12n
0387
(K12n
0387
)
A knot diagram
1
Linearized knot diagam
3 6 10 8 2 12 3 11 4 9 6 7
Solving Sequence
4,9
10
6,11
12 3 2 1 5 8 7
c
9
c
10
c
11
c
3
c
2
c
1
c
5
c
8
c
7
c
4
, c
6
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−2u
14
+ 3u
13
− 7u
12
+ 6u
11
− 13u
10
+ 6u
9
− 12u
8
− 9u
6
− 10u
5
+ u
4
− 13u
3
+ u
2
+ b − 7u − 3,
3u
15
− 9u
14
+ ··· + 2a − 8, u
16
− 3u
15
+ ··· + 2u + 2i
I
u
2
= hu
2
+ b + u + 1, −u
3
+ 2a + u + 2, u
4
+ u
2
+ 2i
I
u
3
= h−u
3
+ au − u
2
+ b + 1, −u
3
a − 2u
2
a + u
3
+ a
2
− 2au − 2u
2
− 1, u
4
+ u
3
+ u
2
+ 1i
I
u
4
= h−u
3
− u
2
+ b + 1, −u
3
− u
2
+ a − u, u
4
+ 1i
I
u
5
= hb − u, a − 1, u
2
+ 1i
I
v
1
= ha, b + 1, v + 1i
* 6 irreducible components of dim
C
= 0, with total 35 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−2u
14
+3u
13
+· · ·+b−3, 3u
15
−9u
14
+· · ·+2a−8, u
16
−3u
15
+· · ·+2u+2i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
10
=
1
u
2
a
6
=
−
3
2
u
15
+
9
2
u
14
+ ··· + 5u + 4
2u
14
− 3u
13
+ ··· + 7u + 3
a
11
=
u
2
+ 1
u
2
a
12
=
−
1
2
u
15
+
1
2
u
14
+ ··· +
3
2
u
3
+ 1
−u
15
+ 2u
14
+ ··· + u + 1
a
3
=
u
u
3
+ u
a
2
=
−
1
2
u
15
+
1
2
u
14
+ ··· − u
2
+ u
−u
15
+ 2u
14
+ ··· + 2u + 1
a
1
=
−
3
2
u
15
+
3
2
u
14
+ ··· +
3
2
u
3
− 3u
2
−4u
15
+ 9u
14
+ ··· + 7u + 5
a
5
=
−u
9
− 2u
7
− 3u
5
− 2u
3
− u
−u
9
− u
7
− u
5
+ u
a
8
=
u
4
+ u
2
+ 1
u
4
a
7
=
−u
8
− u
6
− u
4
+ 1
−u
10
− 2u
8
− 3u
6
− 2u
4
− u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
15
− 8u
14
+ 12u
13
− 18u
12
+ 18u
11
− 28u
10
+ 12u
9
− 16u
8
−
2u
7
− 8u
6
− 30u
5
+ 10u
4
− 20u
3
− 4u
2
− 10u − 20
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
16
+ 31u
15
+ ··· + 18u + 1
c
2
, c
5
, c
6
c
11
, c
12
u
16
+ u
15
+ ··· + 9u
2
− 1
c
3
, c
9
u
16
− 3u
15
+ ··· + 2u + 2
c
4
u
16
+ 15u
15
+ ··· + 1866u + 314
c
7
u
16
− 3u
15
+ ··· + 110u + 50
c
8
, c
10
u
16
− 5u
15
+ ··· + 20u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
16
− 119y
15
+ ··· − 66y + 1
c
2
, c
5
, c
6
c
11
, c
12
y
16
− 31y
15
+ ··· − 18y + 1
c
3
, c
9
y
16
+ 5y
15
+ ··· − 20y + 4
c
4
y
16
− 7y
15
+ ··· − 540404y + 98596
c
7
y
16
− 75y
15
+ ··· + 55500y + 2500
c
8
, c
10
y
16
+ 13y
15
+ ··· − 1008y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.742751 + 0.731255I
a = 0.435126 + 1.127490I
b = −0.501292 + 1.155630I
−3.21485 + 0.54630I −11.15141 − 2.56225I
u = 0.742751 − 0.731255I
a = 0.435126 − 1.127490I
b = −0.501292 − 1.155630I
−3.21485 − 0.54630I −11.15141 + 2.56225I
u = −0.054006 + 0.927701I
a = 0.339113 − 0.403496I
b = 0.356009 + 0.336387I
2.06067 + 1.23650I −2.66728 − 5.86350I
u = −0.054006 − 0.927701I
a = 0.339113 + 0.403496I
b = 0.356009 − 0.336387I
2.06067 − 1.23650I −2.66728 + 5.86350I
u = −0.893186
a = −0.903219
b = 0.806743
−18.7411 −14.2280
u = −0.714194 + 0.883170I
a = 0.230746 + 0.032790I
b = −0.193757 + 0.180370I
−1.49390 + 2.73623I −7.72446 − 2.31094I
u = −0.714194 − 0.883170I
a = 0.230746 − 0.032790I
b = −0.193757 − 0.180370I
−1.49390 − 2.73623I −7.72446 + 2.31094I
u = 0.698495 + 0.969553I
a = −1.020710 − 0.642352I
b = −0.09016 − 1.43831I
−2.49093 − 6.04455I −9.13130 + 8.50305I
u = 0.698495 − 0.969553I
a = −1.020710 + 0.642352I
b = −0.09016 + 1.43831I
−2.49093 + 6.04455I −9.13130 − 8.50305I
u = 0.948967 + 0.727783I
a = −0.62477 − 2.29842I
b = 1.07987 − 2.63582I
16.2883 + 4.2323I −14.03484 − 0.40975I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.948967 − 0.727783I
a = −0.62477 + 2.29842I
b = 1.07987 + 2.63582I
16.2883 − 4.2323I −14.03484 + 0.40975I
u = −0.294487 + 1.168400I
a = −0.864463 + 0.293623I
b = −0.088495 − 1.096510I
−14.6982 + 4.0602I −9.52226 − 2.84200I
u = −0.294487 − 1.168400I
a = −0.864463 − 0.293623I
b = −0.088495 + 1.096510I
−14.6982 − 4.0602I −9.52226 + 2.84200I
u = 0.796357 + 1.060920I
a = 2.12200 + 0.97725I
b = 0.65309 + 3.02951I
17.3459 − 10.6503I −12.77445 + 4.89153I
u = 0.796357 − 1.060920I
a = 2.12200 − 0.97725I
b = 0.65309 − 3.02951I
17.3459 + 10.6503I −12.77445 − 4.89153I
u = −0.354580
a = 0.669122
b = −0.237257
−0.628198 −15.7600
6
II. I
u
2
= hu
2
+ b + u + 1, −u
3
+ 2a + u + 2, u
4
+ u
2
+ 2i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
10
=
1
u
2
a
6
=
1
2
u
3
−
1
2
u − 1
−u
2
− u − 1
a
11
=
u
2
+ 1
u
2
a
12
=
1
2
u
3
+ u
2
−
1
2
u
−u − 1
a
3
=
u
u
3
+ u
a
2
=
1
2
u
3
+
1
2
u − 1
u
3
− u
2
− 1
a
1
=
1
2
u
3
−
1
2
u − 1
−u
2
− u − 1
a
5
=
−u
−u
3
− u
a
8
=
−1
−u
2
− 2
a
7
=
−u
2
− 1
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
− 12
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
11
c
12
(u − 1)
4
c
2
, c
6
(u + 1)
4
c
3
, c
4
, c
7
c
9
u
4
+ u
2
+ 2
c
8
(u
2
+ u + 2)
2
c
10
(u
2
− u + 2)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
11
, c
12
(y −1)
4
c
3
, c
4
, c
7
c
9
(y
2
+ y + 2)
2
c
8
, c
10
(y
2
+ 3y + 4)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.676097 + 0.978318I
a = −2.15417 − 0.28654I
b = −1.17610 − 2.30119I
−4.11234 − 5.33349I −14.0000 + 5.2915I
u = 0.676097 − 0.978318I
a = −2.15417 + 0.28654I
b = −1.17610 + 2.30119I
−4.11234 + 5.33349I −14.0000 − 5.2915I
u = −0.676097 + 0.978318I
a = 0.154169 − 0.286543I
b = 0.176097 + 0.344557I
−4.11234 + 5.33349I −14.0000 − 5.2915I
u = −0.676097 − 0.978318I
a = 0.154169 + 0.286543I
b = 0.176097 − 0.344557I
−4.11234 − 5.33349I −14.0000 + 5.2915I
10
III.
I
u
3
= h−u
3
+au−u
2
+b+1, −u
3
a−2u
2
a+u
3
+a
2
−2au−2u
2
−1, u
4
+u
3
+u
2
+1i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
10
=
1
u
2
a
6
=
a
u
3
− au + u
2
− 1
a
11
=
u
2
+ 1
u
2
a
12
=
u
2
a − u
3
+ au − u
2
+ a + u + 1
u
2
a + au − u
2
+ u + 1
a
3
=
u
u
3
+ u
a
2
=
u
3
− au + 2u
2
− a
−u
3
a − u
2
a − au + u
2
− u − 1
a
1
=
u
2
a + u
3
− au + 3u
2
− a − u + 1
−2u
3
a − u
2
a − 2u
3
− au + 2u
2
− a − 2u − 1
a
5
=
2u
3
+ 1
2u
3
+ 2u
2
+ 2
a
8
=
−u
3
−u
3
− u
2
− 1
a
7
=
−2u
3
−2u
3
− 2u
2
+ u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
2
+ 4u − 10
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
8
+ 13u
7
+ ··· + 889u + 256
c
2
, c
5
, c
6
c
11
, c
12
u
8
+ u
7
− 6u
6
− 4u
5
+ 21u
4
+ 11u
3
− 27u
2
− 5u + 16
c
3
, c
9
(u
4
+ u
3
+ u
2
+ 1)
2
c
4
(u
4
− 5u
3
+ 7u
2
− 2u + 1)
2
c
7
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
c
8
, c
10
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
8
+ 3y
7
+ ··· − 16689y + 65536
c
2
, c
5
, c
6
c
11
, c
12
y
8
− 13y
7
+ ··· − 889y + 256
c
3
, c
9
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
c
4
(y
4
− 11y
3
+ 31y
2
+ 10y + 1)
2
c
7
, c
8
, c
10
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.351808 + 0.720342I
a = −0.560363 + 0.369379I
b = −1.43601 + 0.67423I
−3.07886 − 1.41510I −10.17326 + 4.90874I
u = 0.351808 + 0.720342I
a = −0.03038 + 1.97868I
b = −0.463219 − 0.273703I
−3.07886 − 1.41510I −10.17326 + 4.90874I
u = 0.351808 − 0.720342I
a = −0.560363 − 0.369379I
b = −1.43601 − 0.67423I
−3.07886 + 1.41510I −10.17326 − 4.90874I
u = 0.351808 − 0.720342I
a = −0.03038 − 1.97868I
b = −0.463219 + 0.273703I
−3.07886 + 1.41510I −10.17326 − 4.90874I
u = −0.851808 + 0.911292I
a = 1.15548 − 1.61606I
b = −0.08923 − 2.75519I
−10.08060 + 3.16396I −13.82674 − 2.56480I
u = −0.851808 + 0.911292I
a = −1.56474 + 1.56051I
b = 0.48846 + 2.42955I
−10.08060 + 3.16396I −13.82674 − 2.56480I
u = −0.851808 − 0.911292I
a = 1.15548 + 1.61606I
b = −0.08923 + 2.75519I
−10.08060 − 3.16396I −13.82674 + 2.56480I
u = −0.851808 − 0.911292I
a = −1.56474 − 1.56051I
b = 0.48846 − 2.42955I
−10.08060 − 3.16396I −13.82674 + 2.56480I
14
IV. I
u
4
= h−u
3
− u
2
+ b + 1, −u
3
− u
2
+ a − u, u
4
+ 1i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
10
=
1
u
2
a
6
=
u
3
+ u
2
+ u
u
3
+ u
2
− 1
a
11
=
u
2
+ 1
u
2
a
12
=
−u
3
− u + 1
−u
3
+ 1
a
3
=
u
u
3
+ u
a
2
=
−u
3
− u
2
−u
2
+ u + 1
a
1
=
−u
3
− u
2
− u
−u
3
− u
2
+ 1
a
5
=
u
u
3
+ u
a
8
=
u
2
−1
a
7
=
u
2
+ 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −16
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
(u − 1)
4
c
3
, c
4
, c
7
c
9
u
4
+ 1
c
5
, c
11
, c
12
(u + 1)
4
c
8
, c
10
(u
2
+ 1)
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
11
, c
12
(y −1)
4
c
3
, c
4
, c
7
c
9
(y
2
+ 1)
2
c
8
, c
10
(y + 1)
4
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.707107 + 0.707107I
a = 2.41421I
b = −1.70711 + 1.70711I
−4.93480 −16.0000
u = 0.707107 − 0.707107I
a = − 2.41421I
b = −1.70711 − 1.70711I
−4.93480 −16.0000
u = −0.707107 + 0.707107I
a = 0.414214I
b = −0.292893 − 0.292893I
−4.93480 −16.0000
u = −0.707107 − 0.707107I
a = − 0.414214I
b = −0.292893 + 0.292893I
−4.93480 −16.0000
18
V. I
u
5
= hb − u, a − 1, u
2
+ 1i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
10
=
1
−1
a
6
=
1
u
a
11
=
0
−1
a
12
=
1
u − 1
a
3
=
u
0
a
2
=
u + 1
u
a
1
=
1
u
a
5
=
−u
0
a
8
=
1
1
a
7
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
10
c
11
, c
12
(u − 1)
2
c
2
, c
6
, c
8
(u + 1)
2
c
3
, c
4
, c
7
c
9
u
2
+ 1
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
8
, c
10
c
11
, c
12
(y −1)
2
c
3
, c
4
, c
7
c
9
(y + 1)
2
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = 1.00000
b = 1.000000I
0 −8.00000
u = − 1.000000I
a = 1.00000
b = − 1.000000I
0 −8.00000
22
VI. I
v
1
= ha, b + 1, v + 1i
(i) Arc colorings
a
4
=
−1
0
a
9
=
1
0
a
10
=
1
0
a
6
=
0
−1
a
11
=
1
0
a
12
=
1
1
a
3
=
−1
0
a
2
=
−1
1
a
1
=
0
1
a
5
=
−1
0
a
8
=
1
0
a
7
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
23
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
u − 1
c
3
, c
4
, c
7
c
8
, c
9
, c
10
u
c
5
, c
11
, c
12
u + 1
24
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
11
, c
12
y −1
c
3
, c
4
, c
7
c
8
, c
9
, c
10
y
25
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = −1.00000
−3.28987 −12.0000
26
VII. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
11
)(u
8
+ 13u
7
+ ··· + 889u + 256)(u
16
+ 31u
15
+ ··· + 18u + 1)
c
2
, c
6
((u − 1)
5
)(u + 1)
6
(u
8
+ u
7
+ ··· − 5u + 16)
· (u
16
+ u
15
+ ··· + 9u
2
− 1)
c
3
, c
9
u(u
2
+ 1)(u
4
+ 1)(u
4
+ u
2
+ 2)(u
4
+ u
3
+ u
2
+ 1)
2
(u
16
− 3u
15
+ ··· + 2u + 2)
c
4
u(u
2
+ 1)(u
4
+ 1)(u
4
+ u
2
+ 2)(u
4
− 5u
3
+ 7u
2
− 2u + 1)
2
· (u
16
+ 15u
15
+ ··· + 1866u + 314)
c
5
, c
11
, c
12
((u − 1)
6
)(u + 1)
5
(u
8
+ u
7
+ ··· − 5u + 16)
· (u
16
+ u
15
+ ··· + 9u
2
− 1)
c
7
u(u
2
+ 1)(u
4
+ 1)(u
4
+ u
2
+ 2)(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
· (u
16
− 3u
15
+ ··· + 110u + 50)
c
8
u(u + 1)
2
(u
2
+ 1)
2
(u
2
+ u + 2)
2
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
· (u
16
− 5u
15
+ ··· + 20u + 4)
c
10
u(u − 1)
2
(u
2
+ 1)
2
(u
2
− u + 2)
2
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
· (u
16
− 5u
15
+ ··· + 20u + 4)
27
VIII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
11
)(y
8
+ 3y
7
+ ··· − 16689y + 65536)
· (y
16
− 119y
15
+ ··· − 66y + 1)
c
2
, c
5
, c
6
c
11
, c
12
((y −1)
11
)(y
8
− 13y
7
+ ··· − 889y + 256)(y
16
− 31y
15
+ ··· − 18y + 1)
c
3
, c
9
y(y + 1)
2
(y
2
+ 1)
2
(y
2
+ y + 2)
2
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
· (y
16
+ 5y
15
+ ··· − 20y + 4)
c
4
y(y + 1)
2
(y
2
+ 1)
2
(y
2
+ y + 2)
2
(y
4
− 11y
3
+ 31y
2
+ 10y + 1)
2
· (y
16
− 7y
15
+ ··· − 540404y + 98596)
c
7
y(y + 1)
2
(y
2
+ 1)
2
(y
2
+ y + 2)
2
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
· (y
16
− 75y
15
+ ··· + 55500y + 2500)
c
8
, c
10
y(y −1)
2
(y + 1)
4
(y
2
+ 3y + 4)
2
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
· (y
16
+ 13y
15
+ ··· − 1008y + 16)
28
