12n
0830
(K12n
0830
)
A knot diagram
1
Linearized knot diagam
4 6 12 9 2 10 1 4 6 7 3 7
Solving Sequence
7,12
1
4,8
3 11 10 6 2 5 9
c
12
c
7
c
3
c
11
c
10
c
6
c
2
c
5
c
9
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h7u
11
− 9u
10
− 37u
9
+ 25u
8
+ 84u
7
− 6u
6
− 54u
5
− 45u
4
− 53u
3
+ 27u
2
+ 11b + 46u + 9,
10u
11
− 38u
10
− 12u
9
+ 152u
8
− 12u
7
− 249u
6
+ 36u
5
+ 85u
4
+ 6u
3
+ 180u
2
+ 11a − 82u − 83,
u
12
− 4u
11
+ 15u
9
− 6u
8
− 24u
7
+ 11u
6
+ 8u
5
+ 17u
3
− 14u
2
− 6u + 1i
I
u
2
= hu
3
+ 2u
2
+ b, u
2
+ a + u − 1, u
4
+ 3u
3
+ 2u
2
+ 1i
I
u
3
= hu
2
+ b + a + u − 2, 2u
2
a + a
2
+ au − u
2
− 4a − u + 4, u
3
+ u
2
− 2u − 1i
I
u
4
= h−u
2
+ b + a + u + 2, −2u
2
a + a
2
+ au − u
2
+ 4a + u + 2, u
3
− u
2
− 2u + 1i
* 4 irreducible components of dim
C
= 0, with total 28 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
=
h7u
11
−9u
10
+· · ·+11b+9, 10u
11
−38u
10
+· · ·+11a−83, u
12
−4u
11
+· · ·−6u+1i
(i) Arc colorings
a
7
=
0
u
a
12
=
1
0
a
1
=
1
−u
2
a
4
=
−0.909091u
11
+ 3.45455u
10
+ ··· + 7.45455u + 7.54545
−0.636364u
11
+ 0.818182u
10
+ ··· − 4.18182u − 0.818182
a
8
=
u
−u
3
+ u
a
3
=
−1.54545u
11
+ 4.27273u
10
+ ··· + 3.27273u + 6.72727
−0.636364u
11
+ 0.818182u
10
+ ··· − 4.18182u − 0.818182
a
11
=
−0.818182u
11
+ 3.90909u
10
+ ··· + 13.9091u + 9.09091
0.545455u
11
− 1.27273u
10
+ ··· − 2.27273u − 1.72727
a
10
=
−0.818182u
11
+ 3.90909u
10
+ ··· + 13.9091u + 9.09091
−1.18182u
11
+ 2.09091u
10
+ ··· − 6.90909u − 1.09091
a
6
=
−1.72727u
11
+ 6.36364u
10
+ ··· + 20.3636u + 12.6364
2.81818u
11
− 6.90909u
10
+ ··· + 5.09091u − 4.09091
a
2
=
2.72727u
11
− 8.36364u
10
+ ··· − 10.3636u − 8.63636
−0.0909091u
11
+ 0.545455u
10
+ ··· + 2.54545u + 1.45455
a
5
=
−5.63636u
11
+ 12.8182u
10
+ ··· − 5.18182u + 6.18182
10.6364u
11
− 24.8182u
10
+ ··· + 33.1818u − 7.18182
a
9
=
2.72727u
11
− 8.36364u
10
+ ··· − 9.36364u − 9.63636
−0.0909091u
11
+ 0.545455u
10
+ ··· + 3.54545u + 1.45455
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
5
11
u
11
−
14
11
u
10
+
72
11
u
9
+
67
11
u
8
−
225
11
u
7
−
189
11
u
6
+
224
11
u
5
+
194
11
u
4
+
151
11
u
3
+
53
11
u
2
−
278
11
u−
206
11
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
8
u
12
− u
11
+ ··· + 3u + 1
c
2
, c
5
u
12
+ 6u
10
+ ··· + 6u − 1
c
3
, c
6
, c
9
c
10
, c
11
u
12
+ 2u
11
+ ··· − 5u − 1
c
7
, c
12
u
12
− 4u
11
+ 15u
9
− 6u
8
− 24u
7
+ 11u
6
+ 8u
5
+ 17u
3
− 14u
2
− 6u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
8
y
12
+ 21y
11
+ ··· + 7y + 1
c
2
, c
5
y
12
+ 12y
11
+ ··· − 24y + 1
c
3
, c
6
, c
9
c
10
, c
11
y
12
− 10y
11
+ ··· − 27y + 1
c
7
, c
12
y
12
− 16y
11
+ ··· − 64y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.10971
a = 0.161640
b = 1.43592
−8.31956 −9.86310
u = −0.047522 + 0.875928I
a = 0.550543 − 0.109390I
b = 0.620121 − 0.344417I
−0.93869 + 1.18818I −6.72604 − 6.20651I
u = −0.047522 − 0.875928I
a = 0.550543 + 0.109390I
b = 0.620121 + 0.344417I
−0.93869 − 1.18818I −6.72604 + 6.20651I
u = −1.25634
a = −1.19132
b = 1.21703
−6.81354 −13.0130
u = −1.235650 + 0.562992I
a = −0.167466 + 0.934549I
b = −1.095690 − 0.804498I
3.14055 − 6.45902I −9.32911 + 6.09999I
u = −1.235650 − 0.562992I
a = −0.167466 − 0.934549I
b = −1.095690 + 0.804498I
3.14055 + 6.45902I −9.32911 − 6.09999I
u = −0.405006
a = 1.79428
b = 0.222902
−0.968428 −8.39760
u = 1.68726 + 0.16814I
a = 0.086359 − 0.758415I
b = −0.723528 + 0.260031I
5.78393 + 2.23624I −8.95764 − 2.44896I
u = 1.68726 − 0.16814I
a = 0.086359 + 0.758415I
b = −0.723528 − 0.260031I
5.78393 − 2.23624I −8.95764 + 2.44896I
u = 1.80553 + 0.13825I
a = −0.47270 + 1.37792I
b = 1.46230 − 1.09221I
14.0244 + 9.4961I −8.37495 − 3.79641I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.80553 − 0.13825I
a = −0.47270 − 1.37792I
b = 1.46230 + 1.09221I
14.0244 − 9.4961I −8.37495 + 3.79641I
u = 0.132401
a = 8.24194
b = −1.40225
−11.4695 −21.9510
6
II. I
u
2
= hu
3
+ 2u
2
+ b, u
2
+ a + u − 1, u
4
+ 3u
3
+ 2u
2
+ 1i
(i) Arc colorings
a
7
=
0
u
a
12
=
1
0
a
1
=
1
−u
2
a
4
=
−u
2
− u + 1
−u
3
− 2u
2
a
8
=
u
−u
3
+ u
a
3
=
−u
3
− 3u
2
− u + 1
−u
3
− 2u
2
a
11
=
u
2
+ 2u
−u
3
− u
2
+ u − 1
a
10
=
u
2
+ 2u
u
2
+ u
a
6
=
−u
3
− 2u
2
− u − 1
0
a
2
=
−u
3
− 2u
2
+ 1
−u
3
− 2u
2
a
5
=
−u
3
− 2u
2
u
3
+ u
2
+ 1
a
9
=
u
3
+ 2u
2
+ u
u
2
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
3
+ 5u
2
− 5u − 13
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
4
+ 2u
2
− 3u + 1
c
2
u
4
− u
3
+ 2u
2
− 2u + 1
c
3
, c
9
, c
10
u
4
− u
3
− u
2
+ u + 1
c
5
u
4
+ u
3
+ 2u
2
+ 2u + 1
c
6
, c
11
u
4
+ u
3
− u
2
− u + 1
c
7
u
4
− 3u
3
+ 2u
2
+ 1
c
8
u
4
+ 2u
2
+ 3u + 1
c
12
u
4
+ 3u
3
+ 2u
2
+ 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
8
y
4
+ 4y
3
+ 6y
2
− 5y + 1
c
2
, c
5
y
4
+ 3y
3
+ 2y
2
+ 1
c
3
, c
6
, c
9
c
10
, c
11
y
4
− 3y
3
+ 5y
2
− 3y + 1
c
7
, c
12
y
4
− 5y
3
+ 6y
2
+ 4y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.192440 + 0.547877I
a = 1.070700 − 0.758745I
b = 0.692440 − 0.318148I
−1.74699 + 0.56550I −15.9426 − 2.0994I
u = 0.192440 − 0.547877I
a = 1.070700 + 0.758745I
b = 0.692440 + 0.318148I
−1.74699 − 0.56550I −15.9426 + 2.0994I
u = −1.69244 + 0.31815I
a = −0.070696 + 0.758745I
b = −1.192440 − 0.547877I
5.03685 − 4.62527I −8.05745 + 3.83145I
u = −1.69244 − 0.31815I
a = −0.070696 − 0.758745I
b = −1.192440 + 0.547877I
5.03685 + 4.62527I −8.05745 − 3.83145I
10
III.
I
u
3
= hu
2
+ b + a + u − 2, 2u
2
a + a
2
+ au − u
2
− 4a − u + 4, u
3
+ u
2
− 2u − 1i
(i) Arc colorings
a
7
=
0
u
a
12
=
1
0
a
1
=
1
−u
2
a
4
=
a
−u
2
− a − u + 2
a
8
=
u
u
2
− u − 1
a
3
=
−u
2
− u + 2
−u
2
− a − u + 2
a
11
=
−u
2
a − au + 2u
2
+ 2a + u − 4
−au + u
2
− 1
a
10
=
−u
2
a − au + 2u
2
+ 2a + u − 4
0
a
6
=
−u
2
− a − u + 2
u
a
2
=
−3u
2
− a + 5
u
2
a − a − u + 1
a
5
=
−3u
2
a + au + u
2
+ 2a − 4
u
2
a − au − u
2
+ 3u
a
9
=
−u
2
a − au + 3u
2
+ 2a + u − 4
au − u
2
+ 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −7
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
8
u
6
− u
5
+ 12u
4
− 6u
3
− 7u
2
+ 7u + 7
c
2
, c
5
u
6
+ u
5
+ 9u
4
+ 18u
3
+ 26u
2
+ 29u + 13
c
3
, c
6
, c
9
c
10
, c
11
u
6
+ u
5
+ 3u
4
+ 5u
2
+ 2u + 1
c
7
, c
12
(u
3
+ u
2
− 2u − 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
8
y
6
+ 23y
5
+ 118y
4
− 176y
3
+ 301y
2
− 147y + 49
c
2
, c
5
y
6
+ 17y
5
+ 97y
4
+ 112y
3
− 134y
2
− 165y + 169
c
3
, c
6
, c
9
c
10
, c
11
y
6
+ 5y
5
+ 19y
4
+ 28y
3
+ 31y
2
+ 6y + 1
c
7
, c
12
(y
3
− 5y
2
+ 6y −1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.24698
a = −0.178448 + 1.079920I
b = −0.623490 − 1.079920I
4.69981 −7.00000
u = 1.24698
a = −0.178448 − 1.079920I
b = −0.623490 + 1.079920I
4.69981 −7.00000
u = −0.445042
a = 2.02446 + 0.38542I
b = 0.222521 − 0.385418I
−0.939962 −7.00000
u = −0.445042
a = 2.02446 − 0.38542I
b = 0.222521 + 0.385418I
−0.939962 −7.00000
u = −1.80194
a = −0.34601 + 1.56052I
b = 0.90097 − 1.56052I
15.9794 −7.00000
u = −1.80194
a = −0.34601 − 1.56052I
b = 0.90097 + 1.56052I
15.9794 −7.00000
14
IV.
I
u
4
= h−u
2
+ b + a + u + 2, −2u
2
a + a
2
+ au − u
2
+ 4a + u + 2, u
3
− u
2
− 2u + 1i
(i) Arc colorings
a
7
=
0
u
a
12
=
1
0
a
1
=
1
−u
2
a
4
=
a
u
2
− a − u − 2
a
8
=
u
−u
2
− u + 1
a
3
=
u
2
− u − 2
u
2
− a − u − 2
a
11
=
u
2
a − au + 2u
2
− 2a − u − 4
−au + u
2
− 3
a
10
=
u
2
a − au + 2u
2
− 2a − u − 4
−2
a
6
=
−u
2
− a + u + 2
−2u
2
+ 2a + u + 4
a
2
=
−u
2
+ a + 3
−u
2
a − 2u
2
+ a + u + 3
a
5
=
u
2
a − au + u
2
− 2a − 2
u
2
a + au − u
2
− u
a
9
=
−u
2
a + au − u
2
+ 2a + u + 2
−au − u
2
+ 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −7
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
6
− u
5
+ 2u
3
− 7u
2
+ 5u − 1
c
2
u
6
+ u
5
− u
4
+ 2u
3
+ 2u
2
− 3u − 1
c
3
, c
9
, c
10
u
6
− u
5
− 3u
4
+ 4u
3
+ u
2
− 4u + 1
c
5
u
6
− u
5
− u
4
− 2u
3
+ 2u
2
+ 3u − 1
c
6
, c
11
u
6
+ u
5
− 3u
4
− 4u
3
+ u
2
+ 4u + 1
c
7
(u
3
+ u
2
− 2u − 1)
2
c
8
u
6
+ u
5
− 2u
3
− 7u
2
− 5u − 1
c
12
(u
3
− u
2
− 2u + 1)
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
8
y
6
− y
5
− 10y
4
+ 4y
3
+ 29y
2
− 11y + 1
c
2
, c
5
y
6
− 3y
5
+ y
4
− 4y
3
+ 18y
2
− 13y + 1
c
3
, c
6
, c
9
c
10
, c
11
y
6
− 7y
5
+ 19y
4
− 28y
3
+ 27y
2
− 14y + 1
c
7
, c
12
(y
3
− 5y
2
+ 6y −1)
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −1.24698
a = 1.09156
b = −0.289627
−5.16979 −7.00000
u = −1.24698
a = −0.734668
b = 1.53661
−5.16979 −7.00000
u = 0.445042
a = −0.663777
b = −1.58320
−10.8096 −7.00000
u = 0.445042
a = −3.38514
b = 1.13816
−10.8096 −7.00000
u = 1.80194
a = 0.346011 + 0.659723I
b = −0.900969 − 0.659723I
6.10976 −7.00000
u = 1.80194
a = 0.346011 − 0.659723I
b = −0.900969 + 0.659723I
6.10976 −7.00000
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
4
+ 2u
2
− 3u + 1)(u
6
− u
5
+ 2u
3
− 7u
2
+ 5u − 1)
· (u
6
− u
5
+ ··· + 7u + 7)(u
12
− u
11
+ ··· + 3u + 1)
c
2
(u
4
− u
3
+ 2u
2
− 2u + 1)(u
6
+ u
5
− u
4
+ 2u
3
+ 2u
2
− 3u − 1)
· (u
6
+ u
5
+ ··· + 29u + 13)(u
12
+ 6u
10
+ ··· + 6u − 1)
c
3
, c
9
, c
10
(u
4
− u
3
− u
2
+ u + 1)(u
6
− u
5
− 3u
4
+ 4u
3
+ u
2
− 4u + 1)
· (u
6
+ u
5
+ 3u
4
+ 5u
2
+ 2u + 1)(u
12
+ 2u
11
+ ··· − 5u − 1)
c
5
(u
4
+ u
3
+ 2u
2
+ 2u + 1)(u
6
− u
5
− u
4
− 2u
3
+ 2u
2
+ 3u − 1)
· (u
6
+ u
5
+ ··· + 29u + 13)(u
12
+ 6u
10
+ ··· + 6u − 1)
c
6
, c
11
(u
4
+ u
3
− u
2
− u + 1)(u
6
+ u
5
− 3u
4
− 4u
3
+ u
2
+ 4u + 1)
· (u
6
+ u
5
+ 3u
4
+ 5u
2
+ 2u + 1)(u
12
+ 2u
11
+ ··· − 5u − 1)
c
7
(u
3
+ u
2
− 2u − 1)
4
(u
4
− 3u
3
+ 2u
2
+ 1)
· (u
12
− 4u
11
+ 15u
9
− 6u
8
− 24u
7
+ 11u
6
+ 8u
5
+ 17u
3
− 14u
2
− 6u + 1)
c
8
(u
4
+ 2u
2
+ 3u + 1)(u
6
− u
5
+ 12u
4
− 6u
3
− 7u
2
+ 7u + 7)
· (u
6
+ u
5
− 2u
3
− 7u
2
− 5u − 1)(u
12
− u
11
+ ··· + 3u + 1)
c
12
(u
3
− u
2
− 2u + 1)
2
(u
3
+ u
2
− 2u − 1)
2
(u
4
+ 3u
3
+ 2u
2
+ 1)
· (u
12
− 4u
11
+ 15u
9
− 6u
8
− 24u
7
+ 11u
6
+ 8u
5
+ 17u
3
− 14u
2
− 6u + 1)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
8
(y
4
+ 4y
3
+ 6y
2
− 5y + 1)(y
6
− y
5
− 10y
4
+ 4y
3
+ 29y
2
− 11y + 1)
· (y
6
+ 23y
5
+ 118y
4
− 176y
3
+ 301y
2
− 147y + 49)
· (y
12
+ 21y
11
+ ··· + 7y + 1)
c
2
, c
5
(y
4
+ 3y
3
+ 2y
2
+ 1)(y
6
− 3y
5
+ y
4
− 4y
3
+ 18y
2
− 13y + 1)
· (y
6
+ 17y
5
+ 97y
4
+ 112y
3
− 134y
2
− 165y + 169)
· (y
12
+ 12y
11
+ ··· − 24y + 1)
c
3
, c
6
, c
9
c
10
, c
11
(y
4
− 3y
3
+ 5y
2
− 3y + 1)(y
6
− 7y
5
+ ··· − 14y + 1)
· (y
6
+ 5y
5
+ ··· + 6y + 1)(y
12
− 10y
11
+ ··· − 27y + 1)
c
7
, c
12
(y
3
− 5y
2
+ 6y −1)
4
(y
4
− 5y
3
+ 6y
2
+ 4y + 1)
· (y
12
− 16y
11
+ ··· − 64y + 1)
20
