12n
0370
(K12n
0370
)
A knot diagram
1
Linearized knot diagam
3 6 9 7 2 11 3 1 4 12 7 9
Solving Sequence
3,7 8,11
12 6 2 1 9 5 4 10
c
7
c
11
c
6
c
2
c
1
c
8
c
5
c
4
c
9
c
3
, c
10
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h1.28847 × 10
15
u
11
− 4.74420 × 10
14
u
10
+ ··· + 8.72123 × 10
16
b + 2.22344 × 10
17
,
1.00938 × 10
15
u
11
− 1.97502 × 10
14
u
10
+ ··· + 8.72123 × 10
16
a + 3.30952 × 10
17
,
u
12
− 34u
10
− 10u
9
+ 374u
8
+ 256u
7
− 1158u
6
− 838u
5
+ 1243u
4
+ 1159u
3
+ 685u
2
+ 342u + 61i
I
u
2
= h−491u
9
− 169u
8
+ 138u
7
+ 2797u
6
− 952u
5
− 2978u
4
− 1762u
3
+ 4940u
2
+ 1423b + 1946u − 997,
120u
9
+ 363u
8
− 709u
7
− 501u
6
− 1399u
5
+ 5504u
4
− 1259u
3
− 2839u
2
+ 1423a − 3481u + 4617,
u
10
− u
9
− 5u
7
+ 9u
6
− 3u
4
− 9u
3
+ 9u
2
− u − 1i
* 2 irreducible components of dim
C
= 0, with total 22 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
=
h1.29×10
15
u
11
−4.74×10
14
u
10
+· · ·+8.72×10
16
b+2.22×10
17
, 1.01×10
15
u
11
−
1.98 × 10
14
u
10
+ · · · + 8.72 × 10
16
a + 3.31 × 10
17
, u
12
− 34u
10
+ · · · + 342u + 61i
(i) Arc colorings
a
3
=
0
u
a
7
=
1
0
a
8
=
1
−u
2
a
11
=
−0.0115739u
11
+ 0.00226461u
10
+ ··· − 3.98267u − 3.79478
−0.0147739u
11
+ 0.00543983u
10
+ ··· − 6.01362u − 2.54946
a
12
=
0.00320005u
11
− 0.00317522u
10
+ ··· + 2.03095u − 1.24533
−0.0147739u
11
+ 0.00543983u
10
+ ··· − 6.01362u − 2.54946
a
6
=
−0.0414649u
11
+ 0.0165212u
10
+ ··· − 17.1890u − 6.03303
−0.00596827u
11
+ 0.00228246u
10
+ ··· − 2.07003u − 1.25196
a
2
=
0.0163711u
11
− 0.00810347u
10
+ ··· + 7.17353u + 1.18110
−0.00876668u
11
+ 0.00418598u
10
+ ··· − 3.18088u − 1.53147
a
1
=
0.0163711u
11
− 0.00810347u
10
+ ··· + 7.17353u + 1.18110
−0.0125049u
11
+ 0.00522621u
10
+ ··· − 4.95363u − 2.02578
a
9
=
0.00757115u
11
− 0.00279404u
10
+ ··· + 2.94107u + 2.37499
0.00556736u
11
− 0.00252772u
10
+ ··· + 2.05708u + 0.942635
a
5
=
−0.00964076u
11
+ 0.00285039u
10
+ ··· − 2.23827u − 1.62408
−0.00279404u
11
+ 0.000870939u
10
+ ··· − 0.214346u − 0.461840
a
4
=
−0.00684672u
11
+ 0.00197945u
10
+ ··· − 2.02393u − 1.16224
−0.00279404u
11
+ 0.000870939u
10
+ ··· − 0.214346u − 0.461840
a
10
=
0.00416895u
11
− 0.00269474u
10
+ ··· + 0.812978u + 1.59345
0.00358791u
11
− 0.00224683u
10
+ ··· + 0.877740u + 0.524985
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
8580809856421526
87212327051456087
u
11
+
3421020397474812
87212327051456087
u
10
+···−
4710390701175382035
87212327051456087
u−
26157625863371374
1429710279532067
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
12
+ 10u
10
+ ··· + 3u + 1
c
2
, c
5
u
12
+ 4u
11
+ ··· − u − 1
c
3
, c
4
, c
9
u
12
+ u
11
+ ··· + 4u + 1
c
6
, c
11
u
12
− 8u
11
+ ··· + 10u − 4
c
7
u
12
− 34u
10
+ ··· + 342u + 61
c
8
, c
12
u
12
− u
11
+ ··· + 13u − 1
c
10
u
12
+ 4u
11
+ ··· + 44u + 16
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
12
+ 20y
11
+ ··· − 3y + 1
c
2
, c
5
y
12
+ 10y
10
+ ··· − 3y + 1
c
3
, c
4
, c
9
y
12
− 29y
11
+ ··· − 12y + 1
c
6
, c
11
y
12
− 4y
11
+ ··· − 44y + 16
c
7
y
12
− 68y
11
+ ··· − 33394y + 3721
c
8
, c
12
y
12
+ 33y
11
+ ··· − 269y + 1
c
10
y
12
+ 48y
11
+ ··· + 3216y + 256
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.015982 + 0.502004I
a = −1.33976 + 0.88379I
b = −0.373800 + 0.452174I
−0.313401 + 1.169960I −3.81973 − 5.53143I
u = −0.015982 − 0.502004I
a = −1.33976 − 0.88379I
b = −0.373800 − 0.452174I
−0.313401 − 1.169960I −3.81973 + 5.53143I
u = −0.487209
a = −1.96376
b = 0.690624
−2.57792 8.93780
u = 1.53808 + 0.50690I
a = 0.700750 + 0.138104I
b = 0.742800 − 0.761818I
3.43038 + 0.92181I 0.166703 − 0.576827I
u = 1.53808 − 0.50690I
a = 0.700750 − 0.138104I
b = 0.742800 + 0.761818I
3.43038 − 0.92181I 0.166703 + 0.576827I
u = −1.64293 + 0.28456I
a = 1.107820 − 0.267147I
b = 0.969113 + 0.706030I
2.73449 − 4.65154I −0.40649 + 6.35112I
u = −1.64293 − 0.28456I
a = 1.107820 + 0.267147I
b = 0.969113 − 0.706030I
2.73449 + 4.65154I −0.40649 − 6.35112I
u = −0.299850
a = −2.90979
b = −0.917503
−1.69975 −3.47970
u = −3.60163 + 0.13896I
a = 0.400586 + 0.118619I
b = 1.42147 − 1.15303I
−12.68910 + 1.00394I −1.189954 + 0.108309I
u = −3.60163 − 0.13896I
a = 0.400586 − 0.118619I
b = 1.42147 + 1.15303I
−12.68910 − 1.00394I −1.189954 − 0.108309I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 4.11599 + 0.72988I
a = 0.591968 + 0.319795I
b = 1.35386 − 1.23724I
−12.4077 + 8.7999I −0.97957 − 3.65752I
u = 4.11599 − 0.72988I
a = 0.591968 − 0.319795I
b = 1.35386 + 1.23724I
−12.4077 − 8.7999I −0.97957 + 3.65752I
6
II. I
u
2
= h−491u
9
− 169u
8
+ · · · + 1423b − 997, 120u
9
+ 363u
8
+ · · · +
1423a + 4617, u
10
− u
9
− 5u
7
+ 9u
6
− 3u
4
− 9u
3
+ 9u
2
− u − 1i
(i) Arc colorings
a
3
=
0
u
a
7
=
1
0
a
8
=
1
−u
2
a
11
=
−0.0843289u
9
− 0.255095u
8
+ ··· + 2.44624u − 3.24455
0.345046u
9
+ 0.118763u
8
+ ··· − 1.36753u + 0.700632
a
12
=
−0.429375u
9
− 0.373858u
8
+ ··· + 3.81377u − 3.94519
0.345046u
9
+ 0.118763u
8
+ ··· − 1.36753u + 0.700632
a
6
=
−0.454673u
9
+ 0.849613u
8
+ ··· − 7.55235u + 1.78145
1.04287u
9
− 0.545327u
8
+ ··· + 3.31483u + 0.824315
a
2
=
0.613493u
9
− 0.919185u
8
+ ··· + 7.12860u − 2.12087
−0.781448u
9
+ 0.236121u
8
+ ··· − 2.19817u − 0.666198
a
1
=
0.613493u
9
− 0.919185u
8
+ ··· + 7.12860u − 2.12087
−0.676037u
9
+ 0.304989u
8
+ ··· − 2.50597u − 0.360506
a
9
=
−0.345046u
9
− 0.118763u
8
+ ··· + 1.36753u − 0.700632
−0.436402u
9
+ 0.354884u
8
+ ··· − 3.56571u − 0.965566
a
5
=
−0.0337316u
9
+ 0.297962u
8
+ ··· − 0.821504u − 0.697822
0.463809u
9
− 0.0969782u
8
+ ··· + 1.04568u + 0.345046
a
4
=
−0.497540u
9
+ 0.394940u
8
+ ··· − 1.86718u − 1.04287
0.463809u
9
− 0.0969782u
8
+ ··· + 1.04568u + 0.345046
a
10
=
−0.376669u
9
− 0.339424u
8
+ ··· + 2.65987u − 0.292340
−0.539002u
9
+ 0.594519u
8
+ ··· − 5.10611u − 1.46311
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
2033
1423
u
9
−
3491
1423
u
8
+
1495
1423
u
7
−
10729
1423
u
6
+
25736
1423
u
5
−
14237
1423
u
4
−
2961
1423
u
3
−
18582
1423
u
2
+
34363
1423
u−
16730
1423
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
10
− u
9
+ 6u
8
− 6u
7
+ 9u
6
− 12u
5
− 3u
4
+ 2u
3
+ 6u
2
− 4u + 1
c
2
u
10
+ 3u
9
+ 4u
8
− 5u
6
− 6u
5
− u
4
+ 2u
3
+ 2u
2
− 1
c
3
u
10
− 5u
8
+ 9u
6
+ 2u
5
− 6u
4
− 6u
3
+ 5u + 1
c
4
, c
9
u
10
− 5u
8
+ 9u
6
− 2u
5
− 6u
4
+ 6u
3
− 5u + 1
c
5
u
10
− 3u
9
+ 4u
8
− 5u
6
+ 6u
5
− u
4
− 2u
3
+ 2u
2
− 1
c
6
u
10
− 2u
8
+ 4u
6
− 4u
4
+ u
3
+ 2u
2
− 1
c
7
u
10
− u
9
− 5u
7
+ 9u
6
− 3u
4
− 9u
3
+ 9u
2
− u − 1
c
8
u
10
+ 6u
8
− u
7
+ 15u
6
− 5u
5
+ 18u
4
− 9u
3
+ 9u
2
− 6u + 1
c
10
u
10
− 4u
9
+ ··· − 4u + 1
c
11
u
10
− 2u
8
+ 4u
6
− 4u
4
− u
3
+ 2u
2
− 1
c
12
u
10
+ 6u
8
+ u
7
+ 15u
6
+ 5u
5
+ 18u
4
+ 9u
3
+ 9u
2
+ 6u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
10
+ 11y
9
+ ··· − 4y + 1
c
2
, c
5
y
10
− y
9
+ 6y
8
− 6y
7
+ 9y
6
− 12y
5
− 3y
4
+ 2y
3
+ 6y
2
− 4y + 1
c
3
, c
4
, c
9
y
10
− 10y
9
+ ··· − 25y + 1
c
6
, c
11
y
10
− 4y
9
+ ··· − 4y + 1
c
7
y
10
− y
9
+ ··· − 19y + 1
c
8
, c
12
y
10
+ 12y
9
+ ··· − 18y + 1
c
10
y
10
+ 8y
9
+ ··· + 8y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.811923 + 0.722020I
a = 1.45325 + 0.35022I
b = 0.975490 + 0.644063I
5.24934 − 6.06210I −0.58831 + 6.06500I
u = −0.811923 − 0.722020I
a = 1.45325 − 0.35022I
b = 0.975490 − 0.644063I
5.24934 + 6.06210I −0.58831 − 6.06500I
u = 1.115350 + 0.663499I
a = 0.272302 − 0.257296I
b = 0.724375 − 0.642107I
6.04451 − 1.01363I 0.978414 + 0.079961I
u = 1.115350 − 0.663499I
a = 0.272302 + 0.257296I
b = 0.724375 + 0.642107I
6.04451 + 1.01363I 0.978414 − 0.079961I
u = 1.31175
a = −0.322011
b = −1.13039
1.74489 −3.09540
u = 0.602612 + 0.281923I
a = −1.12138 + 1.00067I
b = −0.995438 − 0.830468I
2.28696 − 3.31057I −1.50855 + 1.46154I
u = 0.602612 − 0.281923I
a = −1.12138 − 1.00067I
b = −0.995438 + 0.830468I
2.28696 + 3.31057I −1.50855 − 1.46154I
u = −0.257810
a = −3.77592
b = 0.809153
−2.87740 −18.8820
u = −0.93301 + 1.57780I
a = −1.055210 + 0.217921I
b = −0.543811 − 0.460848I
5.07972 − 2.66860I 3.60710 + 4.02718I
u = −0.93301 − 1.57780I
a = −1.055210 − 0.217921I
b = −0.543811 + 0.460848I
5.07972 + 2.66860I 3.60710 − 4.02718I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
10
− u
9
+ 6u
8
− 6u
7
+ 9u
6
− 12u
5
− 3u
4
+ 2u
3
+ 6u
2
− 4u + 1)
· (u
12
+ 10u
10
+ ··· + 3u + 1)
c
2
(u
10
+ 3u
9
+ 4u
8
− 5u
6
− 6u
5
− u
4
+ 2u
3
+ 2u
2
− 1)
· (u
12
+ 4u
11
+ ··· − u − 1)
c
3
(u
10
− 5u
8
+ ··· + 5u + 1)(u
12
+ u
11
+ ··· + 4u + 1)
c
4
, c
9
(u
10
− 5u
8
+ ··· − 5u + 1)(u
12
+ u
11
+ ··· + 4u + 1)
c
5
(u
10
− 3u
9
+ 4u
8
− 5u
6
+ 6u
5
− u
4
− 2u
3
+ 2u
2
− 1)
· (u
12
+ 4u
11
+ ··· − u − 1)
c
6
(u
10
− 2u
8
+ ··· + 2u
2
− 1)(u
12
− 8u
11
+ ··· + 10u − 4)
c
7
(u
10
− u
9
− 5u
7
+ 9u
6
− 3u
4
− 9u
3
+ 9u
2
− u − 1)
· (u
12
− 34u
10
+ ··· + 342u + 61)
c
8
(u
10
+ 6u
8
− u
7
+ 15u
6
− 5u
5
+ 18u
4
− 9u
3
+ 9u
2
− 6u + 1)
· (u
12
− u
11
+ ··· + 13u − 1)
c
10
(u
10
− 4u
9
+ ··· − 4u + 1)(u
12
+ 4u
11
+ ··· + 44u + 16)
c
11
(u
10
− 2u
8
+ ··· + 2u
2
− 1)(u
12
− 8u
11
+ ··· + 10u − 4)
c
12
(u
10
+ 6u
8
+ u
7
+ 15u
6
+ 5u
5
+ 18u
4
+ 9u
3
+ 9u
2
+ 6u + 1)
· (u
12
− u
11
+ ··· + 13u − 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
10
+ 11y
9
+ ··· − 4y + 1)(y
12
+ 20y
11
+ ··· − 3y + 1)
c
2
, c
5
(y
10
− y
9
+ 6y
8
− 6y
7
+ 9y
6
− 12y
5
− 3y
4
+ 2y
3
+ 6y
2
− 4y + 1)
· (y
12
+ 10y
10
+ ··· − 3y + 1)
c
3
, c
4
, c
9
(y
10
− 10y
9
+ ··· − 25y + 1)(y
12
− 29y
11
+ ··· − 12y + 1)
c
6
, c
11
(y
10
− 4y
9
+ ··· − 4y + 1)(y
12
− 4y
11
+ ··· − 44y + 16)
c
7
(y
10
− y
9
+ ··· − 19y + 1)(y
12
− 68y
11
+ ··· − 33394y + 3721)
c
8
, c
12
(y
10
+ 12y
9
+ ··· − 18y + 1)(y
12
+ 33y
11
+ ··· − 269y + 1)
c
10
(y
10
+ 8y
9
+ ··· + 8y + 1)(y
12
+ 48y
11
+ ··· + 3216y + 256)
12
