12n
0447
(K12n
0447
)
A knot diagram
1
Linearized knot diagam
3 5 9 11 2 4 12 3 8 5 7 10
Solving Sequence
3,8
9 4
5,10
11 2 6 1 12 7
c
8
c
3
c
9
c
10
c
2
c
5
c
1
c
12
c
7
c
4
, c
6
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−2.24038 × 10
33
u
47
− 1.35892 × 10
32
u
46
+ ··· + 2.30349 × 10
33
b + 1.34156 × 10
34
,
1.73741 × 10
34
u
47
− 3.00948 × 10
33
u
46
+ ··· + 1.15175 × 10
34
a − 1.11706 × 10
35
, u
48
− u
47
+ ··· − 2u + 5i
I
u
2
= h−u
2
a + u
3
+ b + a − u, a
2
u
2
+ a
3
+ 1, u
4
− u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 60 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−2.24×10
33
u
47
−1.36×10
32
u
46
+· · ·+2.30×10
33
b+1.34×10
34
, 1.74×
10
34
u
47
−3.01×10
33
u
46
+· · ·+1.15×10
34
a−1.12×10
35
, u
48
−u
47
+· · ·−2u+5i
(i) Arc colorings
a
3
=
0
u
a
8
=
1
0
a
9
=
1
u
2
a
4
=
−u
−u
3
+ u
a
5
=
−1.50850u
47
+ 0.261297u
46
+ ··· + 4.48572u + 9.69889
0.972603u
47
+ 0.0589938u
46
+ ··· − 0.659198u − 5.82402
a
10
=
−u
2
+ 1
u
2
a
11
=
1.16480u
47
− 0.192201u
46
+ ··· − 4.27812u − 2.98881
1.24721u
47
− 0.462790u
46
+ ··· − 6.68188u − 7.54251
a
2
=
1.76087u
47
+ 0.00355222u
46
+ ··· − 4.42873u − 9.58615
0.817561u
47
− 0.0937876u
46
+ ··· − 5.25658u − 6.61698
a
6
=
−1.61232u
47
+ 0.251322u
46
+ ··· + 3.62117u + 8.11692
−0.821845u
47
+ 0.0692617u
46
+ ··· + 6.06620u + 5.24702
a
1
=
1.76087u
47
+ 0.00355222u
46
+ ··· − 4.42873u − 9.58615
−0.673929u
47
− 0.140698u
46
+ ··· + 0.0189389u + 2.20515
a
12
=
1.78030u
47
+ 0.0974277u
46
+ ··· − 4.61437u − 9.37957
0.553189u
47
− 0.0489697u
46
+ ··· − 3.34948u − 4.95558
a
7
=
−2.55276u
47
+ 0.387556u
46
+ ··· + 8.31343u + 13.4329
−0.486776u
47
+ 0.132833u
46
+ ··· + 4.46775u + 3.95207
(ii) Obstruction class = −1
(iii) Cusp Shapes = −0.327963u
47
− 0.0729656u
46
+ ··· − 4.23858u − 0.466106
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
48
+ 53u
47
+ ··· − 27678u + 289
c
2
, c
5
u
48
+ 3u
47
+ ··· − 142u + 17
c
3
, c
8
u
48
− u
47
+ ··· − 2u + 5
c
4
, c
10
u
48
− u
47
+ ··· − 8u + 1
c
6
u
48
+ 5u
47
+ ··· − 7742u + 26561
c
7
, c
11
u
48
− u
47
+ ··· − 14u + 1
c
9
u
48
+ 29u
47
+ ··· − 36u + 25
c
12
u
48
+ 3u
47
+ ··· + 4284478u + 1826857
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
48
− 111y
47
+ ··· − 298879486y + 83521
c
2
, c
5
y
48
+ 53y
47
+ ··· − 27678y + 289
c
3
, c
8
y
48
− 29y
47
+ ··· + 36y + 25
c
4
, c
10
y
48
+ 13y
47
+ ··· + 14y + 1
c
6
y
48
+ 31y
47
+ ··· + 12208109238y + 705486721
c
7
, c
11
y
48
− 39y
47
+ ··· − 86y + 1
c
9
y
48
− 13y
47
+ ··· − 28896y + 625
c
12
y
48
+ 41y
47
+ ··· + 44880385612764y + 3337406498449
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.108352 + 0.991513I
a = −1.65182 − 0.00446I
b = 1.36066 + 0.53287I
−7.23468 − 3.54967I −1.53238 + 2.33493I
u = −0.108352 − 0.991513I
a = −1.65182 + 0.00446I
b = 1.36066 − 0.53287I
−7.23468 + 3.54967I −1.53238 − 2.33493I
u = 0.174873 + 0.981711I
a = 1.74425 + 0.00932I
b = −1.41834 + 0.60453I
−2.92990 + 8.29102I 2.36664 − 4.88040I
u = 0.174873 − 0.981711I
a = 1.74425 − 0.00932I
b = −1.41834 − 0.60453I
−2.92990 − 8.29102I 2.36664 + 4.88040I
u = 0.030174 + 0.983574I
a = 1.55380 − 0.03101I
b = −1.287640 + 0.465293I
−3.56945 − 1.27655I 1.43121 + 0.83260I
u = 0.030174 − 0.983574I
a = 1.55380 + 0.03101I
b = −1.287640 − 0.465293I
−3.56945 + 1.27655I 1.43121 − 0.83260I
u = 0.881157 + 0.424614I
a = −1.141540 − 0.360327I
b = 0.731724 − 0.770961I
3.03026 + 0.73910I −0.764157 + 0.955160I
u = 0.881157 − 0.424614I
a = −1.141540 + 0.360327I
b = 0.731724 + 0.770961I
3.03026 − 0.73910I −0.764157 − 0.955160I
u = −0.955184 + 0.061016I
a = −0.450503 + 1.000600I
b = −1.139020 − 0.816551I
−0.935829 + 0.351963I −4.28024 + 0.58826I
u = −0.955184 − 0.061016I
a = −0.450503 − 1.000600I
b = −1.139020 + 0.816551I
−0.935829 − 0.351963I −4.28024 − 0.58826I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.819695 + 0.488889I
a = 0.108560 − 0.099280I
b = 0.818403 − 0.513954I
1.72059 − 2.04540I 7.89567 + 4.02405I
u = 0.819695 − 0.488889I
a = 0.108560 + 0.099280I
b = 0.818403 + 0.513954I
1.72059 + 2.04540I 7.89567 − 4.02405I
u = −0.895933 + 0.584550I
a = 0.571984 − 0.664326I
b = −0.915407 − 0.330137I
−1.17884 + 2.22567I −5.86884 − 3.07805I
u = −0.895933 − 0.584550I
a = 0.571984 + 0.664326I
b = −0.915407 + 0.330137I
−1.17884 − 2.22567I −5.86884 + 3.07805I
u = 1.079750 + 0.223592I
a = 0.106506 + 0.739409I
b = 1.081550 − 0.643185I
1.98776 − 3.98499I −0.58611 + 4.25564I
u = 1.079750 − 0.223592I
a = 0.106506 − 0.739409I
b = 1.081550 + 0.643185I
1.98776 + 3.98499I −0.58611 − 4.25564I
u = 0.665759 + 0.587232I
a = −0.797687 − 0.156330I
b = 0.764248 − 0.205453I
2.86945 + 0.64829I 1.178037 − 0.420823I
u = 0.665759 − 0.587232I
a = −0.797687 + 0.156330I
b = 0.764248 + 0.205453I
2.86945 − 0.64829I 1.178037 + 0.420823I
u = 0.896239 + 0.683994I
a = −0.334230 − 0.804354I
b = 0.831578 − 0.212398I
2.31381 − 5.68856I 0.35737 + 7.14815I
u = 0.896239 − 0.683994I
a = −0.334230 + 0.804354I
b = 0.831578 + 0.212398I
2.31381 + 5.68856I 0.35737 − 7.14815I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.129410 + 0.115901I
a = 0.528176 + 0.194049I
b = 0.823353 − 0.308996I
−2.25793 + 0.22669I −3.88130 + 0.48636I
u = −1.129410 − 0.115901I
a = 0.528176 − 0.194049I
b = 0.823353 + 0.308996I
−2.25793 − 0.22669I −3.88130 − 0.48636I
u = 0.841927 + 0.080741I
a = 0.86370 − 1.44512I
b = 1.24610 + 1.12698I
3.83049 − 3.07305I 1.41041 + 1.51798I
u = 0.841927 − 0.080741I
a = 0.86370 + 1.44512I
b = 1.24610 − 1.12698I
3.83049 + 3.07305I 1.41041 − 1.51798I
u = −0.994669 + 0.588297I
a = 0.296924 + 0.239330I
b = −0.984134 − 0.447751I
4.59952 + 2.16567I 4.10775 − 1.86694I
u = −0.994669 − 0.588297I
a = 0.296924 − 0.239330I
b = −0.984134 + 0.447751I
4.59952 − 2.16567I 4.10775 + 1.86694I
u = −1.112110 + 0.405732I
a = 0.779825 + 1.100780I
b = 1.22762 − 1.41031I
2.60733 + 6.94232I 0.84068 − 7.46411I
u = −1.112110 − 0.405732I
a = 0.779825 − 1.100780I
b = 1.22762 + 1.41031I
2.60733 − 6.94232I 0.84068 + 7.46411I
u = −0.555750 + 0.577343I
a = −0.416557 − 0.803655I
b = −0.515683 − 0.475483I
5.85592 + 2.51809I 7.55361 − 3.16560I
u = −0.555750 − 0.577343I
a = −0.416557 + 0.803655I
b = −0.515683 + 0.475483I
5.85592 − 2.51809I 7.55361 + 3.16560I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.163420 + 0.303926I
a = −0.536341 + 0.659705I
b = −1.100960 − 0.856931I
−3.28855 − 4.12116I −4.45892 + 6.35434I
u = 1.163420 − 0.303926I
a = −0.536341 − 0.659705I
b = −1.100960 + 0.856931I
−3.28855 + 4.12116I −4.45892 − 6.35434I
u = 1.262720 + 0.577522I
a = 0.268798 + 1.336830I
b = −2.09613 − 0.86670I
−6.2609 − 13.9095I 0
u = 1.262720 − 0.577522I
a = 0.268798 − 1.336830I
b = −2.09613 + 0.86670I
−6.2609 + 13.9095I 0
u = −1.343520 + 0.367319I
a = 0.348682 − 1.226170I
b = −1.204320 + 0.348043I
−7.84517 − 3.63200I 0
u = −1.343520 − 0.367319I
a = 0.348682 + 1.226170I
b = −1.204320 − 0.348043I
−7.84517 + 3.63200I 0
u = 1.304590 + 0.505937I
a = 0.147430 + 1.094530I
b = −1.88535 − 0.79417I
−7.51956 − 4.03416I 0
u = 1.304590 − 0.505937I
a = 0.147430 − 1.094530I
b = −1.88535 + 0.79417I
−7.51956 + 4.03416I 0
u = −1.318780 + 0.470886I
a = 0.389956 − 1.214920I
b = −1.162380 + 0.271439I
−7.78759 + 6.42660I 0
u = −1.318780 − 0.470886I
a = 0.389956 + 1.214920I
b = −1.162380 − 0.271439I
−7.78759 − 6.42660I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.288130 + 0.550367I
a = −0.228557 + 1.218180I
b = 2.00350 − 0.82275I
−10.8644 + 9.0813I 0
u = −1.288130 − 0.550367I
a = −0.228557 − 1.218180I
b = 2.00350 + 0.82275I
−10.8644 − 9.0813I 0
u = 1.340310 + 0.422305I
a = −0.370934 − 1.220450I
b = 1.180220 + 0.312167I
−11.84160 − 1.40433I 0
u = 1.340310 − 0.422305I
a = −0.370934 + 1.220450I
b = 1.180220 − 0.312167I
−11.84160 + 1.40433I 0
u = −0.222716 + 0.468775I
a = −2.01359 − 1.24143I
b = 0.560309 + 1.188780I
5.14299 − 3.26209I 6.73158 + 4.68534I
u = −0.222716 − 0.468775I
a = −2.01359 + 1.24143I
b = 0.560309 − 1.188780I
5.14299 + 3.26209I 6.73158 − 4.68534I
u = −0.036037 + 0.440473I
a = 0.933153 − 0.958140I
b = −0.419907 + 0.597866I
0.077905 + 1.136720I 0.96258 − 6.02333I
u = −0.036037 − 0.440473I
a = 0.933153 + 0.958140I
b = −0.419907 − 0.597866I
0.077905 − 1.136720I 0.96258 + 6.02333I
9
II. I
u
2
= h−u
2
a + u
3
+ b + a − u, a
2
u
2
+ a
3
+ 1, u
4
− u
2
+ 1i
(i) Arc colorings
a
3
=
0
u
a
8
=
1
0
a
9
=
1
u
2
a
4
=
−u
−u
3
+ u
a
5
=
a
u
2
a − u
3
− a + u
a
10
=
−u
2
+ 1
u
2
a
11
=
−u
3
a − u
2
+ 1
au
a
2
=
−a
2
u
−u
3
a
2
+ a
2
u − a + u
a
6
=
−a
2
u
2
+ a
2
− u
2
+ a
a
2
u
2
+ a
2
u − u
3
− a + u + 1
a
1
=
−a
2
u
a
2
u − a + u
a
12
=
u
3
a
2
− 2a
2
u − u
2
a + u
3
−u
3
a
2
+ a
2
u
a
7
=
u
3
a
2
− a
2
u
2
− u
2
a + a
2
− u
2
+ 2a + u
a
2
u
2
+ u
2
a − a + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
a + 4u
2
− 4a
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
3
− u
2
+ 2u − 1)
4
c
2
, c
5
(u
6
+ u
4
+ 2u
2
+ 1)
2
c
3
, c
8
(u
4
− u
2
+ 1)
3
c
4
, c
10
(u
2
+ 1)
6
c
6
u
12
− 8u
11
+ ··· − 40u + 25
c
7
, c
11
(u
6
− 3u
4
+ 2u
2
+ 1)
2
c
9
(u
2
+ u + 1)
6
c
12
u
12
+ 4u
11
+ ··· − 90u + 25
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
3
+ 3y
2
+ 2y −1)
4
c
2
, c
5
(y
3
+ y
2
+ 2y + 1)
4
c
3
, c
8
(y
2
− y + 1)
6
c
4
, c
10
(y + 1)
12
c
6
y
12
+ 6y
11
+ ··· + 2100y + 625
c
7
, c
11
(y
3
− 3y
2
+ 2y + 1)
4
c
9
(y
2
+ y + 1)
6
c
12
y
12
− 30y
10
+ ··· − 4050y + 625
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.866025 + 0.500000I
a = −1.083790 − 0.387453I
b = 1.74346 − 1.24486I
4.66906 + 0.79824I 5.50976 + 0.48465I
u = 0.866025 + 0.500000I
a = 0.206350 − 1.132320I
b = 1.74346 + 0.24486I
4.66906 − 4.85801I 5.50976 + 6.44355I
u = 0.866025 + 0.500000I
a = 0.377439 + 0.653743I
b = 0.111148 − 0.500000I
0.53148 − 2.02988I −1.01951 + 3.46410I
u = 0.866025 − 0.500000I
a = −1.083790 + 0.387453I
b = 1.74346 + 1.24486I
4.66906 − 0.79824I 5.50976 − 0.48465I
u = 0.866025 − 0.500000I
a = 0.206350 + 1.132320I
b = 1.74346 − 0.24486I
4.66906 + 4.85801I 5.50976 − 6.44355I
u = 0.866025 − 0.500000I
a = 0.377439 − 0.653743I
b = 0.111148 + 0.500000I
0.53148 + 2.02988I −1.01951 − 3.46410I
u = −0.866025 + 0.500000I
a = −1.083790 + 0.387453I
b = 0.011413 + 0.244862I
4.66906 − 0.79824I 5.50976 − 0.48465I
u = −0.866025 + 0.500000I
a = 0.206350 + 1.132320I
b = 0.011413 − 1.244860I
4.66906 + 4.85801I 5.50976 − 6.44355I
u = −0.866025 + 0.500000I
a = 0.377439 − 0.653743I
b = −1.62090 − 0.50000I
0.53148 + 2.02988I −1.01951 − 3.46410I
u = −0.866025 − 0.500000I
a = −1.083790 − 0.387453I
b = 0.011413 − 0.244862I
4.66906 + 0.79824I 5.50976 + 0.48465I
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.866025 − 0.500000I
a = 0.206350 − 1.132320I
b = 0.011413 + 1.244860I
4.66906 − 4.85801I 5.50976 + 6.44355I
u = −0.866025 − 0.500000I
a = 0.377439 + 0.653743I
b = −1.62090 + 0.50000I
0.53148 − 2.02988I −1.01951 + 3.46410I
14
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
3
− u
2
+ 2u − 1)
4
)(u
48
+ 53u
47
+ ··· − 27678u + 289)
c
2
, c
5
((u
6
+ u
4
+ 2u
2
+ 1)
2
)(u
48
+ 3u
47
+ ··· − 142u + 17)
c
3
, c
8
((u
4
− u
2
+ 1)
3
)(u
48
− u
47
+ ··· − 2u + 5)
c
4
, c
10
((u
2
+ 1)
6
)(u
48
− u
47
+ ··· − 8u + 1)
c
6
(u
12
− 8u
11
+ ··· − 40u + 25)(u
48
+ 5u
47
+ ··· − 7742u + 26561)
c
7
, c
11
((u
6
− 3u
4
+ 2u
2
+ 1)
2
)(u
48
− u
47
+ ··· − 14u + 1)
c
9
((u
2
+ u + 1)
6
)(u
48
+ 29u
47
+ ··· − 36u + 25)
c
12
(u
12
+ 4u
11
+ ··· − 90u + 25)
· (u
48
+ 3u
47
+ ··· + 4284478u + 1826857)
15
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
3
+ 3y
2
+ 2y −1)
4
)(y
48
− 111y
47
+ ··· − 2.98879 × 10
8
y + 83521)
c
2
, c
5
((y
3
+ y
2
+ 2y + 1)
4
)(y
48
+ 53y
47
+ ··· − 27678y + 289)
c
3
, c
8
((y
2
− y + 1)
6
)(y
48
− 29y
47
+ ··· + 36y + 25)
c
4
, c
10
((y + 1)
12
)(y
48
+ 13y
47
+ ··· + 14y + 1)
c
6
(y
12
+ 6y
11
+ ··· + 2100y + 625)
· (y
48
+ 31y
47
+ ··· + 12208109238y + 705486721)
c
7
, c
11
((y
3
− 3y
2
+ 2y + 1)
4
)(y
48
− 39y
47
+ ··· − 86y + 1)
c
9
((y
2
+ y + 1)
6
)(y
48
− 13y
47
+ ··· − 28896y + 625)
c
12
(y
12
− 30y
10
+ ··· − 4050y + 625)
· (y
48
+ 41y
47
+ ··· + 44880385612764y + 3337406498449)
16
