12n
0275
(K12n
0275
)
A knot diagram
1
Linearized knot diagam
3 4 8 2 9 4 11 6 3 12 7 10
Solving Sequence
7,11 3,8
4 12 2 5 6 10 1 9
c
7
c
3
c
11
c
2
c
4
c
6
c
10
c
12
c
9
c
1
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−28980693473u
29
− 23872677880u
28
+ ··· + 214600733306b − 182162526839,
− 563591323347u
29
+ 759883303462u
28
+ ··· + 214600733306a − 449269653153,
u
30
− u
29
+ ··· + 3u + 1i
I
u
2
= h−u
5
a − 2u
5
− u
3
a − u
2
a − u
3
− au − u
2
+ 2b − 2u − 1,
u
5
a + u
4
a + 3u
5
+ u
4
+ u
2
a + u
3
+ a
2
+ u
2
+ 2a + 6u + 1, u
6
+ u
4
+ 2u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 42 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−2.90×10
10
u
29
−2.39×10
10
u
28
+· · ·+2.15×10
11
b−1.82×10
11
, −5.64×
10
11
u
29
+7.60×10
11
u
28
+· · ·+2.15×10
11
a−4.49×10
11
, u
30
−u
29
+· · ·+3u+1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
3
=
2.62623u
29
− 3.54092u
28
+ ··· + 12.0914u + 2.09351
0.135045u
29
+ 0.111242u
28
+ ··· + 0.745217u + 0.848844
a
8
=
1
u
2
a
4
=
2.84774u
29
− 3.81213u
28
+ ··· + 12.7188u + 2.02767
−0.0194466u
29
+ 0.399642u
28
+ ··· + 0.672837u + 0.898553
a
12
=
−u
u
a
2
=
1.18157u
29
− 2.05139u
28
+ ··· + 1.94907u + 0.834389
−0.135455u
29
− 0.0144421u
28
+ ··· − 1.18940u − 1.25661
a
5
=
1.09680u
29
− 0.576198u
28
+ ··· + 6.48933u + 4.12117
−0.313730u
29
+ 0.598271u
28
+ ··· − 1.65860u − 0.520599
a
6
=
−0.947509u
29
+ 0.761849u
28
+ ··· − 8.67784u − 0.427524
−0.0270939u
29
− 0.328296u
28
+ ··· + 0.399856u − 1.39992
a
10
=
−u
3
u
3
+ u
a
1
=
−u
5
− u
u
5
+ u
3
+ u
a
9
=
1.54614u
29
− 2.33858u
28
+ ··· + 6.58387u + 1.61709
−0.255461u
29
+ 0.288919u
28
+ ··· − 1.75987u − 0.674753
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
116649735226
107300366653
u
29
+
148588842691
107300366653
u
28
+ ··· −
788548417729
107300366653
u +
226312667156
107300366653
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
30
+ 47u
29
+ ··· + 191u + 1
c
2
, c
4
u
30
− 5u
29
+ ··· + 11u + 1
c
3
u
30
− u
29
+ ··· + 3u + 1
c
5
, c
8
u
30
− u
29
+ ··· − 95u + 25
c
6
u
30
+ 3u
29
+ ··· + 861u + 649
c
7
, c
11
u
30
+ u
29
+ ··· − 3u + 1
c
9
u
30
+ 3u
29
+ ··· − 401099u + 75377
c
10
, c
12
u
30
− 5u
29
+ ··· − 9u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
30
− 121y
29
+ ··· + 51727y + 1
c
2
, c
4
y
30
+ 47y
29
+ ··· + 191y + 1
c
3
y
30
− 5y
29
+ ··· + 11y + 1
c
5
, c
8
y
30
+ y
29
+ ··· − 3175y + 625
c
6
y
30
+ 33y
29
+ ··· + 2246675y + 421201
c
7
, c
11
y
30
+ 5y
29
+ ··· + 9y + 1
c
9
y
30
+ 89y
29
+ ··· + 64272198739y + 5681692129
c
10
, c
12
y
30
+ 45y
29
+ ··· + 81y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.194178 + 0.861301I
a = −1.021630 − 0.362095I
b = 0.263501 + 0.422304I
0.69640 − 1.72034I 1.49037 + 4.91074I
u = 0.194178 − 0.861301I
a = −1.021630 + 0.362095I
b = 0.263501 − 0.422304I
0.69640 + 1.72034I 1.49037 − 4.91074I
u = −0.938260 + 0.627104I
a = −0.499497 + 0.926953I
b = 1.52482 + 0.04891I
−6.16879 − 1.21008I −2.96584 + 0.88236I
u = −0.938260 − 0.627104I
a = −0.499497 − 0.926953I
b = 1.52482 − 0.04891I
−6.16879 + 1.21008I −2.96584 − 0.88236I
u = −0.423018 + 0.756823I
a = 1.89502 − 0.68357I
b = −0.189341 − 0.657186I
2.22190 + 4.31885I 5.85919 − 8.87008I
u = −0.423018 − 0.756823I
a = 1.89502 + 0.68357I
b = −0.189341 + 0.657186I
2.22190 − 4.31885I 5.85919 + 8.87008I
u = −0.149957 + 0.853854I
a = 0.90254 − 1.96897I
b = −0.496999 + 0.995959I
3.54151 − 0.51606I 10.41425 − 1.32912I
u = −0.149957 − 0.853854I
a = 0.90254 + 1.96897I
b = −0.496999 − 0.995959I
3.54151 + 0.51606I 10.41425 + 1.32912I
u = 0.765379 + 0.893759I
a = 0.159784 + 1.137040I
b = −1.46119 − 0.25226I
−1.42212 − 2.89990I 2.16966 + 2.50499I
u = 0.765379 − 0.893759I
a = 0.159784 − 1.137040I
b = −1.46119 + 0.25226I
−1.42212 + 2.89990I 2.16966 − 2.50499I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.882441 + 0.785868I
a = 0.28444 + 1.44916I
b = −0.835514 + 0.010950I
−4.43991 − 4.87528I −1.40537 + 4.69553I
u = 0.882441 − 0.785868I
a = 0.28444 − 1.44916I
b = −0.835514 − 0.010950I
−4.43991 + 4.87528I −1.40537 − 4.69553I
u = 0.633178 + 0.448369I
a = −0.204887 − 0.469950I
b = 0.234278 + 0.488949I
−0.96186 − 1.37588I −2.99068 + 3.21474I
u = 0.633178 − 0.448369I
a = −0.204887 + 0.469950I
b = 0.234278 − 0.488949I
−0.96186 + 1.37588I −2.99068 − 3.21474I
u = 0.736913 + 0.984710I
a = 0.273772 + 0.001505I
b = −1.135180 − 0.258754I
−3.71029 − 1.15996I −1.23536 + 0.81989I
u = 0.736913 − 0.984710I
a = 0.273772 − 0.001505I
b = −1.135180 + 0.258754I
−3.71029 + 1.15996I −1.23536 − 0.81989I
u = −0.647311 + 1.068760I
a = −0.99194 + 1.30440I
b = 1.71967 − 0.46050I
−4.58052 + 7.07309I −0.56937 − 6.52418I
u = −0.647311 − 1.068760I
a = −0.99194 − 1.30440I
b = 1.71967 + 0.46050I
−4.58052 − 7.07309I −0.56937 + 6.52418I
u = 1.022110 + 0.895164I
a = −0.555012 − 0.809078I
b = 2.44769 − 0.22730I
−16.7193 + 4.9302I −0.87414 − 1.79412I
u = 1.022110 − 0.895164I
a = −0.555012 + 0.809078I
b = 2.44769 + 0.22730I
−16.7193 − 4.9302I −0.87414 + 1.79412I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.919420 + 1.045210I
a = −0.57144 − 1.81078I
b = 2.49834 + 0.28520I
−16.2013 − 12.0291I −0.14700 + 6.09920I
u = 0.919420 − 1.045210I
a = −0.57144 + 1.81078I
b = 2.49834 − 0.28520I
−16.2013 + 12.0291I −0.14700 − 6.09920I
u = −1.011130 + 0.959398I
a = 0.19777 − 1.52281I
b = −1.98241 + 0.56395I
−17.2554 + 2.6942I −1.35658 − 2.19435I
u = −1.011130 − 0.959398I
a = 0.19777 + 1.52281I
b = −1.98241 − 0.56395I
−17.2554 − 2.6942I −1.35658 + 2.19435I
u = −0.968194 + 1.019150I
a = 0.859717 − 0.564264I
b = −2.08559 − 0.52091I
−17.0465 + 4.5510I −1.15103 − 1.97489I
u = −0.968194 − 1.019150I
a = 0.859717 + 0.564264I
b = −2.08559 + 0.52091I
−17.0465 − 4.5510I −1.15103 + 1.97489I
u = −0.213905 + 0.490541I
a = −1.65502 + 0.37124I
b = −0.548040 + 0.708878I
1.59074 − 1.61799I 1.307109 − 0.397413I
u = −0.213905 − 0.490541I
a = −1.65502 − 0.37124I
b = −0.548040 − 0.708878I
1.59074 + 1.61799I 1.307109 + 0.397413I
u = −0.301843 + 0.271696I
a = 2.42638 + 1.75700I
b = −0.454022 − 0.958650I
1.49855 + 2.22404I −0.54522 − 4.55103I
u = −0.301843 − 0.271696I
a = 2.42638 − 1.75700I
b = −0.454022 + 0.958650I
1.49855 − 2.22404I −0.54522 + 4.55103I
7
II.
I
u
2
= h−u
5
a − 2u
5
+ · · · + 2b − 1, u
5
a + 3u
5
+ · · · + 2a + 1, u
6
+ u
4
+ 2u
2
+ 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
3
=
a
1
2
u
5
a + u
5
+ ··· + u +
1
2
a
8
=
1
u
2
a
4
=
1
2
u
5
a + u
5
+ ··· + a +
1
2
1
2
u
5
a +
1
2
u
5
+ ··· + u
2
+
1
2
a
12
=
−u
u
a
2
=
−2u
5
−
1
2
u
4
+ ··· +
1
2
a − 2
1
2
u
4
a +
3
2
u
5
+ ··· + 2u + 1
a
5
=
u
5
+ u
−u
5
− u
3
− u
a
6
=
1
2
u
5
a +
7
2
u
5
+ ··· +
1
2
a +
5
2
u
5
a −
3
2
u
5
+ ··· +
1
2
a −
5
2
u
a
10
=
−u
3
u
3
+ u
a
1
=
−u
5
− u
u
5
+ u
3
+ u
a
9
=
3
2
u
5
a +
1
2
u
5
+ ··· +
1
2
a −
3
2
−u
5
a −
5
2
u
5
+ ··· +
1
2
a − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
5
a − 2u
3
a + 4u
4
− 2u
2
a + 2u
3
− 2au + 2u
2
+ 10
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
2
− u + 1)
6
c
2
(u
2
+ u + 1)
6
c
3
(u
4
− u
2
+ 1)
3
c
5
, c
8
(u
2
+ 1)
6
c
6
u
12
− 6u
11
+ ··· + 2u + 1
c
7
, c
11
(u
6
+ u
4
+ 2u
2
+ 1)
2
c
9
u
12
+ 2u
11
+ ··· + 4u + 1
c
10
(u
3
+ u
2
+ 2u + 1)
4
c
12
(u
3
− u
2
+ 2u − 1)
4
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y
2
+ y + 1)
6
c
3
(y
2
− y + 1)
6
c
5
, c
8
(y + 1)
12
c
6
y
12
+ 12y
11
+ ··· + 6y + 1
c
7
, c
11
(y
3
+ y
2
+ 2y + 1)
4
c
9
y
12
− 12y
11
+ ··· − 6y + 1
c
10
, c
12
(y
3
+ 3y
2
+ 2y −1)
4
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.744862 + 0.877439I
a = 0.850078 − 0.184922I
b = −0.807141 + 0.650946I
−1.37919 − 0.79824I 2.49024 − 0.48465I
u = 0.744862 + 0.877439I
a = −0.227778 + 1.317500I
b = −0.807141 − 1.081110I
−1.37919 − 4.85801I 2.49024 + 6.44355I
u = 0.744862 − 0.877439I
a = 0.850078 + 0.184922I
b = −0.807141 − 0.650946I
−1.37919 + 0.79824I 2.49024 + 0.48465I
u = 0.744862 − 0.877439I
a = −0.227778 − 1.317500I
b = −0.807141 + 1.081110I
−1.37919 + 4.85801I 2.49024 − 6.44355I
u = −0.744862 + 0.877439I
a = 0.317499 + 0.772222I
b = 1.80714 − 1.08111I
−1.37919 + 0.79824I 2.49024 + 0.48465I
u = −0.744862 + 0.877439I
a = −1.18492 + 1.85008I
b = 1.80714 + 0.65095I
−1.37919 + 4.85801I 2.49024 − 6.44355I
u = −0.744862 − 0.877439I
a = 0.317499 − 0.772222I
b = 1.80714 + 1.08111I
−1.37919 − 0.79824I 2.49024 − 0.48465I
u = −0.744862 − 0.877439I
a = −1.18492 − 1.85008I
b = 1.80714 − 0.65095I
−1.37919 − 4.85801I 2.49024 + 6.44355I
u = 0.754878I
a = 0.64233 − 1.64233I
b = 0.50000 + 1.43587I
2.75839 + 2.02988I 9.01951 − 3.46410I
u = 0.754878I
a = −2.39721 + 1.39721I
b = 0.500000 − 0.296185I
2.75839 − 2.02988I 9.01951 + 3.46410I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = − 0.754878I
a = 0.64233 + 1.64233I
b = 0.50000 − 1.43587I
2.75839 − 2.02988I 9.01951 + 3.46410I
u = − 0.754878I
a = −2.39721 − 1.39721I
b = 0.500000 + 0.296185I
2.75839 + 2.02988I 9.01951 − 3.46410I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
6
)(u
30
+ 47u
29
+ ··· + 191u + 1)
c
2
((u
2
+ u + 1)
6
)(u
30
− 5u
29
+ ··· + 11u + 1)
c
3
((u
4
− u
2
+ 1)
3
)(u
30
− u
29
+ ··· + 3u + 1)
c
4
((u
2
− u + 1)
6
)(u
30
− 5u
29
+ ··· + 11u + 1)
c
5
, c
8
((u
2
+ 1)
6
)(u
30
− u
29
+ ··· − 95u + 25)
c
6
(u
12
− 6u
11
+ ··· + 2u + 1)(u
30
+ 3u
29
+ ··· + 861u + 649)
c
7
, c
11
((u
6
+ u
4
+ 2u
2
+ 1)
2
)(u
30
+ u
29
+ ··· − 3u + 1)
c
9
(u
12
+ 2u
11
+ ··· + 4u + 1)(u
30
+ 3u
29
+ ··· − 401099u + 75377)
c
10
((u
3
+ u
2
+ 2u + 1)
4
)(u
30
− 5u
29
+ ··· − 9u + 1)
c
12
((u
3
− u
2
+ 2u − 1)
4
)(u
30
− 5u
29
+ ··· − 9u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
6
)(y
30
− 121y
29
+ ··· + 51727y + 1)
c
2
, c
4
((y
2
+ y + 1)
6
)(y
30
+ 47y
29
+ ··· + 191y + 1)
c
3
((y
2
− y + 1)
6
)(y
30
− 5y
29
+ ··· + 11y + 1)
c
5
, c
8
((y + 1)
12
)(y
30
+ y
29
+ ··· − 3175y + 625)
c
6
(y
12
+ 12y
11
+ ··· + 6y + 1)(y
30
+ 33y
29
+ ··· + 2246675y + 421201)
c
7
, c
11
((y
3
+ y
2
+ 2y + 1)
4
)(y
30
+ 5y
29
+ ··· + 9y + 1)
c
9
(y
12
− 12y
11
+ ··· − 6y + 1)
· (y
30
+ 89y
29
+ ··· + 64272198739y + 5681692129)
c
10
, c
12
((y
3
+ 3y
2
+ 2y −1)
4
)(y
30
+ 45y
29
+ ··· + 81y + 1)
14
