12a
1147
(K12a
1147
)
A knot diagram
1
Linearized knot diagam
4 8 9 12 1 10 11 3 2 7 6 5
Solving Sequence
4,12
5 1 2
6,9
10 3 8 11 7
c
4
c
12
c
1
c
5
c
9
c
3
c
8
c
11
c
7
c
2
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
22
− u
21
+ ··· + 2b + 1, u
22
+ u
21
+ ··· + 2a − 1, u
24
+ u
23
+ ··· + u
2
+ 1i
I
u
2
= h663778u
39
+ 3468564u
38
+ ··· + 4497023b + 12669528,
1460252u
39
+ 3892416u
38
+ ··· + 4497023a + 44643349, u
40
+ u
39
+ ··· + 10u − 1i
I
u
3
= hb, a + 1, u + 1i
I
u
4
= hb + a − 1, a
2
− 2a − 1, u − 1i
* 4 irreducible components of dim
C
= 0, with total 67 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
22
−u
21
+· · ·+2b +1, u
22
+u
21
+· · ·+2a −1, u
24
+u
23
+· · ·+u
2
+1i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
u
a
5
=
1
u
2
a
1
=
−u
−u
3
+ u
a
2
=
u
3
− 2u
−u
3
+ u
a
6
=
−u
2
+ 1
−u
4
+ 2u
2
a
9
=
−
1
2
u
22
−
1
2
u
21
+ ··· − 2u +
1
2
1
2
u
22
+
1
2
u
21
+ ··· − u −
1
2
a
10
=
u
3
− 2u
1
2
u
22
+
1
2
u
21
+ ··· − u −
1
2
a
3
=
−
1
2
u
22
−
1
2
u
21
+ ··· − u +
1
2
−
1
2
u
23
+
11
2
u
21
+ ··· +
3
2
u +
1
2
a
8
=
u
6
− 3u
4
+ 2u
2
+ 1
−
1
2
u
23
−
1
2
u
22
+ ··· + 4u
2
+
1
2
u
a
11
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
7
=
−u
4
+ u
2
+ 1
−
1
2
u
23
−
1
2
u
22
+ ··· + 3u
2
+
1
2
u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −2u
23
−u
22
+23u
21
+8u
20
−113u
19
−20u
18
+300u
17
−9u
16
−430u
15
+142u
14
+219u
13
−
271u
12
+253u
11
+142u
10
−424u
9
+149u
8
+116u
7
−175u
6
+131u
5
−14u
4
−65u
3
+44u
2
−8u−1
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
11
u
24
− 3u
23
+ ··· + 16u − 16
c
2
, c
3
, c
8
u
24
− 3u
23
+ ··· + 2u − 2
c
4
, c
5
, c
6
c
7
, c
10
, c
12
u
24
+ u
23
+ ··· + u
2
+ 1
c
9
u
24
+ 9u
23
+ ··· − 38u − 46
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
11
y
24
+ 17y
23
+ ··· − 1280y + 256
c
2
, c
3
, c
8
y
24
− 23y
23
+ ··· + 28y + 4
c
4
, c
5
, c
6
c
7
, c
10
, c
12
y
24
− 23y
23
+ ··· + 2y + 1
c
9
y
24
− 11y
23
+ ··· + 7020y + 2116
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.064601 + 0.857743I
a = 3.00242 + 0.79694I
b = −1.47055 + 0.22609I
11.05360 − 5.17653I 3.95325 + 3.49150I
u = 0.064601 − 0.857743I
a = 3.00242 − 0.79694I
b = −1.47055 − 0.22609I
11.05360 + 5.17653I 3.95325 − 3.49150I
u = −0.034624 + 0.810902I
a = −1.048330 − 0.309670I
b = 0.448897 + 0.626898I
4.86252 + 2.05888I 0.88457 − 3.56826I
u = −0.034624 − 0.810902I
a = −1.048330 + 0.309670I
b = 0.448897 − 0.626898I
4.86252 − 2.05888I 0.88457 + 3.56826I
u = 1.24908
a = −1.63637
b = 1.51781
−0.270498 −8.82770
u = 1.259130 + 0.350678I
a = 1.67906 + 0.25248I
b = −1.50623 − 0.17059I
3.67444 − 3.51286I −3.38395 + 3.71816I
u = 1.259130 − 0.350678I
a = 1.67906 − 0.25248I
b = −1.50623 + 0.17059I
3.67444 + 3.51286I −3.38395 − 3.71816I
u = −1.315770 + 0.340499I
a = −0.177134 + 0.134393I
b = 0.594644 − 0.588222I
−3.18817 + 6.18598I −7.02896 − 2.89473I
u = −1.315770 − 0.340499I
a = −0.177134 − 0.134393I
b = 0.594644 + 0.588222I
−3.18817 − 6.18598I −7.02896 + 2.89473I
u = −1.384040 + 0.044740I
a = 0.091228 + 0.281923I
b = −0.942704 + 0.288972I
−8.06493 + 0.19408I −10.28074 + 0.61912I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.384040 − 0.044740I
a = 0.091228 − 0.281923I
b = −0.942704 − 0.288972I
−8.06493 − 0.19408I −10.28074 − 0.61912I
u = 1.346140 + 0.369990I
a = −0.959184 − 0.578715I
b = 0.398944 − 0.727533I
−3.88120 − 10.62900I −8.08796 + 8.14735I
u = 1.346140 − 0.369990I
a = −0.959184 + 0.578715I
b = 0.398944 + 0.727533I
−3.88120 + 10.62900I −8.08796 − 8.14735I
u = 1.400780 + 0.126559I
a = 0.372750 + 0.796952I
b = −0.137322 + 0.736533I
−10.61170 − 4.02452I −13.52498 + 4.12532I
u = 1.400780 − 0.126559I
a = 0.372750 − 0.796952I
b = −0.137322 − 0.736533I
−10.61170 + 4.02452I −13.52498 − 4.12532I
u = −1.352420 + 0.399114I
a = 1.62910 − 1.77474I
b = −1.46757 − 0.27259I
2.1308 + 14.2744I −4.23619 − 8.14865I
u = −1.352420 − 0.399114I
a = 1.62910 + 1.77474I
b = −1.46757 + 0.27259I
2.1308 − 14.2744I −4.23619 + 8.14865I
u = −1.40349 + 0.18980I
a = −0.55401 + 1.37979I
b = 1.301280 + 0.292342I
−6.12604 + 7.74233I −8.51765 − 6.24570I
u = −1.40349 − 0.18980I
a = −0.55401 − 1.37979I
b = 1.301280 − 0.292342I
−6.12604 − 7.74233I −8.51765 + 6.24570I
u = 0.271980 + 0.498934I
a = −2.01833 − 1.68837I
b = 1.348350 − 0.118156I
4.60442 − 2.61585I 2.73829 + 6.23776I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.271980 − 0.498934I
a = −2.01833 + 1.68837I
b = 1.348350 + 0.118156I
4.60442 + 2.61585I 2.73829 − 6.23776I
u = 0.454758
a = 0.0591891
b = −1.34147
3.38497 −0.780610
u = −0.204200 + 0.280203I
a = 0.771021 − 0.484359I
b = −0.155898 − 0.381963I
−0.123351 + 0.761222I −3.71153 − 9.11663I
u = −0.204200 − 0.280203I
a = 0.771021 + 0.484359I
b = −0.155898 + 0.381963I
−0.123351 − 0.761222I −3.71153 + 9.11663I
7
II. I
u
2
= h6.64 × 10
5
u
39
+ 3.47 × 10
6
u
38
+ · · · + 4.50 × 10
6
b + 1.27 × 10
7
, 1.46 ×
10
6
u
39
+3.89×10
6
u
38
+· · · +4.50×10
6
a+4.46×10
7
, u
40
+u
39
+· · · +10u −1i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
u
a
5
=
1
u
2
a
1
=
−u
−u
3
+ u
a
2
=
u
3
− 2u
−u
3
+ u
a
6
=
−u
2
+ 1
−u
4
+ 2u
2
a
9
=
−0.324715u
39
− 0.865554u
38
+ ··· + 25.4189u − 9.92731
−0.147604u
39
− 0.771302u
38
+ ··· + 11.3275u − 2.81731
a
10
=
−0.502758u
39
− 1.22710u
38
+ ··· + 23.4208u − 10.1653
−0.0239736u
39
− 0.611436u
38
+ ··· + 13.0357u − 2.77196
a
3
=
2.56209u
39
+ 1.79705u
38
+ ··· − 39.2267u + 15.9467
1.45613u
39
+ 0.698967u
38
+ ··· − 2.49928u + 2.49602
a
8
=
1.46016u
39
+ 3.14955u
38
+ ··· − 50.0991u + 14.1572
0.192619u
39
+ 0.247031u
38
+ ··· − 7.76383u + 2.21612
a
11
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
7
=
1.65385u
39
+ 3.16578u
38
+ ··· − 41.7713u + 13.1425
−0.193691u
39
− 0.0162321u
38
+ ··· − 8.32777u + 2.01469
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
8450436
4497023
u
39
+
11603704
4497023
u
38
+ ··· −
184433680
4497023
u +
28189534
4497023
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
11
(u
20
− 3u
19
+ ··· − 12u + 1)
2
c
2
, c
3
, c
8
(u
20
+ u
19
+ ··· − 2u − 1)
2
c
4
, c
5
, c
6
c
7
, c
10
, c
12
u
40
+ u
39
+ ··· + 10u − 1
c
9
(u
20
− 3u
19
+ ··· + 2u + 5)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
11
(y
20
+ 17y
19
+ ··· − 62y + 1)
2
c
2
, c
3
, c
8
(y
20
− 19y
19
+ ··· − 2y + 1)
2
c
4
, c
5
, c
6
c
7
, c
10
, c
12
y
40
− 29y
39
+ ··· − 60y + 1
c
9
(y
20
− 7y
19
+ ··· − 274y + 25)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.923862
a = 1.32744
b = −1.38920
3.24334 1.89980
u = 0.111900 + 0.892848I
a = −2.88979 − 0.73183I
b = 1.46202 − 0.24989I
6.73027 − 9.64430I −0.34532 + 6.20543I
u = 0.111900 − 0.892848I
a = −2.88979 + 0.73183I
b = 1.46202 + 0.24989I
6.73027 + 9.64430I −0.34532 − 6.20543I
u = 0.759025 + 0.475822I
a = −1.48000 − 0.53937I
b = 1.218960 + 0.103071I
−1.34713 + 0.58469I −6.79795 − 0.00910I
u = 0.759025 − 0.475822I
a = −1.48000 + 0.53937I
b = 1.218960 − 0.103071I
−1.34713 − 0.58469I −6.79795 + 0.00910I
u = −1.13904
a = −0.0164448
b = 0.432245
−2.31303 −1.06120
u = −0.118681 + 0.840736I
a = 0.962099 + 0.331136I
b = −0.403387 − 0.672553I
0.72067 + 6.27316I −3.89985 − 6.54347I
u = −0.118681 − 0.840736I
a = 0.962099 − 0.331136I
b = −0.403387 + 0.672553I
0.72067 − 6.27316I −3.89985 + 6.54347I
u = −1.128490 + 0.400676I
a = −0.025745 + 0.174387I
b = 0.380611 − 0.584774I
−2.37392 − 1.80448I −7.17537 + 3.70058I
u = −1.128490 − 0.400676I
a = −0.025745 − 0.174387I
b = 0.380611 + 0.584774I
−2.37392 + 1.80448I −7.17537 − 3.70058I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.014873 + 0.802003I
a = −3.15229 − 0.90911I
b = 1.47490 − 0.19643I
7.52808 − 0.63661I 0.960350 − 0.169887I
u = 0.014873 − 0.802003I
a = −3.15229 + 0.90911I
b = 1.47490 + 0.19643I
7.52808 + 0.63661I 0.960350 + 0.169887I
u = 0.067576 + 0.777860I
a = 1.154830 + 0.297440I
b = −0.506351 − 0.571230I
1.14846 − 2.14390I −2.54408 + 0.24308I
u = 0.067576 − 0.777860I
a = 1.154830 − 0.297440I
b = −0.506351 + 0.571230I
1.14846 + 2.14390I −2.54408 − 0.24308I
u = 0.405495 + 0.666361I
a = 2.09904 + 0.99095I
b = −1.287780 + 0.198735I
−0.30488 − 4.84109I −4.36837 + 6.37981I
u = 0.405495 − 0.666361I
a = 2.09904 − 0.99095I
b = −1.287780 − 0.198735I
−0.30488 + 4.84109I −4.36837 − 6.37981I
u = 1.168250 + 0.467812I
a = 1.64680 + 0.34227I
b = −1.44525 − 0.22406I
3.49387 + 4.79919I −3.30190 − 3.09464I
u = 1.168250 − 0.467812I
a = 1.64680 − 0.34227I
b = −1.44525 + 0.22406I
3.49387 − 4.79919I −3.30190 + 3.09464I
u = 1.221280 + 0.315797I
a = −1.117640 − 0.858852I
b = 0.380611 − 0.584774I
−2.37392 − 1.80448I −7.17537 + 3.70058I
u = 1.221280 − 0.315797I
a = −1.117640 + 0.858852I
b = 0.380611 + 0.584774I
−2.37392 + 1.80448I −7.17537 − 3.70058I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.506013 + 0.529581I
a = −0.492988 − 0.064584I
b = 0.084750 + 0.594489I
−4.54605 + 1.94645I −10.94680 − 4.81876I
u = −0.506013 − 0.529581I
a = −0.492988 + 0.064584I
b = 0.084750 − 0.594489I
−4.54605 − 1.94645I −10.94680 + 4.81876I
u = 1.210060 + 0.408349I
a = −1.65887 − 0.29795I
b = 1.47490 + 0.19643I
7.52808 + 0.63661I − 60.960350 + 0.10I
u = 1.210060 − 0.408349I
a = −1.65887 + 0.29795I
b = 1.47490 − 0.19643I
7.52808 − 0.63661I − 60.960350 + 0.10I
u = 1.280890 + 0.069261I
a = −0.308042 − 1.258930I
b = 0.084750 − 0.594489I
−4.54605 − 1.94645I −10.94680 + 4.81876I
u = 1.280890 − 0.069261I
a = −0.308042 + 1.258930I
b = 0.084750 + 0.594489I
−4.54605 + 1.94645I −10.94680 − 4.81876I
u = −1.238550 + 0.356207I
a = 0.111095 − 0.148235I
b = −0.506351 + 0.571230I
1.14846 + 2.14390I −2.54408 + 0.I
u = −1.238550 − 0.356207I
a = 0.111095 + 0.148235I
b = −0.506351 − 0.571230I
1.14846 − 2.14390I −2.54408 + 0.I
u = −1.288630 + 0.060039I
a = 1.00326 + 1.02639I
b = 1.218960 + 0.103071I
−1.34713 + 0.58469I −6.79795 + 0.I
u = −1.288630 − 0.060039I
a = 1.00326 − 1.02639I
b = 1.218960 − 0.103071I
−1.34713 − 0.58469I −6.79795 + 0.I
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.317540 + 0.151393I
a = −0.06745 − 1.74399I
b = −1.287780 − 0.198735I
−0.30488 + 4.84109I −4.00000 − 6.37981I
u = −1.317540 − 0.151393I
a = −0.06745 + 1.74399I
b = −1.287780 + 0.198735I
−0.30488 − 4.84109I −4.00000 + 6.37981I
u = −1.281360 + 0.353972I
a = 1.52306 − 2.23919I
b = −1.44525 − 0.22406I
3.49387 + 4.79919I −4.00000 − 3.09464I
u = −1.281360 − 0.353972I
a = 1.52306 + 2.23919I
b = −1.44525 + 0.22406I
3.49387 − 4.79919I −4.00000 + 3.09464I
u = 1.293540 + 0.359734I
a = 1.030770 + 0.670930I
b = −0.403387 + 0.672553I
0.72067 − 6.27316I −4.00000 + 6.54347I
u = 1.293540 − 0.359734I
a = 1.030770 − 0.670930I
b = −0.403387 − 0.672553I
0.72067 + 6.27316I −4.00000 − 6.54347I
u = −1.317700 + 0.387037I
a = −1.62992 + 1.96256I
b = 1.46202 + 0.24989I
6.73027 + 9.64430I 0. − 6.20543I
u = −1.317700 − 0.387037I
a = −1.62992 − 1.96256I
b = 1.46202 − 0.24989I
6.73027 − 9.64430I 0. + 6.20543I
u = 0.405383
a = −1.98705
b = 0.432245
−2.31303 −1.06120
u = 0.137952
a = −6.74035
b = −1.38920
3.24334 1.89980
14
III. I
u
3
= hb, a + 1, u + 1i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
−1
a
5
=
1
1
a
1
=
1
0
a
2
=
1
0
a
6
=
0
1
a
9
=
−1
0
a
10
=
−1
0
a
3
=
1
0
a
8
=
−1
0
a
11
=
0
−1
a
7
=
−1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
8
, c
9
, c
11
u
c
4
, c
5
, c
10
u + 1
c
6
, c
7
, c
12
u − 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
8
, c
9
, c
11
y
c
4
, c
5
, c
6
c
7
, c
10
, c
12
y −1
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000
b = 0
−3.28987 −12.0000
18
IV. I
u
4
= hb + a − 1, a
2
− 2a − 1, u − 1i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
1
a
5
=
1
1
a
1
=
−1
0
a
2
=
−1
0
a
6
=
0
1
a
9
=
a
−a + 1
a
10
=
1
−a + 1
a
3
=
−a
2
a
8
=
−1
a − 1
a
11
=
0
1
a
7
=
−1
a
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
11
u
2
c
2
, c
3
, c
8
c
9
u
2
− 2
c
4
, c
5
, c
10
(u − 1)
2
c
6
, c
7
, c
12
(u + 1)
2
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
11
y
2
c
2
, c
3
, c
8
c
9
(y −2)
2
c
4
, c
5
, c
6
c
7
, c
10
, c
12
(y −1)
2
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −0.414214
b = 1.41421
1.64493 −4.00000
u = 1.00000
a = 2.41421
b = −1.41421
1.64493 −4.00000
22
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
11
u
3
(u
20
− 3u
19
+ ··· − 12u + 1)
2
(u
24
− 3u
23
+ ··· + 16u − 16)
c
2
, c
3
, c
8
u(u
2
− 2)(u
20
+ u
19
+ ··· − 2u − 1)
2
(u
24
− 3u
23
+ ··· + 2u − 2)
c
4
, c
5
, c
10
((u − 1)
2
)(u + 1)(u
24
+ u
23
+ ··· + u
2
+ 1)(u
40
+ u
39
+ ··· + 10u − 1)
c
6
, c
7
, c
12
(u − 1)(u + 1)
2
(u
24
+ u
23
+ ··· + u
2
+ 1)(u
40
+ u
39
+ ··· + 10u − 1)
c
9
u(u
2
− 2)(u
20
− 3u
19
+ ··· + 2u + 5)
2
(u
24
+ 9u
23
+ ··· − 38u − 46)
23
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
11
y
3
(y
20
+ 17y
19
+ ··· − 62y + 1)
2
(y
24
+ 17y
23
+ ··· − 1280y + 256)
c
2
, c
3
, c
8
y(y −2)
2
(y
20
− 19y
19
+ ··· − 2y + 1)
2
(y
24
− 23y
23
+ ··· + 28y + 4)
c
4
, c
5
, c
6
c
7
, c
10
, c
12
((y −1)
3
)(y
24
− 23y
23
+ ··· + 2y + 1)(y
40
− 29y
39
+ ··· − 60y + 1)
c
9
y(y −2)
2
(y
20
− 7y
19
+ ··· − 274y + 25)
2
· (y
24
− 11y
23
+ ··· + 7020y + 2116)
24
