12n
0371
(K12n
0371
)
A knot diagram
1
Linearized knot diagam
3 6 8 10 2 11 5 12 4 6 10 8
Solving Sequence
2,6 3,11
7 1 5 8 10 4 12 9
c
2
c
6
c
1
c
5
c
7
c
10
c
4
c
12
c
8
c
3
, c
9
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h2u
13
− 7u
12
+ 11u
11
− 7u
10
+ 10u
9
− 33u
8
+ 54u
7
− 34u
6
+ u
5
+ 5u
4
+ u
3
+ 4u
2
+ b − 5u + 1,
2u
13
− 7u
12
+ 11u
11
− 7u
10
+ 10u
9
− 33u
8
+ 54u
7
− 34u
6
+ u
5
+ 5u
4
+ u
3
+ 4u
2
+ a − 6u + 1,
u
15
− 4u
14
+ 7u
13
− 5u
12
+ 4u
11
− 16u
10
+ 33u
9
− 25u
8
− 4u
7
+ 17u
6
− 6u
5
− u
4
− 2u
3
+ 2u
2
+ u − 1i
I
u
2
= hu
9
+ 3u
8
+ 4u
7
− 4u
5
− 4u
4
+ b + u, u
9
+ 3u
8
+ 4u
7
− 4u
5
− 4u
4
+ a, u
10
+ 3u
9
+ 4u
8
− 4u
6
− 4u
5
+ u
3
+ u
2
− 1i
* 2 irreducible components of dim
C
= 0, with total 25 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
= h2u
13
−7u
12
+· · ·+b+1, 2u
13
−7u
12
+· · ·+a+1, u
15
−4u
14
+· · ·+u−1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
11
=
−2u
13
+ 7u
12
+ ··· + 6u − 1
−2u
13
+ 7u
12
+ ··· + 5u − 1
a
7
=
−3u
13
+ 11u
12
+ ··· + 7u − 2
u
14
− 6u
13
+ ··· − 6u
2
+ 6u
a
1
=
−u
2
+ 1
−u
4
a
5
=
u
u
a
8
=
−3u
13
+ 11u
12
+ ··· + 7u − 3
u
14
− 6u
13
+ ··· + 6u − 1
a
10
=
−2u
13
+ 7u
12
+ ··· + 6u − 1
−u
14
+ u
13
+ ··· + 7u − 3
a
4
=
u
11
− 2u
10
+ u
9
+ 5u
7
− 10u
6
+ 5u
5
+ u
3
− u
2
+ 1
u
13
− 2u
12
+ 2u
11
− u
10
+ 5u
9
− 10u
8
+ 10u
7
− 5u
6
+ u
5
− u
4
+ u
3
+ u
2
a
12
=
−u
12
+ 3u
11
+ ··· − 2u
2
+ 1
−u
14
+ 3u
13
+ ··· − 3u
4
+ u
3
a
9
=
u
13
− 5u
12
+ ··· − 6u + 3
u
14
− 2u
13
+ ··· − 7u + 5
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
13
−14u
12
+ 19u
11
−7u
10
+ 13u
9
−68u
8
+ 98u
7
−35u
6
−33u
5
+ 23u
3
+ 3u
2
−14u −6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
15
+ 2u
14
+ ··· + 5u + 1
c
2
, c
5
u
15
+ 4u
14
+ ··· + u + 1
c
3
u
15
− 10u
13
+ ··· + 4u + 1
c
4
, c
7
, c
9
u
15
− u
14
+ ··· + 2u + 1
c
6
, c
10
u
15
+ 14u
14
+ ··· + 240u + 32
c
8
, c
12
u
15
+ 4u
14
+ ··· + 9u + 1
c
11
u
15
− 10u
14
+ ··· + 12032u − 1024
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
15
+ 30y
14
+ ··· + 5y −1
c
2
, c
5
y
15
− 2y
14
+ ··· + 5y −1
c
3
y
15
− 20y
14
+ ··· − 2y −1
c
4
, c
7
, c
9
y
15
− 35y
14
+ ··· − 6y −1
c
6
, c
10
y
15
− 10y
14
+ ··· + 12032y −1024
c
8
, c
12
y
15
+ 24y
14
+ ··· + 89y −1
c
11
y
15
+ 78y
14
+ ··· + 66256896y −1048576
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.712549 + 0.633880I
a = 0.274451 − 1.191860I
b = −0.43810 − 1.82574I
−1.05212 − 4.72304I −2.71866 + 7.35913I
u = 0.712549 − 0.633880I
a = 0.274451 + 1.191860I
b = −0.43810 + 1.82574I
−1.05212 + 4.72304I −2.71866 − 7.35913I
u = 0.815797 + 0.485623I
a = 1.042200 + 0.113566I
b = 0.226400 − 0.372057I
−1.57751 + 0.34385I −3.36524 − 0.94481I
u = 0.815797 − 0.485623I
a = 1.042200 − 0.113566I
b = 0.226400 + 0.372057I
−1.57751 − 0.34385I −3.36524 + 0.94481I
u = 0.766118
a = 0.453886
b = −0.312232
−1.00577 −12.0370
u = −0.635896 + 0.026087I
a = −1.32826 − 1.25555I
b = −0.69236 − 1.28163I
−3.88717 + 2.88458I −6.82119 + 2.09374I
u = −0.635896 − 0.026087I
a = −1.32826 + 1.25555I
b = −0.69236 + 1.28163I
−3.88717 − 2.88458I −6.82119 − 2.09374I
u = −0.289639 + 0.547664I
a = −0.006426 + 1.347820I
b = 0.283213 + 0.800152I
1.30871 + 0.90771I 3.11296 − 2.32958I
u = −0.289639 − 0.547664I
a = −0.006426 − 1.347820I
b = 0.283213 − 0.800152I
1.30871 − 0.90771I 3.11296 + 2.32958I
u = 1.07267 + 0.99560I
a = 0.05227 − 1.45647I
b = −1.02040 − 2.45206I
−11.1785 − 9.6087I −3.50462 + 4.00725I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.07267 − 0.99560I
a = 0.05227 + 1.45647I
b = −1.02040 + 2.45206I
−11.1785 + 9.6087I −3.50462 − 4.00725I
u = 0.99469 + 1.08421I
a = 1.49910 − 0.14023I
b = 0.504408 − 1.224440I
−10.84830 + 1.98719I −3.37763 − 0.18602I
u = 0.99469 − 1.08421I
a = 1.49910 + 0.14023I
b = 0.504408 + 1.224440I
−10.84830 − 1.98719I −3.37763 + 0.18602I
u = −1.05323 + 1.04849I
a = −0.760279 − 0.943123I
b = 0.29295 − 1.99161I
10.46600 + 3.87226I −4.30716 − 1.63349I
u = −1.05323 − 1.04849I
a = −0.760279 + 0.943123I
b = 0.29295 + 1.99161I
10.46600 − 3.87226I −4.30716 + 1.63349I
6
II. I
u
2
= hu
9
+ 3u
8
+ 4u
7
− 4u
5
− 4u
4
+ b + u, u
9
+ 3u
8
+ 4u
7
− 4u
5
− 4u
4
+
a, u
10
+ 3u
9
+ 4u
8
− 4u
6
− 4u
5
+ u
3
+ u
2
− 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
11
=
−u
9
− 3u
8
− 4u
7
+ 4u
5
+ 4u
4
−u
9
− 3u
8
− 4u
7
+ 4u
5
+ 4u
4
− u
a
7
=
u
9
+ 3u
8
+ 4u
7
− 3u
5
− 2u
4
+ u
3
− u
2
− u
u
9
+ 3u
8
+ 4u
7
− 3u
5
− 3u
4
− u
2
+ u
a
1
=
−u
2
+ 1
−u
4
a
5
=
u
u
a
8
=
u
9
+ 3u
8
+ 4u
7
+ u
6
− 2u
5
− 2u
4
− u
3
− u
2
− u
u
9
+ 3u
8
+ 4u
7
+ u
6
− 2u
5
− 3u
4
− 2u
3
− u
2
+ u
a
10
=
−u
9
− 3u
8
− 4u
7
+ 4u
5
+ 4u
4
−u
9
− 3u
8
− 4u
7
+ 4u
5
+ 5u
4
+ u
3
− 2u
a
4
=
−u
7
− 2u
6
− u
5
+ 3u
4
+ 3u
3
− u
−u
9
− 2u
8
− 2u
7
+ 2u
6
+ 3u
5
+ 3u
4
− u
3
− u
a
12
=
−u
8
− 3u
7
− 3u
6
+ 2u
5
+ 6u
4
+ 3u
3
− 2u
2
− 2u
u
6
+ 2u
5
+ u
4
− u
3
− u
2
− u − 1
a
9
=
−u
8
− 3u
7
− 3u
6
+ 2u
5
+ 5u
4
+ u
3
− 2u
2
+ 1
−u
8
− 2u
7
− 2u
6
+ u
5
+ 2u
4
+ 2u
3
+ u
2
− 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
9
− 6u
7
− 13u
6
− 3u
5
+ 11u
4
+ 16u
3
+ 5u
2
+ u − 8
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
10
− u
9
+ 8u
8
− 8u
7
+ 12u
6
− 10u
5
− 8u
4
+ 7u
3
+ u
2
− 2u + 1
c
2
u
10
+ 3u
9
+ 4u
8
− 4u
6
− 4u
5
+ u
3
+ u
2
− 1
c
3
u
10
− u
9
+ 3u
8
− 5u
6
+ 5u
5
− 11u
4
+ 10u
3
− 5u
2
+ 5u − 1
c
4
, c
7
u
10
+ u
8
+ 3u
7
+ 5u
5
+ 10u
4
+ 13u
3
+ 11u
2
+ 7u + 1
c
5
u
10
− 3u
9
+ 4u
8
− 4u
6
+ 4u
5
− u
3
+ u
2
− 1
c
6
u
10
− 5u
8
+ 3u
7
+ 10u
6
− 8u
5
− 6u
4
+ 8u
3
+ 2u
2
− 3u − 1
c
8
u
10
+ 3u
9
+ 3u
8
+ u
7
− 4u
6
− 9u
5
− 6u
4
+ 3u
3
+ 11u
2
+ 2u − 1
c
9
u
10
+ u
8
− 3u
7
− 5u
5
+ 10u
4
− 13u
3
+ 11u
2
− 7u + 1
c
10
u
10
− 5u
8
− 3u
7
+ 10u
6
+ 8u
5
− 6u
4
− 8u
3
+ 2u
2
+ 3u − 1
c
11
u
10
− 10u
9
+ ··· − 13u + 1
c
12
u
10
− 3u
9
+ 3u
8
− u
7
− 4u
6
+ 9u
5
− 6u
4
− 3u
3
+ 11u
2
− 2u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
10
+ 15y
9
+ ··· − 2y + 1
c
2
, c
5
y
10
− y
9
+ 8y
8
− 8y
7
+ 12y
6
− 10y
5
− 8y
4
+ 7y
3
+ y
2
− 2y + 1
c
3
y
10
+ 5y
9
− y
8
− 42y
7
− 31y
6
+ 63y
5
+ 65y
4
− 30y
3
− 53y
2
− 15y + 1
c
4
, c
7
, c
9
y
10
+ 2y
9
+ y
8
+ 11y
7
+ 12y
6
− 79y
5
− 70y
4
− 19y
3
− 41y
2
− 27y + 1
c
6
, c
10
y
10
− 10y
9
+ ··· − 13y + 1
c
8
, c
12
y
10
− 3y
9
− 5y
8
+ 17y
7
+ 2y
6
+ 13y
5
− 8y
4
− 97y
3
+ 121y
2
− 26y + 1
c
11
y
10
− 10y
9
+ ··· − 41y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.957717
a = 0.830787
b = −0.126929
−0.427947 4.64810
u = −0.857224
a = 1.04417
b = 1.90139
−4.84848 −11.3010
u = −0.138927 + 0.799162I
a = −0.54714 + 1.79172I
b = −0.408212 + 0.992555I
−1.32406 + 2.21202I −2.09245 − 1.71917I
u = −0.138927 − 0.799162I
a = −0.54714 − 1.79172I
b = −0.408212 − 0.992555I
−1.32406 − 2.21202I −2.09245 + 1.71917I
u = −0.931721 + 0.885936I
a = −0.284841 − 0.228994I
b = 0.646880 − 1.114930I
6.28538 + 3.27846I −3.95412 − 2.52602I
u = −0.931721 − 0.885936I
a = −0.284841 + 0.228994I
b = 0.646880 + 1.114930I
6.28538 − 3.27846I −3.95412 + 2.52602I
u = 0.602982 + 0.323142I
a = −0.42625 + 1.40330I
b = −1.02923 + 1.08016I
−3.73438 − 3.52182I −4.49714 + 9.15215I
u = 0.602982 − 0.323142I
a = −0.42625 − 1.40330I
b = −1.02923 − 1.08016I
−3.73438 + 3.52182I −4.49714 − 9.15215I
u = −1.08258 + 1.10501I
a = −0.679245 − 0.825748I
b = 0.40334 − 1.93075I
11.28090 + 4.04196I 5.87002 − 3.50031I
u = −1.08258 − 1.10501I
a = −0.679245 + 0.825748I
b = 0.40334 + 1.93075I
11.28090 − 4.04196I 5.87002 + 3.50031I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
10
− u
9
+ 8u
8
− 8u
7
+ 12u
6
− 10u
5
− 8u
4
+ 7u
3
+ u
2
− 2u + 1)
· (u
15
+ 2u
14
+ ··· + 5u + 1)
c
2
(u
10
+ 3u
9
+ ··· + u
2
− 1)(u
15
+ 4u
14
+ ··· + u + 1)
c
3
(u
10
− u
9
+ 3u
8
− 5u
6
+ 5u
5
− 11u
4
+ 10u
3
− 5u
2
+ 5u − 1)
· (u
15
− 10u
13
+ ··· + 4u + 1)
c
4
, c
7
(u
10
+ u
8
+ 3u
7
+ 5u
5
+ 10u
4
+ 13u
3
+ 11u
2
+ 7u + 1)
· (u
15
− u
14
+ ··· + 2u + 1)
c
5
(u
10
− 3u
9
+ ··· + u
2
− 1)(u
15
+ 4u
14
+ ··· + u + 1)
c
6
(u
10
− 5u
8
+ 3u
7
+ 10u
6
− 8u
5
− 6u
4
+ 8u
3
+ 2u
2
− 3u − 1)
· (u
15
+ 14u
14
+ ··· + 240u + 32)
c
8
(u
10
+ 3u
9
+ 3u
8
+ u
7
− 4u
6
− 9u
5
− 6u
4
+ 3u
3
+ 11u
2
+ 2u − 1)
· (u
15
+ 4u
14
+ ··· + 9u + 1)
c
9
(u
10
+ u
8
− 3u
7
− 5u
5
+ 10u
4
− 13u
3
+ 11u
2
− 7u + 1)
· (u
15
− u
14
+ ··· + 2u + 1)
c
10
(u
10
− 5u
8
− 3u
7
+ 10u
6
+ 8u
5
− 6u
4
− 8u
3
+ 2u
2
+ 3u − 1)
· (u
15
+ 14u
14
+ ··· + 240u + 32)
c
11
(u
10
− 10u
9
+ ··· − 13u + 1)(u
15
− 10u
14
+ ··· + 12032u − 1024)
c
12
(u
10
− 3u
9
+ 3u
8
− u
7
− 4u
6
+ 9u
5
− 6u
4
− 3u
3
+ 11u
2
− 2u − 1)
· (u
15
+ 4u
14
+ ··· + 9u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
10
+ 15y
9
+ ··· − 2y + 1)(y
15
+ 30y
14
+ ··· + 5y −1)
c
2
, c
5
(y
10
− y
9
+ 8y
8
− 8y
7
+ 12y
6
− 10y
5
− 8y
4
+ 7y
3
+ y
2
− 2y + 1)
· (y
15
− 2y
14
+ ··· + 5y −1)
c
3
(y
10
+ 5y
9
− y
8
− 42y
7
− 31y
6
+ 63y
5
+ 65y
4
− 30y
3
− 53y
2
− 15y + 1)
· (y
15
− 20y
14
+ ··· − 2y −1)
c
4
, c
7
, c
9
(y
10
+ 2y
9
+ y
8
+ 11y
7
+ 12y
6
− 79y
5
− 70y
4
− 19y
3
− 41y
2
− 27y + 1)
· (y
15
− 35y
14
+ ··· − 6y −1)
c
6
, c
10
(y
10
− 10y
9
+ ··· − 13y + 1)(y
15
− 10y
14
+ ··· + 12032y −1024)
c
8
, c
12
(y
10
− 3y
9
− 5y
8
+ 17y
7
+ 2y
6
+ 13y
5
− 8y
4
− 97y
3
+ 121y
2
− 26y + 1)
· (y
15
+ 24y
14
+ ··· + 89y −1)
c
11
(y
10
− 10y
9
+ ··· − 41y + 1)
· (y
15
+ 78y
14
+ ··· + 66256896y −1048576)
12
