12n
0638
(K12n
0638
)
A knot diagram
1
Linearized knot diagam
3 8 9 11 1 10 2 7 5 7 9 5
Solving Sequence
2,7
8 3
5,9
10 11 1 4 6 12
c
7
c
2
c
8
c
9
c
10
c
1
c
4
c
6
c
12
c
3
, c
5
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
14
− 3u
13
+ ··· + 2b − 2, 7u
14
+ 29u
13
+ ··· + 4a + 24, u
15
+ 5u
14
+ ··· + 18u + 4i
I
u
2
= hu
10
+ u
9
− u
8
+ 4u
6
+ 3u
5
− 2u
4
+ 3u
2
+ b + 2u, u
10
+ 3u
6
+ u
5
+ u
4
− u
3
+ 2u
2
+ a + 2u + 1,
u
11
+ u
10
− u
9
− u
8
+ 4u
7
+ 4u
6
− 2u
5
− 3u
4
+ 3u
3
+ 4u
2
− 1i
I
u
3
= hb + 2, a + 1, u − 1i
I
u
4
= ha
3
+ 2a
2
+ 3b + a + 5, a
4
+ a
3
+ 2a
2
+ 4a + 1, u − 1i
* 4 irreducible components of dim
C
= 0, with total 31 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−u
14
−3u
13
+· · ·+2b−2, 7u
14
+29u
13
+· · ·+4a+24, u
15
+5u
14
+· · ·+18u+4i
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
3
=
−u
−u
3
+ u
a
5
=
−
7
4
u
14
−
29
4
u
13
+ ··· −
105
4
u − 6
1
2
u
14
+
3
2
u
13
+ ··· +
11
2
u + 1
a
9
=
−u
2
+ 1
u
2
a
10
=
−
1
2
u
14
− 2u
13
+ ··· −
9
2
u −
1
2
1
2
u
14
+
3
2
u
13
+ ··· + 4u
2
+
1
2
u
a
11
=
−u
14
−
7
2
u
13
+ ··· − 5u −
1
2
1
2
u
14
+
3
2
u
13
+ ··· + 4u
2
+
1
2
u
a
1
=
u
3
u
5
− u
3
+ u
a
4
=
u
7
− 2u
5
+ 2u
3
− 2u
−u
7
+ u
5
− 2u
3
+ u
a
6
=
5
4
u
14
+
23
4
u
13
+ ··· +
79
4
u + 6
−
1
2
u
14
−
3
2
u
13
+ ··· −
9
2
u − 1
a
12
=
−
1
2
u
14
− 2u
13
+ ··· −
9
2
u −
3
2
1
2
u
14
+
3
2
u
13
+ ··· + 3u
2
+
3
2
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −9u
14
− 39u
13
− 62u
12
+ 12u
11
+ 189u
10
+ 271u
9
+ 58u
8
−
259u
7
− 277u
6
+ 33u
5
+ 223u
4
+ 70u
3
− 138u
2
− 142u − 54
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
15
+ 5u
14
+ ··· + 108u + 16
c
2
, c
7
u
15
+ 5u
14
+ ··· + 18u + 4
c
3
u
15
− 7u
14
+ ··· − 7974u + 2196
c
4
, c
6
, c
10
u
15
− 3u
14
+ ··· + 2u + 1
c
5
, c
9
, c
12
u
15
+ 2u
14
+ ··· − 3u − 1
c
11
u
15
+ 8u
14
+ ··· + 26u + 2
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
15
+ 11y
14
+ ··· + 4464y −256
c
2
, c
7
y
15
− 5y
14
+ ··· + 108y −16
c
3
y
15
− 89y
14
+ ··· + 102923820y −4822416
c
4
, c
6
, c
10
y
15
− 41y
14
+ ··· + 36y −1
c
5
, c
9
, c
12
y
15
− 28y
14
+ ··· − 11y −1
c
11
y
15
− 38y
14
+ ··· + 200y −4
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.606388 + 0.721644I
a = 1.39179 + 0.62558I
b = −0.960938 − 0.688802I
−0.178243 − 0.909766I −10.00645 + 2.94587I
u = −0.606388 − 0.721644I
a = 1.39179 − 0.62558I
b = −0.960938 + 0.688802I
−0.178243 + 0.909766I −10.00645 − 2.94587I
u = 1.08047
a = −0.675053
b = −1.51745
−5.54081 −14.3560
u = 0.746431 + 0.514902I
a = 0.082980 − 0.242831I
b = 0.397865 − 0.145660I
1.19505 − 1.99555I −7.66777 + 5.97030I
u = 0.746431 − 0.514902I
a = 0.082980 + 0.242831I
b = 0.397865 + 0.145660I
1.19505 + 1.99555I −7.66777 − 5.97030I
u = −1.021710 + 0.661454I
a = −0.06650 − 1.96813I
b = −1.33073 + 0.86334I
−1.39940 + 6.23344I −10.94430 − 8.75401I
u = −1.021710 − 0.661454I
a = −0.06650 + 1.96813I
b = −1.33073 − 0.86334I
−1.39940 − 6.23344I −10.94430 + 8.75401I
u = −0.518806 + 1.107840I
a = −1.210170 − 0.046109I
b = 1.78546 + 0.11853I
−10.31580 − 3.71425I −10.43409 + 0.73580I
u = −0.518806 − 1.107840I
a = −1.210170 + 0.046109I
b = 1.78546 − 0.11853I
−10.31580 + 3.71425I −10.43409 − 0.73580I
u = −0.931933 + 0.895825I
a = −0.466639 + 0.398594I
b = 0.712541 + 0.015963I
9.82516 + 3.30608I −14.5483 − 3.5573I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.931933 − 0.895825I
a = −0.466639 − 0.398594I
b = 0.712541 − 0.015963I
9.82516 − 3.30608I −14.5483 + 3.5573I
u = −1.21080 + 0.76031I
a = −0.31276 + 1.60256I
b = 1.83309 − 0.17932I
−12.5061 + 10.4262I −11.73103 − 4.47783I
u = −1.21080 − 0.76031I
a = −0.31276 − 1.60256I
b = 1.83309 + 0.17932I
−12.5061 − 10.4262I −11.73103 + 4.47783I
u = 1.46326
a = 0.512604
b = 1.84763
−18.0985 −13.9500
u = −0.457334
a = 0.825053
b = −0.204761
−0.594889 −17.0300
6
II. I
u
2
= hu
10
+ u
9
− u
8
+ 4u
6
+ 3u
5
− 2u
4
+ 3u
2
+ b + 2u, u
10
+ 3u
6
+ u
5
+
u
4
− u
3
+ 2u
2
+ a + 2u + 1, u
11
+ u
10
+ · · · + 4u
2
− 1i
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
3
=
−u
−u
3
+ u
a
5
=
−u
10
− 3u
6
− u
5
− u
4
+ u
3
− 2u
2
− 2u − 1
−u
10
− u
9
+ u
8
− 4u
6
− 3u
5
+ 2u
4
− 3u
2
− 2u
a
9
=
−u
2
+ 1
u
2
a
10
=
−u
10
+ 2u
8
− 5u
6
+ 6u
4
− 6u
2
− u + 4
u
10
+ u
9
− u
8
− u
7
+ 4u
6
+ 3u
5
− 2u
4
− 2u
3
+ 3u
2
+ 2u
a
11
=
−2u
10
− u
9
+ 3u
8
+ u
7
− 9u
6
− 3u
5
+ 8u
4
+ 2u
3
− 9u
2
− 3u + 4
u
10
+ u
9
− u
8
− u
7
+ 4u
6
+ 3u
5
− 2u
4
− 2u
3
+ 3u
2
+ 2u
a
1
=
u
3
u
5
− u
3
+ u
a
4
=
u
7
− 2u
5
+ 2u
3
− 2u
−u
7
+ u
5
− 2u
3
+ u
a
6
=
−2u
10
− u
9
+ u
8
− 7u
6
− 3u
5
+ u
4
+ u
3
− 5u
2
− 3u
−u
10
− u
9
+ u
8
+ u
7
− 4u
6
− 4u
5
+ 2u
4
+ 2u
3
− 3u
2
− 3u
a
12
=
−u
10
+ 2u
8
− 5u
6
+ 6u
4
+ u
3
− 5u
2
− u + 3
u
10
+ u
9
− u
8
− u
7
+ 4u
6
+ 4u
5
− 2u
4
− 3u
3
+ 2u
2
+ 3u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
10
+ 2u
7
+ 8u
6
− 2u
5
+ u
3
+ 10u
2
+ u − 10
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
11
− 3u
10
+ ··· + 8u − 1
c
2
u
11
− u
10
− u
9
+ u
8
+ 4u
7
− 4u
6
− 2u
5
+ 3u
4
+ 3u
3
− 4u
2
+ 1
c
3
u
11
− u
10
+ 7u
9
+ u
8
− 2u
7
+ 5u
6
+ 39u
5
− 31u
4
+ 12u
3
− u
2
− 2u + 1
c
4
, c
10
u
11
+ 2u
10
+ 2u
9
+ 2u
8
− u
7
− 3u
6
− u
5
+ 2u
3
+ 3u
2
+ u + 1
c
5
, c
9
u
11
+ u
10
+ 3u
9
+ 2u
8
− u
6
− 3u
5
− u
4
+ 2u
3
+ 2u
2
+ 2u + 1
c
6
u
11
− 2u
10
+ 2u
9
− 2u
8
− u
7
+ 3u
6
− u
5
+ 2u
3
− 3u
2
+ u − 1
c
7
u
11
+ u
10
− u
9
− u
8
+ 4u
7
+ 4u
6
− 2u
5
− 3u
4
+ 3u
3
+ 4u
2
− 1
c
8
u
11
+ 3u
10
+ ··· + 8u + 1
c
11
u
11
− 11u
10
+ ··· − 24u + 9
c
12
u
11
− u
10
+ 3u
9
− 2u
8
+ u
6
− 3u
5
+ u
4
+ 2u
3
− 2u
2
+ 2u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
11
+ 13y
10
+ ··· + 20y −1
c
2
, c
7
y
11
− 3y
10
+ ··· + 8y −1
c
3
y
11
+ 13y
10
+ ··· + 6y −1
c
4
, c
6
, c
10
y
11
− 6y
9
+ 2y
8
+ 13y
7
− 9y
6
− 15y
5
+ 8y
4
+ 8y
3
− 5y
2
− 5y −1
c
5
, c
9
, c
12
y
11
+ 5y
10
+ 5y
9
− 8y
8
− 8y
7
+ 15y
6
+ 9y
5
− 13y
4
− 2y
3
+ 6y
2
− 1
c
11
y
11
− 23y
10
+ ··· − 324y −81
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.859595 + 0.621070I
a = 0.264591 − 0.511619I
b = 1.184910 − 0.173635I
5.06364 − 2.43633I −9.98510 + 2.91167I
u = 0.859595 − 0.621070I
a = 0.264591 + 0.511619I
b = 1.184910 + 0.173635I
5.06364 + 2.43633I −9.98510 − 2.91167I
u = −0.715758 + 0.795244I
a = 1.83523 − 0.23082I
b = −1.61321 − 0.43685I
−1.149260 + 0.247570I −13.50982 − 0.73342I
u = −0.715758 − 0.795244I
a = 1.83523 + 0.23082I
b = −1.61321 + 0.43685I
−1.149260 − 0.247570I −13.50982 + 0.73342I
u = −0.791184 + 0.262463I
a = −0.50598 + 1.77609I
b = 0.389923 − 0.338442I
3.12519 + 1.08690I −6.47529 − 6.28285I
u = −0.791184 − 0.262463I
a = −0.50598 − 1.77609I
b = 0.389923 + 0.338442I
3.12519 − 1.08690I −6.47529 + 6.28285I
u = −1.006190 + 0.705559I
a = 0.60734 − 2.06814I
b = −1.77582 + 0.58284I
−2.06494 + 5.42980I −15.7370 − 3.3620I
u = −1.006190 − 0.705559I
a = 0.60734 + 2.06814I
b = −1.77582 − 0.58284I
−2.06494 − 5.42980I −15.7370 + 3.3620I
u = 0.925242 + 0.874685I
a = −0.038280 + 0.149800I
b = −0.412394 + 0.056790I
10.30640 − 3.24156I 2.36799 + 1.55443I
u = 0.925242 − 0.874685I
a = −0.038280 − 0.149800I
b = −0.412394 − 0.056790I
10.30640 + 3.24156I 2.36799 − 1.55443I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.456590
a = −2.32582
b = −1.54682
−4.24309 −7.32160
11
III. I
u
3
= hb + 2, a + 1, u − 1i
(i) Arc colorings
a
2
=
0
1
a
7
=
1
0
a
8
=
1
1
a
3
=
−1
0
a
5
=
−1
−2
a
9
=
0
1
a
10
=
−1
−1
a
11
=
0
−1
a
1
=
1
1
a
4
=
−1
−1
a
6
=
0
−1
a
12
=
0
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −24
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
7
, c
9
, c
10
u − 1
c
2
, c
3
, c
6
c
8
, c
12
u + 1
c
11
u
13
(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
8
, c
9
c
10
, c
12
y −1
c
11
y
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = −2.00000
−6.57974 −24.0000
15
IV. I
u
4
= ha
3
+ 2a
2
+ 3b + a + 5, a
4
+ a
3
+ 2a
2
+ 4a + 1, u − 1i
(i) Arc colorings
a
2
=
0
1
a
7
=
1
0
a
8
=
1
1
a
3
=
−1
0
a
5
=
a
−
1
3
a
3
−
2
3
a
2
−
1
3
a −
5
3
a
9
=
0
1
a
10
=
−a
2
1
3
a
3
−
1
3
a
2
+
1
3
a +
2
3
a
11
=
−
1
3
a
3
−
2
3
a
2
−
1
3
a −
2
3
1
3
a
3
−
1
3
a
2
+
1
3
a +
2
3
a
1
=
1
1
a
4
=
−1
−1
a
6
=
1
3
a
3
+
2
3
a
2
+
7
3
a +
5
3
a
a
12
=
1
3
a
3
+
2
3
a
2
+
1
3
a +
2
3
−
2
3
a
3
−
1
3
a
2
−
2
3
a −
4
3
(ii) Obstruction class = −1
(iii) Cusp Shapes = −14
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
(u + 1)
4
c
2
, c
3
, c
7
(u − 1)
4
c
4
, c
6
, c
10
u
4
+ u
3
− 2u − 1
c
5
, c
9
, c
12
u
4
− u
3
+ 2u
2
− 4u + 1
c
11
(u
2
− u − 1)
2
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
7
, c
8
(y −1)
4
c
4
, c
6
, c
10
y
4
− y
3
+ 2y
2
− 4y + 1
c
5
, c
9
, c
12
y
4
+ 3y
3
− 2y
2
− 12y + 1
c
11
(y
2
− 3y + 1)
2
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.33107
b = −1.61803
−5.59278 −14.0000
u = 1.00000
a = 0.30902 + 1.58825I
b = 0.618034
2.30291 −14.0000
u = 1.00000
a = 0.30902 − 1.58825I
b = 0.618034
2.30291 −14.0000
u = 1.00000
a = −0.286961
b = −1.61803
−5.59278 −14.0000
19
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)(u + 1)
4
(u
11
− 3u
10
+ ··· + 8u − 1)(u
15
+ 5u
14
+ ··· + 108u + 16)
c
2
(u − 1)
4
(u + 1)
· (u
11
− u
10
− u
9
+ u
8
+ 4u
7
− 4u
6
− 2u
5
+ 3u
4
+ 3u
3
− 4u
2
+ 1)
· (u
15
+ 5u
14
+ ··· + 18u + 4)
c
3
(u − 1)
4
(u + 1)
· (u
11
− u
10
+ 7u
9
+ u
8
− 2u
7
+ 5u
6
+ 39u
5
− 31u
4
+ 12u
3
− u
2
− 2u + 1)
· (u
15
− 7u
14
+ ··· − 7974u + 2196)
c
4
, c
10
(u − 1)(u
4
+ u
3
− 2u − 1)
· (u
11
+ 2u
10
+ 2u
9
+ 2u
8
− u
7
− 3u
6
− u
5
+ 2u
3
+ 3u
2
+ u + 1)
· (u
15
− 3u
14
+ ··· + 2u + 1)
c
5
, c
9
(u − 1)(u
4
− u
3
+ 2u
2
− 4u + 1)
· (u
11
+ u
10
+ 3u
9
+ 2u
8
− u
6
− 3u
5
− u
4
+ 2u
3
+ 2u
2
+ 2u + 1)
· (u
15
+ 2u
14
+ ··· − 3u − 1)
c
6
(u + 1)(u
4
+ u
3
− 2u − 1)
· (u
11
− 2u
10
+ 2u
9
− 2u
8
− u
7
+ 3u
6
− u
5
+ 2u
3
− 3u
2
+ u − 1)
· (u
15
− 3u
14
+ ··· + 2u + 1)
c
7
(u − 1)
5
(u
11
+ u
10
− u
9
− u
8
+ 4u
7
+ 4u
6
− 2u
5
− 3u
4
+ 3u
3
+ 4u
2
− 1)
· (u
15
+ 5u
14
+ ··· + 18u + 4)
c
8
((u + 1)
5
)(u
11
+ 3u
10
+ ··· + 8u + 1)(u
15
+ 5u
14
+ ··· + 108u + 16)
c
11
u(u
2
− u − 1)
2
(u
11
− 11u
10
+ ··· − 24u + 9)
· (u
15
+ 8u
14
+ ··· + 26u + 2)
c
12
(u + 1)(u
4
− u
3
+ 2u
2
− 4u + 1)
· (u
11
− u
10
+ 3u
9
− 2u
8
+ u
6
− 3u
5
+ u
4
+ 2u
3
− 2u
2
+ 2u − 1)
· (u
15
+ 2u
14
+ ··· − 3u − 1)
20
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
8
((y −1)
5
)(y
11
+ 13y
10
+ ··· + 20y −1)
· (y
15
+ 11y
14
+ ··· + 4464y −256)
c
2
, c
7
((y −1)
5
)(y
11
− 3y
10
+ ··· + 8y −1)(y
15
− 5y
14
+ ··· + 108y −16)
c
3
((y −1)
5
)(y
11
+ 13y
10
+ ··· + 6y −1)
· (y
15
− 89y
14
+ ··· + 102923820y −4822416)
c
4
, c
6
, c
10
(y −1)(y
4
− y
3
+ 2y
2
− 4y + 1)
· (y
11
− 6y
9
+ 2y
8
+ 13y
7
− 9y
6
− 15y
5
+ 8y
4
+ 8y
3
− 5y
2
− 5y −1)
· (y
15
− 41y
14
+ ··· + 36y −1)
c
5
, c
9
, c
12
(y −1)(y
4
+ 3y
3
− 2y
2
− 12y + 1)
· (y
11
+ 5y
10
+ 5y
9
− 8y
8
− 8y
7
+ 15y
6
+ 9y
5
− 13y
4
− 2y
3
+ 6y
2
− 1)
· (y
15
− 28y
14
+ ··· − 11y −1)
c
11
y(y
2
− 3y + 1)
2
(y
11
− 23y
10
+ ··· − 324y −81)
· (y
15
− 38y
14
+ ··· + 200y −4)
21
