12n
0282
(K12n
0282
)
A knot diagram
1
Linearized knot diagam
3 6 7 8 10 2 12 6 1 8 4 10
Solving Sequence
3,6
2 7
1,10
5 9 8 4 12 11
c
2
c
6
c
1
c
5
c
9
c
8
c
4
c
12
c
11
c
3
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
13
+ 3u
12
+ 9u
11
+ 16u
10
+ 26u
9
+ 32u
8
+ 33u
7
+ 28u
6
+ 17u
5
+ 9u
4
+ u
3
+ u
2
+ b + 1,
u
15
+ 5u
14
+ ··· + 2a + 4, u
16
+ 5u
15
+ ··· + 4u + 2i
I
u
2
= hu
11
− 2u
10
+ 5u
9
− 7u
8
+ 9u
7
− 11u
6
+ 10u
5
− 10u
4
+ 6u
3
− 4u
2
+ b + 3u − 1,
u
11
+ 2u
9
+ 2u
8
− u
7
+ 4u
6
− 6u
5
+ 5u
4
− 8u
3
+ 3u
2
+ 2a − 2u + 3,
u
12
− 2u
11
+ 6u
10
− 8u
9
+ 13u
8
− 14u
7
+ 16u
6
− 15u
5
+ 12u
4
− 9u
3
+ 6u
2
− 3u + 2i
I
u
3
= h−au − u
2
+ b + u − 1, u
2
a + a
2
+ 5u
2
+ a − 2u + 7, u
3
− u
2
+ 2u − 1i
* 3 irreducible components of dim
C
= 0, with total 34 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
13
+3u
12
+· · ·+b+1, u
15
+5u
14
+· · ·+2a+4, u
16
+5u
15
+· · ·+4u+2i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
7
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
2
a
10
=
−
1
2
u
15
−
5
2
u
14
+ ··· −
5
2
u − 2
−u
13
− 3u
12
+ ··· − u
2
− 1
a
5
=
−
1
2
u
15
−
3
2
u
14
+ ··· − u
2
−
1
2
u
−u
15
− 4u
14
+ ··· − u − 1
a
9
=
1
2
u
15
+
5
2
u
14
+ ··· +
1
2
u + 1
u
14
+ 4u
13
+ ··· + 2u + 1
a
8
=
1
2
u
15
+
5
2
u
14
+ ··· +
1
2
u + 1
−u
14
− u
13
+ ··· + u + 1
a
4
=
u
4
+ u
2
+ 1
u
6
+ 2u
4
+ u
2
a
12
=
−
1
2
u
15
−
5
2
u
14
+ ··· − 2u
2
−
3
2
u
−u
13
− 3u
12
+ ··· − u − 1
a
11
=
−
5
2
u
15
−
25
2
u
14
+ ··· −
15
2
u − 5
u
14
− 4u
13
+ ··· − 3u − 5
(ii) Obstruction class = −1
(iii) Cusp Shapes = 3u
15
+ 14u
14
+ 49u
13
+ 118u
12
+ 232u
11
+ 373u
10
+ 501u
9
+
574u
8
+ 544u
7
+ 435u
6
+ 280u
5
+ 141u
4
+ 62u
3
+ 25u
2
+ 20u + 14
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
16
+ 11u
15
+ ··· + 12u + 4
c
2
, c
6
u
16
− 5u
15
+ ··· − 4u + 2
c
3
u
16
+ 5u
15
+ ··· − 4u + 10
c
4
, c
5
, c
11
u
16
+ 13u
14
+ ··· − u + 1
c
7
u
16
+ 9u
15
+ ··· + 24u + 8
c
8
u
16
+ u
15
+ ··· − 487u + 889
c
9
, c
12
u
16
− 2u
15
+ ··· − 17u + 1
c
10
u
16
− 17u
15
+ ··· − 52u + 10
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
16
− 9y
15
+ ··· + 632y + 16
c
2
, c
6
y
16
+ 11y
15
+ ··· + 12y + 4
c
3
y
16
− 41y
15
+ ··· + 764y + 100
c
4
, c
5
, c
11
y
16
+ 26y
15
+ ··· + 7y + 1
c
7
y
16
+ 3y
15
+ ··· + 288y + 64
c
8
y
16
+ 109y
15
+ ··· − 11168313y + 790321
c
9
, c
12
y
16
+ 34y
15
+ ··· − 47y + 1
c
10
y
16
− 49y
15
+ ··· + 36y + 100
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.478305 + 1.028310I
a = 0.050929 + 0.582193I
b = 0.623036 + 0.226095I
−0.64871 − 3.08703I 2.33099 + 0.59290I
u = −0.478305 − 1.028310I
a = 0.050929 − 0.582193I
b = 0.623036 − 0.226095I
−0.64871 + 3.08703I 2.33099 − 0.59290I
u = −1.136630 + 0.036859I
a = 0.25201 − 1.80758I
b = 0.21982 − 2.06384I
−19.1158 − 4.5602I 1.43322 + 1.94202I
u = −1.136630 − 0.036859I
a = 0.25201 + 1.80758I
b = 0.21982 + 2.06384I
−19.1158 + 4.5602I 1.43322 − 1.94202I
u = 0.065300 + 1.174500I
a = 0.511213 + 0.476614I
b = 0.526402 − 0.631543I
−3.55568 − 0.83919I −1.52217 + 2.21836I
u = 0.065300 − 1.174500I
a = 0.511213 − 0.476614I
b = 0.526402 + 0.631543I
−3.55568 + 0.83919I −1.52217 − 2.21836I
u = 0.284247 + 1.175830I
a = −0.458912 − 0.456538I
b = −0.406366 + 0.669372I
−1.92269 + 4.32175I −1.05745 − 3.72733I
u = 0.284247 − 1.175830I
a = −0.458912 + 0.456538I
b = −0.406366 − 0.669372I
−1.92269 − 4.32175I −1.05745 + 3.72733I
u = −0.462160 + 0.504593I
a = −0.810441 − 0.051513I
b = −0.400547 + 0.385135I
0.900558 − 0.884001I 7.79246 + 5.53646I
u = −0.462160 − 0.504593I
a = −0.810441 + 0.051513I
b = −0.400547 − 0.385135I
0.900558 + 0.884001I 7.79246 − 5.53646I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.59084 + 1.38518I
a = 1.216170 − 0.591623I
b = −0.10095 − 2.03417I
16.1808 − 1.5728I −0.910122 + 0.801977I
u = −0.59084 − 1.38518I
a = 1.216170 + 0.591623I
b = −0.10095 + 2.03417I
16.1808 + 1.5728I −0.910122 − 0.801977I
u = 0.364622 + 0.333050I
a = −0.375430 − 0.860631I
b = −0.149743 + 0.438842I
0.56988 − 1.44730I 4.97392 + 6.26939I
u = 0.364622 − 0.333050I
a = −0.375430 + 0.860631I
b = −0.149743 − 0.438842I
0.56988 + 1.44730I 4.97392 − 6.26939I
u = −0.54623 + 1.41233I
a = −1.38554 + 0.31520I
b = −0.31165 + 2.12901I
15.8163 − 10.5372I −1.04085 + 4.44511I
u = −0.54623 − 1.41233I
a = −1.38554 − 0.31520I
b = −0.31165 − 2.12901I
15.8163 + 10.5372I −1.04085 − 4.44511I
6
II.
I
u
2
= hu
11
−2u
10
+· · · +b−1, u
11
+2u
9
+· · · +2a+3, u
12
−2u
11
+· · · −3u+2i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
7
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
2
a
10
=
−
1
2
u
11
− u
9
+ ··· + u −
3
2
−u
11
+ 2u
10
+ ··· − 3u + 1
a
5
=
−
3
2
u
11
+ 3u
10
+ ··· − 5u +
5
2
u
10
− 2u
9
+ 5u
8
− 6u
7
+ 8u
6
− 8u
5
+ 8u
4
− 7u
3
+ 4u
2
− u + 3
a
9
=
−
1
2
u
11
− u
9
+ ··· +
1
2
u
2
−
1
2
−u
11
+ 2u
10
+ ··· − 3u + 1
a
8
=
−
1
2
u
11
− u
9
+ ··· +
1
2
u
2
−
1
2
−u
11
+ 3u
10
+ ··· − 5u + 3
a
4
=
u
4
+ u
2
+ 1
u
6
+ 2u
4
+ u
2
a
12
=
1
2
u
11
+ u
9
+ ··· +
1
2
u
2
+
3
2
u
11
− 2u
10
+ 5u
9
− 6u
8
+ 8u
7
− 8u
6
+ 8u
5
− 7u
4
+ 4u
3
− 2u
2
+ 2u − 1
a
11
=
1
2
u
11
+ u
9
+ ··· −
1
2
u
2
+
1
2
u
11
− 3u
10
+ ··· + 3u − 3
(ii) Obstruction class = 1
(iii) Cusp Shapes
= u
11
− 5u
10
+ 10u
9
− 21u
8
+ 24u
7
− 31u
6
+ 28u
5
− 25u
4
+ 22u
3
− 12u
2
+ 8u − 4
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
12
− 8u
11
+ ··· − 15u + 4
c
2
u
12
− 2u
11
+ ··· − 3u + 2
c
3
u
12
+ 2u
11
− 2u
9
+ 6u
8
+ 10u
7
+ 5u
6
+ 2u
5
+ 11u
4
+ u
3
+ 4u
2
+ 5u + 2
c
4
, c
11
u
12
+ 8u
10
+ ··· − 2u + 1
c
5
u
12
+ 8u
10
+ ··· + 2u + 1
c
6
u
12
+ 2u
11
+ ··· + 3u + 2
c
7
u
12
+ 2u
11
+ ··· + 2u + 1
c
8
u
12
+ 5u
11
+ ··· − 7u
2
+ 1
c
9
u
12
− 2u
11
+ ··· − 2u + 1
c
10
u
12
+ 14u
11
+ ··· + 495u + 80
c
12
u
12
+ 2u
11
+ ··· + 2u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
12
− 4y
11
+ ··· + 15y + 16
c
2
, c
6
y
12
+ 8y
11
+ ··· + 15y + 4
c
3
y
12
− 4y
11
+ ··· − 9y + 4
c
4
, c
5
, c
11
y
12
+ 16y
11
+ ··· + 2y + 1
c
7
y
12
+ 4y
11
+ ··· + 8y + 1
c
8
y
12
− 13y
11
+ ··· − 14y + 1
c
9
, c
12
y
12
+ 8y
11
+ ··· + 4y + 1
c
10
y
12
− 8y
11
+ ··· + 3615y + 6400
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.249672 + 0.959195I
a = −1.68135 − 0.45806I
b = 0.01959 − 1.72711I
−7.94302 + 1.00045I −2.72933 − 0.10711I
u = 0.249672 − 0.959195I
a = −1.68135 + 0.45806I
b = 0.01959 + 1.72711I
−7.94302 − 1.00045I −2.72933 + 0.10711I
u = −0.429646 + 0.953539I
a = −0.073134 − 0.459239I
b = 0.469324 + 0.127574I
−0.45876 − 4.18304I 4.58234 + 6.10453I
u = −0.429646 − 0.953539I
a = −0.073134 + 0.459239I
b = 0.469324 − 0.127574I
−0.45876 + 4.18304I 4.58234 − 6.10453I
u = 0.839161 + 0.302874I
a = 0.461054 + 1.306310I
b = −0.008748 + 1.235850I
−2.36446 − 1.00466I 2.85304 + 0.63873I
u = 0.839161 − 0.302874I
a = 0.461054 − 1.306310I
b = −0.008748 − 1.235850I
−2.36446 + 1.00466I 2.85304 − 0.63873I
u = −0.484489 + 0.716111I
a = 0.556723 − 0.092910I
b = −0.203192 + 0.443689I
0.268555 + 0.420031I 1.039574 + 0.824157I
u = −0.484489 − 0.716111I
a = 0.556723 + 0.092910I
b = −0.203192 − 0.443689I
0.268555 − 0.420031I 1.039574 − 0.824157I
u = 0.581682 + 1.140840I
a = 1.010720 + 0.383459I
b = 0.150456 + 1.376120I
−4.83530 + 6.22925I 0.60260 − 5.56850I
u = 0.581682 − 1.140840I
a = 1.010720 − 0.383459I
b = 0.150456 − 1.376120I
−4.83530 − 6.22925I 0.60260 + 5.56850I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.243620 + 1.359490I
a = −1.024010 + 0.130899I
b = −0.42743 − 1.36024I
−7.69609 + 2.59197I −0.34822 − 1.94419I
u = 0.243620 − 1.359490I
a = −1.024010 − 0.130899I
b = −0.42743 + 1.36024I
−7.69609 − 2.59197I −0.34822 + 1.94419I
11
III.
I
u
3
= h−au − u
2
+ b + u − 1, u
2
a + a
2
+ 5u
2
+ a − 2u + 7, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
7
=
u
u
2
− u + 1
a
1
=
u
2
+ 1
u
2
a
10
=
a
au + u
2
− u + 1
a
5
=
u
2
a − au + 3u
2
+ a − 3u + 5
3
a
9
=
−au − u
2
+ a − 1
−au + a − u + 1
a
8
=
−au − u
2
+ a − 1
au
a
4
=
−u + 2
u
a
12
=
−au − u
2
+ a − u − 1
au − u
2
+ u − 1
a
11
=
u
2
a − 3au − u
2
+ a − u − 1
−u
2
a + au − u
2
+ u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
2
− 4u + 2
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
3
+ 3u
2
+ 2u − 1)
2
c
2
, c
6
(u
3
+ u
2
+ 2u + 1)
2
c
3
(u
3
− u
2
+ 1)
2
c
4
, c
5
, c
11
u
6
− u
5
+ 8u
4
− 2u
3
+ 24u
2
+ 23
c
7
(u − 1)
6
c
8
u
6
− 5u
5
− 6u
4
+ 30u
3
+ 78u
2
+ 70u + 23
c
9
, c
12
u
6
+ 5u
5
+ 18u
4
+ 28u
3
+ 42u
2
+ 30u + 25
c
10
(u
3
+ 5u
2
+ 10u + 7)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
3
− 5y
2
+ 10y − 1)
2
c
2
, c
6
(y
3
+ 3y
2
+ 2y − 1)
2
c
3
(y
3
− y
2
+ 2y − 1)
2
c
4
, c
5
, c
11
y
6
+ 15y
5
+ 108y
4
+ 426y
3
+ 944y
2
+ 1104y + 529
c
7
(y − 1)
6
c
8
y
6
− 37y
5
+ 492y
4
− 1090y
3
+ 1608y
2
− 1312y + 529
c
9
, c
12
y
6
+ 11y
5
+ 128y
4
+ 478y
3
+ 984y
2
+ 1200y + 625
c
10
(y
3
− 5y
2
+ 30y − 49)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = −1.007880 − 0.138006I
b = −0.91382 − 2.09199I
−9.60386 + 2.82812I −5.50976 − 2.97945I
u = 0.215080 + 1.307140I
a = 1.67024 − 0.42427I
b = 0.036382 + 1.347130I
−9.60386 + 2.82812I −5.50976 − 2.97945I
u = 0.215080 − 1.307140I
a = −1.007880 + 0.138006I
b = −0.91382 + 2.09199I
−9.60386 − 2.82812I −5.50976 + 2.97945I
u = 0.215080 − 1.307140I
a = 1.67024 + 0.42427I
b = 0.036382 − 1.347130I
−9.60386 − 2.82812I −5.50976 + 2.97945I
u = 0.569840
a = −0.66236 + 2.65428I
b = 0.37744 + 1.51251I
−5.46628 1.01950
u = 0.569840
a = −0.66236 − 2.65428I
b = 0.37744 − 1.51251I
−5.46628 1.01950
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
3
+ 3u
2
+ 2u − 1)
2
)(u
12
− 8u
11
+ ··· − 15u + 4)
· (u
16
+ 11u
15
+ ··· + 12u + 4)
c
2
((u
3
+ u
2
+ 2u + 1)
2
)(u
12
− 2u
11
+ ··· − 3u + 2)
· (u
16
− 5u
15
+ ··· − 4u + 2)
c
3
(u
3
− u
2
+ 1)
2
· (u
12
+ 2u
11
− 2u
9
+ 6u
8
+ 10u
7
+ 5u
6
+ 2u
5
+ 11u
4
+ u
3
+ 4u
2
+ 5u + 2)
· (u
16
+ 5u
15
+ ··· − 4u + 10)
c
4
, c
11
(u
6
− u
5
+ 8u
4
− 2u
3
+ 24u
2
+ 23)(u
12
+ 8u
10
+ ··· − 2u + 1)
· (u
16
+ 13u
14
+ ··· − u + 1)
c
5
(u
6
− u
5
+ 8u
4
− 2u
3
+ 24u
2
+ 23)(u
12
+ 8u
10
+ ··· + 2u + 1)
· (u
16
+ 13u
14
+ ··· − u + 1)
c
6
((u
3
+ u
2
+ 2u + 1)
2
)(u
12
+ 2u
11
+ ··· + 3u + 2)
· (u
16
− 5u
15
+ ··· − 4u + 2)
c
7
((u − 1)
6
)(u
12
+ 2u
11
+ ··· + 2u + 1)(u
16
+ 9u
15
+ ··· + 24u + 8)
c
8
(u
6
− 5u
5
+ ··· + 70u + 23)(u
12
+ 5u
11
+ ··· − 7u
2
+ 1)
· (u
16
+ u
15
+ ··· − 487u + 889)
c
9
(u
6
+ 5u
5
+ ··· + 30u + 25)(u
12
− 2u
11
+ ··· − 2u + 1)
· (u
16
− 2u
15
+ ··· − 17u + 1)
c
10
((u
3
+ 5u
2
+ 10u + 7)
2
)(u
12
+ 14u
11
+ ··· + 495u + 80)
· (u
16
− 17u
15
+ ··· − 52u + 10)
c
12
(u
6
+ 5u
5
+ ··· + 30u + 25)(u
12
+ 2u
11
+ ··· + 2u + 1)
· (u
16
− 2u
15
+ ··· − 17u + 1)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
3
− 5y
2
+ 10y − 1)
2
)(y
12
− 4y
11
+ ··· + 15y + 16)
· (y
16
− 9y
15
+ ··· + 632y + 16)
c
2
, c
6
((y
3
+ 3y
2
+ 2y − 1)
2
)(y
12
+ 8y
11
+ ··· + 15y + 4)
· (y
16
+ 11y
15
+ ··· + 12y + 4)
c
3
((y
3
− y
2
+ 2y − 1)
2
)(y
12
− 4y
11
+ ··· − 9y + 4)
· (y
16
− 41y
15
+ ··· + 764y + 100)
c
4
, c
5
, c
11
(y
6
+ 15y
5
+ 108y
4
+ 426y
3
+ 944y
2
+ 1104y + 529)
· (y
12
+ 16y
11
+ ··· + 2y + 1)(y
16
+ 26y
15
+ ··· + 7y + 1)
c
7
((y − 1)
6
)(y
12
+ 4y
11
+ ··· + 8y + 1)(y
16
+ 3y
15
+ ··· + 288y + 64)
c
8
(y
6
− 37y
5
+ 492y
4
− 1090y
3
+ 1608y
2
− 1312y + 529)
· (y
12
− 13y
11
+ ··· − 14y + 1)
· (y
16
+ 109y
15
+ ··· − 11168313y + 790321)
c
9
, c
12
(y
6
+ 11y
5
+ 128y
4
+ 478y
3
+ 984y
2
+ 1200y + 625)
· (y
12
+ 8y
11
+ ··· + 4y + 1)(y
16
+ 34y
15
+ ··· − 47y + 1)
c
10
((y
3
− 5y
2
+ 30y − 49)
2
)(y
12
− 8y
11
+ ··· + 3615y + 6400)
· (y
16
− 49y
15
+ ··· + 36y + 100)
17
