12a
0270
(K12a
0270
)
A knot diagram
1
Linearized knot diagam
3 6 8 7 11 2 4 1 12 5 10 9
Solving Sequence
1,8 4,9
3 2 7 5 6 12 10 11
c
8
c
3
c
1
c
7
c
4
c
6
c
12
c
9
c
11
c
2
, c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−798268399u
41
+ 6250449929u
40
+ ··· + 10357680512b − 2584646788,
− 358149411u
41
+ 2967411165u
40
+ ··· + 5178840256a − 14354766596,
u
42
− 8u
41
+ ··· + 19u + 4i
I
u
2
= hu
4
a
2
− 3u
3
a
2
+ 3u
4
a + 4a
2
u
2
+ 2u
4
− 5a
2
u + 3u
2
a − 6u
3
+ a
2
+ 3au + 8u
2
+ 3b − 3a − 10u + 2,
2u
4
a
2
− 2u
3
a
2
+ 3u
4
a + 8a
2
u
2
− 2u
3
a + a
3
− 6a
2
u + 11u
2
a + 6a
2
− 5au + 10a + u,
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1i
I
u
3
= h−u
3
+ u
2
+ b + a − 3u + 2, −2u
3
a + 2u
2
a − 2u
3
+ a
2
− 6au + u
2
+ 4a − 5u + 2, u
4
− u
3
+ 3u
2
− 2u + 1i
* 3 irreducible components of dim
C
= 0, with total 65 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−7.98 × 10
8
u
41
+ 6.25 × 10
9
u
40
+ · · · + 1.04× 10
10
b − 2.58 ×10
9
, −3.58 ×
10
8
u
41
+2.97×10
9
u
40
+· · ·+5.18×10
9
a−1.44×10
10
, u
42
−8u
41
+· · ·+19u+4i
(i) Arc colorings
a
1
=
0
u
a
8
=
1
0
a
4
=
0.0691563u
41
− 0.572988u
40
+ ··· + 3.90826u + 2.77181
0.0770702u
41
− 0.603460u
40
+ ··· − 0.703314u + 0.249539
a
9
=
1
u
2
a
3
=
0.146226u
41
− 1.17645u
40
+ ··· + 3.20495u + 3.02135
0.0770702u
41
− 0.603460u
40
+ ··· − 0.703314u + 0.249539
a
2
=
0.0816169u
41
− 0.664619u
40
+ ··· + 6.14207u + 0.645428
0.0288129u
41
− 0.231921u
40
+ ··· + 1.24875u + 0.105032
a
7
=
−0.0196220u
41
+ 0.147935u
40
+ ··· − 6.11036u + 0.584750
−0.00663609u
41
+ 0.0909429u
40
+ ··· + 1.05555u + 0.165094
a
5
=
0.0115304u
41
− 0.109146u
40
+ ··· + 7.16076u + 1.81175
0.0576259u
41
− 0.463842u
40
+ ··· − 0.502505u + 0.210065
a
6
=
−0.0525162u
41
+ 0.477755u
40
+ ··· − 5.10962u − 1.50031
−0.0197372u
41
+ 0.196786u
40
+ ··· + 0.707841u − 0.276625
a
12
=
u
u
3
+ u
a
10
=
u
2
+ 1
u
4
+ 2u
2
a
11
=
u
3
+ 2u
u
5
+ 3u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes =
1891553825
2589420128
u
41
−
14308919975
2589420128
u
40
+ ··· +
33847466689
2589420128
u −
1329586633
647355032
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
42
+ 13u
41
+ ··· + 147u + 4
c
2
, c
6
u
42
− u
41
+ ··· − 11u + 2
c
3
, c
4
, c
7
u
42
− u
41
+ ··· − 17u + 2
c
5
, c
10
u
42
− 2u
41
+ ··· − u + 2
c
8
, c
9
, c
11
c
12
u
42
+ 8u
41
+ ··· − 19u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
42
+ 41y
41
+ ··· − 1601y + 16
c
2
, c
6
y
42
+ 13y
41
+ ··· + 147y + 4
c
3
, c
4
, c
7
y
42
+ 49y
41
+ ··· + 163y + 4
c
5
, c
10
y
42
− 8y
41
+ ··· + 19y + 4
c
8
, c
9
, c
11
c
12
y
42
+ 52y
41
+ ··· − 593y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.199296 + 0.987147I
a = −1.16462 − 1.63691I
b = −0.07090 + 1.52196I
9.23309 − 0.66880I 0. + 2.09820I
u = −0.199296 − 0.987147I
a = −1.16462 + 1.63691I
b = −0.07090 − 1.52196I
9.23309 + 0.66880I 0. − 2.09820I
u = 0.838964 + 0.514549I
a = 0.469470 − 0.436996I
b = −0.043564 + 1.403040I
2.90089 − 0.20752I −6.26688 + 0.I
u = 0.838964 − 0.514549I
a = 0.469470 + 0.436996I
b = −0.043564 − 1.403040I
2.90089 + 0.20752I −6.26688 + 0.I
u = 0.915478 + 0.320876I
a = −0.717082 + 0.350239I
b = −0.16817 − 1.42645I
2.31734 − 5.40050I −8.00000 + 6.13743I
u = 0.915478 − 0.320876I
a = −0.717082 − 0.350239I
b = −0.16817 + 1.42645I
2.31734 + 5.40050I −8.00000 − 6.13743I
u = 0.450384 + 1.002590I
a = −0.392271 + 0.489442I
b = −0.762687 − 0.383581I
0.60208 − 6.73872I 0
u = 0.450384 − 1.002590I
a = −0.392271 − 0.489442I
b = −0.762687 + 0.383581I
0.60208 + 6.73872I 0
u = −0.302006 + 0.836923I
a = 1.67864 + 1.44506I
b = 0.22550 − 1.51897I
8.07529 + 5.48503I −0.70916 − 2.91852I
u = −0.302006 − 0.836923I
a = 1.67864 − 1.44506I
b = 0.22550 + 1.51897I
8.07529 − 5.48503I −0.70916 + 2.91852I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.060244 + 0.788682I
a = 0.605228 + 0.925856I
b = 0.693973 − 0.485377I
1.48615 + 2.15029I −3.71859 − 3.19032I
u = −0.060244 − 0.788682I
a = 0.605228 − 0.925856I
b = 0.693973 + 0.485377I
1.48615 − 2.15029I −3.71859 + 3.19032I
u = 0.509270 + 0.583804I
a = −0.220296 + 1.029270I
b = −0.264495 + 0.109372I
−1.98852 − 1.21898I −14.0512 + 3.6292I
u = 0.509270 − 0.583804I
a = −0.220296 − 1.029270I
b = −0.264495 − 0.109372I
−1.98852 + 1.21898I −14.0512 − 3.6292I
u = 0.588709 + 1.120860I
a = −1.07852 + 1.38460I
b = −0.26708 − 1.48568I
6.68238 − 10.46650I 0
u = 0.588709 − 1.120860I
a = −1.07852 − 1.38460I
b = −0.26708 + 1.48568I
6.68238 + 10.46650I 0
u = 0.701442 + 0.208159I
a = −0.779517 − 0.388802I
b = −0.556875 − 0.244623I
−3.10427 − 2.84153I −15.5735 + 6.4385I
u = 0.701442 − 0.208159I
a = −0.779517 + 0.388802I
b = −0.556875 + 0.244623I
−3.10427 + 2.84153I −15.5735 − 6.4385I
u = 0.439199 + 1.197000I
a = 0.69981 − 1.54440I
b = 0.13274 + 1.47292I
8.27660 − 4.56148I 0
u = 0.439199 − 1.197000I
a = 0.69981 + 1.54440I
b = 0.13274 − 1.47292I
8.27660 + 4.56148I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.220775 + 0.575673I
a = −0.342696 − 0.818451I
b = 0.139249 + 0.831769I
1.40137 − 1.70947I −0.06449 + 5.59686I
u = 0.220775 − 0.575673I
a = −0.342696 + 0.818451I
b = 0.139249 − 0.831769I
1.40137 + 1.70947I −0.06449 − 5.59686I
u = 0.11412 + 1.52011I
a = 0.001825 − 1.058650I
b = 0.013978 + 1.115610I
8.31413 − 3.12469I 0
u = 0.11412 − 1.52011I
a = 0.001825 + 1.058650I
b = 0.013978 − 1.115610I
8.31413 + 3.12469I 0
u = −0.453394 + 0.093145I
a = −0.29248 + 1.41477I
b = 0.09646 + 1.48180I
5.82303 − 2.87870I 0.05391 + 2.88471I
u = −0.453394 − 0.093145I
a = −0.29248 − 1.41477I
b = 0.09646 − 1.48180I
5.82303 + 2.87870I 0.05391 − 2.88471I
u = 0.09947 + 1.57211I
a = −0.005582 + 0.932064I
b = −0.027777 − 0.190004I
5.22550 − 3.18763I 0
u = 0.09947 − 1.57211I
a = −0.005582 − 0.932064I
b = −0.027777 + 0.190004I
5.22550 + 3.18763I 0
u = −0.01154 + 1.67140I
a = −0.049530 + 0.747153I
b = 0.911882 − 0.498339I
10.24920 + 2.39200I 0
u = −0.01154 − 1.67140I
a = −0.049530 − 0.747153I
b = 0.911882 + 0.498339I
10.24920 − 2.39200I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.08120 + 1.67874I
a = 0.80016 + 1.97320I
b = 0.33059 − 1.55890I
16.9456 + 6.9601I 0
u = −0.08120 − 1.67874I
a = 0.80016 − 1.97320I
b = 0.33059 + 1.55890I
16.9456 − 6.9601I 0
u = 0.12364 + 1.71287I
a = 0.056715 + 0.692130I
b = −0.920998 − 0.474791I
10.09190 − 9.06080I 0
u = 0.12364 − 1.71287I
a = 0.056715 − 0.692130I
b = −0.920998 + 0.474791I
10.09190 + 9.06080I 0
u = −0.03891 + 1.71786I
a = −0.51740 − 2.10632I
b = −0.20263 + 1.61169I
18.9080 + 0.2132I 0
u = −0.03891 − 1.71786I
a = −0.51740 + 2.10632I
b = −0.20263 − 1.61169I
18.9080 − 0.2132I 0
u = 0.17216 + 1.75098I
a = −0.74841 + 1.91623I
b = −0.34100 − 1.54865I
16.6556 − 13.6886I 0
u = 0.17216 − 1.75098I
a = −0.74841 − 1.91623I
b = −0.34100 + 1.54865I
16.6556 + 13.6886I 0
u = 0.12140 + 1.76169I
a = 0.47312 − 2.07005I
b = 0.21928 + 1.60308I
18.7265 − 6.9926I 0
u = 0.12140 − 1.76169I
a = 0.47312 + 2.07005I
b = 0.21928 − 1.60308I
18.7265 + 6.9926I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.148412 + 0.075498I
a = 1.14844 + 3.46411I
b = 0.362523 + 0.421697I
−0.422743 − 1.307230I −4.54925 + 5.05506I
u = −0.148412 − 0.075498I
a = 1.14844 − 3.46411I
b = 0.362523 − 0.421697I
−0.422743 + 1.307230I −4.54925 − 5.05506I
9
II. I
u
2
= hu
4
a
2
+ 3u
4
a + · · · − 3a + 2, 2u
4
a
2
+ 3u
4
a + · · · + 6a
2
+ 10a, u
5
−
u
4
+ 4u
3
− 3u
2
+ 3u − 1i
(i) Arc colorings
a
1
=
0
u
a
8
=
1
0
a
4
=
a
−
1
3
u
4
a
2
− u
4
a + ··· + a −
2
3
a
9
=
1
u
2
a
3
=
−
1
3
u
4
a
2
− u
4
a + ··· + 2a −
2
3
−
1
3
u
4
a
2
− u
4
a + ··· + a −
2
3
a
2
=
a
−
1
3
u
4
a
2
− u
4
a + ··· + a −
2
3
a
7
=
1
3
u
4
a
2
+
2
3
u
4
+ ··· + a +
2
3
−
2
3
u
4
a
2
+ u
4
a + ··· + a +
2
3
a
5
=
−u
u
a
6
=
1
0
a
12
=
u
u
3
+ u
a
10
=
u
2
+ 1
u
4
+ 2u
2
a
11
=
u
3
+ 2u
u
4
− u
3
+ 3u
2
− 2u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
4
+ 4u
3
− 16u
2
+ 12u − 14
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
15
+ 10u
14
+ ··· + 3u − 1
c
2
, c
3
, c
4
c
6
, c
7
u
15
+ 5u
13
+ ··· + 3u + 1
c
5
, c
10
(u
5
+ u
4
− u
2
+ u + 1)
3
c
8
, c
9
, c
11
c
12
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)
3
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
15
− 10y
14
+ ··· + 47y − 1
c
2
, c
3
, c
4
c
6
, c
7
y
15
+ 10y
14
+ ··· + 3y − 1
c
5
, c
10
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y − 1)
3
c
8
, c
9
, c
11
c
12
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y − 1)
3
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.233677 + 0.885557I
a = −0.387789 − 0.623465I
b = −0.497623 + 0.756574I
1.81981 − 2.21397I −3.11432 + 4.22289I
u = 0.233677 + 0.885557I
a = 0.085680 − 0.388688I
b = 0.555046 + 0.543774I
1.81981 − 2.21397I −3.11432 + 4.22289I
u = 0.233677 + 0.885557I
a = −0.25505 + 3.12360I
b = −0.057423 − 1.300350I
1.81981 − 2.21397I −3.11432 + 4.22289I
u = 0.233677 − 0.885557I
a = −0.387789 + 0.623465I
b = −0.497623 − 0.756574I
1.81981 + 2.21397I −3.11432 − 4.22289I
u = 0.233677 − 0.885557I
a = 0.085680 + 0.388688I
b = 0.555046 − 0.543774I
1.81981 + 2.21397I −3.11432 − 4.22289I
u = 0.233677 − 0.885557I
a = −0.25505 − 3.12360I
b = −0.057423 + 1.300350I
1.81981 + 2.21397I −3.11432 − 4.22289I
u = 0.416284
a = −0.0435290
b = 0.366895
−0.882183 −11.6090
u = 0.416284
a = −2.38044 + 1.97405I
b = −0.183448 − 1.049270I
−0.882183 −11.6090
u = 0.416284
a = −2.38044 − 1.97405I
b = −0.183448 + 1.049270I
−0.882183 −11.6090
u = 0.05818 + 1.69128I
a = 0.091113 − 0.799543I
b = −0.778812 + 0.748610I
10.95830 − 3.33174I −2.08126 + 2.36228I
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.05818 + 1.69128I
a = −0.117137 − 0.758678I
b = 0.789470 + 0.718695I
10.95830 − 3.33174I −2.08126 + 2.36228I
u = 0.05818 + 1.69128I
a = −0.01461 + 2.73936I
b = −0.01066 − 1.46731I
10.95830 − 3.33174I −2.08126 + 2.36228I
u = 0.05818 − 1.69128I
a = 0.091113 + 0.799543I
b = −0.778812 − 0.748610I
10.95830 + 3.33174I −2.08126 − 2.36228I
u = 0.05818 − 1.69128I
a = −0.117137 + 0.758678I
b = 0.789470 − 0.718695I
10.95830 + 3.33174I −2.08126 − 2.36228I
u = 0.05818 − 1.69128I
a = −0.01461 − 2.73936I
b = −0.01066 + 1.46731I
10.95830 + 3.33174I −2.08126 − 2.36228I
14
III.
I
u
3
= h−u
3
+u
2
+b+a−3u+2, −2u
3
a−2u
3
+· · ·+4a+2, u
4
−u
3
+3u
2
−2u+1i
(i) Arc colorings
a
1
=
0
u
a
8
=
1
0
a
4
=
a
u
3
− u
2
− a + 3u − 2
a
9
=
1
u
2
a
3
=
u
3
− u
2
+ 3u − 2
u
3
− u
2
− a + 3u − 2
a
2
=
u
3
− u
2
+ 3u − 2
u
3
− u
2
− a + 4u − 2
a
7
=
u
3
a − u
2
a + 2u
3
+ 3au − u
2
− 2a + 5u − 1
1
a
5
=
−u
3
+ u
2
+ a − 3u + 2
0
a
6
=
0
−au − 1
a
12
=
u
u
3
+ u
a
10
=
u
2
+ 1
u
3
− u
2
+ 2u − 1
a
11
=
u
3
+ 2u
u
3
− u
2
+ 2u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
3
− 4u
2
+ 12u − 12
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
8
c
2
, c
3
, c
4
c
6
, c
7
(u
2
+ 1)
4
c
5
, c
10
u
8
− u
6
+ 3u
4
− 2u
2
+ 1
c
8
, c
9
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
11
, c
12
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y − 1)
8
c
2
, c
3
, c
4
c
6
, c
7
(y + 1)
8
c
5
, c
10
(y
4
− y
3
+ 3y
2
− 2y + 1)
2
c
8
, c
9
, c
11
c
12
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.395123 + 0.506844I
a = −0.956685 + 0.227186I
b = 1.000000I
−0.21101 − 1.41510I −7.82674 + 4.90874I
u = 0.395123 + 0.506844I
a = −0.95668 + 2.22719I
b = − 1.000000I
−0.21101 − 1.41510I −7.82674 + 4.90874I
u = 0.395123 − 0.506844I
a = −0.956685 − 0.227186I
b = − 1.000000I
−0.21101 + 1.41510I −7.82674 − 4.90874I
u = 0.395123 − 0.506844I
a = −0.95668 − 2.22719I
b = 1.000000I
−0.21101 + 1.41510I −7.82674 − 4.90874I
u = 0.10488 + 1.55249I
a = −0.043315 − 0.358800I
b = 1.000000I
6.79074 − 3.16396I −4.17326 + 2.56480I
u = 0.10488 + 1.55249I
a = −0.04332 + 1.64120I
b = − 1.000000I
6.79074 − 3.16396I −4.17326 + 2.56480I
u = 0.10488 − 1.55249I
a = −0.043315 + 0.358800I
b = − 1.000000I
6.79074 + 3.16396I −4.17326 − 2.56480I
u = 0.10488 − 1.55249I
a = −0.04332 − 1.64120I
b = 1.000000I
6.79074 + 3.16396I −4.17326 − 2.56480I
18
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
15
+ 10u
14
+ ··· + 3u − 1)(u
42
+ 13u
41
+ ··· + 147u + 4)
c
2
, c
6
((u
2
+ 1)
4
)(u
15
+ 5u
13
+ ··· + 3u + 1)(u
42
− u
41
+ ··· − 11u + 2)
c
3
, c
4
, c
7
((u
2
+ 1)
4
)(u
15
+ 5u
13
+ ··· + 3u + 1)(u
42
− u
41
+ ··· − 17u + 2)
c
5
, c
10
((u
5
+ u
4
− u
2
+ u + 1)
3
)(u
8
− u
6
+ 3u
4
− 2u
2
+ 1)(u
42
− 2u
41
+ ··· − u + 2)
c
8
, c
9
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)
3
· (u
42
+ 8u
41
+ ··· − 19u + 4)
c
11
, c
12
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)
3
· (u
42
+ 8u
41
+ ··· − 19u + 4)
19
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y − 1)
8
)(y
15
− 10y
14
+ ··· + 47y − 1)(y
42
+ 41y
41
+ ··· − 1601y + 16)
c
2
, c
6
((y + 1)
8
)(y
15
+ 10y
14
+ ··· + 3y − 1)(y
42
+ 13y
41
+ ··· + 147y + 4)
c
3
, c
4
, c
7
((y + 1)
8
)(y
15
+ 10y
14
+ ··· + 3y − 1)(y
42
+ 49y
41
+ ··· + 163y + 4)
c
5
, c
10
(y
4
− y
3
+ 3y
2
− 2y + 1)
2
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y − 1)
3
· (y
42
− 8y
41
+ ··· + 19y + 4)
c
8
, c
9
, c
11
c
12
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y − 1)
3
· (y
42
+ 52y
41
+ ··· − 593y + 16)
20
