12n
0170
(K12n
0170
)
A knot diagram
1
Linearized knot diagam
3 5 9 2 9 11 12 3 1 7 6 10
Solving Sequence
7,10
11 6 12 8 1
3,9
4 5 2
c
10
c
6
c
11
c
7
c
12
c
9
c
3
c
5
c
2
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−2u
46
+ 4u
45
+ ··· + b + 2, 2u
45
− 2u
44
+ ··· + a − 1, u
47
− 2u
46
+ ··· − 12u
2
+ 1i
I
u
2
= hb − u, a − u − 1, u
3
+ 2u − 1i
I
u
3
= hu
3
+ b + 2u + 1, a + 1, u
4
+ u
3
+ 2u
2
+ 2u + 1i
* 3 irreducible components of dim
C
= 0, with total 54 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−2u
46
+4u
45
+· · ·+b+2, 2u
45
−2u
44
+· · ·+a−1, u
47
−2u
46
+· · ·−12u
2
+1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
6
=
−u
u
3
+ u
a
12
=
u
2
+ 1
−u
4
− 2u
2
a
8
=
u
5
+ 2u
3
+ u
−u
7
− 3u
5
− 2u
3
+ u
a
1
=
−u
4
− u
2
+ 1
−u
4
− 2u
2
a
3
=
−2u
45
+ 2u
44
+ ··· − 5u + 1
2u
46
− 4u
45
+ ··· − 3u − 2
a
9
=
u
8
+ 3u
6
+ u
4
− 2u
2
+ 1
u
8
+ 4u
6
+ 4u
4
a
4
=
−4u
45
+ 4u
44
+ ··· + 16u
2
− 5u
4u
46
− 8u
45
+ ··· − 4u − 4
a
5
=
−u
19
− 8u
17
− 24u
15
− 30u
13
− 7u
11
+ 10u
9
− 4u
7
− 6u
5
+ 3u
3
− 2u
−u
19
− 9u
17
− 32u
15
− 55u
13
− 43u
11
− 9u
9
− 4u
5
+ u
3
+ u
a
2
=
−u
45
+ u
44
+ ··· − 4u + 2
u
46
− 2u
45
+ ··· − 2u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
46
+ 8u
45
+ ··· + 38u + 5
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
47
+ 14u
46
+ ··· − 5u + 1
c
2
, c
4
u
47
− 8u
46
+ ··· − 5u + 1
c
3
, c
8
u
47
+ u
46
+ ··· + 192u + 128
c
5
u
47
− 2u
46
+ ··· + 2u + 1
c
6
, c
10
, c
11
u
47
− 2u
46
+ ··· − 12u
2
+ 1
c
7
u
47
+ 2u
46
+ ··· + 96u + 72
c
9
, c
12
u
47
+ 8u
46
+ ··· + 112u − 49
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
47
+ 46y
46
+ ··· + 83y −1
c
2
, c
4
y
47
− 14y
46
+ ··· − 5y −1
c
3
, c
8
y
47
+ 45y
46
+ ··· − 167936y −16384
c
5
y
47
− 52y
46
+ ··· + 24y −1
c
6
, c
10
, c
11
y
47
+ 44y
46
+ ··· + 24y −1
c
7
y
47
+ 12y
46
+ ··· + 50832y −5184
c
9
, c
12
y
47
+ 32y
46
+ ··· + 170128y −2401
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.240850 + 1.172380I
a = 0.76218 + 1.23522I
b = 0.934577 + 0.402639I
4.61957 − 0.02295I 0
u = −0.240850 − 1.172380I
a = 0.76218 − 1.23522I
b = 0.934577 − 0.402639I
4.61957 + 0.02295I 0
u = 0.708507 + 0.363703I
a = −0.94806 + 1.22755I
b = 2.37910 + 1.16766I
3.42438 + 9.90306I 2.22004 − 7.67510I
u = 0.708507 − 0.363703I
a = −0.94806 − 1.22755I
b = 2.37910 − 1.16766I
3.42438 − 9.90306I 2.22004 + 7.67510I
u = −0.635055 + 0.455943I
a = 0.114980 + 0.251136I
b = 0.0894725 + 0.0904282I
−4.12081 − 2.09104I 4.57556 + 3.64684I
u = −0.635055 − 0.455943I
a = 0.114980 − 0.251136I
b = 0.0894725 − 0.0904282I
−4.12081 + 2.09104I 4.57556 − 3.64684I
u = 0.517341 + 0.581926I
a = −1.35583 − 1.46783I
b = −1.70532 + 1.15194I
2.58809 − 5.73384I 0.56569 + 2.04831I
u = 0.517341 − 0.581926I
a = −1.35583 + 1.46783I
b = −1.70532 − 1.15194I
2.58809 + 5.73384I 0.56569 − 2.04831I
u = 0.109843 + 1.219880I
a = 0.767710 + 0.740963I
b = 0.864402 + 0.041075I
−2.27528 + 2.11283I 0
u = 0.109843 − 1.219880I
a = 0.767710 − 0.740963I
b = 0.864402 − 0.041075I
−2.27528 − 2.11283I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.698777 + 0.324988I
a = 1.001160 − 0.887981I
b = −2.00444 − 1.18278I
4.64870 + 3.20376I 4.16524 − 3.31906I
u = 0.698777 − 0.324988I
a = 1.001160 + 0.887981I
b = −2.00444 + 1.18278I
4.64870 − 3.20376I 4.16524 + 3.31906I
u = −0.256980 + 1.225330I
a = −0.62010 − 1.29363I
b = −0.881559 − 0.519345I
4.25025 − 6.95397I 0
u = −0.256980 − 1.225330I
a = −0.62010 + 1.29363I
b = −0.881559 + 0.519345I
4.25025 + 6.95397I 0
u = 0.435156 + 0.599476I
a = 1.31533 + 1.27222I
b = 1.34439 − 0.94929I
3.57234 + 0.73807I 1.82651 − 2.67731I
u = 0.435156 − 0.599476I
a = 1.31533 − 1.27222I
b = 1.34439 + 0.94929I
3.57234 − 0.73807I 1.82651 + 2.67731I
u = −0.037205 + 1.270570I
a = −1.61889 − 0.61214I
b = −1.283830 + 0.388483I
−4.85814 − 0.97601I 0
u = −0.037205 − 1.270570I
a = −1.61889 + 0.61214I
b = −1.283830 − 0.388483I
−4.85814 + 0.97601I 0
u = −0.623399 + 0.342153I
a = −0.426734 + 0.510909I
b = −0.111541 + 0.262625I
−1.47489 − 3.82342I 0.98515 + 6.99857I
u = −0.623399 − 0.342153I
a = −0.426734 − 0.510909I
b = −0.111541 − 0.262625I
−1.47489 + 3.82342I 0.98515 − 6.99857I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.701377 + 0.026834I
a = −0.073148 + 1.342220I
b = −0.017726 + 0.554191I
8.08746 − 3.45650I 7.23123 + 2.83028I
u = −0.701377 − 0.026834I
a = −0.073148 − 1.342220I
b = −0.017726 − 0.554191I
8.08746 + 3.45650I 7.23123 − 2.83028I
u = 0.568091 + 0.371758I
a = −2.10914 − 0.10547I
b = 0.57523 + 2.44925I
−3.30787 + 1.75612I 1.06262 − 3.54613I
u = 0.568091 − 0.371758I
a = −2.10914 + 0.10547I
b = 0.57523 − 2.44925I
−3.30787 − 1.75612I 1.06262 + 3.54613I
u = 0.155614 + 1.341930I
a = −0.415404 + 0.937335I
b = 0.224512 + 0.844236I
−3.43598 + 2.59417I 0
u = 0.155614 − 1.341930I
a = −0.415404 − 0.937335I
b = 0.224512 − 0.844236I
−3.43598 − 2.59417I 0
u = −0.480171 + 0.404521I
a = 0.848787 − 0.023160I
b = 0.311223 − 0.154426I
−1.96248 + 0.31457I −1.52612 + 0.64426I
u = −0.480171 − 0.404521I
a = 0.848787 + 0.023160I
b = 0.311223 + 0.154426I
−1.96248 − 0.31457I −1.52612 − 0.64426I
u = −0.19689 + 1.43311I
a = −0.446867 + 0.559009I
b = −0.063667 + 0.550788I
−7.79399 − 2.25837I 0
u = −0.19689 − 1.43311I
a = −0.446867 − 0.559009I
b = −0.063667 − 0.550788I
−7.79399 + 2.25837I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.12786 + 1.44594I
a = −1.76062 + 0.79907I
b = −0.81689 + 1.44748I
−2.88567 + 2.62103I 0
u = 0.12786 − 1.44594I
a = −1.76062 − 0.79907I
b = −0.81689 − 1.44748I
−2.88567 − 2.62103I 0
u = −0.23877 + 1.43350I
a = 0.045007 − 0.588992I
b = −0.205747 − 0.403699I
−7.17228 − 6.98944I 0
u = −0.23877 − 1.43350I
a = 0.045007 + 0.588992I
b = −0.205747 + 0.403699I
−7.17228 + 6.98944I 0
u = 0.21933 + 1.43707I
a = 1.26805 − 3.51350I
b = −0.98988 − 3.16929I
−9.10435 + 4.67464I 0
u = 0.21933 − 1.43707I
a = 1.26805 + 3.51350I
b = −0.98988 + 3.16929I
−9.10435 − 4.67464I 0
u = 0.26857 + 1.43228I
a = 0.89236 + 2.94088I
b = 2.26858 + 1.61505I
−0.98075 + 6.72660I 0
u = 0.26857 − 1.43228I
a = 0.89236 − 2.94088I
b = 2.26858 − 1.61505I
−0.98075 − 6.72660I 0
u = 0.518516 + 0.082438I
a = 0.604605 + 0.167780I
b = −0.557380 − 0.379635I
1.061820 + 0.206928I 9.18626 − 0.93345I
u = 0.518516 − 0.082438I
a = 0.604605 − 0.167780I
b = −0.557380 + 0.379635I
1.061820 − 0.206928I 9.18626 + 0.93345I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.26958 + 1.45051I
a = −1.41479 − 3.13096I
b = −2.74488 − 1.48581I
−2.40424 + 13.46930I 0
u = 0.26958 − 1.45051I
a = −1.41479 + 3.13096I
b = −2.74488 + 1.48581I
−2.40424 − 13.46930I 0
u = 0.15911 + 1.47665I
a = 2.44634 − 0.79574I
b = 1.32702 − 1.81501I
−4.02537 − 3.37401I 0
u = 0.15911 − 1.47665I
a = 2.44634 + 0.79574I
b = 1.32702 + 1.81501I
−4.02537 + 3.37401I 0
u = −0.22773 + 1.47482I
a = −0.232276 − 0.098321I
b = −0.194212 + 0.027487I
−10.35500 − 5.24252I 0
u = −0.22773 − 1.47482I
a = −0.232276 + 0.098321I
b = −0.194212 − 0.027487I
−10.35500 + 5.24252I 0
u = −0.235762
a = 2.71069
b = 0.517152
−1.27831 −10.8230
9
II. I
u
2
= hb − u, a − u − 1, u
3
+ 2u − 1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
6
=
−u
−u + 1
a
12
=
u
2
+ 1
−u
a
8
=
u
2
+ u
u
2
a
1
=
u
2
− u + 1
−u
a
3
=
u + 1
u
a
9
=
u
2
+ u
u
2
a
4
=
u + 1
u
a
5
=
−u
2
+ u − 1
u
a
2
=
u
2
+ 2
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 7u
2
+ 5u + 6
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
8
u
3
c
4
(u + 1)
3
c
5
, c
6
, c
9
u
3
+ 2u + 1
c
7
u
3
+ 3u
2
+ 5u + 2
c
10
, c
11
, c
12
u
3
+ 2u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
3
c
3
, c
8
y
3
c
5
, c
6
, c
9
c
10
, c
11
, c
12
y
3
+ 4y
2
+ 4y −1
c
7
y
3
+ y
2
+ 13y −4
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.22670 + 1.46771I
a = 0.77330 + 1.46771I
b = −0.22670 + 1.46771I
−11.08570 − 5.13794I −9.85299 + 2.68036I
u = −0.22670 − 1.46771I
a = 0.77330 − 1.46771I
b = −0.22670 − 1.46771I
−11.08570 + 5.13794I −9.85299 − 2.68036I
u = 0.453398
a = 1.45340
b = 0.453398
−0.857735 9.70600
13
III. I
u
3
= hu
3
+ b + 2u + 1, a + 1, u
4
+ u
3
+ 2u
2
+ 2u + 1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
6
=
−u
u
3
+ u
a
12
=
u
2
+ 1
u
3
+ 2u + 1
a
8
=
u
3
+ 2u + 1
−u
3
− u
2
− u − 2
a
1
=
u
3
+ u
2
+ 2u + 2
u
3
+ 2u + 1
a
3
=
−1
−u
3
− 2u − 1
a
9
=
u
3
+ 2u + 1
−u
3
− u
2
− u − 2
a
4
=
−1
−u
3
− 2u − 1
a
5
=
−u
3
− u
2
− 2u − 2
−u
3
− 2u − 1
a
2
=
u
3
+ u
2
+ 2u + 1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
3
− 2u
2
+ 2u − 5
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
4
c
3
, c
8
u
4
c
4
(u + 1)
4
c
5
, c
6
, c
9
u
4
− u
3
+ 2u
2
− 2u + 1
c
7
(u
2
− u + 1)
2
c
10
, c
11
, c
12
u
4
+ u
3
+ 2u
2
+ 2u + 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
8
y
4
c
5
, c
6
, c
9
c
10
, c
11
, c
12
y
4
+ 3y
3
+ 2y
2
+ 1
c
7
(y
2
+ y + 1)
2
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.621744 + 0.440597I
a = −1.00000
b = 0.121744 − 1.306620I
−4.93480 − 2.02988I −6.26314 + 3.25323I
u = −0.621744 − 0.440597I
a = −1.00000
b = 0.121744 + 1.306620I
−4.93480 + 2.02988I −6.26314 − 3.25323I
u = 0.121744 + 1.306620I
a = −1.00000
b = −0.621744 − 0.440597I
−4.93480 + 2.02988I −3.23686 − 4.54099I
u = 0.121744 − 1.306620I
a = −1.00000
b = −0.621744 + 0.440597I
−4.93480 − 2.02988I −3.23686 + 4.54099I
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
7
)(u
47
+ 14u
46
+ ··· − 5u + 1)
c
2
((u − 1)
7
)(u
47
− 8u
46
+ ··· − 5u + 1)
c
3
, c
8
u
7
(u
47
+ u
46
+ ··· + 192u + 128)
c
4
((u + 1)
7
)(u
47
− 8u
46
+ ··· − 5u + 1)
c
5
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
47
− 2u
46
+ ··· + 2u + 1)
c
6
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
47
− 2u
46
+ ··· − 12u
2
+ 1)
c
7
((u
2
− u + 1)
2
)(u
3
+ 3u
2
+ 5u + 2)(u
47
+ 2u
46
+ ··· + 96u + 72)
c
9
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
47
+ 8u
46
+ ··· + 112u − 49)
c
10
, c
11
(u
3
+ 2u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)(u
47
− 2u
46
+ ··· − 12u
2
+ 1)
c
12
(u
3
+ 2u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)(u
47
+ 8u
46
+ ··· + 112u − 49)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
7
)(y
47
+ 46y
46
+ ··· + 83y −1)
c
2
, c
4
((y −1)
7
)(y
47
− 14y
46
+ ··· − 5y −1)
c
3
, c
8
y
7
(y
47
+ 45y
46
+ ··· − 167936y −16384)
c
5
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
47
− 52y
46
+ ··· + 24y −1)
c
6
, c
10
, c
11
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
47
+ 44y
46
+ ··· + 24y −1)
c
7
((y
2
+ y + 1)
2
)(y
3
+ y
2
+ 13y −4)(y
47
+ 12y
46
+ ··· + 50832y −5184)
c
9
, c
12
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)
· (y
47
+ 32y
46
+ ··· + 170128y −2401)
19
