12a
0718
(K12a
0718
)
A knot diagram
1
Linearized knot diagam
3 8 9 10 11 12 2 1 4 7 6 5
Solving Sequence
7,12
6 11 5 1 10 4 9 3 2 8
c
6
c
11
c
5
c
12
c
10
c
4
c
9
c
3
c
1
c
8
c
2
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
69
− 2u
68
+ ··· + 2u
2
− 1i
I
u
2
= hu + 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
= hu
69
− 2u
68
+ · · · + 2u
2
− 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
6
=
1
u
2
a
11
=
−u
−u
3
+ u
a
5
=
−u
2
+ 1
−u
4
+ 2u
2
a
1
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
10
=
u
3
− 2u
−u
3
+ u
a
4
=
u
10
− 5u
8
+ 8u
6
− 3u
4
− 3u
2
+ 1
−u
10
+ 4u
8
− 5u
6
+ 3u
2
a
9
=
−u
17
+ 8u
15
− 25u
13
+ 36u
11
− 17u
9
− 12u
7
+ 12u
5
+ 2u
3
− 3u
u
17
− 7u
15
+ 19u
13
− 22u
11
+ 3u
9
+ 14u
7
− 6u
5
− 4u
3
+ u
a
3
=
−u
24
+ 11u
22
+ ··· − 6u
2
+ 1
u
24
− 10u
22
+ ··· − 2u
4
+ 4u
2
a
2
=
−u
55
+ 24u
53
+ ··· − 28u
5
+ 12u
3
u
55
− 23u
53
+ ··· − 2u
3
+ u
a
8
=
−u
29
+ 12u
27
+ ··· − 2u
3
− 3u
−u
31
+ 13u
29
+ ··· − 8u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
68
− 116u
66
+ ··· − 8u + 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
69
+ 30u
68
+ ··· + 4u + 1
c
2
, c
7
u
69
+ 2u
68
+ ··· − 2u
2
+ 1
c
3
, c
4
, c
9
u
69
− 35u
67
+ ··· + 16u + 1
c
5
, c
6
, c
11
u
69
+ 2u
68
+ ··· − 2u
2
+ 1
c
8
u
69
+ 3u
68
+ ··· − 8u − 1
c
10
, c
12
u
69
− 3u
68
+ ··· + 4u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
69
+ 18y
68
+ ··· − 28y −1
c
2
, c
7
y
69
− 30y
68
+ ··· + 4y −1
c
3
, c
4
, c
9
y
69
− 70y
68
+ ··· + 100y −1
c
5
, c
6
, c
11
y
69
− 58y
68
+ ··· + 4y −1
c
8
y
69
− 3y
68
+ ··· + 4y −1
c
10
, c
12
y
69
+ 33y
68
+ ··· − 12y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.957601 + 0.300766I
5.05918 + 6.89076I 6.00000 − 4.16482I
u = −0.957601 − 0.300766I
5.05918 − 6.89076I 6.00000 + 4.16482I
u = −0.970924 + 0.140812I
1.64199 + 0.00698I 6.00000 + 0.I
u = −0.970924 − 0.140812I
1.64199 − 0.00698I 6.00000 + 0.I
u = 0.933707 + 0.288136I
6.89512 − 1.63728I 11.63888 + 0.I
u = 0.933707 − 0.288136I
6.89512 + 1.63728I 11.63888 + 0.I
u = 0.857602 + 0.279930I
7.01858 + 1.34841I 12.02587 − 1.24080I
u = 0.857602 − 0.279930I
7.01858 − 1.34841I 12.02587 + 1.24080I
u = −0.825401 + 0.288872I
5.28751 − 6.59810I 9.41471 + 6.14577I
u = −0.825401 − 0.288872I
5.28751 + 6.59810I 9.41471 − 6.14577I
u = −0.189683 + 0.774659I
2.65904 − 10.93250I 5.56084 + 8.27245I
u = −0.189683 − 0.774659I
2.65904 + 10.93250I 5.56084 − 8.27245I
u = 0.193338 + 0.767498I
4.55534 + 5.62792I 8.38609 − 3.88248I
u = 0.193338 − 0.767498I
4.55534 − 5.62792I 8.38609 + 3.88248I
u = 1.184060 + 0.293256I
−1.37585 − 2.62316I 0
u = 1.184060 − 0.293256I
−1.37585 + 2.62316I 0
u = 0.205061 + 0.747895I
4.87077 + 2.53274I 8.88434 − 3.60320I
u = 0.205061 − 0.747895I
4.87077 − 2.53274I 8.88434 + 3.60320I
u = −0.177328 + 0.752440I
−0.80018 − 3.75250I 2.22528 + 3.72594I
u = −0.177328 − 0.752440I
−0.80018 + 3.75250I 2.22528 − 3.72594I
u = −0.211423 + 0.738250I
3.23850 + 2.75221I 6.50930 − 1.22477I
u = −0.211423 − 0.738250I
3.23850 − 2.75221I 6.50930 + 1.22477I
u = 0.080892 + 0.759659I
−4.71738 + 6.46211I 0.26695 − 7.64034I
u = 0.080892 − 0.759659I
−4.71738 − 6.46211I 0.26695 + 7.64034I
u = −1.211110 + 0.265826I
1.02259 − 1.39020I 0
u = −1.211110 − 0.265826I
1.02259 + 1.39020I 0
u = 0.037081 + 0.752751I
−5.81603 − 0.56590I −2.61211 + 0.31548I
u = 0.037081 − 0.752751I
−5.81603 + 0.56590I −2.61211 − 0.31548I
u = −0.075375 + 0.730057I
−2.40624 − 2.22308I 3.89696 + 3.92721I
u = −0.075375 − 0.730057I
−2.40624 + 2.22308I 3.89696 − 3.92721I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.229420 + 0.304406I
−2.15788 + 4.39261I 0
u = 1.229420 − 0.304406I
−2.15788 − 4.39261I 0
u = 1.310820 + 0.046148I
5.92654 + 0.72025I 0
u = 1.310820 − 0.046148I
5.92654 − 0.72025I 0
u = −1.286920 + 0.315645I
−1.69307 − 3.29241I 0
u = −1.286920 − 0.315645I
−1.69307 + 3.29241I 0
u = −1.322760 + 0.090992I
4.47049 − 5.27183I 0
u = −1.322760 − 0.090992I
4.47049 + 5.27183I 0
u = −1.307600 + 0.227331I
3.03142 − 0.82065I 0
u = −1.307600 − 0.227331I
3.03142 + 0.82065I 0
u = 1.319690 + 0.261661I
3.51103 + 5.18896I 0
u = 1.319690 − 0.261661I
3.51103 − 5.18896I 0
u = 1.312500 + 0.307397I
1.94259 + 5.98180I 0
u = 1.312500 − 0.307397I
1.94259 − 5.98180I 0
u = −1.314070 + 0.323446I
−0.34941 − 10.37660I 0
u = −1.314070 − 0.323446I
−0.34941 + 10.37660I 0
u = −0.088647 + 0.610634I
−0.91571 − 1.95235I 6.00572 + 4.84634I
u = −0.088647 − 0.610634I
−0.91571 + 1.95235I 6.00572 − 4.84634I
u = 1.367020 + 0.315964I
4.07913 + 7.62771I 0
u = 1.367020 − 0.315964I
4.07913 − 7.62771I 0
u = 1.378610 + 0.304843I
8.26841 + 1.03300I 0
u = 1.378610 − 0.304843I
8.26841 − 1.03300I 0
u = −1.377770 + 0.309975I
9.87821 − 6.36933I 0
u = −1.377770 − 0.309975I
9.87821 + 6.36933I 0
u = −1.375830 + 0.320526I
9.51830 − 9.56822I 0
u = −1.375830 − 0.320526I
9.51830 + 9.56822I 0
u = 1.375220 + 0.324354I
7.6078 + 14.9103I 0
u = 1.375220 − 0.324354I
7.6078 − 14.9103I 0
u = 1.41865
8.31659 0
u = −1.43187 + 0.00706I
13.96300 − 1.63595I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.43187 − 0.00706I
13.96300 + 1.63595I 0
u = 1.43193 + 0.01291I
12.19380 + 7.00395I 0
u = 1.43193 − 0.01291I
12.19380 − 7.00395I 0
u = 0.399202 + 0.279360I
−0.70238 + 4.07184I 6.74606 − 8.81132I
u = 0.399202 − 0.279360I
−0.70238 − 4.07184I 6.74606 + 8.81132I
u = 0.198393 + 0.391221I
−1.31494 − 1.68072I 3.43317 − 0.28369I
u = 0.198393 − 0.391221I
−1.31494 + 1.68072I 3.43317 + 0.28369I
u = −0.399566 + 0.111347I
0.839509 − 0.175599I 12.54321 + 2.20743I
u = −0.399566 − 0.111347I
0.839509 + 0.175599I 12.54321 − 2.20743I
7
II. I
u
2
= hu + 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
−1
a
6
=
1
1
a
11
=
1
0
a
5
=
0
1
a
1
=
0
−1
a
10
=
1
0
a
4
=
−1
1
a
9
=
0
1
a
3
=
−1
0
a
2
=
1
−1
a
8
=
0
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u + 1
c
2
, c
3
, c
4
c
5
, c
6
, c
7
c
9
, c
11
u − 1
c
8
, c
10
, c
12
u
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
9
, c
11
y −1
c
8
, c
10
, c
12
y
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
1.64493 6.00000
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u + 1)(u
69
+ 30u
68
+ ··· + 4u + 1)
c
2
, c
7
(u − 1)(u
69
+ 2u
68
+ ··· − 2u
2
+ 1)
c
3
, c
4
, c
9
(u − 1)(u
69
− 35u
67
+ ··· + 16u + 1)
c
5
, c
6
, c
11
(u − 1)(u
69
+ 2u
68
+ ··· − 2u
2
+ 1)
c
8
u(u
69
+ 3u
68
+ ··· − 8u − 1)
c
10
, c
12
u(u
69
− 3u
68
+ ··· + 4u − 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)(y
69
+ 18y
68
+ ··· − 28y −1)
c
2
, c
7
(y −1)(y
69
− 30y
68
+ ··· + 4y −1)
c
3
, c
4
, c
9
(y −1)(y
69
− 70y
68
+ ··· + 100y −1)
c
5
, c
6
, c
11
(y −1)(y
69
− 58y
68
+ ··· + 4y −1)
c
8
y(y
69
− 3y
68
+ ··· + 4y −1)
c
10
, c
12
y(y
69
+ 33y
68
+ ··· − 12y −1)
13
