12n
0171
(K12n
0171
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 11 10 4 5 12 6 7 9
Solving Sequence
5,11 2,6
4 10 7 8 12 3 1 9
c
5
c
4
c
10
c
6
c
7
c
11
c
3
c
1
c
9
c
2
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
23
+ u
22
+ ··· + 6u
2
+ b, −u
21
+ u
20
+ ··· + a + 2, u
26
− 2u
25
+ ··· + 2u − 1i
I
u
2
= hb + 1, u
2
+ a + u + 3, u
3
+ 2u − 1i
I
u
3
= hb + 1, u
3
+ a + u + 2, u
4
+ u
3
+ 2u
2
+ 2u + 1i
* 3 irreducible components of dim
C
= 0, with total 33 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
23
+u
22
+· · ·+6u
2
+b, −u
21
+u
20
+· · ·+a+2, u
26
−2u
25
+· · ·+2u−1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
2
=
u
21
− u
20
+ ··· + 6u − 2
u
23
− u
22
+ ··· + 12u
3
− 6u
2
a
6
=
1
u
2
a
4
=
−u
22
+ u
21
+ ··· + 6u − 1
−u
22
+ u
21
+ ··· − 6u
2
+ u
a
10
=
u
u
3
+ u
a
7
=
u
2
+ 1
u
4
+ 2u
2
a
8
=
u
11
+ 6u
9
+ 12u
7
+ 8u
5
+ u
3
+ 2u
u
11
+ 5u
9
+ 8u
7
+ 3u
5
− u
3
+ u
a
12
=
−u
5
− 2u
3
− u
−u
7
− 3u
5
− 2u
3
+ u
a
3
=
−u
23
+ u
22
+ ··· + 6u − 2
u
23
− u
22
+ ··· + 12u
3
− 6u
2
a
1
=
u
13
+ 6u
11
+ 13u
9
+ 12u
7
+ 6u
5
+ 4u
3
+ u
u
15
+ 7u
13
+ 18u
11
+ 19u
9
+ 6u
7
+ 2u
5
+ 4u
3
− u
a
9
=
u
9
+ 4u
7
+ 5u
5
+ 2u
3
+ u
u
11
+ 5u
9
+ 8u
7
+ 3u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
25
+ 8u
24
− 59u
23
+ 102u
22
− 370u
21
+ 552u
20
− 1282u
19
+
1632u
18
− 2664u
17
+ 2824u
16
− 3388u
15
+ 2869u
14
− 2682u
13
+ 1765u
12
− 1574u
11
+
986u
10
− 1012u
9
+ 713u
8
− 547u
7
+ 276u
6
− 182u
5
+ 78u
4
− 102u
3
+ 80u
2
− 8u − 13
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
26
+ 30u
24
+ ··· + 25u + 1
c
2
, c
4
u
26
− 8u
25
+ ··· + 9u − 1
c
3
, c
7
u
26
− u
25
+ ··· − 64u + 128
c
5
, c
6
, c
10
u
26
+ 2u
25
+ ··· − 2u − 1
c
8
u
26
+ 2u
25
+ ··· + 3088u − 11981
c
9
, c
12
u
26
− 2u
25
+ ··· − 7u
2
+ 1
c
11
u
26
− 2u
25
+ ··· + 48u − 72
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
26
+ 60y
25
+ ··· − 337y + 1
c
2
, c
4
y
26
+ 30y
24
+ ··· − 25y + 1
c
3
, c
7
y
26
− 45y
25
+ ··· − 258048y + 16384
c
5
, c
6
, c
10
y
26
+ 26y
25
+ ··· − 14y + 1
c
8
y
26
+ 122y
25
+ ··· − 4243693030y + 143544361
c
9
, c
12
y
26
+ 38y
25
+ ··· − 14y + 1
c
11
y
26
+ 18y
25
+ ··· − 16272y + 5184
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.703140 + 0.538371I
a = 0.595244 − 1.218420I
b = 1.13151 − 1.20043I
14.5729 + 1.9359I −4.36941 + 0.69311I
u = 0.703140 − 0.538371I
a = 0.595244 + 1.218420I
b = 1.13151 + 1.20043I
14.5729 − 1.9359I −4.36941 − 0.69311I
u = 0.729197 + 0.493462I
a = 2.28117 − 0.26467I
b = 1.17041 + 1.16109I
14.4224 − 6.7123I −4.71846 + 4.71456I
u = 0.729197 − 0.493462I
a = 2.28117 + 0.26467I
b = 1.17041 − 1.16109I
14.4224 + 6.7123I −4.71846 − 4.71456I
u = −0.657044 + 0.360115I
a = 1.94829 − 0.09632I
b = 0.522022 − 0.742639I
3.09785 + 3.69296I −4.58596 − 5.59657I
u = −0.657044 − 0.360115I
a = 1.94829 + 0.09632I
b = 0.522022 + 0.742639I
3.09785 − 3.69296I −4.58596 + 5.59657I
u = −0.532790 + 0.522258I
a = 0.281675 + 0.540170I
b = 0.345233 + 0.836005I
3.71321 + 0.25250I −2.90666 − 1.69873I
u = −0.532790 − 0.522258I
a = 0.281675 − 0.540170I
b = 0.345233 − 0.836005I
3.71321 − 0.25250I −2.90666 + 1.69873I
u = 0.146766 + 1.279190I
a = 1.289160 − 0.159208I
b = 0.339771 + 0.227061I
2.95015 − 2.37770I −2.22960 + 4.04579I
u = 0.146766 − 1.279190I
a = 1.289160 + 0.159208I
b = 0.339771 − 0.227061I
2.95015 + 2.37770I −2.22960 − 4.04579I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.041859 + 1.351780I
a = −1.30338 − 1.15705I
b = −1.156640 + 0.215585I
2.15651 + 1.07901I −3.64779 + 0.86156I
u = −0.041859 − 1.351780I
a = −1.30338 + 1.15705I
b = −1.156640 − 0.215585I
2.15651 − 1.07901I −3.64779 − 0.86156I
u = 0.108685 + 1.405560I
a = 0.007384 + 1.006780I
b = −0.587676 − 0.621059I
4.56460 − 2.76012I −2.16196 + 3.94765I
u = 0.108685 − 1.405560I
a = 0.007384 − 1.006780I
b = −0.587676 + 0.621059I
4.56460 + 2.76012I −2.16196 − 3.94765I
u = −0.24567 + 1.43288I
a = 1.45979 + 0.68729I
b = 0.662222 − 0.763929I
8.83594 + 6.98292I −1.16198 − 5.80218I
u = −0.24567 − 1.43288I
a = 1.45979 − 0.68729I
b = 0.662222 + 0.763929I
8.83594 − 6.98292I −1.16198 + 5.80218I
u = 0.512846
a = 1.47619
b = 0.224405
−1.00355 −9.54280
u = −0.17183 + 1.48536I
a = −0.125401 − 0.252038I
b = 0.312586 + 1.039960I
10.21460 + 2.79653I 0. − 1.62269I
u = −0.17183 − 1.48536I
a = −0.125401 + 0.252038I
b = 0.312586 − 1.039960I
10.21460 − 2.79653I 0. + 1.62269I
u = 0.25787 + 1.50930I
a = 1.32769 − 1.23506I
b = 1.21947 + 1.15279I
−18.5465 − 10.3199I −1.57367 + 4.71876I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.25787 − 1.50930I
a = 1.32769 + 1.23506I
b = 1.21947 − 1.15279I
−18.5465 + 10.3199I −1.57367 − 4.71876I
u = 0.23540 + 1.52332I
a = −0.334114 − 0.224598I
b = 1.12696 − 1.25945I
−18.1670 − 1.4877I −1.145266 + 0.723281I
u = 0.23540 − 1.52332I
a = −0.334114 + 0.224598I
b = 1.12696 + 1.25945I
−18.1670 + 1.4877I −1.145266 − 0.723281I
u = 0.368033 + 0.267400I
a = −0.272224 + 1.261050I
b = −0.657600 − 0.268641I
−0.775564 − 1.043300I −8.38623 + 6.25558I
u = 0.368033 − 0.267400I
a = −0.272224 − 1.261050I
b = −0.657600 + 0.268641I
−0.775564 + 1.043300I −8.38623 − 6.25558I
u = −0.312651
a = −4.78676
b = −1.08095
−2.08164 2.25490
7
II. I
u
2
= hb + 1, u
2
+ a + u + 3, u
3
+ 2u − 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
2
=
−u
2
− u − 3
−1
a
6
=
1
u
2
a
4
=
−u
2
− u − 2
−1
a
10
=
u
−u + 1
a
7
=
u
2
+ 1
u
a
8
=
u
2
+ 1
u
a
12
=
−u
2
− u
u
2
a
3
=
−u
2
− u − 2
−1
a
1
=
−1
0
a
9
=
u
2
− u + 1
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −7u
2
− 5u − 18
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
7
u
3
c
4
(u + 1)
3
c
5
, c
6
, c
9
u
3
+ 2u − 1
c
8
, c
10
, c
12
u
3
+ 2u + 1
c
11
u
3
+ 3u
2
+ 5u + 2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
3
c
3
, c
7
y
3
c
5
, c
6
, c
8
c
9
, c
10
, c
12
y
3
+ 4y
2
+ 4y −1
c
11
y
3
+ y
2
+ 13y −4
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.22670 + 1.46771I
a = −0.670516 − 0.802255I
b = −1.00000
7.79580 + 5.13794I −2.14701 − 2.68036I
u = −0.22670 − 1.46771I
a = −0.670516 + 0.802255I
b = −1.00000
7.79580 − 5.13794I −2.14701 + 2.68036I
u = 0.453398
a = −3.65897
b = −1.00000
−2.43213 −21.7060
11
III. I
u
3
= hb + 1, u
3
+ a + u + 2, u
4
+ u
3
+ 2u
2
+ 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
2
=
−u
3
− u − 2
−1
a
6
=
1
u
2
a
4
=
−u
3
− u − 1
−1
a
10
=
u
u
3
+ u
a
7
=
u
2
+ 1
−u
3
− 2u − 1
a
8
=
u
2
+ 1
−u
3
− 2u − 1
a
12
=
−u
3
− 2u − 1
−u
3
− u
2
− u − 2
a
3
=
−u
3
− u − 1
−1
a
1
=
−1
0
a
9
=
u
3
+ u
2
+ 2u + 2
−u
3
− 2u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
3
+ 2u
2
− 2u − 7
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
4
c
3
, c
7
u
4
c
4
(u + 1)
4
c
5
, c
6
, c
9
u
4
+ u
3
+ 2u
2
+ 2u + 1
c
8
, c
10
, c
12
u
4
− u
3
+ 2u
2
− 2u + 1
c
11
(u
2
− u + 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
7
y
4
c
5
, c
6
, c
8
c
9
, c
10
, c
12
y
4
+ 3y
3
+ 2y
2
+ 1
c
11
(y
2
+ y + 1)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.621744 + 0.440597I
a = −1.50000 − 0.86603I
b = −1.00000
1.64493 + 2.02988I −5.73686 − 3.25323I
u = −0.621744 − 0.440597I
a = −1.50000 + 0.86603I
b = −1.00000
1.64493 − 2.02988I −5.73686 + 3.25323I
u = 0.121744 + 1.306620I
a = −1.50000 + 0.86603I
b = −1.00000
1.64493 − 2.02988I −8.76314 + 4.54099I
u = 0.121744 − 1.306620I
a = −1.50000 − 0.86603I
b = −1.00000
1.64493 + 2.02988I −8.76314 − 4.54099I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
7
)(u
26
+ 30u
24
+ ··· + 25u + 1)
c
2
((u − 1)
7
)(u
26
− 8u
25
+ ··· + 9u − 1)
c
3
, c
7
u
7
(u
26
− u
25
+ ··· − 64u + 128)
c
4
((u + 1)
7
)(u
26
− 8u
25
+ ··· + 9u − 1)
c
5
, c
6
(u
3
+ 2u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)(u
26
+ 2u
25
+ ··· − 2u − 1)
c
8
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
26
+ 2u
25
+ ··· + 3088u − 11981)
c
9
(u
3
+ 2u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)(u
26
− 2u
25
+ ··· − 7u
2
+ 1)
c
10
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
26
+ 2u
25
+ ··· − 2u − 1)
c
11
((u
2
− u + 1)
2
)(u
3
+ 3u
2
+ 5u + 2)(u
26
− 2u
25
+ ··· + 48u − 72)
c
12
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
26
− 2u
25
+ ··· − 7u
2
+ 1)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
7
)(y
26
+ 60y
25
+ ··· − 337y + 1)
c
2
, c
4
((y −1)
7
)(y
26
+ 30y
24
+ ··· − 25y + 1)
c
3
, c
7
y
7
(y
26
− 45y
25
+ ··· − 258048y + 16384)
c
5
, c
6
, c
10
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
26
+ 26y
25
+ ··· − 14y + 1)
c
8
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)
· (y
26
+ 122y
25
+ ··· − 4243693030y + 143544361)
c
9
, c
12
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
26
+ 38y
25
+ ··· − 14y + 1)
c
11
((y
2
+ y + 1)
2
)(y
3
+ y
2
+ 13y −4)(y
26
+ 18y
25
+ ··· − 16272y + 5184)
17
