12n
0546
(K12n
0546
)
A knot diagram
1
Linearized knot diagam
3 5 9 10 2 11 5 12 4 7 3 8
Solving Sequence
4,9
10 5 3 2
1,12
8 7 11 6
c
9
c
4
c
3
c
2
c
1
c
8
c
7
c
11
c
6
c
5
, c
10
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
18
− 2u
17
+ ··· + b − 1, −3u
18
+ 5u
17
+ ··· + 2a + 3, u
19
− 3u
18
+ ··· + u + 2i
I
u
2
= h3u
9
a + 3u
9
+ ··· − a + 7, 2u
9
a + 3u
9
+ ··· + 3a + 9,
u
10
+ u
9
− 5u
8
− 4u
7
+ 8u
6
+ 3u
5
− 5u
4
+ 2u
3
+ 3u
2
+ u − 1i
I
u
3
= h−u
7
+ 4u
5
− 4u
3
+ b, u
7
− 3u
5
+ u
4
+ u
3
− 3u
2
+ a + 2u + 1, u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 47 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
18
−2u
17
+· · ·+b−1, −3u
18
+5u
17
+· · ·+2a+3, u
19
−3u
18
+· · ·+u+2i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
5
=
u
−u
3
+ u
a
3
=
−u
u
a
2
=
u
5
− 2u
3
− u
−u
7
+ 3u
5
− 2u
3
+ u
a
1
=
−u
9
+ 4u
7
− 3u
5
− 2u
3
− u
u
9
− 5u
7
+ 7u
5
− 2u
3
+ u
a
12
=
3
2
u
18
−
5
2
u
17
+ ··· −
5
2
u −
3
2
−u
18
+ 2u
17
+ ··· + 2u + 1
a
8
=
−
1
2
u
18
+
1
2
u
17
+ ··· −
1
2
u +
1
2
u
18
− u
17
+ ··· − 2u − 1
a
7
=
3
2
u
18
−
5
2
u
17
+ ··· −
5
2
u −
3
2
−2u
18
+ 3u
17
+ ··· + 3u + 3
a
11
=
−
1
2
u
18
+
1
2
u
17
+ ··· −
1
2
u +
1
2
u
18
− u
17
+ ··· − u
2
− 1
a
6
=
−u
9
+ 4u
7
− 3u
5
− 2u
3
− u
u
11
− 5u
9
+ 8u
7
− 5u
5
+ 3u
3
− u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
18
+ 4u
17
+ 38u
16
− 28u
15
− 150u
14
+ 56u
13
+ 322u
12
+
22u
11
−400u
10
−208u
9
+ 252u
8
+ 260u
7
−22u
6
−128u
5
−46u
4
+ 12u
3
+ 16u
2
+ 12u + 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
19
+ 15u
18
+ ··· + 5217u − 64
c
2
, c
5
u
19
+ 3u
18
+ ··· + 55u − 8
c
3
, c
4
, c
9
u
19
+ 3u
18
+ ··· + u − 2
c
6
, c
8
, c
10
c
12
u
19
+ 2u
17
+ ··· − 4u
2
− 1
c
7
, c
11
u
19
+ 4u
18
+ ··· − 20u + 7
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
19
− 21y
18
+ ··· + 25516865y −4096
c
2
, c
5
y
19
+ 15y
18
+ ··· + 5217y −64
c
3
, c
4
, c
9
y
19
− 21y
18
+ ··· − 3y −4
c
6
, c
8
, c
10
c
12
y
19
+ 4y
18
+ ··· − 8y −1
c
7
, c
11
y
19
+ 20y
18
+ ··· − 818y −49
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.580262 + 0.658546I
a = −1.37393 + 1.42710I
b = −0.88981 − 1.12332I
−6.36763 − 9.41239I 2.24569 + 7.34910I
u = −0.580262 − 0.658546I
a = −1.37393 − 1.42710I
b = −0.88981 + 1.12332I
−6.36763 + 9.41239I 2.24569 − 7.34910I
u = −0.430386 + 0.693344I
a = 0.0599143 − 0.0949401I
b = 0.92934 − 1.06686I
−6.81456 + 4.87246I 1.03546 − 1.73049I
u = −0.430386 − 0.693344I
a = 0.0599143 + 0.0949401I
b = 0.92934 + 1.06686I
−6.81456 − 4.87246I 1.03546 + 1.73049I
u = 0.736476 + 0.343968I
a = −0.27984 − 1.85203I
b = −0.603953 + 0.709870I
1.05767 + 4.20616I 4.94312 − 8.38858I
u = 0.736476 − 0.343968I
a = −0.27984 + 1.85203I
b = −0.603953 − 0.709870I
1.05767 − 4.20616I 4.94312 + 8.38858I
u = −0.609189 + 0.322877I
a = 0.730128 − 0.841386I
b = −0.206405 + 0.511932I
1.17956 − 0.90815I 4.85874 + 1.33440I
u = −0.609189 − 0.322877I
a = 0.730128 + 0.841386I
b = −0.206405 − 0.511932I
1.17956 + 0.90815I 4.85874 − 1.33440I
u = −1.39358
a = 0.388481
b = −0.879243
3.32984 1.51140
u = 1.45691 + 0.21847I
a = 0.750156 + 0.704154I
b = −0.975115 − 0.981574I
−0.73866 − 1.59783I 4.12138 + 1.54661I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.45691 − 0.21847I
a = 0.750156 − 0.704154I
b = −0.975115 + 0.981574I
−0.73866 + 1.59783I 4.12138 − 1.54661I
u = 0.094599 + 0.501708I
a = 0.412228 − 0.106777I
b = 0.620442 + 0.453397I
−0.91663 − 1.23074I −2.16741 + 3.03279I
u = 0.094599 − 0.501708I
a = 0.412228 + 0.106777I
b = 0.620442 − 0.453397I
−0.91663 + 1.23074I −2.16741 − 3.03279I
u = 1.55341 + 0.21071I
a = 0.57402 + 2.23414I
b = 0.84741 − 1.16742I
0.68963 + 12.60340I 5.72673 − 6.88576I
u = 1.55341 − 0.21071I
a = 0.57402 − 2.23414I
b = 0.84741 + 1.16742I
0.68963 − 12.60340I 5.72673 + 6.88576I
u = 1.57600 + 0.10063I
a = −0.61810 − 1.45739I
b = 0.193132 + 0.647212I
8.62226 + 2.51189I 5.75786 + 1.20779I
u = 1.57600 − 0.10063I
a = −0.61810 + 1.45739I
b = 0.193132 − 0.647212I
8.62226 − 2.51189I 5.75786 − 1.20779I
u = −1.60077 + 0.07995I
a = −0.19881 − 2.15189I
b = 0.524578 + 0.835989I
9.02564 − 5.69404I 7.72276 + 7.13841I
u = −1.60077 − 0.07995I
a = −0.19881 + 2.15189I
b = 0.524578 − 0.835989I
9.02564 + 5.69404I 7.72276 − 7.13841I
6
II.
I
u
2
= h3u
9
a + 3u
9
+· · · − a + 7, 2u
9
a + 3u
9
+· · · + 3a + 9, u
10
+u
9
+· · · + u − 1i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
5
=
u
−u
3
+ u
a
3
=
−u
u
a
2
=
u
5
− 2u
3
− u
−u
7
+ 3u
5
− 2u
3
+ u
a
1
=
−u
9
+ 4u
7
− 3u
5
− 2u
3
− u
u
9
− 5u
7
+ 7u
5
− 2u
3
+ u
a
12
=
a
−
3
8
u
9
a −
3
8
u
9
+ ··· +
1
8
a −
7
8
a
8
=
−
3
8
u
9
a −
3
8
u
9
+ ··· −
7
8
a −
23
8
−
3
8
u
9
a −
3
8
u
9
+ ··· +
1
8
a +
9
8
a
7
=
−2u
9
− 2u
8
+ 8u
7
+ 8u
6
− 8u
5
− 8u
4
+ 2u
3
+ u
2
− a − 6u − 3
−
3
8
u
9
a +
13
8
u
9
+ ··· +
1
8
a −
7
8
a
11
=
3
8
u
9
a +
3
8
u
9
+ ··· +
7
8
a −
1
8
−
3
4
u
9
a −
3
4
u
9
+ ··· +
1
4
a −
3
4
a
6
=
−u
9
+ 4u
7
− 3u
5
− 2u
3
− u
u
9
− u
8
− 4u
7
+ 5u
6
+ 3u
5
− 7u
4
+ 2u
3
+ 2u
2
+ u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
8
− 20u
6
+ 28u
4
− 4u
3
− 8u
2
+ 8u + 10
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
10
+ 13u
9
+ ··· − 7u + 1)
2
c
2
, c
5
(u
10
+ u
9
+ 7u
8
+ 6u
7
+ 16u
6
+ 11u
5
+ 13u
4
+ 6u
3
+ 3u
2
+ u − 1)
2
c
3
, c
4
, c
9
(u
10
− u
9
− 5u
8
+ 4u
7
+ 8u
6
− 3u
5
− 5u
4
− 2u
3
+ 3u
2
− u − 1)
2
c
6
, c
8
, c
10
c
12
u
20
+ u
19
+ ··· − u + 2
c
7
, c
11
u
20
+ u
19
+ ··· + 637u + 1708
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
10
− 31y
9
+ ··· − 107y + 1)
2
c
2
, c
5
(y
10
+ 13y
9
+ ··· − 7y + 1)
2
c
3
, c
4
, c
9
(y
10
− 11y
9
+ ··· − 7y + 1)
2
c
6
, c
8
, c
10
c
12
y
20
+ 7y
19
+ ··· + 35y + 4
c
7
, c
11
y
20
− y
19
+ ··· − 641473y + 2917264
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.510102 + 0.680941I
a = 1.03901 + 0.99672I
b = 1.030620 − 0.877048I
−7.43042 + 2.28632I 0.39779 − 2.91176I
u = 0.510102 + 0.680941I
a = 0.139978 − 0.495824I
b = −1.064860 − 0.795059I
−7.43042 + 2.28632I 0.39779 − 2.91176I
u = 0.510102 − 0.680941I
a = 1.03901 − 0.99672I
b = 1.030620 + 0.877048I
−7.43042 − 2.28632I 0.39779 + 2.91176I
u = 0.510102 − 0.680941I
a = 0.139978 + 0.495824I
b = −1.064860 + 0.795059I
−7.43042 − 2.28632I 0.39779 + 2.91176I
u = −0.449833 + 0.459351I
a = −0.646997 − 0.974683I
b = −0.210455 − 0.300293I
1.43061 − 1.60532I 0.94346 + 5.03395I
u = −0.449833 + 0.459351I
a = 1.22866 − 0.93250I
b = 0.057050 + 1.133970I
1.43061 − 1.60532I 0.94346 + 5.03395I
u = −0.449833 − 0.459351I
a = −0.646997 + 0.974683I
b = −0.210455 + 0.300293I
1.43061 + 1.60532I 0.94346 − 5.03395I
u = −0.449833 − 0.459351I
a = 1.22866 + 0.93250I
b = 0.057050 − 1.133970I
1.43061 + 1.60532I 0.94346 − 5.03395I
u = 1.50079 + 0.11328I
a = 0.415489 − 0.034479I
b = 0.528203 − 0.415736I
7.87146 + 3.55946I 5.64226 − 4.06361I
u = 1.50079 + 0.11328I
a = −0.70682 − 2.20481I
b = −0.107369 + 1.258670I
7.87146 + 3.55946I 5.64226 − 4.06361I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.50079 − 0.11328I
a = 0.415489 + 0.034479I
b = 0.528203 + 0.415736I
7.87146 − 3.55946I 5.64226 + 4.06361I
u = 1.50079 − 0.11328I
a = −0.70682 + 2.20481I
b = −0.107369 − 1.258670I
7.87146 − 3.55946I 5.64226 + 4.06361I
u = −1.50960
a = 0.93389 + 2.29683I
b = 0.317795 − 1.141480I
10.4232 10.0490
u = −1.50960
a = 0.93389 − 2.29683I
b = 0.317795 + 1.141480I
10.4232 10.0490
u = −1.51481 + 0.22020I
a = −0.956858 + 0.125386I
b = 1.093790 − 0.701969I
−0.80829 − 5.55652I 3.79190 + 2.88175I
u = −1.51481 + 0.22020I
a = −0.21158 + 1.73958I
b = −0.985922 − 0.960677I
−0.80829 − 5.55652I 3.79190 + 2.88175I
u = −1.51481 − 0.22020I
a = −0.956858 − 0.125386I
b = 1.093790 + 0.701969I
−0.80829 + 5.55652I 3.79190 − 2.88175I
u = −1.51481 − 0.22020I
a = −0.21158 − 1.73958I
b = −0.985922 + 0.960677I
−0.80829 + 5.55652I 3.79190 − 2.88175I
u = 0.417104
a = −2.73478 + 2.69676I
b = −0.158842 − 1.037160I
3.89939 12.4010
u = 0.417104
a = −2.73478 − 2.69676I
b = −0.158842 + 1.037160I
3.89939 12.4010
11
III. I
u
3
=
h−u
7
+4u
5
−4u
3
+b, u
7
−3u
5
+u
4
+u
3
−3u
2
+a+2u+1, u
8
−5u
6
+7u
4
−2u
2
+1i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
5
=
u
−u
3
+ u
a
3
=
−u
u
a
2
=
u
5
− 2u
3
− u
−u
7
+ 3u
5
− 2u
3
+ u
a
1
=
−u
7
+ 4u
5
− 4u
3
0
a
12
=
−u
7
+ 3u
5
− u
4
− u
3
+ 3u
2
− 2u − 1
u
7
− 4u
5
+ 4u
3
a
8
=
u
6
− 4u
4
− u
3
+ 4u
2
+ 2u − 1
1
a
7
=
u
7
− 3u
5
− u
4
+ u
3
+ 3u
2
+ 2u − 1
−u
7
+ u
6
+ 2u
5
− 3u
4
+ u
2
+ u
a
11
=
u
6
− 4u
4
+ u
3
+ 4u
2
− 2u − 1
−u
6
− u
5
+ 3u
4
+ 2u
3
− u
2
a
6
=
u
7
− 4u
5
+ 4u
3
−u
7
+ 3u
5
− 2u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
6
+ 16u
4
− 16u
2
+ 12
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
2
(u
4
− u
3
+ u
2
+ 1)
2
c
3
, c
4
, c
9
u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1
c
5
(u
4
+ u
3
+ u
2
+ 1)
2
c
6
, c
8
, c
10
c
12
(u
2
+ 1)
4
c
7
u
8
− 6u
7
+ 20u
6
− 52u
5
+ 97u
4
− 112u
3
+ 87u
2
− 62u + 29
c
11
u
8
+ 6u
7
+ 20u
6
+ 52u
5
+ 97u
4
+ 112u
3
+ 87u
2
+ 62u + 29
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
c
2
, c
5
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
c
3
, c
4
, c
9
(y
4
− 5y
3
+ 7y
2
− 2y + 1)
2
c
6
, c
8
, c
10
c
12
(y + 1)
8
c
7
, c
11
y
8
+ 4y
7
− 30y
6
+ 6y
5
+ 555y
4
− 954y
3
− 693y
2
+ 1202y + 841
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.506844 + 0.395123I
a = −1.77461 + 0.07756I
b = 1.000000I
3.07886 + 1.41510I 8.17326 − 4.90874I
u = 0.506844 − 0.395123I
a = −1.77461 − 0.07756I
b = − 1.000000I
3.07886 − 1.41510I 8.17326 + 4.90874I
u = −0.506844 + 0.395123I
a = 0.67976 − 2.16419I
b = 1.000000I
3.07886 − 1.41510I 8.17326 + 4.90874I
u = −0.506844 − 0.395123I
a = 0.67976 + 2.16419I
b = − 1.000000I
3.07886 + 1.41510I 8.17326 − 4.90874I
u = 1.55249 + 0.10488I
a = −0.09378 − 2.54234I
b = 1.000000I
10.08060 + 3.16396I 11.82674 − 2.56480I
u = 1.55249 − 0.10488I
a = −0.09378 + 2.54234I
b = − 1.000000I
10.08060 − 3.16396I 11.82674 + 2.56480I
u = −1.55249 + 0.10488I
a = 1.18862 − 1.37103I
b = 1.000000I
10.08060 − 3.16396I 11.82674 + 2.56480I
u = −1.55249 − 0.10488I
a = 1.18862 + 1.37103I
b = − 1.000000I
10.08060 + 3.16396I 11.82674 − 2.56480I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
4
− u
3
+ 3u
2
− 2u + 1)
2
)(u
10
+ 13u
9
+ ··· − 7u + 1)
2
· (u
19
+ 15u
18
+ ··· + 5217u − 64)
c
2
(u
4
− u
3
+ u
2
+ 1)
2
· (u
10
+ u
9
+ 7u
8
+ 6u
7
+ 16u
6
+ 11u
5
+ 13u
4
+ 6u
3
+ 3u
2
+ u − 1)
2
· (u
19
+ 3u
18
+ ··· + 55u − 8)
c
3
, c
4
, c
9
(u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1)
· (u
10
− u
9
− 5u
8
+ 4u
7
+ 8u
6
− 3u
5
− 5u
4
− 2u
3
+ 3u
2
− u − 1)
2
· (u
19
+ 3u
18
+ ··· + u − 2)
c
5
(u
4
+ u
3
+ u
2
+ 1)
2
· (u
10
+ u
9
+ 7u
8
+ 6u
7
+ 16u
6
+ 11u
5
+ 13u
4
+ 6u
3
+ 3u
2
+ u − 1)
2
· (u
19
+ 3u
18
+ ··· + 55u − 8)
c
6
, c
8
, c
10
c
12
((u
2
+ 1)
4
)(u
19
+ 2u
17
+ ··· − 4u
2
− 1)(u
20
+ u
19
+ ··· − u + 2)
c
7
(u
8
− 6u
7
+ 20u
6
− 52u
5
+ 97u
4
− 112u
3
+ 87u
2
− 62u + 29)
· (u
19
+ 4u
18
+ ··· − 20u + 7)(u
20
+ u
19
+ ··· + 637u + 1708)
c
11
(u
8
+ 6u
7
+ 20u
6
+ 52u
5
+ 97u
4
+ 112u
3
+ 87u
2
+ 62u + 29)
· (u
19
+ 4u
18
+ ··· − 20u + 7)(u
20
+ u
19
+ ··· + 637u + 1708)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
)(y
10
− 31y
9
+ ··· − 107y + 1)
2
· (y
19
− 21y
18
+ ··· + 25516865y − 4096)
c
2
, c
5
((y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
)(y
10
+ 13y
9
+ ··· − 7y + 1)
2
· (y
19
+ 15y
18
+ ··· + 5217y − 64)
c
3
, c
4
, c
9
((y
4
− 5y
3
+ 7y
2
− 2y + 1)
2
)(y
10
− 11y
9
+ ··· − 7y + 1)
2
· (y
19
− 21y
18
+ ··· − 3y − 4)
c
6
, c
8
, c
10
c
12
((y + 1)
8
)(y
19
+ 4y
18
+ ··· − 8y − 1)(y
20
+ 7y
19
+ ··· + 35y + 4)
c
7
, c
11
(y
8
+ 4y
7
− 30y
6
+ 6y
5
+ 555y
4
− 954y
3
− 693y
2
+ 1202y + 841)
· (y
19
+ 20y
18
+ ··· − 818y − 49)
· (y
20
− y
19
+ ··· − 641473y + 2917264)
17
