I
a = 3.08474 − 0.58209I
b = −1.381130 − 0.226724I
1.84996 + 3.15841I 9.47520 + 9.12090I
u = 0.654308
a = 3.62045
b = −1.65895
−2.22969 67.2950
u = 0.588102 + 0.171365I
a = −2.84290 − 0.61949I
b = 1.353460 − 0.186397I
1.92105 + 2.39883I 3.66261 − 13.09776I
u = 0.588102 − 0.171365I
a = −2.84290 + 0.61949I
b = 1.353460 + 0.186397I
1.92105 − 2.39883I 3.66261 + 13.09776I
u = −0.320160 + 0.383969I
a = 1.020170 − 0.857951I
b = −0.61660 − 1.40696I
4.53679 + 2.47928I −0.68036 − 6.13798I
u = −0.320160 − 0.383969I
a = 1.020170 + 0.857951I
b = −0.61660 + 1.40696I
4.53679 − 2.47928I −0.68036 + 6.13798I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.490007
a = 2.37907
b = 1.24966
−2.28886 9.23230
u = −0.114853 + 0.474161I
a = −1.27092 − 1.05752I
b = −0.161350 + 0.312486I
1.14890 − 1.37318I 0.36469 + 1.97759I
u = −0.114853 − 0.474161I
a = −1.27092 + 1.05752I
b = −0.161350 − 0.312486I
1.14890 + 1.37318I 0.36469 − 1.97759I
u = 1.53031 + 0.01709I
a = 0.00320 + 2.92027I
b = −0.08616 + 2.87859I
−1.61422 − 3.29569I 0
u = 1.53031 − 0.01709I
a = 0.00320 − 2.92027I
b = −0.08616 − 2.87859I
−1.61422 + 3.29569I 0
u = 1.54794 + 0.07507I
a = −0.887875 − 0.463306I
b = −0.137758 − 0.593721I
−3.01572 − 1.95455I 0
u = 1.54794 − 0.07507I
a = −0.887875 + 0.463306I
b = −0.137758 + 0.593721I
−3.01572 + 1.95455I 0
u = −0.211055 + 0.373484I
a = −1.43515 + 0.89082I
b = 0.69504 + 1.26985I
4.11116 − 3.52834I −1.181145 − 0.111847I
u = −0.211055 − 0.373484I
a = −1.43515 − 0.89082I
b = 0.69504 − 1.26985I
4.11116 + 3.52834I −1.181145 + 0.111847I
u = 1.58169
a = 1.57467
b = 1.01714
−9.58431 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.58216 + 0.03642I
a = 1.85708 + 0.98106I
b = 3.32141 + 0.99145I
−5.54467 − 1.67662I 0
u = −1.58216 − 0.03642I
a = 1.85708 − 0.98106I
b = 3.32141 − 0.99145I
−5.54467 + 1.67662I 0
u = 1.58698 + 0.08661I
a = 0.905854 + 0.707004I
b = 0.165459 + 0.939592I
−4.56560 − 7.81339I 0
u = 1.58698 − 0.08661I
a = 0.905854 − 0.707004I
b = 0.165459 − 0.939592I
−4.56560 + 7.81339I 0
u = −1.58072 + 0.17789I
a = 0.586609 − 1.128980I
b = 0.46188 − 1.34959I
−7.17267 + 2.64825I 0
u = −1.58072 − 0.17789I
a = 0.586609 + 1.128980I
b = 0.46188 + 1.34959I
−7.17267 − 2.64825I 0
u = −1.60470 + 0.04263I
a = −2.06138 − 0.83331I
b = −3.78591 − 0.82320I
−5.95012 + 3.87436I 0
u = −1.60470 − 0.04263I
a = −2.06138 + 0.83331I
b = −3.78591 + 0.82320I
−5.95012 − 3.87436I 0
u = 1.61720 + 0.10797I
a = −0.04874 + 2.47310I
b = −0.48218 + 2.99880I
−8.61560 − 6.27258I 0
u = 1.61720 − 0.10797I
a = −0.04874 − 2.47310I
b = −0.48218 − 2.99880I
−8.61560 + 6.27258I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.62182 + 0.05898I
a = 0.73604 − 1.67622I
b = 1.04916 − 2.39235I
−9.23214 + 1.34124I 0
u = −1.62182 − 0.05898I
a = 0.73604 + 1.67622I
b = 1.04916 + 2.39235I
−9.23214 − 1.34124I 0
u = −1.61031 + 0.20630I
a = −0.489420 + 1.128370I
b = −0.21406 + 1.41657I
−8.02726 + 8.18238I 0
u = −1.61031 − 0.20630I
a = −0.489420 − 1.128370I
b = −0.21406 − 1.41657I
−8.02726 − 8.18238I 0
u = −1.62518
a = −3.46972
b = −5.99252
−10.2714 0
u = 1.62760 + 0.06695I
a = 0.28314 − 2.51515I
b = 0.48132 − 3.01262I
−11.98070 − 1.70600I 0
u = 1.62760 − 0.06695I
a = 0.28314 + 2.51515I
b = 0.48132 + 3.01262I
−11.98070 + 1.70600I 0
u = 1.63827 + 0.17196I
a = 0.15508 + 2.31534I
b = −0.57948 + 3.08235I
−5.46017 − 10.12170I 0
u = 1.63827 − 0.17196I
a = 0.15508 − 2.31534I
b = −0.57948 − 3.08235I
−5.46017 + 10.12170I 0
u = 1.65503 + 0.12700I
a = 0.01629 − 2.29305I
b = 0.47525 − 3.05821I
−13.0774 − 9.7323I 0
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.65503 − 0.12700I
a = 0.01629 + 2.29305I
b = 0.47525 + 3.05821I
−13.0774 + 9.7323I 0
u = 1.65248 + 0.17831I
a = −0.16298 − 2.26633I
b = 0.57652 − 3.12315I
−6.5902 − 16.2413I 0
u = 1.65248 − 0.17831I
a = −0.16298 + 2.26633I
b = 0.57652 + 3.12315I
−6.5902 + 16.2413I 0
u = −1.67198 + 0.13267I
a = −0.42070 + 1.35733I
b = −0.28865 + 2.00232I
−12.88690 + 2.88771I 0
u = −1.67198 − 0.13267I
a = −0.42070 − 1.35733I
b = −0.28865 − 2.00232I
−12.88690 − 2.88771I 0
u = −1.72195 + 0.03637I
a = −0.13162 + 1.56155I
b = −0.10120 + 2.52020I
−10.21030 − 2.29473I 0
u = −1.72195 − 0.03637I
a = −0.13162 − 1.56155I
b = −0.10120 − 2.52020I
−10.21030 + 2.29473I 0
u = 0.159298 + 0.045390I
a = −3.50317 − 1.96300I
b = 0.632024 + 0.233803I
−1.37886 − 0.33747I −8.08157 + 0.11304I
u = 0.159298 − 0.045390I
a = −3.50317 + 1.96300I
b = 0.632024 − 0.233803I
−1.37886 + 0.33747I −8.08157 − 0.11304I
12
II. I
u
2
= hb, a
3
+ a
2
+ 2a + 1, u − 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
1
a
8
=
1
1
a
12
=
−1
0
a
4
=
a
0
a
9
=
0
1
a
5
=
a
a
a
1
=
−1
0
a
6
=
−a
2
+ 1
−a
2
a
10
=
1
1
a
3
=
2a
a
a
2
=
−2a
2
− 1
−a
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −7a
2
− 5a − 17
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
3
− u
2
+ 2u − 1
c
2
u
3
+ u
2
− 1
c
3
, c
4
, c
6
u
3
+ u
2
+ 2u + 1
c
5
u
3
− u
2
+ 1
c
7
, c
8
, c
9
(u − 1)
3
c
10
, c
11
(u + 1)
3
c
12
u
3
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
6
y
3
+ 3y
2
+ 2y −1
c
2
, c
5
y
3
− y
2
+ 2y −1
c
7
, c
8
, c
9
c
10
, c
11
(y −1)
3
c
12
y
3
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −0.215080 + 1.307140I
b = 0
1.37919 + 2.82812I −4.28809 − 2.59975I
u = 1.00000
a = −0.215080 − 1.307140I
b = 0
1.37919 − 2.82812I −4.28809 + 2.59975I
u = 1.00000
a = −0.569840
b = 0
−2.75839 −16.4240
16
III. I
u
3
= hb + u, a + u + 1, u
2
+ u − 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u + 1
a
12
=
−u
−u + 1
a
4
=
−u − 1
−u
a
9
=
u
u
a
5
=
−2
−1
a
1
=
1
1
a
6
=
−1
−1
a
10
=
u
u
a
3
=
−2
−1
a
2
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −17
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
12
(u − 1)
2
c
3
, c
4
, c
10
c
11
u
2
− u − 1
c
5
, c
6
(u + 1)
2
c
7
, c
8
u
2
+ u − 1
c
9
u
2
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
12
(y −1)
2
c
3
, c
4
, c
7
c
8
, c
10
, c
11
y
2
− 3y + 1
c
9
y
2
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −1.61803
b = −0.618034
−2.63189 −17.0000
u = −1.61803
a = 0.618034
b = 1.61803
−10.5276 −17.0000
20
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
2
)(u
3
− u
2
+ 2u − 1)(u
79
+ 26u
78
+ ··· + 39u + 1)
c
2
((u − 1)
2
)(u
3
+ u
2
− 1)(u
79
+ 4u
78
+ ··· − 13u − 1)
c
3
(u
2
− u − 1)(u
3
+ u
2
+ 2u + 1)(u
79
+ u
78
+ ··· − 1072u + 71)
c
4
(u
2
− u − 1)(u
3
+ u
2
+ 2u + 1)(u
79
+ 3u
78
+ ··· + 108u + 43)
c
5
((u + 1)
2
)(u
3
− u
2
+ 1)(u
79
+ 4u
78
+ ··· − 13u − 1)
c
6
((u + 1)
2
)(u
3
+ u
2
+ 2u + 1)(u
79
+ 26u
78
+ ··· + 39u + 1)
c
7
, c
8
((u − 1)
3
)(u
2
+ u − 1)(u
79
+ 5u
78
+ ··· + 9u + 1)
c
9
u
2
(u − 1)
3
(u
79
− 4u
78
+ ··· − 16u + 4)
c
10
, c
11
((u + 1)
3
)(u
2
− u − 1)(u
79
+ 5u
78
+ ··· + 9u + 1)
c
12
u
3
(u − 1)
2
(u
79
+ 7u
78
+ ··· + 20u − 8)
21
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
((y −1)
2
)(y
3
+ 3y
2
+ 2y −1)(y
79
+ 58y
78
+ ··· − 337y −1)
c
2
, c
5
((y −1)
2
)(y
3
− y
2
+ 2y −1)(y
79
− 26y
78
+ ··· + 39y −1)
c
3
(y
2
− 3y + 1)(y
3
+ 3y
2
+ 2y −1)(y
79
− 75y
78
+ ··· + 614980y −5041)
c
4
(y
2
− 3y + 1)(y
3
+ 3y
2
+ 2y −1)(y
79
− 87y
78
+ ··· + 1126052y −1849)
c
7
, c
8
, c
10
c
11
((y −1)
3
)(y
2
− 3y + 1)(y
79
− 93y
78
+ ··· + 81y −1)
c
9
y
2
(y −1)
3
(y
79
− 18y
78
+ ··· + 488y −16)
c
12
y
3
(y −1)
2
(y
79
+ 19y
78
+ ··· + 720y −64)
22
