12n
0107
(K12n
0107
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 10 4 11 12 6 7 8 10
Solving Sequence
7,10
11 8 12
1,4
3 6 5 2 9
c
10
c
7
c
11
c
12
c
3
c
6
c
5
c
2
c
9
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h47441368u
25
+ 57693789u
24
+ ··· + 104373924b + 18568285,
120201295u
25
+ 398061213u
24
+ ··· + 104373924a + 783424087, u
26
+ 5u
25
+ ··· + 14u − 1i
I
u
2
= h2a
2
u + a
2
− au + b − a + 2u, a
3
− a
2
u + a
2
− 2au + 4a − 2u + 3, u
2
− u − 1i
I
u
3
= hb + 1, a, u + 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
= h4.74 × 10
7
u
25
+ 5 .77 × 10
7
u
24
+ · · · + 1.04 × 10
8
b + 1.86 × 10
7
, 1.20 ×
10
8
u
25
+3.98×10
8
u
24
+· · ·+1.04×10
8
a+7.83×10
8
, u
26
+5u
25
+· · ·+14u−1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
8
=
−u
−u
3
+ u
a
12
=
−u
2
+ 1
−u
4
+ 2u
2
a
1
=
u
4
− 3u
2
+ 1
−u
4
+ 2u
2
a
4
=
−1.15164u
25
− 3.81380u
24
+ ··· − 24.4464u − 7.50594
−0.454533u
25
− 0.552761u
24
+ ··· + 13.0629u − 0.177902
a
3
=
−1.15164u
25
− 3.81380u
24
+ ··· − 24.4464u − 7.50594
3.33329u
25
+ 11.3654u
24
+ ··· + 41.4363u − 2.12231
a
6
=
−0.633480u
25
− 1.36564u
24
+ ··· + 2.69053u + 4.69952
−1.23184u
25
− 3.55184u
24
+ ··· − 6.61306u + 0.205455
a
5
=
0.598359u
25
+ 2.18620u
24
+ ··· + 9.30359u + 4.49406
−1.23184u
25
− 3.55184u
24
+ ··· − 6.61306u + 0.205455
a
2
=
−0.598359u
25
− 2.18620u
24
+ ··· − 9.30359u − 4.49406
−0.203032u
25
+ 0.469119u
24
+ ··· + 15.6624u − 0.533217
a
9
=
u
3
− 2u
u
5
− 3u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
92088909
17395654
u
25
−
188693588
8697827
u
24
+ ··· −
885802456
8697827
u −
15795407
8697827
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
26
+ 20u
25
+ ··· + 79u + 1
c
2
, c
4
u
26
− 4u
25
+ ··· − 11u − 1
c
3
, c
6
u
26
− 3u
25
+ ··· − 6u + 2
c
5
, c
9
u
26
− 2u
25
+ ··· − 96u + 64
c
7
, c
8
, c
10
c
11
u
26
+ 5u
25
+ ··· + 14u − 1
c
12
u
26
− 21u
25
+ ··· + 52120u + 337
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
26
− 24y
25
+ ··· − 6127y + 1
c
2
, c
4
y
26
− 20y
25
+ ··· − 79y + 1
c
3
, c
6
y
26
− 3y
25
+ ··· − 40y + 4
c
5
, c
9
y
26
+ 34y
25
+ ··· − 95232y + 4096
c
7
, c
8
, c
10
c
11
y
26
− 39y
25
+ ··· − 304y + 1
c
12
y
26
− 123y
25
+ ··· − 2285596764y + 113569
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.476607 + 0.919429I
a = −0.688440 − 0.733293I
b = 0.227587 + 1.287750I
−5.18299 + 2.95142I −14.7696 − 4.0162I
u = −0.476607 − 0.919429I
a = −0.688440 + 0.733293I
b = 0.227587 − 1.287750I
−5.18299 − 2.95142I −14.7696 + 4.0162I
u = −0.757036
a = −0.441343
b = 0.637723
−1.34161 −6.51520
u = 1.223650 + 0.232594I
a = −0.924838 − 0.484695I
b = 0.280314 − 1.358420I
−5.77000 − 3.25214I −11.73752 + 3.41900I
u = 1.223650 − 0.232594I
a = −0.924838 + 0.484695I
b = 0.280314 + 1.358420I
−5.77000 + 3.25214I −11.73752 − 3.41900I
u = −1.329480 + 0.241157I
a = 0.277950 + 0.611388I
b = 0.308061 + 0.000558I
−3.29395 − 1.34186I −11.30499 + 4.69401I
u = −1.329480 − 0.241157I
a = 0.277950 − 0.611388I
b = 0.308061 − 0.000558I
−3.29395 + 1.34186I −11.30499 − 4.69401I
u = 1.369250 + 0.095489I
a = 0.500507 − 0.680570I
b = 0.10931 − 1.70719I
−9.37037 − 1.40410I −14.8448 + 0.4920I
u = 1.369250 − 0.095489I
a = 0.500507 + 0.680570I
b = 0.10931 + 1.70719I
−9.37037 + 1.40410I −14.8448 − 0.4920I
u = 1.273730 + 0.536724I
a = 0.988931 + 0.204426I
b = −0.48656 + 1.60886I
−10.60190 − 7.95034I −14.2939 + 5.5201I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.273730 − 0.536724I
a = 0.988931 − 0.204426I
b = −0.48656 − 1.60886I
−10.60190 + 7.95034I −14.2939 − 5.5201I
u = −0.586682 + 0.167108I
a = −0.045237 − 0.711969I
b = −0.71246 − 2.34576I
−2.72200 + 0.36882I −3.94523 + 10.24837I
u = −0.586682 − 0.167108I
a = −0.045237 + 0.711969I
b = −0.71246 + 2.34576I
−2.72200 − 0.36882I −3.94523 − 10.24837I
u = −0.348677 + 0.367916I
a = 1.247060 + 0.371784I
b = −0.209352 − 0.864493I
−0.636376 + 1.127340I −7.20662 − 6.11077I
u = −0.348677 − 0.367916I
a = 1.247060 − 0.371784I
b = −0.209352 + 0.864493I
−0.636376 − 1.127340I −7.20662 + 6.11077I
u = 0.424219 + 0.095685I
a = 0.07818 + 2.89428I
b = −0.1185550 − 0.0289313I
2.35620 + 2.67700I 3.89989 + 1.35809I
u = 0.424219 − 0.095685I
a = 0.07818 − 2.89428I
b = −0.1185550 + 0.0289313I
2.35620 − 2.67700I 3.89989 − 1.35809I
u = 1.63681
a = 0.377195
b = −1.87474
−9.79249 1.55260
u = −1.80599 + 0.06787I
a = 0.735548 − 0.651435I
b = −0.23521 − 1.59914I
−16.9214 + 4.6752I 0
u = −1.80599 − 0.06787I
a = 0.735548 + 0.651435I
b = −0.23521 + 1.59914I
−16.9214 − 4.6752I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.81333 + 0.15530I
a = −0.732203 + 0.586510I
b = 0.50599 + 1.87679I
17.8851 + 11.1272I 0
u = −1.81333 − 0.15530I
a = −0.732203 − 0.586510I
b = 0.50599 − 1.87679I
17.8851 − 11.1272I 0
u = −1.83864 + 0.02244I
a = −0.672414 − 0.695929I
b = −0.16075 − 1.49337I
18.0411 + 1.9797I 0
u = −1.83864 − 0.02244I
a = −0.672414 + 0.695929I
b = −0.16075 + 1.49337I
18.0411 − 1.9797I 0
u = 1.87792
a = −0.655106
b = −0.356336
−16.1049 −16.4270
u = 0.0594263
a = −8.81083
b = 0.576606
−1.19028 −8.21100
7
II.
I
u
2
= h2a
2
u+a
2
−au+b −a + 2u, a
3
−a
2
u+a
2
−2au+4a −2u + 3, u
2
−u−1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
u + 1
a
8
=
−u
−u − 1
a
12
=
−u
−u
a
1
=
0
−u
a
4
=
a
−2a
2
u − a
2
+ au + a − 2u
a
3
=
a
−2a
2
u − a
2
− 2u
a
6
=
a
2
u
0
a
5
=
a
2
u
0
a
2
=
a
2
u
−2a
2
u − a
2
− 2u
a
9
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −19a
2
u − 13a
2
+ 9au + a − 8u − 29
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
4
(u
3
− u
2
+ 1)
2
c
5
, c
9
u
6
c
6
(u
3
+ u
2
+ 2u + 1)
2
c
7
, c
8
(u
2
+ u − 1)
3
c
10
, c
11
, c
12
(u
2
− u − 1)
3
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
4
(y
3
− y
2
+ 2y −1)
2
c
5
, c
9
y
6
c
7
, c
8
, c
10
c
11
, c
12
(y
2
− 3y + 1)
3
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = −0.922021
b = 1.08457
−2.10041 −20.9180
u = −0.618034
a = −0.34801 + 2.11500I
b = 0.075747 + 0.460350I
2.03717 + 2.82812I −16.9959 − 7.7984I
u = −0.618034
a = −0.34801 − 2.11500I
b = 0.075747 − 0.460350I
2.03717 − 2.82812I −16.9959 + 7.7984I
u = 1.61803
a = 0.132927 + 0.807858I
b = −0.198308 + 1.205210I
−5.85852 − 2.82812I −12.10059 + 3.17745I
u = 1.61803
a = 0.132927 − 0.807858I
b = −0.198308 − 1.205210I
−5.85852 + 2.82812I −12.10059 − 3.17745I
u = 1.61803
a = 0.352181
b = −2.83945
−9.99610 −41.8890
11
III. I
u
3
= hb + 1, a, u + 1i
(i) Arc colorings
a
7
=
0
−1
a
10
=
1
0
a
11
=
1
1
a
8
=
1
0
a
12
=
0
1
a
1
=
−1
1
a
4
=
0
−1
a
3
=
0
−1
a
6
=
0
−1
a
5
=
1
−1
a
2
=
−1
0
a
9
=
1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
7
, c
8
u − 1
c
3
, c
6
u
c
4
, c
9
, c
10
c
11
, c
12
u + 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
7
, c
8
c
9
, c
10
, c
11
c
12
y −1
c
3
, c
6
y
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0
b = −1.00000
−3.28987 −12.0000
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)(u
3
− u
2
+ 2u − 1)
2
(u
26
+ 20u
25
+ ··· + 79u + 1)
c
2
(u − 1)(u
3
+ u
2
− 1)
2
(u
26
− 4u
25
+ ··· − 11u − 1)
c
3
u(u
3
− u
2
+ 2u − 1)
2
(u
26
− 3u
25
+ ··· − 6u + 2)
c
4
(u + 1)(u
3
− u
2
+ 1)
2
(u
26
− 4u
25
+ ··· − 11u − 1)
c
5
u
6
(u − 1)(u
26
− 2u
25
+ ··· − 96u + 64)
c
6
u(u
3
+ u
2
+ 2u + 1)
2
(u
26
− 3u
25
+ ··· − 6u + 2)
c
7
, c
8
(u − 1)(u
2
+ u − 1)
3
(u
26
+ 5u
25
+ ··· + 14u − 1)
c
9
u
6
(u + 1)(u
26
− 2u
25
+ ··· − 96u + 64)
c
10
, c
11
(u + 1)(u
2
− u − 1)
3
(u
26
+ 5u
25
+ ··· + 14u − 1)
c
12
(u + 1)(u
2
− u − 1)
3
(u
26
− 21u
25
+ ··· + 52120u + 337)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)(y
3
+ 3y
2
+ 2y −1)
2
(y
26
− 24y
25
+ ··· − 6127y + 1)
c
2
, c
4
(y −1)(y
3
− y
2
+ 2y −1)
2
(y
26
− 20y
25
+ ··· − 79y + 1)
c
3
, c
6
y(y
3
+ 3y
2
+ 2y −1)
2
(y
26
− 3y
25
+ ··· − 40y + 4)
c
5
, c
9
y
6
(y −1)(y
26
+ 34y
25
+ ··· − 95232y + 4096)
c
7
, c
8
, c
10
c
11
(y −1)(y
2
− 3y + 1)
3
(y
26
− 39y
25
+ ··· − 304y + 1)
c
12
(y −1)(y
2
− 3y + 1)
3
(y
26
− 123y
25
+ ··· − 2.28560 × 10
9
y + 113569)
17
