12n
0565
(K12n
0565
)
A knot diagram
1
Linearized knot diagam
3 7 9 8 10 11 2 4 12 6 7 9
Solving Sequence
7,11 3,12
2 8 1 6 10 5 4 9
c
11
c
2
c
7
c
1
c
6
c
10
c
5
c
4
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h5u
23
− 65u
21
+ ··· + 4b + 8, −u
23
+ 14u
21
+ ··· + 4a − 4, u
24
+ 2u
23
+ ··· − u + 2i
I
u
2
= h−u
6
+ u
5
+ 3u
4
− 3u
3
− 2u
2
+ b + u, u
6
− 4u
4
+ 4u
2
+ a, u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1i
I
u
3
= ha
2
u + a
2
+ 2au + b + 2a, a
3
− au + 2a + u − 2, u
2
− u − 1i
* 3 irreducible components of dim
C
= 0, with total 38 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
=
h5u
23
−65u
21
+· · ·+4b+8, −u
23
+14u
21
+· · ·+4a−4, u
24
+2u
23
+· · ·−u+2i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
3
=
1
4
u
23
−
7
2
u
21
+ ··· +
1
4
u + 1
−
5
4
u
23
+
65
4
u
21
+ ··· +
5
2
u − 2
a
12
=
1
−u
2
a
2
=
1
4
u
23
−
7
2
u
21
+ ··· +
1
4
u + 1
−
3
4
u
23
+
39
4
u
21
+ ··· +
3
2
u − 1
a
8
=
1
4
u
19
−
11
4
u
17
+ ··· +
3
4
u − 1
−
1
4
u
21
+
11
4
u
19
+ ··· −
1
2
u
2
+
1
2
u
a
1
=
u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1
−u
10
+ 4u
8
− 3u
6
− 2u
4
− u
2
a
6
=
−u
u
a
10
=
−u
2
+ 1
u
2
a
5
=
u
3
− 2u
−u
3
+ u
a
4
=
1
2
u
13
− 4u
11
+ ··· −
3
2
u +
1
2
−
1
4
u
20
+
11
4
u
18
+ ··· −
5
4
u
2
+ u
a
9
=
u
4
− 3u
2
+ 1
−u
6
+ 2u
4
+ u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 2u
23
−28u
21
+ 164u
19
+ 2u
18
−520u
17
−22u
16
+ 968u
15
+ 96u
14
−1076u
13
−210u
12
+
672u
11
+ 242u
10
− 116u
9
− 148u
8
− 170u
7
+ 50u
6
+ 116u
5
− 2u
4
− 38u
3
− 2u
2
+ 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
24
+ u
23
+ ··· − 61u + 4
c
2
, c
7
u
24
+ u
23
+ ··· + 3u + 2
c
3
, c
4
, c
8
u
24
+ u
23
+ ··· + 9u + 2
c
5
, c
6
, c
10
c
11
u
24
− 2u
23
+ ··· + u + 2
c
9
, c
12
u
24
+ 8u
23
+ ··· + 5111u + 1016
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
24
+ 53y
23
+ ··· − 1345y + 16
c
2
, c
7
y
24
+ y
23
+ ··· − 61y + 4
c
3
, c
4
, c
8
y
24
+ 37y
23
+ ··· − 77y + 4
c
5
, c
6
, c
10
c
11
y
24
− 28y
23
+ ··· + 19y + 4
c
9
, c
12
y
24
− 16y
23
+ ··· − 9677345y + 1032256
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.831943 + 0.541956I
a = 1.49407 − 0.41444I
b = −2.17525 + 1.60820I
11.55510 − 8.08338I 8.69208 + 5.77253I
u = −0.831943 − 0.541956I
a = 1.49407 + 0.41444I
b = −2.17525 − 1.60820I
11.55510 + 8.08338I 8.69208 − 5.77253I
u = 0.922316 + 0.467245I
a = −0.58520 + 1.32150I
b = 2.12119 − 1.04366I
12.25690 + 0.16906I 9.76193 − 1.32433I
u = 0.922316 − 0.467245I
a = −0.58520 − 1.32150I
b = 2.12119 + 1.04366I
12.25690 − 0.16906I 9.76193 + 1.32433I
u = 0.792849 + 0.338620I
a = 1.099400 + 0.306166I
b = −1.77343 − 1.20094I
2.74098 + 4.13108I 9.27324 − 6.99625I
u = 0.792849 − 0.338620I
a = 1.099400 − 0.306166I
b = −1.77343 + 1.20094I
2.74098 − 4.13108I 9.27324 + 6.99625I
u = −0.061235 + 0.729145I
a = 1.29723 − 1.54591I
b = 0.134722 − 0.625129I
9.24115 + 3.84022I 5.77183 − 1.94083I
u = −0.061235 − 0.729145I
a = 1.29723 + 1.54591I
b = 0.134722 + 0.625129I
9.24115 − 3.84022I 5.77183 + 1.94083I
u = −0.576976 + 0.440713I
a = −0.357196 − 0.355358I
b = 0.897638 + 0.531756I
1.59762 − 1.32160I 9.98992 + 3.60168I
u = −0.576976 − 0.440713I
a = −0.357196 + 0.355358I
b = 0.897638 − 0.531756I
1.59762 + 1.32160I 9.98992 − 3.60168I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.463900 + 0.410678I
a = −1.09909 − 1.07076I
b = 0.643004 + 0.398425I
−1.88644 + 1.49317I −2.47681 − 5.21553I
u = 0.463900 − 0.410678I
a = −1.09909 + 1.07076I
b = 0.643004 − 0.398425I
−1.88644 − 1.49317I −2.47681 + 5.21553I
u = −1.54266 + 0.08137I
a = −0.146402 + 0.665748I
b = 1.51464 − 1.48863I
4.87581 − 3.07258I 2.46496 + 3.31099I
u = −1.54266 − 0.08137I
a = −0.146402 − 0.665748I
b = 1.51464 + 1.48863I
4.87581 + 3.07258I 2.46496 − 3.31099I
u = 1.54979 + 0.13776I
a = −0.331233 − 0.120821I
b = 1.82744 − 0.86788I
8.73438 + 3.47321I 12.37420 − 2.64740I
u = 1.54979 − 0.13776I
a = −0.331233 + 0.120821I
b = 1.82744 + 0.86788I
8.73438 − 3.47321I 12.37420 + 2.64740I
u = −0.043929 + 0.434651I
a = 1.31585 + 0.89975I
b = 0.193330 + 0.366457I
0.33398 − 1.43694I 3.82594 + 4.60037I
u = −0.043929 − 0.434651I
a = 1.31585 − 0.89975I
b = 0.193330 − 0.366457I
0.33398 + 1.43694I 3.82594 − 4.60037I
u = −1.64402 + 0.09119I
a = 0.613245 − 0.374905I
b = −4.52678 + 1.79716I
11.18040 − 5.75443I 10.48816 + 4.61050I
u = −1.64402 − 0.09119I
a = 0.613245 + 0.374905I
b = −4.52678 − 1.79716I
11.18040 + 5.75443I 10.48816 − 4.61050I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.65555 + 0.16059I
a = 0.679203 + 0.691735I
b = −4.69788 − 2.45987I
−19.4184 + 10.7998I 10.40821 − 4.70031I
u = 1.65555 − 0.16059I
a = 0.679203 − 0.691735I
b = −4.69788 + 2.45987I
−19.4184 − 10.7998I 10.40821 + 4.70031I
u = −1.68364 + 0.12349I
a = −0.729891 − 0.590434I
b = 4.34137 + 2.97651I
−18.1825 − 2.4647I 11.42634 + 0.60606I
u = −1.68364 − 0.12349I
a = −0.729891 + 0.590434I
b = 4.34137 − 2.97651I
−18.1825 + 2.4647I 11.42634 − 0.60606I
7
II. I
u
2
=
h−u
6
+u
5
+3u
4
−3u
3
−2u
2
+b+u, u
6
−4u
4
+4u
2
+a, u
8
−5u
6
+7u
4
−2u
2
+1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
3
=
−u
6
+ 4u
4
− 4u
2
u
6
− u
5
− 3u
4
+ 3u
3
+ 2u
2
− u
a
12
=
1
−u
2
a
2
=
−u
6
+ 4u
4
− 4u
2
−u
5
+ 3u
3
− u + 1
a
8
=
u
7
− 5u
5
+ 7u
3
− 2u
−u
7
+ 4u
5
− u
4
− 4u
3
+ 2u
2
+ u
a
1
=
0
−u
6
+ 3u
4
− 2u
2
+ 1
a
6
=
−u
u
a
10
=
−u
2
+ 1
u
2
a
5
=
u
3
− 2u
−u
3
+ u
a
4
=
−u
6
+ 4u
4
+ u
3
− 4u
2
− 2u
u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
a
9
=
u
4
− 3u
2
+ 1
−u
6
+ 2u
4
+ u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
6
+ 16u
4
− 16u
2
+ 8
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
8
c
2
, c
3
, c
4
c
7
, c
8
(u
2
+ 1)
4
c
5
, c
6
, c
10
c
11
u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1
c
9
(u
4
− u
3
+ u
2
+ 1)
2
c
12
(u
4
+ u
3
+ u
2
+ 1)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y −1)
8
c
2
, c
3
, c
4
c
7
, c
8
(y + 1)
8
c
5
, c
6
, c
10
c
11
(y
4
− 5y
3
+ 7y
2
− 2y + 1)
2
c
9
, c
12
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.506844 + 0.395123I
a = −0.95668 − 1.22719I
b = −0.115465 + 0.858652I
−0.21101 + 1.41510I 4.17326 − 4.90874I
u = 0.506844 − 0.395123I
a = −0.95668 + 1.22719I
b = −0.115465 − 0.858652I
−0.21101 − 1.41510I 4.17326 + 4.90874I
u = −0.506844 + 0.395123I
a = −0.95668 + 1.22719I
b = 1.325220 − 0.155036I
−0.21101 − 1.41510I 4.17326 + 4.90874I
u = −0.506844 − 0.395123I
a = −0.95668 − 1.22719I
b = 1.325220 + 0.155036I
−0.21101 + 1.41510I 4.17326 − 4.90874I
u = 1.55249 + 0.10488I
a = −0.043315 − 0.641200I
b = 1.80642 + 0.70068I
6.79074 + 3.16396I 7.82674 − 2.56480I
u = 1.55249 − 0.10488I
a = −0.043315 + 0.641200I
b = 1.80642 − 0.70068I
6.79074 − 3.16396I 7.82674 + 2.56480I
u = −1.55249 + 0.10488I
a = −0.043315 + 0.641200I
b = −0.01617 − 2.40430I
6.79074 − 3.16396I 7.82674 + 2.56480I
u = −1.55249 − 0.10488I
a = −0.043315 − 0.641200I
b = −0.01617 + 2.40430I
6.79074 + 3.16396I 7.82674 − 2.56480I
11
III. I
u
3
= ha
2
u + a
2
+ 2au + b + 2a, a
3
− au + 2a + u − 2, u
2
− u − 1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
3
=
a
−a
2
u − a
2
− 2au − 2a
a
12
=
1
−u − 1
a
2
=
a
−a
2
u − a
2
− au − a
a
8
=
a
2
u
−2a
2
u − a
2
+ au
a
1
=
1
−2u − 2
a
6
=
−u
u
a
10
=
−u
u + 1
a
5
=
1
−u − 1
a
4
=
a
−a
2
u − a
2
− au − a
a
9
=
0
−u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 10
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
6
+ 4u
5
+ 6u
4
+ u
3
− 5u
2
− 3u + 1
c
2
, c
3
, c
4
c
7
, c
8
u
6
+ 2u
4
+ u
3
+ u
2
+ u − 1
c
5
, c
6
, c
9
c
10
, c
11
, c
12
(u
2
+ u − 1)
3
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
6
− 4y
5
+ 18y
4
− 35y
3
+ 43y
2
− 19y + 1
c
2
, c
3
, c
4
c
7
, c
8
y
6
+ 4y
5
+ 6y
4
+ y
3
− 5y
2
− 3y + 1
c
5
, c
6
, c
9
c
10
, c
11
, c
12
(y
2
− 3y + 1)
3
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 0.802554
b = −0.859119
0.986960 10.0000
u = −0.618034
a = −0.40128 + 1.76100I
b = 1.42956 − 0.80545I
0.986960 10.0000
u = −0.618034
a = −0.40128 − 1.76100I
b = 1.42956 + 0.80545I
0.986960 10.0000
u = 1.61803
a = −0.277125 + 0.782535I
b = 2.85317 − 2.96191I
8.88264 10.0000
u = 1.61803
a = −0.277125 − 0.782535I
b = 2.85317 + 2.96191I
8.88264 10.0000
u = 1.61803
a = 0.554250
b = −3.70633
8.88264 10.0000
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
6
+ 4u
5
+ ··· − 3u + 1)(u
24
+ u
23
+ ··· − 61u + 4)
c
2
, c
7
((u
2
+ 1)
4
)(u
6
+ 2u
4
+ ··· + u − 1)(u
24
+ u
23
+ ··· + 3u + 2)
c
3
, c
4
, c
8
((u
2
+ 1)
4
)(u
6
+ 2u
4
+ ··· + u − 1)(u
24
+ u
23
+ ··· + 9u + 2)
c
5
, c
6
, c
10
c
11
((u
2
+ u − 1)
3
)(u
8
− 5u
6
+ ··· − 2u
2
+ 1)(u
24
− 2u
23
+ ··· + u + 2)
c
9
((u
2
+ u − 1)
3
)(u
4
− u
3
+ u
2
+ 1)
2
(u
24
+ 8u
23
+ ··· + 5111u + 1016)
c
12
((u
2
+ u − 1)
3
)(u
4
+ u
3
+ u
2
+ 1)
2
(u
24
+ 8u
23
+ ··· + 5111u + 1016)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)
8
(y
6
− 4y
5
+ 18y
4
− 35y
3
+ 43y
2
− 19y + 1)
· (y
24
+ 53y
23
+ ··· − 1345y + 16)
c
2
, c
7
((y + 1)
8
)(y
6
+ 4y
5
+ ··· − 3y + 1)(y
24
+ y
23
+ ··· − 61y + 4)
c
3
, c
4
, c
8
(y + 1)
8
(y
6
+ 4y
5
+ 6y
4
+ y
3
− 5y
2
− 3y + 1)
· (y
24
+ 37y
23
+ ··· − 77y + 4)
c
5
, c
6
, c
10
c
11
(y
2
− 3y + 1)
3
(y
4
− 5y
3
+ 7y
2
− 2y + 1)
2
· (y
24
− 28y
23
+ ··· + 19y + 4)
c
9
, c
12
(y
2
− 3y + 1)
3
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
· (y
24
− 16y
23
+ ··· − 9677345y + 1032256)
17
