12n
0148
(K12n
0148
)
A knot diagram
1
Linearized knot diagam
3 4 9 2 10 3 12 11 6 4 7 8
Solving Sequence
7,11
12 8
3,9
4 1 2 5 6 10
c
11
c
7
c
8
c
3
c
12
c
2
c
4
c
6
c
10
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h5402057u
19
− 3583441u
18
+ ··· + 1977478b + 13308358,
− 639093u
19
+ 2151139u
18
+ ··· + 1977478a − 11355740, u
20
− u
19
+ ··· + 8u − 1i
I
u
2
= hu
5
a + 2u
4
a − 4u
3
a − 5u
4
− 3u
2
a + au + 10u
2
+ 5b − 3a,
u
5
a − 2u
5
− 3u
3
a + u
2
a + 6u
3
+ a
2
+ 2au + u
2
− a − 6u − 1, u
6
− 3u
4
+ 2u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 32 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
= h5.40 × 10
6
u
19
− 3.58 × 10
6
u
18
+ · · · + 1.98 × 10
6
b + 1.33 × 10
7
, −6.39 ×
10
5
u
19
+ 2.15 × 10
6
u
18
+ · · · + 1.98 × 10
6
a − 1.14 × 10
7
, u
20
− u
19
+ · · · + 8u − 1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
8
=
u
−u
3
+ u
a
3
=
0.323186u
19
− 1.08782u
18
+ ··· − 21.2507u + 5.74254
−2.73179u
19
+ 1.81213u
18
+ ··· + 39.4687u − 6.72997
a
9
=
−u
3
+ 2u
−u
3
+ u
a
4
=
−0.119500u
19
− 0.378541u
18
+ ··· − 7.19615u + 2.43256
−3.10782u
19
+ 2.21574u
18
+ ··· + 47.4853u − 8.50446
a
1
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
2.81060u
19
− 2.56427u
18
+ ··· − 52.8262u + 10.8039
−3.18615u
19
+ 1.40304u
18
+ ··· + 33.4356u − 4.59062
a
5
=
4.68409u
19
− 2.27049u
18
+ ··· − 53.3439u + 8.58394
1.97808u
19
− 0.907299u
18
+ ··· − 22.6536u + 3.12055
a
6
=
−3.69057u
19
+ 2.49256u
18
+ ··· + 52.9545u − 9.28098
1.03914u
19
− 0.150948u
18
+ ··· − 5.14484u + 0.559226
a
10
=
−2.78422u
19
+ 0.960462u
18
+ ··· + 26.8596u − 3.11259
−2.92442u
19
+ 1.93725u
18
+ ··· + 42.5149u − 7.01762
(ii) Obstruction class = −1
(iii) Cusp Shapes =
8158475
988739
u
19
−
7089886
988739
u
18
+ ··· −
162706955
988739
u +
47565543
988739
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
20
+ 43u
19
+ ··· − 20046u + 625
c
2
, c
4
u
20
− u
19
+ ··· − 214u + 25
c
3
u
20
− u
19
+ ··· + 18u − 5
c
5
, c
9
u
20
− u
19
+ ··· + 4u − 1
c
6
u
20
+ 3u
19
+ ··· − 70u + 1
c
7
, c
11
, c
12
u
20
− u
19
+ ··· + 8u − 1
c
8
u
20
+ 3u
19
+ ··· − 162u + 17
c
10
u
20
− u
19
+ ··· − 564u − 2209
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
20
− 253y
19
+ ··· − 251953366y + 390625
c
2
, c
4
y
20
+ 43y
19
+ ··· − 20046y + 625
c
3
y
20
− y
19
+ ··· − 214y + 25
c
5
, c
9
y
20
+ 41y
19
+ ··· + 84y + 1
c
6
y
20
+ 49y
19
+ ··· − 2266y + 1
c
7
, c
11
, c
12
y
20
− 15y
19
+ ··· − 16y + 1
c
8
y
20
+ 45y
19
+ ··· − 6660y + 289
c
10
y
20
+ 89y
19
+ ··· − 52304702y + 4879681
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.823692 + 0.671587I
a = 0.380830 − 0.847218I
b = 0.227244 + 0.074641I
−3.84702 − 2.60363I 6.95480 + 2.97260I
u = −0.823692 − 0.671587I
a = 0.380830 + 0.847218I
b = 0.227244 − 0.074641I
−3.84702 + 2.60363I 6.95480 − 2.97260I
u = −0.101123 + 1.192090I
a = −2.77141 + 0.53695I
b = −2.30625 + 0.07618I
17.7040 − 4.4434I 6.63619 + 1.99809I
u = −0.101123 − 1.192090I
a = −2.77141 − 0.53695I
b = −2.30625 − 0.07618I
17.7040 + 4.4434I 6.63619 − 1.99809I
u = 1.153440 + 0.367601I
a = −0.297249 − 0.756563I
b = −0.989133 + 0.311759I
0.86870 + 1.47310I 9.24158 − 0.66486I
u = 1.153440 − 0.367601I
a = −0.297249 + 0.756563I
b = −0.989133 − 0.311759I
0.86870 − 1.47310I 9.24158 + 0.66486I
u = −1.24490
a = −1.13302
b = −0.227826
5.11345 18.8050
u = 1.281610 + 0.133928I
a = −1.172220 − 0.435450I
b = −0.804234 + 0.979321I
2.04381 + 2.88061I 12.72572 − 3.09919I
u = 1.281610 − 0.133928I
a = −1.172220 + 0.435450I
b = −0.804234 − 0.979321I
2.04381 − 2.88061I 12.72572 + 3.09919I
u = −1.296610 + 0.228232I
a = 0.146274 + 0.961667I
b = −0.862052 − 0.640482I
3.01054 − 4.88727I 15.6926 + 6.5772I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.296610 − 0.228232I
a = 0.146274 − 0.961667I
b = −0.862052 + 0.640482I
3.01054 + 4.88727I 15.6926 − 6.5772I
u = 0.238874 + 0.462215I
a = 0.0284078 − 0.0494443I
b = 0.868979 − 0.326831I
−1.40207 + 2.19140I 8.77126 − 4.72251I
u = 0.238874 − 0.462215I
a = 0.0284078 + 0.0494443I
b = 0.868979 + 0.326831I
−1.40207 − 2.19140I 8.77126 + 4.72251I
u = 0.452767 + 0.174316I
a = −0.37480 − 1.75664I
b = 0.957475 − 0.420375I
−1.43242 + 2.12619I 10.10877 − 2.97976I
u = 0.452767 − 0.174316I
a = −0.37480 + 1.75664I
b = 0.957475 + 0.420375I
−1.43242 − 2.12619I 10.10877 + 2.97976I
u = −1.40293 + 0.65488I
a = 1.40779 − 0.91448I
b = 2.25453 − 0.06910I
−17.7758 − 2.1213I 8.50196 + 0.80172I
u = −1.40293 − 0.65488I
a = 1.40779 + 0.91448I
b = 2.25453 + 0.06910I
−17.7758 + 2.1213I 8.50196 − 0.80172I
u = 1.46039 + 0.53064I
a = 1.21678 + 1.71521I
b = 2.39771 + 0.19059I
−16.8226 + 10.5550I 9.32128 − 4.53620I
u = 1.46039 − 0.53064I
a = 1.21678 − 1.71521I
b = 2.39771 − 0.19059I
−16.8226 − 10.5550I 9.32128 + 4.53620I
u = 0.319454
a = 1.00420
b = −0.260725
0.583408 17.2870
6
II.
I
u
2
= hu
5
a + 2u
4
a + · · · + 5b − 3a, u
5
a − 2u
5
+ · · · − a − 1, u
6
− 3u
4
+ 2u
2
+ 1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
8
=
u
−u
3
+ u
a
3
=
a
−
1
5
u
5
a −
2
5
u
4
a + ··· −
1
5
au +
3
5
a
a
9
=
−u
3
+ 2u
−u
3
+ u
a
4
=
−
1
5
u
5
a −
2
5
u
4
a + ··· +
8
5
a − 1
−u
4
a + u
2
a + a
a
1
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
−u
5
+ 3u
3
− 2u
2
− 2u + 2
−
1
5
u
5
a − u
5
+ ··· −
2
5
a − u
a
5
=
u
2
− 1
−u
4
+ 2u
2
a
6
=
u
3
a + u
3
− au − 2u
2
− a − u + 2
−u
5
a + u
4
a − u
5
+ 2u
3
a + u
4
− u
2
a + 2u
3
− 2u
2
− a
a
10
=
−u
5
a + u
4
a + 2u
3
a + u
4
− 2u
2
a − 3u
3
− au − 2u
2
+ 6u
u
4
a + u
4
− u
2
a − 2u
3
− au − u
2
+ 2u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
8
5
u
5
a −
16
5
u
4
a +
12
5
u
3
a − 4u
4
+
24
5
u
2
a −
8
5
au + 8u
2
+
4
5
a + 8
7
(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
9
(u
2
+ 1)
6
c
6
u
12
+ 6u
11
+ ··· + 70u + 25
c
7
, c
11
, c
12
(u
6
− 3u
4
+ 2u
2
+ 1)
2
c
8
(u
6
+ u
4
+ 2u
2
+ 1)
2
c
10
u
12
+ 2u
11
+ ··· + 40u + 25
8
(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
9
(y + 1)
12
c
6
y
12
+ 4y
11
+ ··· − 850y + 625
c
7
, c
11
, c
12
(y
3
− 3y
2
+ 2y + 1)
4
c
8
(y
3
+ y
2
+ 2y + 1)
4
c
10
y
12
− 4y
11
+ ··· + 850y + 625
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.307140 + 0.215080I
a = 0.634341 − 0.270971I
b = −0.011413 + 0.244862I
1.37919 + 4.85801I 9.50976 − 6.44355I
u = 1.307140 + 0.215080I
a = −0.551838 − 0.413870I
b = −1.74346 + 0.24486I
1.37919 + 0.79824I 9.50976 + 0.48465I
u = 1.307140 − 0.215080I
a = 0.634341 + 0.270971I
b = −0.011413 − 0.244862I
1.37919 − 4.85801I 9.50976 + 6.44355I
u = 1.307140 − 0.215080I
a = −0.551838 + 0.413870I
b = −1.74346 − 0.24486I
1.37919 − 0.79824I 9.50976 − 0.48465I
u = −1.307140 + 0.215080I
a = −0.32280 + 1.43855I
b = −1.74346 − 1.24486I
1.37919 − 0.79824I 9.50976 − 0.48465I
u = −1.307140 + 0.215080I
a = −1.08442 − 0.99883I
b = −0.011413 − 1.244860I
1.37919 − 4.85801I 9.50976 + 6.44355I
u = −1.307140 − 0.215080I
a = −0.32280 − 1.43855I
b = −1.74346 + 1.24486I
1.37919 + 0.79824I 9.50976 + 0.48465I
u = −1.307140 − 0.215080I
a = −1.08442 + 0.99883I
b = −0.011413 + 1.244860I
1.37919 + 4.85801I 9.50976 − 6.44355I
u = 0.569840I
a = −0.85741 − 2.02468I
b = −0.111148 − 0.500000I
−2.75839 − 2.02988I 2.98049 + 3.46410I
u = 0.569840I
a = 2.18213 + 0.26980I
b = 1.62090 − 0.50000I
−2.75839 + 2.02988I 2.98049 − 3.46410I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = − 0.569840I
a = −0.85741 + 2.02468I
b = −0.111148 + 0.500000I
−2.75839 + 2.02988I 2.98049 − 3.46410I
u = − 0.569840I
a = 2.18213 − 0.26980I
b = 1.62090 + 0.50000I
−2.75839 − 2.02988I 2.98049 + 3.46410I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
6
)(u
20
+ 43u
19
+ ··· − 20046u + 625)
c
2
((u
2
+ u + 1)
6
)(u
20
− u
19
+ ··· − 214u + 25)
c
3
((u
4
− u
2
+ 1)
3
)(u
20
− u
19
+ ··· + 18u − 5)
c
4
((u
2
− u + 1)
6
)(u
20
− u
19
+ ··· − 214u + 25)
c
5
, c
9
((u
2
+ 1)
6
)(u
20
− u
19
+ ··· + 4u − 1)
c
6
(u
12
+ 6u
11
+ ··· + 70u + 25)(u
20
+ 3u
19
+ ··· − 70u + 1)
c
7
, c
11
, c
12
((u
6
− 3u
4
+ 2u
2
+ 1)
2
)(u
20
− u
19
+ ··· + 8u − 1)
c
8
((u
6
+ u
4
+ 2u
2
+ 1)
2
)(u
20
+ 3u
19
+ ··· − 162u + 17)
c
10
(u
12
+ 2u
11
+ ··· + 40u + 25)(u
20
− u
19
+ ··· − 564u − 2209)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
6
)(y
20
− 253y
19
+ ··· − 2.51953 × 10
8
y + 390625)
c
2
, c
4
((y
2
+ y + 1)
6
)(y
20
+ 43y
19
+ ··· − 20046y + 625)
c
3
((y
2
− y + 1)
6
)(y
20
− y
19
+ ··· − 214y + 25)
c
5
, c
9
((y + 1)
12
)(y
20
+ 41y
19
+ ··· + 84y + 1)
c
6
(y
12
+ 4y
11
+ ··· − 850y + 625)(y
20
+ 49y
19
+ ··· − 2266y + 1)
c
7
, c
11
, c
12
((y
3
− 3y
2
+ 2y + 1)
4
)(y
20
− 15y
19
+ ··· − 16y + 1)
c
8
((y
3
+ y
2
+ 2y + 1)
4
)(y
20
+ 45y
19
+ ··· − 6660y + 289)
c
10
(y
12
− 4y
11
+ ··· + 850y + 625)
· (y
20
+ 89y
19
+ ··· − 52304702y + 4879681)
13
