12a
0037
(K12a
0037
)
A knot diagram
1
Linearized knot diagam
3 5 6 10 2 12 1 11 4 9 8 7
Solving Sequence
1,8
7 12
4,6
3 2 5 11 9 10
c
7
c
12
c
6
c
3
c
1
c
5
c
11
c
8
c
10
c
2
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h3u
67
+ 5u
66
+ ··· + b − 2, 11u
67
+ 20u
66
+ ··· + 2a − 9, u
68
+ 3u
67
+ ··· − u − 1i
I
u
2
= hb, a
2
+ a + 1, u − 1i
* 2 irreducible components of dim
C
= 0, with total 70 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
= h3u
67
+5u
66
+· · ·+b−2, 11u
67
+20u
66
+· · ·+2a−9, u
68
+3u
67
+· · ·−u−1i
(i) Arc colorings
a
1
=
0
u
a
8
=
1
0
a
7
=
1
−u
2
a
12
=
u
−u
3
+ u
a
4
=
−
11
2
u
67
− 10u
66
+ ··· − 2u +
9
2
−3u
67
− 5u
66
+ ··· − 11u
2
+ 2
a
6
=
−u
2
+ 1
u
4
− 2u
2
a
3
=
−
7
2
u
67
− 6u
66
+ ··· − 3u +
5
2
1
2
u
67
+ u
66
+ ··· − u −
1
2
a
2
=
−
1
2
u
67
− u
66
+ ··· + 8u −
1
2
1
2
u
67
+ u
66
+ ··· + 2u −
1
2
a
5
=
−
5
2
u
67
− 4u
66
+ ··· − u +
5
2
6u
67
+ 11u
66
+ ··· − u − 5
a
11
=
−u
3
+ 2u
−u
3
+ u
a
9
=
u
6
− 3u
4
+ 2u
2
+ 1
u
6
− 2u
4
+ u
2
a
10
=
−u
9
+ 4u
7
− 5u
5
+ 3u
−u
9
+ 3u
7
− 3u
5
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 14u
67
+ 19u
66
+ ··· − 9u − 17
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
68
+ 30u
67
+ ··· − 2u + 1
c
2
, c
5
u
68
+ 2u
67
+ ··· + 6u + 1
c
3
u
68
− 2u
67
+ ··· − 36u + 9
c
4
, c
9
u
68
+ u
67
+ ··· − 8u − 4
c
6
, c
7
, c
12
u
68
− 3u
67
+ ··· + u − 1
c
8
, c
10
, c
11
u
68
+ 15u
67
+ ··· + 152u + 16
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
68
+ 18y
67
+ ··· − 62y + 1
c
2
, c
5
y
68
+ 30y
67
+ ··· − 2y + 1
c
3
y
68
+ 6y
67
+ ··· + 4014y + 81
c
4
, c
9
y
68
− 15y
67
+ ··· − 152y + 16
c
6
, c
7
, c
12
y
68
− 53y
67
+ ··· − 13y + 1
c
8
, c
10
, c
11
y
68
+ 73y
67
+ ··· − 2592y + 256
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.895522 + 0.351354I
a = 0.535572 − 0.466640I
b = 0.419390 − 0.166116I
−2.40466 + 3.24426I −12.79487 + 0.I
u = 0.895522 − 0.351354I
a = 0.535572 + 0.466640I
b = 0.419390 + 0.166116I
−2.40466 − 3.24426I −12.79487 + 0.I
u = 1.030920 + 0.245142I
a = −0.144942 + 0.300433I
b = −0.536018 + 0.056317I
−0.929775 − 0.694625I 0
u = 1.030920 − 0.245142I
a = −0.144942 − 0.300433I
b = −0.536018 − 0.056317I
−0.929775 + 0.694625I 0
u = 0.057827 + 0.908622I
a = −1.03946 − 2.62331I
b = −0.97261 − 2.56092I
7.88380 − 10.29650I −4.69943 + 7.27910I
u = 0.057827 − 0.908622I
a = −1.03946 + 2.62331I
b = −0.97261 + 2.56092I
7.88380 + 10.29650I −4.69943 − 7.27910I
u = 0.042720 + 0.903115I
a = 0.65870 + 2.72612I
b = 0.70317 + 2.64498I
9.72190 − 4.90561I −2.02313 + 2.74922I
u = 0.042720 − 0.903115I
a = 0.65870 − 2.72612I
b = 0.70317 − 2.64498I
9.72190 + 4.90561I −2.02313 − 2.74922I
u = 0.003428 + 0.888270I
a = −0.34321 + 2.84573I
b = −0.00404 + 2.75361I
9.88596 − 1.57474I −1.66767 + 2.29520I
u = 0.003428 − 0.888270I
a = −0.34321 − 2.84573I
b = −0.00404 − 2.75361I
9.88596 + 1.57474I −1.66767 − 2.29520I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.013729 + 0.881119I
a = 0.78129 − 2.82252I
b = 0.31193 − 2.74554I
8.18285 + 3.81988I −4.00036 − 2.43227I
u = −0.013729 − 0.881119I
a = 0.78129 + 2.82252I
b = 0.31193 + 2.74554I
8.18285 − 3.81988I −4.00036 + 2.43227I
u = 0.040273 + 0.862504I
a = −0.13886 − 1.98365I
b = −0.31232 − 2.13385I
4.10183 − 3.09420I −7.72899 + 2.66205I
u = 0.040273 − 0.862504I
a = −0.13886 + 1.98365I
b = −0.31232 + 2.13385I
4.10183 + 3.09420I −7.72899 − 2.66205I
u = −1.186300 + 0.127926I
a = −0.90918 − 1.24102I
b = 0.436322 − 0.125183I
−2.19445 − 0.93220I 0
u = −1.186300 − 0.127926I
a = −0.90918 + 1.24102I
b = 0.436322 + 0.125183I
−2.19445 + 0.93220I 0
u = 1.187500 + 0.181105I
a = −0.228112 + 0.140426I
b = −1.199570 + 0.450826I
−1.34633 − 1.22738I 0
u = 1.187500 − 0.181105I
a = −0.228112 − 0.140426I
b = −1.199570 − 0.450826I
−1.34633 + 1.22738I 0
u = −1.213540 + 0.182772I
a = 0.391833 + 1.256010I
b = −0.311997 + 0.100144I
−1.54755 + 4.00408I 0
u = −1.213540 − 0.182772I
a = 0.391833 − 1.256010I
b = −0.311997 − 0.100144I
−1.54755 − 4.00408I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.247790 + 0.069716I
a = 0.262311 − 0.505966I
b = 0.81938 − 1.63973I
−4.41264 + 1.01414I 0
u = 1.247790 − 0.069716I
a = 0.262311 + 0.505966I
b = 0.81938 + 1.63973I
−4.41264 − 1.01414I 0
u = 0.626055 + 0.376261I
a = 0.779550 + 0.334297I
b = 0.620215 − 0.009222I
−3.00525 − 2.85440I −15.4568 + 5.4193I
u = 0.626055 − 0.376261I
a = 0.779550 − 0.334297I
b = 0.620215 + 0.009222I
−3.00525 + 2.85440I −15.4568 − 5.4193I
u = 1.260630 + 0.174920I
a = 0.502698 − 0.114267I
b = 1.88463 − 0.84447I
−3.25504 − 5.41820I 0
u = 1.260630 − 0.174920I
a = 0.502698 + 0.114267I
b = 1.88463 + 0.84447I
−3.25504 + 5.41820I 0
u = 0.260329 + 0.674892I
a = 0.059835 − 1.200510I
b = 0.036602 − 0.171621I
−0.54570 − 7.12975I −8.28357 + 9.67470I
u = 0.260329 − 0.674892I
a = 0.059835 + 1.200510I
b = 0.036602 + 0.171621I
−0.54570 + 7.12975I −8.28357 − 9.67470I
u = 1.239520 + 0.401715I
a = 1.098620 − 0.433220I
b = −0.64232 − 1.99545I
0.39705 − 1.43860I 0
u = 1.239520 − 0.401715I
a = 1.098620 + 0.433220I
b = −0.64232 + 1.99545I
0.39705 + 1.43860I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.232810 + 0.456922I
a = 1.83683 − 0.03858I
b = 0.38520 − 2.25306I
4.26036 + 5.42868I 0
u = 1.232810 − 0.456922I
a = 1.83683 + 0.03858I
b = 0.38520 + 2.25306I
4.26036 − 5.42868I 0
u = −1.32040
a = −0.354230
b = 0.972963
−6.07475 0
u = 1.245710 + 0.445929I
a = −1.76468 + 0.31816I
b = −0.09620 + 2.45581I
6.00508 + 0.09093I 0
u = 1.245710 − 0.445929I
a = −1.76468 − 0.31816I
b = −0.09620 − 2.45581I
6.00508 − 0.09093I 0
u = −1.313060 + 0.227651I
a = 0.198042 + 0.731027I
b = 0.288228 + 0.423614I
−3.29836 + 5.64762I 0
u = −1.313060 − 0.227651I
a = 0.198042 − 0.731027I
b = 0.288228 − 0.423614I
−3.29836 − 5.64762I 0
u = −1.266290 + 0.417661I
a = −1.99000 − 0.31891I
b = 0.23911 − 2.26817I
4.29886 + 0.82820I 0
u = −1.266290 − 0.417661I
a = −1.99000 + 0.31891I
b = 0.23911 + 2.26817I
4.29886 − 0.82820I 0
u = 1.276200 + 0.421764I
a = −1.50037 + 0.93745I
b = 0.67185 + 2.85731I
5.93444 − 3.11164I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.276200 − 0.421764I
a = −1.50037 − 0.93745I
b = 0.67185 − 2.85731I
5.93444 + 3.11164I 0
u = 0.194231 + 0.624177I
a = 0.200700 + 0.974073I
b = 0.166804 − 0.050748I
1.39398 − 2.63459I −3.83138 + 5.30336I
u = 0.194231 − 0.624177I
a = 0.200700 − 0.974073I
b = 0.166804 + 0.050748I
1.39398 + 2.63459I −3.83138 − 5.30336I
u = −1.281690 + 0.420235I
a = 1.86247 + 0.63495I
b = −0.54582 + 2.40594I
5.89281 + 6.25632I 0
u = −1.281690 − 0.420235I
a = 1.86247 − 0.63495I
b = −0.54582 − 2.40594I
5.89281 − 6.25632I 0
u = −1.339060 + 0.171690I
a = −0.296922 − 0.787136I
b = 0.164287 − 0.826587I
−7.45261 + 2.99913I 0
u = −1.339060 − 0.171690I
a = −0.296922 + 0.787136I
b = 0.164287 + 0.826587I
−7.45261 − 2.99913I 0
u = 1.288500 + 0.412732I
a = 1.35888 − 1.17047I
b = −1.01417 − 2.98905I
4.13145 − 8.45277I 0
u = 1.288500 − 0.412732I
a = 1.35888 + 1.17047I
b = −1.01417 + 2.98905I
4.13145 + 8.45277I 0
u = −1.305290 + 0.396523I
a = −1.154370 − 0.714946I
b = 1.13010 − 1.94509I
−0.09743 + 7.61245I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.305290 − 0.396523I
a = −1.154370 + 0.714946I
b = 1.13010 + 1.94509I
−0.09743 − 7.61245I 0
u = −1.364380 + 0.036473I
a = 0.061668 + 0.259510I
b = −1.252540 + 0.333167I
−9.10989 + 3.71218I 0
u = −1.364380 − 0.036473I
a = 0.061668 − 0.259510I
b = −1.252540 − 0.333167I
−9.10989 − 3.71218I 0
u = −1.347280 + 0.239850I
a = −0.322191 − 0.625658I
b = −0.620089 − 0.700964I
−5.60593 + 10.34200I 0
u = −1.347280 − 0.239850I
a = −0.322191 + 0.625658I
b = −0.620089 + 0.700964I
−5.60593 − 10.34200I 0
u = 0.352806 + 0.523351I
a = −0.429659 − 1.178270I
b = −0.416900 − 0.230397I
−2.23370 − 0.62660I −12.61670 + 3.83528I
u = 0.352806 − 0.523351I
a = −0.429659 + 1.178270I
b = −0.416900 + 0.230397I
−2.23370 + 0.62660I −12.61670 − 3.83528I
u = −1.313460 + 0.421883I
a = 1.43757 + 1.23124I
b = −1.29435 + 2.56292I
5.48960 + 9.64527I 0
u = −1.313460 − 0.421883I
a = 1.43757 − 1.23124I
b = −1.29435 − 2.56292I
5.48960 − 9.64527I 0
u = −1.324500 + 0.421925I
a = −1.24973 − 1.42112I
b = 1.58602 − 2.58289I
3.5636 + 15.0559I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.324500 − 0.421925I
a = −1.24973 + 1.42112I
b = 1.58602 + 2.58289I
3.5636 − 15.0559I 0
u = 0.013858 + 0.538053I
a = 0.93729 + 1.22313I
b = 0.676344 − 0.109112I
2.08272 − 1.40752I −0.96646 + 3.79117I
u = 0.013858 − 0.538053I
a = 0.93729 − 1.22313I
b = 0.676344 + 0.109112I
2.08272 + 1.40752I −0.96646 − 3.79117I
u = −0.099088 + 0.477326I
a = −1.27945 − 1.92708I
b = −0.871503 − 0.181323I
0.87080 + 3.06361I −3.02049 − 3.15038I
u = −0.099088 − 0.477326I
a = −1.27945 + 1.92708I
b = −0.871503 + 0.181323I
0.87080 − 3.06361I −3.02049 + 3.15038I
u = 0.440851
a = 0.0522081
b = −0.459473
−0.831706 −11.9310
u = −0.189182 + 0.115386I
a = −0.02171 − 3.59531I
b = −0.205879 − 0.531302I
−0.30583 − 1.79467I −2.23312 + 3.53008I
u = −0.189182 − 0.115386I
a = −0.02171 + 3.59531I
b = −0.205879 + 0.531302I
−0.30583 + 1.79467I −2.23312 − 3.53008I
11
II. I
u
2
= hb, a
2
+ a + 1, u − 1i
(i) Arc colorings
a
1
=
0
1
a
8
=
1
0
a
7
=
1
−1
a
12
=
1
0
a
4
=
a
0
a
6
=
0
−1
a
3
=
a
−a
a
2
=
a + 1
−a
a
5
=
a
0
a
11
=
1
0
a
9
=
1
0
a
10
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4a − 11
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
u
2
− u + 1
c
2
u
2
+ u + 1
c
4
, c
8
, c
9
c
10
, c
11
u
2
c
6
, c
7
(u − 1)
2
c
12
(u + 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
y
2
+ y + 1
c
4
, c
8
, c
9
c
10
, c
11
y
2
c
6
, c
7
, c
12
(y −1)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −0.500000 + 0.866025I
b = 0
−1.64493 + 2.02988I −9.00000 − 3.46410I
u = 1.00000
a = −0.500000 − 0.866025I
b = 0
−1.64493 − 2.02988I −9.00000 + 3.46410I
15
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)(u
68
+ 30u
67
+ ··· − 2u + 1)
c
2
(u
2
+ u + 1)(u
68
+ 2u
67
+ ··· + 6u + 1)
c
3
(u
2
− u + 1)(u
68
− 2u
67
+ ··· − 36u + 9)
c
4
, c
9
u
2
(u
68
+ u
67
+ ··· − 8u − 4)
c
5
(u
2
− u + 1)(u
68
+ 2u
67
+ ··· + 6u + 1)
c
6
, c
7
((u − 1)
2
)(u
68
− 3u
67
+ ··· + u − 1)
c
8
, c
10
, c
11
u
2
(u
68
+ 15u
67
+ ··· + 152u + 16)
c
12
((u + 1)
2
)(u
68
− 3u
67
+ ··· + u − 1)
16
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
2
+ y + 1)(y
68
+ 18y
67
+ ··· − 62y + 1)
c
2
, c
5
(y
2
+ y + 1)(y
68
+ 30y
67
+ ··· − 2y + 1)
c
3
(y
2
+ y + 1)(y
68
+ 6y
67
+ ··· + 4014y + 81)
c
4
, c
9
y
2
(y
68
− 15y
67
+ ··· − 152y + 16)
c
6
, c
7
, c
12
((y −1)
2
)(y
68
− 53y
67
+ ··· − 13y + 1)
c
8
, c
10
, c
11
y
2
(y
68
+ 73y
67
+ ··· − 2592y + 256)
17
