12n
0306
(K12n
0306
)
A knot diagram
1
Linearized knot diagam
3 6 7 9 2 5 12 11 5 7 8 10
Solving Sequence
2,5
6 3
7,10
11 1 9 4 8 12
c
5
c
2
c
6
c
10
c
1
c
9
c
4
c
8
c
12
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−3u
25
− 12u
24
+ ··· + 2b + 3, u
25
+ 10u
24
+ ··· + 4a + 11, u
26
+ 4u
25
+ ··· − u − 1i
I
u
2
= hb, u
2
+ a − u, u
3
− u
2
+ 1i
I
u
3
= hb, −u
2
a + a
2
+ 2au + u
2
− a − 2u + 2, u
3
− u
2
+ 1i
* 3 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−3u
25
−12u
24
+· · ·+2b+3, u
25
+10u
24
+· · ·+4a+11, u
26
+4u
25
+· · ·−u−1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
3
+ u
a
7
=
−u
2
+ 1
u
2
a
10
=
−
1
4
u
25
−
5
2
u
24
+ ··· −
13
2
u −
11
4
3
2
u
25
+ 6u
24
+ ··· + u −
3
2
a
11
=
−
1
2
u
25
−
9
4
u
24
+ ··· −
15
2
u −
17
4
1
4
u
24
+
5
4
u
23
+ ··· +
7
4
u
2
+
3
4
u
a
1
=
u
3
u
5
− u
3
+ u
a
9
=
−
7
4
u
25
−
17
2
u
24
+ ··· −
15
2
u −
5
4
3
2
u
25
+ 6u
24
+ ··· + u −
3
2
a
4
=
u
7
− 2u
5
+ 2u
3
− 2u
−u
7
+ u
5
− 2u
3
+ u
a
8
=
−
3
4
u
25
−
13
4
u
24
+ ··· −
7
4
u +
1
4
1
4
u
25
+ u
24
+ ··· + 3u
2
−
1
4
a
12
=
1
4
u
23
+
3
4
u
22
+ ··· +
9
4
u +
5
4
−u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes =
11
4
u
25
+
21
2
u
24
+ ··· +
17
4
u −
35
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
26
+ 4u
25
+ ··· + 15u + 1
c
2
, c
5
u
26
+ 4u
25
+ ··· − u − 1
c
3
u
26
− 4u
25
+ ··· − 103464u − 31428
c
4
, c
9
u
26
− u
25
+ ··· − 3456u
2
+ 512
c
7
, c
8
, c
11
u
26
− 4u
25
+ ··· + 7u − 1
c
10
u
26
+ 4u
25
+ ··· + 889u − 193
c
12
u
26
+ 28u
24
+ ··· + 25u + 3
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
26
+ 40y
25
+ ··· − 15y + 1
c
2
, c
5
y
26
− 4y
25
+ ··· − 15y + 1
c
3
y
26
+ 124y
25
+ ··· − 27391370184y + 987719184
c
4
, c
9
y
26
− 49y
25
+ ··· − 3538944y + 262144
c
7
, c
8
, c
11
y
26
+ 28y
25
+ ··· − 23y + 1
c
10
y
26
+ 28y
25
+ ··· − 417831y + 37249
c
12
y
26
+ 56y
25
+ ··· − 1231y + 9
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.927978 + 0.281302I
a = −0.892022 + 0.688694I
b = −0.624177 + 0.789814I
2.80822 + 0.17134I −6.35112 − 1.45434I
u = −0.927978 − 0.281302I
a = −0.892022 − 0.688694I
b = −0.624177 − 0.789814I
2.80822 − 0.17134I −6.35112 + 1.45434I
u = 0.736454 + 0.746707I
a = 0.855774 + 0.275171I
b = −0.860539 + 0.488199I
3.28913 − 1.44124I −2.49548 + 1.39542I
u = 0.736454 − 0.746707I
a = 0.855774 − 0.275171I
b = −0.860539 − 0.488199I
3.28913 + 1.44124I −2.49548 − 1.39542I
u = 0.930739 + 0.665116I
a = −0.109280 − 0.822401I
b = 0.882348 + 0.149927I
2.63703 − 3.89810I −3.35232 + 6.23910I
u = 0.930739 − 0.665116I
a = −0.109280 + 0.822401I
b = 0.882348 − 0.149927I
2.63703 + 3.89810I −3.35232 − 6.23910I
u = −0.828014
a = 0.505037
b = 0.430120
−1.35925 −5.94650
u = 0.670758 + 0.970438I
a = −1.198940 − 0.023490I
b = 1.59537 − 1.11430I
10.51200 − 0.45901I −0.702444 + 1.109804I
u = 0.670758 − 0.970438I
a = −1.198940 + 0.023490I
b = 1.59537 + 1.11430I
10.51200 + 0.45901I −0.702444 − 1.109804I
u = −0.437740 + 0.645989I
a = −0.332493 + 0.449600I
b = 0.029303 + 1.123950I
4.72486 + 3.33852I −2.36603 − 3.49962I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.437740 − 0.645989I
a = −0.332493 − 0.449600I
b = 0.029303 − 1.123950I
4.72486 − 3.33852I −2.36603 + 3.49962I
u = 1.079900 + 0.695708I
a = −0.362002 + 1.203280I
b = −1.56097 − 0.59580I
9.04679 − 5.69366I −2.13177 + 3.88502I
u = 1.079900 − 0.695708I
a = −0.362002 − 1.203280I
b = −1.56097 + 0.59580I
9.04679 + 5.69366I −2.13177 − 3.88502I
u = −0.953135 + 0.981373I
a = 1.41685 − 1.08331I
b = −2.31682 + 0.09546I
14.7661 + 0.7496I −4.35446 + 0.19083I
u = −0.953135 − 0.981373I
a = 1.41685 + 1.08331I
b = −2.31682 − 0.09546I
14.7661 − 0.7496I −4.35446 − 0.19083I
u = −0.996483 + 0.952652I
a = −1.54076 + 1.18418I
b = 2.26473 + 0.34444I
14.6174 + 6.3423I −4.67499 − 4.46303I
u = −0.996483 − 0.952652I
a = −1.54076 − 1.18418I
b = 2.26473 − 0.34444I
14.6174 − 6.3423I −4.67499 + 4.46303I
u = −0.918139 + 1.031730I
a = −1.21694 + 1.06831I
b = 2.52632 − 0.56908I
−17.5237 − 3.4354I −1.66037 + 0.61088I
u = −0.918139 − 1.031730I
a = −1.21694 − 1.06831I
b = 2.52632 + 0.56908I
−17.5237 + 3.4354I −1.66037 − 0.61088I
u = −1.046110 + 0.939916I
a = 1.58869 − 1.30078I
b = −2.33420 − 0.78739I
−17.9709 + 10.6421I −2.17770 − 4.87168I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.046110 − 0.939916I
a = 1.58869 + 1.30078I
b = −2.33420 + 0.78739I
−17.9709 − 10.6421I −2.17770 + 4.87168I
u = 0.547038 + 0.202189I
a = 2.44519 + 0.75423I
b = −0.603638 + 0.076459I
2.36049 − 3.21762I 0.10564 + 7.29080I
u = 0.547038 − 0.202189I
a = 2.44519 − 0.75423I
b = −0.603638 − 0.076459I
2.36049 + 3.21762I 0.10564 − 7.29080I
u = −0.482360 + 0.313885I
a = 0.443368 − 0.835553I
b = 0.021358 − 0.703916I
−0.662371 + 1.026000I −7.80706 − 6.48797I
u = −0.482360 − 0.313885I
a = 0.443368 + 0.835553I
b = 0.021358 + 0.703916I
−0.662371 − 1.026000I −7.80706 + 6.48797I
u = 0.422126
a = −2.69989
b = 0.531699
−1.56806 −4.11730
7
II. I
u
2
= hb, u
2
+ a − u, u
3
− u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
2
+ u + 1
a
7
=
−u
2
+ 1
u
2
a
10
=
−u
2
+ u
0
a
11
=
−2u
2
+ 2u + 1
u
2
− 1
a
1
=
u
2
− 1
−u
2
a
9
=
−u
2
+ u
0
a
4
=
1
0
a
8
=
−2u
2
− u
u + 1
a
12
=
u
2
− 2
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
2
+ 9u − 11
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
7
c
8
u
3
− u
2
+ 2u − 1
c
2
u
3
+ u
2
− 1
c
4
, c
9
u
3
c
5
, c
10
, c
12
u
3
− u
2
+ 1
c
6
, c
11
u
3
+ u
2
+ 2u + 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
7
, c
8
, c
11
y
3
+ 3y
2
+ 2y −1
c
2
, c
5
, c
10
c
12
y
3
− y
2
+ 2y −1
c
4
, c
9
y
3
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = 0.662359 − 0.562280I
b = 0
6.04826 − 5.65624I −3.31813 + 5.39661I
u = 0.877439 − 0.744862I
a = 0.662359 + 0.562280I
b = 0
6.04826 + 5.65624I −3.31813 − 5.39661I
u = −0.754878
a = −1.32472
b = 0
−2.22691 −18.3640
11
III. I
u
3
= hb, −u
2
a + a
2
+ 2au + u
2
− a − 2u + 2, u
3
− u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
2
+ u + 1
a
7
=
−u
2
+ 1
u
2
a
10
=
a
0
a
11
=
au + 2a
u
2
a − au − a
a
1
=
u
2
− 1
−u
2
a
9
=
a
0
a
4
=
1
0
a
8
=
u
2
a + au + u
2
− a − 2u + 3
−u
2
a + u
2
− u
a
12
=
au + 2u
2
− a − u
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
2
a + au − a + 3u − 7
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
7
c
8
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
4
, c
9
u
6
c
5
, c
10
, c
12
(u
3
− u
2
+ 1)
2
c
6
, c
11
(u
3
+ u
2
+ 2u + 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
7
, c
8
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
5
, c
10
c
12
(y
3
− y
2
+ 2y −1)
2
c
4
, c
9
y
6
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = −0.447279 − 0.744862I
b = 0
6.04826 −2.00317 + 0.50299I
u = 0.877439 + 0.744862I
a = −0.092519 + 0.562280I
b = 0
1.91067 − 2.82812I −6.28492 + 2.09676I
u = 0.877439 − 0.744862I
a = −0.447279 + 0.744862I
b = 0
6.04826 −2.00317 − 0.50299I
u = 0.877439 − 0.744862I
a = −0.092519 − 0.562280I
b = 0
1.91067 + 2.82812I −6.28492 − 2.09676I
u = −0.754878
a = 1.53980 + 1.30714I
b = 0
1.91067 − 2.82812I −10.21191 − 0.80415I
u = −0.754878
a = 1.53980 − 1.30714I
b = 0
1.91067 + 2.82812I −10.21191 + 0.80415I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
3
− u
2
+ 2u − 1)
3
)(u
26
+ 4u
25
+ ··· + 15u + 1)
c
2
((u
3
+ u
2
− 1)
3
)(u
26
+ 4u
25
+ ··· − u − 1)
c
3
((u
3
− u
2
+ 2u − 1)
3
)(u
26
− 4u
25
+ ··· − 103464u − 31428)
c
4
, c
9
u
9
(u
26
− u
25
+ ··· − 3456u
2
+ 512)
c
5
((u
3
− u
2
+ 1)
3
)(u
26
+ 4u
25
+ ··· − u − 1)
c
6
((u
3
+ u
2
+ 2u + 1)
3
)(u
26
+ 4u
25
+ ··· + 15u + 1)
c
7
, c
8
((u
3
− u
2
+ 2u − 1)
3
)(u
26
− 4u
25
+ ··· + 7u − 1)
c
10
((u
3
− u
2
+ 1)
3
)(u
26
+ 4u
25
+ ··· + 889u − 193)
c
11
((u
3
+ u
2
+ 2u + 1)
3
)(u
26
− 4u
25
+ ··· + 7u − 1)
c
12
((u
3
− u
2
+ 1)
3
)(u
26
+ 28u
24
+ ··· + 25u + 3)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
((y
3
+ 3y
2
+ 2y −1)
3
)(y
26
+ 40y
25
+ ··· − 15y + 1)
c
2
, c
5
((y
3
− y
2
+ 2y −1)
3
)(y
26
− 4y
25
+ ··· − 15y + 1)
c
3
(y
3
+ 3y
2
+ 2y −1)
3
· (y
26
+ 124y
25
+ ··· − 27391370184y + 987719184)
c
4
, c
9
y
9
(y
26
− 49y
25
+ ··· − 3538944y + 262144)
c
7
, c
8
, c
11
((y
3
+ 3y
2
+ 2y −1)
3
)(y
26
+ 28y
25
+ ··· − 23y + 1)
c
10
((y
3
− y
2
+ 2y −1)
3
)(y
26
+ 28y
25
+ ··· − 417831y + 37249)
c
12
((y
3
− y
2
+ 2y −1)
3
)(y
26
+ 56y
25
+ ··· − 1231y + 9)
17
