12n
0296
(K12n
0296
)
A knot diagram
1
Linearized knot diagam
3 6 7 9 10 2 12 4 11 5 7 11
Solving Sequence
5,11
10
2,6
3 7 12 1 9 4 8
c
10
c
5
c
2
c
6
c
11
c
1
c
9
c
4
c
8
c
3
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h508219866971u
31
− 100745166922u
30
+ ··· + 2157681620548b − 1727274605244,
− 659556719114u
31
+ 420249241465u
30
+ ··· + 2157681620548a − 609787719508,
u
32
− u
31
+ ··· + 12u − 4i
I
u
2
= h−au − u
2
+ b − 1, −u
3
a + 2u
2
a + 3u
3
+ 2a
2
+ 2au + u
2
+ 2a + 2u − 2, u
4
+ 2u
2
+ 2i
I
v
1
= ha, b + v, v
2
+ v + 1i
* 3 irreducible components of dim
C
= 0, with total 42 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
= h5.08×10
11
u
31
−1.01×10
11
u
30
+· · ·+2.16×10
12
b−1.73×10
12
, −6.60×
10
11
u
31
+4.20×10
11
u
30
+· · ·+2.16×10
12
a−6.10×10
11
, u
32
−u
31
+· · ·+12u−4i
(i) Arc colorings
a
5
=
0
u
a
11
=
1
0
a
10
=
1
u
2
a
2
=
0.305678u
31
− 0.194769u
30
+ ··· + 0.947602u + 0.282612
−0.235540u
31
+ 0.0466914u
30
+ ··· + 0.152492u + 0.800523
a
6
=
u
u
3
+ u
a
3
=
0.569624u
31
− 0.540602u
30
+ ··· − 0.831433u + 0.738674
−0.374580u
31
+ 0.0319571u
30
+ ··· + 0.411896u + 0.929033
a
7
=
−0.0263408u
31
− 0.0396638u
30
+ ··· − 0.0912398u − 0.0158334
0.286928u
31
− 0.815573u
30
+ ··· − 6.66152u + 2.51686
a
12
=
−0.111217u
31
− 0.545439u
30
+ ··· − 5.88693u + 3.81525
−0.346449u
31
+ 0.596107u
30
+ ··· + 3.53802u − 0.422190
a
1
=
0.235232u
31
− 1.14155u
30
+ ··· − 9.42495u + 4.23744
−0.346449u
31
+ 0.596107u
30
+ ··· + 3.53802u − 0.422190
a
9
=
u
2
+ 1
u
2
a
4
=
u
5
+ 2u
3
+ u
u
5
+ u
3
+ u
a
8
=
−u
8
− 3u
6
− 3u
4
+ 1
−u
8
− 2u
6
− 2u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
518677209154
539420405137
u
31
−
1641358644799
539420405137
u
30
+ ··· −
8769457355210
539420405137
u +
688678562270
539420405137
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
32
+ 24u
31
+ ··· + 16u + 1
c
2
, c
6
u
32
− 2u
31
+ ··· + 6u + 1
c
3
u
32
+ 2u
31
+ ··· + 742u + 173
c
4
, c
8
u
32
− u
31
+ ··· + 20u − 4
c
5
, c
10
u
32
+ u
31
+ ··· − 12u − 4
c
7
, c
11
u
32
+ 3u
31
+ ··· + 43u − 13
c
9
u
32
+ 21u
31
+ ··· + 80u + 16
c
12
u
32
+ 53u
31
+ ··· + 1745u + 169
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
32
− 24y
31
+ ··· − 816y + 1
c
2
, c
6
y
32
+ 24y
31
+ ··· + 16y + 1
c
3
y
32
− 72y
31
+ ··· + 762852y + 29929
c
4
, c
8
y
32
− 51y
31
+ ··· − 112y + 16
c
5
, c
10
y
32
+ 21y
31
+ ··· + 80y + 16
c
7
, c
11
y
32
− 53y
31
+ ··· − 1745y + 169
c
9
y
32
− 15y
31
+ ··· + 256y + 256
c
12
y
32
− 133y
31
+ ··· − 9350077y + 28561
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.01428
a = −0.536855
b = 0.0938357
−12.7729 −5.31340
u = −0.300579 + 0.918264I
a = 0.391932 − 0.210423I
b = −0.161520 + 0.335427I
−0.55082 − 1.63457I −3.27047 + 3.85284I
u = −0.300579 − 0.918264I
a = 0.391932 + 0.210423I
b = −0.161520 − 0.335427I
−0.55082 + 1.63457I −3.27047 − 3.85284I
u = 1.043470 + 0.101095I
a = −0.26843 − 1.93217I
b = −0.57959 − 2.65577I
−17.2641 − 6.3638I −7.47410 + 2.59633I
u = 1.043470 − 0.101095I
a = −0.26843 + 1.93217I
b = −0.57959 + 2.65577I
−17.2641 + 6.3638I −7.47410 − 2.59633I
u = 0.123488 + 1.046200I
a = −0.560397 + 1.014520I
b = 0.578687 − 0.837647I
−3.34462 + 2.78018I −9.66898 − 3.45316I
u = 0.123488 − 1.046200I
a = −0.560397 − 1.014520I
b = 0.578687 + 0.837647I
−3.34462 − 2.78018I −9.66898 + 3.45316I
u = −0.880708 + 0.236833I
a = −0.59959 + 1.89659I
b = 0.17577 + 2.14910I
−5.35743 − 0.84578I −8.20870 + 1.07921I
u = −0.880708 − 0.236833I
a = −0.59959 − 1.89659I
b = 0.17577 − 2.14910I
−5.35743 + 0.84578I −8.20870 − 1.07921I
u = 0.419631 + 1.045310I
a = −1.39528 − 0.99960I
b = −0.207451 − 1.209000I
−2.17363 + 5.75346I −5.15068 − 8.16213I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.419631 − 1.045310I
a = −1.39528 + 0.99960I
b = −0.207451 + 1.209000I
−2.17363 − 5.75346I −5.15068 + 8.16213I
u = −0.011003 + 1.154310I
a = 1.50102 − 0.43416I
b = 0.532426 + 0.445478I
−4.29088 − 1.34269I −11.00178 + 0.73571I
u = −0.011003 − 1.154310I
a = 1.50102 + 0.43416I
b = 0.532426 − 0.445478I
−4.29088 + 1.34269I −11.00178 − 0.73571I
u = 0.429441 + 1.086430I
a = −0.368309 − 1.007620I
b = 1.129540 − 0.239167I
−4.20427 + 3.60564I −10.41579 − 4.53089I
u = 0.429441 − 1.086430I
a = −0.368309 + 1.007620I
b = 1.129540 + 0.239167I
−4.20427 − 3.60564I −10.41579 + 4.53089I
u = 0.300263 + 0.761792I
a = 0.357934 + 0.706332I
b = 0.690935 + 1.189790I
−2.49558 − 0.98889I −10.28745 − 0.57316I
u = 0.300263 − 0.761792I
a = 0.357934 − 0.706332I
b = 0.690935 − 1.189790I
−2.49558 + 0.98889I −10.28745 + 0.57316I
u = −0.384747 + 0.600251I
a = 0.721932 + 0.539648I
b = −0.302394 + 0.210503I
0.25012 − 1.51862I 0.08529 + 4.58805I
u = −0.384747 − 0.600251I
a = 0.721932 − 0.539648I
b = −0.302394 − 0.210503I
0.25012 + 1.51862I 0.08529 − 4.58805I
u = −0.613650 + 1.166290I
a = 1.41978 − 0.58897I
b = 0.11410 − 2.43615I
−8.05657 − 4.56260I −10.63691 + 3.18178I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.613650 − 1.166290I
a = 1.41978 + 0.58897I
b = 0.11410 + 2.43615I
−8.05657 + 4.56260I −10.63691 − 3.18178I
u = −0.356156 + 1.331570I
a = −1.82084 − 0.21954I
b = −0.07721 + 2.01088I
−10.24520 − 5.08725I −11.32137 + 3.44892I
u = −0.356156 − 1.331570I
a = −1.82084 + 0.21954I
b = −0.07721 − 2.01088I
−10.24520 + 5.08725I −11.32137 − 3.44892I
u = 0.459215 + 0.354759I
a = 0.80233 + 2.03995I
b = −0.151979 + 0.783354I
−0.25749 − 2.03582I −0.07050 + 3.37549I
u = 0.459215 − 0.354759I
a = 0.80233 − 2.03995I
b = −0.151979 − 0.783354I
−0.25749 + 2.03582I −0.07050 − 3.37549I
u = −0.50730 + 1.33114I
a = −0.533965 + 0.005339I
b = 0.0323333 − 0.1210510I
−16.9136 − 5.4099I −8.22953 + 2.64698I
u = −0.50730 − 1.33114I
a = −0.533965 − 0.005339I
b = 0.0323333 + 0.1210510I
−16.9136 + 5.4099I −8.22953 − 2.64698I
u = 0.56860 + 1.30790I
a = 1.95984 + 0.66591I
b = −0.66128 + 2.62930I
18.4866 + 12.1069I −9.82586 − 5.57772I
u = 0.56860 − 1.30790I
a = 1.95984 − 0.66591I
b = −0.66128 − 2.62930I
18.4866 − 12.1069I −9.82586 + 5.57772I
u = 0.541663
a = 0.857323
b = 0.704419
−1.42188 −6.40410
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.44634 + 1.39008I
a = −1.268180 + 0.456140I
b = −0.51150 − 2.59045I
17.4567 − 1.0678I −10.66447 + 0.I
u = 0.44634 − 1.39008I
a = −1.268180 − 0.456140I
b = −0.51150 + 2.59045I
17.4567 + 1.0678I −10.66447 + 0.I
8
II. I
u
2
= h−au − u
2
+ b − 1, −u
3
a + 3u
3
+ · · · + 2a − 2, u
4
+ 2u
2
+ 2i
(i) Arc colorings
a
5
=
0
u
a
11
=
1
0
a
10
=
1
u
2
a
2
=
a
au + u
2
+ 1
a
6
=
u
u
3
+ u
a
3
=
u
3
a + u
2
a − u
2
+ 3a − 2
−u
3
a + u
2
a − au + 1
a
7
=
u
3
a −
1
2
u
3
+ au − u
2
+ a − 2
1
a
12
=
u
3
a −
1
2
u
3
+ au − u
2
+ a − 1
1
a
1
=
u
3
a −
1
2
u
3
+ au − u
2
+ a − 2
1
a
9
=
u
2
+ 1
u
2
a
4
=
−u
−u
3
− u
a
8
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
3
a + 4u
2
a + 4u
3
− 4au − 4u
2
+ 4u − 12
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u
2
− u + 1)
4
c
3
, c
6
(u
2
+ u + 1)
4
c
4
, c
8
(u
4
− 2u
2
+ 2)
2
c
5
, c
10
(u
4
+ 2u
2
+ 2)
2
c
7
, c
12
(u + 1)
8
c
9
(u
2
− 2u + 2)
4
c
11
(u − 1)
8
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
6
(y
2
+ y + 1)
4
c
4
, c
8
(y
2
− 2y + 2)
4
c
5
, c
10
(y
2
+ 2y + 2)
4
c
7
, c
11
, c
12
(y −1)
8
c
9
(y
2
+ 4)
4
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.455090 + 1.098680I
a = 0.922841 − 0.931556I
b = 1.44346 + 1.58997I
−4.11234 + 1.63398I −10.00000 − 0.53590I
u = 0.455090 + 1.098680I
a = −2.15482 − 1.48893I
b = 0.65522 − 2.04506I
−4.11234 + 5.69375I −10.0000 − 7.46410I
u = 0.455090 − 1.098680I
a = 0.922841 + 0.931556I
b = 1.44346 − 1.58997I
−4.11234 − 1.63398I −10.00000 + 0.53590I
u = 0.455090 − 1.098680I
a = −2.15482 + 1.48893I
b = 0.65522 + 2.04506I
−4.11234 − 5.69375I −10.0000 + 7.46410I
u = −0.455090 + 1.098680I
a = 0.809210 + 0.068444I
b = −0.443461 − 0.142082I
−4.11234 − 1.63398I −10.00000 + 0.53590I
u = −0.455090 + 1.098680I
a = 0.422767 − 0.488925I
b = 0.344777 − 0.313008I
−4.11234 − 5.69375I −10.00000 + 7.46410I
u = −0.455090 − 1.098680I
a = 0.809210 − 0.068444I
b = −0.443461 + 0.142082I
−4.11234 + 1.63398I −10.00000 − 0.53590I
u = −0.455090 − 1.098680I
a = 0.422767 + 0.488925I
b = 0.344777 + 0.313008I
−4.11234 + 5.69375I −10.00000 − 7.46410I
12
III. I
v
1
= ha, b + v, v
2
+ v + 1i
(i) Arc colorings
a
5
=
v
0
a
11
=
1
0
a
10
=
1
0
a
2
=
0
−v
a
6
=
v
0
a
3
=
−1
−v
a
7
=
v
−1
a
12
=
−v + 1
1
a
1
=
−v
1
a
9
=
1
0
a
4
=
v
0
a
8
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4v − 8
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
6
u
2
− u + 1
c
2
u
2
+ u + 1
c
4
, c
5
, c
8
c
9
, c
10
u
2
c
7
(u − 1)
2
c
11
, c
12
(u + 1)
2
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
6
y
2
+ y + 1
c
4
, c
5
, c
8
c
9
, c
10
y
2
c
7
, c
11
, c
12
(y − 1)
2
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −0.500000 + 0.866025I
a = 0
b = 0.500000 − 0.866025I
−1.64493 + 2.02988I −6.00000 − 3.46410I
v = −0.500000 − 0.866025I
a = 0
b = 0.500000 + 0.866025I
−1.64493 − 2.02988I −6.00000 + 3.46410I
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
5
)(u
32
+ 24u
31
+ ··· + 16u + 1)
c
2
((u
2
− u + 1)
4
)(u
2
+ u + 1)(u
32
− 2u
31
+ ··· + 6u + 1)
c
3
(u
2
− u + 1)(u
2
+ u + 1)
4
(u
32
+ 2u
31
+ ··· + 742u + 173)
c
4
, c
8
u
2
(u
4
− 2u
2
+ 2)
2
(u
32
− u
31
+ ··· + 20u − 4)
c
5
, c
10
u
2
(u
4
+ 2u
2
+ 2)
2
(u
32
+ u
31
+ ··· − 12u − 4)
c
6
(u
2
− u + 1)(u
2
+ u + 1)
4
(u
32
− 2u
31
+ ··· + 6u + 1)
c
7
((u − 1)
2
)(u + 1)
8
(u
32
+ 3u
31
+ ··· + 43u − 13)
c
9
u
2
(u
2
− 2u + 2)
4
(u
32
+ 21u
31
+ ··· + 80u + 16)
c
11
((u − 1)
8
)(u + 1)
2
(u
32
+ 3u
31
+ ··· + 43u − 13)
c
12
((u + 1)
10
)(u
32
+ 53u
31
+ ··· + 1745u + 169)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
5
)(y
32
− 24y
31
+ ··· − 816y + 1)
c
2
, c
6
((y
2
+ y + 1)
5
)(y
32
+ 24y
31
+ ··· + 16y + 1)
c
3
((y
2
+ y + 1)
5
)(y
32
− 72y
31
+ ··· + 762852y + 29929)
c
4
, c
8
y
2
(y
2
− 2y + 2)
4
(y
32
− 51y
31
+ ··· − 112y + 16)
c
5
, c
10
y
2
(y
2
+ 2y + 2)
4
(y
32
+ 21y
31
+ ··· + 80y + 16)
c
7
, c
11
((y − 1)
10
)(y
32
− 53y
31
+ ··· − 1745y + 169)
c
9
y
2
(y
2
+ 4)
4
(y
32
− 15y
31
+ ··· + 256y + 256)
c
12
((y − 1)
10
)(y
32
− 133y
31
+ ··· − 9350077y + 28561)
18
