12n
0153
(K12n
0153
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 9 11 3 5 6 12 7 10
Solving Sequence
6,11 3,7
8 4 12 10 1 9 5 2
c
6
c
7
c
3
c
11
c
10
c
12
c
9
c
5
c
2
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
13
− u
12
+ 2u
11
+ 3u
10
− 2u
9
− 4u
8
+ 4u
6
+ 2u
5
− 2u
4
− u
3
+ 2u
2
+ b + u − 1,
− 2u
13
− 2u
12
+ 4u
11
+ 7u
10
− 4u
9
− 11u
8
− u
7
+ 12u
6
+ 6u
5
− 6u
4
− 4u
3
+ 4u
2
+ a + 3u − 1,
u
14
+ 2u
13
− u
12
− 6u
11
− 2u
10
+ 8u
9
+ 7u
8
− 6u
7
− 10u
6
+ 6u
4
− 4u
2
− u + 1i
I
u
2
= hu
7
− u
5
+ 2u
3
+ b − u + 1, u
7
− u
5
+ u
4
+ 2u
3
− u
2
+ a + 2, u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 22 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−u
13
−u
12
+· · ·+b−1, −2u
13
−2u
12
+· · ·+a−1, u
14
+2u
13
+· · ·−u+1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
2u
13
+ 2u
12
+ ··· − 3u + 1
u
13
+ u
12
+ ··· − u + 1
a
7
=
1
u
2
a
8
=
−3u
13
− 4u
12
+ ··· + 5u − 2
−2u
13
− 4u
12
+ ··· + 4u − 2
a
4
=
−2u
13
− 4u
12
+ ··· + 3u − 2
2u
13
+ u
12
+ ··· + 2u + 1
a
12
=
−u
−u
3
+ u
a
10
=
u
3
u
5
− u
3
+ u
a
1
=
−u
5
− u
−u
7
+ u
5
− 2u
3
+ u
a
9
=
u
5
+ u
u
5
− u
3
+ u
a
5
=
−u
10
+ u
8
− 2u
6
+ u
4
− u
2
+ 1
−u
10
+ 2u
8
− 3u
6
+ 2u
4
− u
2
a
2
=
5u
13
+ 5u
12
+ ··· − 6u + 3
u
13
+ u
12
− 2u
11
− u
10
+ 2u
9
+ u
7
+ 2u
6
− 3u
5
− 2u
4
+ 3u
3
− 2u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −7u
13
−6u
12
+18u
11
+25u
10
−24u
9
−44u
8
+9u
7
+53u
6
+13u
5
−35u
4
−13u
3
+23u
2
+7u−21
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
14
+ 35u
13
+ ··· + 57u + 1
c
2
, c
4
u
14
− 9u
13
+ ··· + u − 1
c
3
, c
7
u
14
− 7u
13
+ ··· − 640u − 256
c
5
, c
8
, c
9
u
14
+ 2u
13
+ ··· + 3u + 1
c
6
, c
11
u
14
− 2u
13
+ ··· + u + 1
c
10
, c
12
u
14
+ 6u
13
+ ··· + 9u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
14
− 215y
13
+ ··· − 1173y + 1
c
2
, c
4
y
14
− 35y
13
+ ··· − 57y + 1
c
3
, c
7
y
14
− 75y
13
+ ··· + 16384y + 65536
c
5
, c
8
, c
9
y
14
− 30y
13
+ ··· − 9y + 1
c
6
, c
11
y
14
− 6y
13
+ ··· − 9y + 1
c
10
, c
12
y
14
+ 6y
13
+ ··· − 25y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.959410 + 0.328783I
a = 2.22180 + 0.56610I
b = 1.191800 + 0.163474I
−3.28458 + 1.19495I −18.0412 − 3.1465I
u = −0.959410 − 0.328783I
a = 2.22180 − 0.56610I
b = 1.191800 − 0.163474I
−3.28458 − 1.19495I −18.0412 + 3.1465I
u = −0.501889 + 0.920209I
a = 0.725724 + 0.027363I
b = 3.28288 − 0.17435I
18.7096 − 2.3664I −13.94239 + 0.06300I
u = −0.501889 − 0.920209I
a = 0.725724 − 0.027363I
b = 3.28288 + 0.17435I
18.7096 + 2.3664I −13.94239 − 0.06300I
u = −0.853744 + 0.641916I
a = −0.410449 − 0.466723I
b = −0.596688 − 0.171568I
1.83462 + 2.50408I −6.20303 − 3.70135I
u = −0.853744 − 0.641916I
a = −0.410449 + 0.466723I
b = −0.596688 + 0.171568I
1.83462 − 2.50408I −6.20303 + 3.70135I
u = 1.014210 + 0.562829I
a = 1.61553 − 1.07680I
b = 1.036730 + 0.627532I
−1.62931 − 4.65799I −15.4888 + 5.2954I
u = 1.014210 − 0.562829I
a = 1.61553 + 1.07680I
b = 1.036730 − 0.627532I
−1.62931 + 4.65799I −15.4888 − 5.2954I
u = 0.589347 + 0.525928I
a = −0.333608 + 0.150120I
b = 0.644384 − 0.529402I
−0.335782 + 0.137583I −12.53131 − 0.75433I
u = 0.589347 − 0.525928I
a = −0.333608 − 0.150120I
b = 0.644384 + 0.529402I
−0.335782 − 0.137583I −12.53131 + 0.75433I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.25934
a = 4.69391
b = 3.12501
12.1268 −18.9430
u = −1.128420 + 0.686699I
a = 2.14185 + 3.38010I
b = 3.21915 + 0.28216I
16.7915 + 8.2751I −16.0152 − 4.1669I
u = −1.128420 − 0.686699I
a = 2.14185 − 3.38010I
b = 3.21915 − 0.28216I
16.7915 − 8.2751I −16.0152 + 4.1669I
u = 0.420479
a = −0.615608
b = 0.318491
−0.632046 −15.6130
6
II. I
u
2
= hu
7
− u
5
+ 2u
3
+ b − u + 1, u
7
− u
5
+ u
4
+ 2u
3
− u
2
+ a + 2, u
8
−
u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
−u
7
+ u
5
− u
4
− 2u
3
+ u
2
− 2
−u
7
+ u
5
− 2u
3
+ u − 1
a
7
=
1
u
2
a
8
=
1
u
2
a
4
=
−u
7
+ u
5
− u
4
− 2u
3
+ u
2
− 2
−u
7
+ u
5
− 2u
3
+ u − 1
a
12
=
−u
−u
3
+ u
a
10
=
u
3
u
5
− u
3
+ u
a
1
=
−u
5
− u
−u
7
+ u
5
− 2u
3
+ u
a
9
=
u
5
+ u
u
5
− u
3
+ u
a
5
=
u
5
+ u
u
7
− u
5
+ 2u
3
− u
a
2
=
−u
7
− u
4
− 2u
3
+ u
2
− u − 2
−2u
7
+ 2u
5
− 4u
3
+ 2u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
7
− u
6
+ 5u
5
− 5u
3
+ u
2
+ 4u − 17
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
8
c
3
, c
7
u
8
c
4
(u + 1)
8
c
5
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
c
6
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
c
8
, c
9
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
c
10
u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1
c
11
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1
c
12
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
8
c
3
, c
7
y
8
c
5
, c
8
, c
9
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
c
6
, c
11
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
c
10
, c
12
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.570868 + 0.730671I
a = −0.805639 − 0.183365I
b = 0.320534 − 0.633953I
−2.68559 + 1.13123I −13.47926 − 0.84929I
u = 0.570868 − 0.730671I
a = −0.805639 + 0.183365I
b = 0.320534 + 0.633953I
−2.68559 − 1.13123I −13.47926 + 0.84929I
u = −0.855237 + 0.665892I
a = −0.189481 − 1.310380I
b = −1.54709 − 0.16160I
0.51448 + 2.57849I −14.5054 − 3.2330I
u = −0.855237 − 0.665892I
a = −0.189481 + 1.310380I
b = −1.54709 + 0.16160I
0.51448 − 2.57849I −14.5054 + 3.2330I
u = −1.09818
a = 0.729394
b = 0.879647
−8.14766 −19.4520
u = 1.031810 + 0.655470I
a = 0.708845 − 0.169402I
b = 0.679246 + 0.851242I
−4.02461 − 6.44354I −15.2754 + 5.9053I
u = 1.031810 − 0.655470I
a = 0.708845 + 0.169402I
b = 0.679246 − 0.851242I
−4.02461 + 6.44354I −15.2754 − 5.9053I
u = 0.603304
a = −2.15684
b = −0.785038
−2.48997 −15.0280
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
14
+ 35u
13
+ ··· + 57u + 1)
c
2
((u − 1)
8
)(u
14
− 9u
13
+ ··· + u − 1)
c
3
, c
7
u
8
(u
14
− 7u
13
+ ··· − 640u − 256)
c
4
((u + 1)
8
)(u
14
− 9u
13
+ ··· + u − 1)
c
5
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
14
+ 2u
13
+ ··· + 3u + 1)
c
6
(u
8
− u
7
+ ··· + 2u − 1)(u
14
− 2u
13
+ ··· + u + 1)
c
8
, c
9
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)(u
14
+ 2u
13
+ ··· + 3u + 1)
c
10
(u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
· (u
14
+ 6u
13
+ ··· + 9u + 1)
c
11
(u
8
+ u
7
+ ··· − 2u − 1)(u
14
− 2u
13
+ ··· + u + 1)
c
12
(u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
· (u
14
+ 6u
13
+ ··· + 9u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
8
)(y
14
− 215y
13
+ ··· − 1173y + 1)
c
2
, c
4
((y −1)
8
)(y
14
− 35y
13
+ ··· − 57y + 1)
c
3
, c
7
y
8
(y
14
− 75y
13
+ ··· + 16384y + 65536)
c
5
, c
8
, c
9
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
14
− 30y
13
+ ··· − 9y + 1)
c
6
, c
11
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
14
− 6y
13
+ ··· − 9y + 1)
c
10
, c
12
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
14
+ 6y
13
+ ··· − 25y + 1)
12
