12n
0640
(K12n
0640
)
A knot diagram
1
Linearized knot diagam
3 8 12 9 11 10 2 5 3 5 6 9
Solving Sequence
5,10
11 6 7
3,12
9 4 8 2 1
c
10
c
5
c
6
c
11
c
9
c
4
c
8
c
2
c
1
c
3
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h9.89972 × 10
17
u
23
− 1.91877 × 10
15
u
22
+ ··· + 4.06813 × 10
18
b − 2.49679 × 10
18
,
8.63753 × 10
17
u
23
− 1.70899 × 10
18
u
22
+ ··· + 4.06813 × 10
18
a − 1.68305 × 10
19
, u
24
− 10u
22
+ ··· + 6u + 1i
I
u
2
= hu
11
− 7u
9
+ 17u
7
+ u
6
− 15u
5
− 3u
4
+ 2u
2
+ b + 3u,
u
9
− u
8
− 6u
7
+ 6u
6
+ 11u
5
− 11u
4
− 5u
3
+ 6u
2
+ a − 3u + 1,
u
12
− 8u
10
+ 24u
8
+ u
7
− 32u
6
− 4u
5
+ 15u
4
+ 5u
3
+ 2u
2
− 2u − 1i
I
u
3
= hb, a − 1, u − 1i
* 3 irreducible components of dim
C
= 0, with total 37 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
=
h9.90×10
17
u
23
−1.92×10
15
u
22
+· · ·+4.07×10
18
b−2.50×10
18
, 8.64×10
17
u
23
−
1.71 × 10
18
u
22
+ · · · + 4.07 × 10
18
a − 1.68 × 10
19
, u
24
− 10u
22
+ · · · + 6u + 1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
6
=
−u
−u
3
+ u
a
7
=
u
3
− 2u
−u
3
+ u
a
3
=
−0.212322u
23
+ 0.420093u
22
+ ··· + 3.09567u + 4.13716
−0.243348u
23
+ 0.000471658u
22
+ ··· + 1.81524u + 0.613744
a
12
=
−u
2
+ 1
−u
4
+ 2u
2
a
9
=
−1.25860u
23
+ 0.336558u
22
+ ··· − 22.2342u − 4.64549
−0.0840693u
23
+ 0.188815u
22
+ ··· − 7.10991u − 1.40341
a
4
=
−0.483379u
23
+ 0.505798u
22
+ ··· − 2.99675u + 2.78091
−0.208694u
23
− 0.00110386u
22
+ ··· + 1.55323u + 0.479053
a
8
=
−1.25860u
23
+ 0.336558u
22
+ ··· − 22.2342u − 4.64549
−0.226293u
23
+ 0.0998004u
22
+ ··· − 6.34916u − 1.06686
a
2
=
−1.51744u
23
+ 0.547755u
22
+ ··· − 19.4211u − 3.44143
−0.320558u
23
+ 0.0501945u
22
+ ··· − 5.21590u − 0.630497
a
1
=
1.40524u
23
− 0.275084u
22
+ ··· + 14.6942u + 2.32021
0.344813u
23
− 0.0300623u
22
+ ··· − 3.26431u − 0.145094
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
18808788651478118843
4068132824722451189
u
23
+
899618609945270896
4068132824722451189
u
22
+ ··· −
136381012325864299426
4068132824722451189
u −
118763470922061227713
4068132824722451189
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
24
+ 4u
23
+ ··· + 1789u + 1
c
2
, c
7
u
24
+ 2u
23
+ ··· + 47u + 1
c
3
u
24
− 3u
23
+ ··· + 14u + 1
c
4
, c
8
u
24
+ 3u
23
+ ··· + 80u + 19
c
5
, c
10
, c
11
u
24
− 10u
22
+ ··· + 6u + 1
c
6
u
24
+ 3u
23
+ ··· + 696u + 37
c
9
u
24
− u
23
+ ··· + 40u − 8
c
12
u
24
+ 2u
23
+ ··· + 5u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
24
+ 44y
23
+ ··· − 3160089y + 1
c
2
, c
7
y
24
− 4y
23
+ ··· − 1789y + 1
c
3
y
24
+ 13y
23
+ ··· − 214y + 1
c
4
, c
8
y
24
+ 29y
23
+ ··· − 5754y + 361
c
5
, c
10
, c
11
y
24
− 20y
23
+ ··· − 26y + 1
c
6
y
24
+ 47y
23
+ ··· − 161850y + 1369
c
9
y
24
+ 23y
23
+ ··· − 4064y + 64
c
12
y
24
− 60y
23
+ ··· − 93y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.797828 + 0.678753I
a = −1.02319 + 1.10302I
b = −0.55673 − 1.65487I
3.88067 − 2.62654I −11.07995 + 2.70331I
u = 0.797828 − 0.678753I
a = −1.02319 − 1.10302I
b = −0.55673 + 1.65487I
3.88067 + 2.62654I −11.07995 − 2.70331I
u = 1.05078
a = 0.906274
b = 0.133759
−4.93398 −18.0800
u = 1.21774
a = 1.31377
b = 1.26448
−6.34892 −11.4760
u = −1.117180 + 0.558107I
a = 0.264601 + 0.177693I
b = −0.642330 − 1.001630I
0.11807 + 2.01122I −11.63298 − 1.47216I
u = −1.117180 − 0.558107I
a = 0.264601 − 0.177693I
b = −0.642330 + 1.001630I
0.11807 − 2.01122I −11.63298 + 1.47216I
u = 1.282720 + 0.266061I
a = −0.417258 + 1.231710I
b = −0.313111 − 0.558407I
−2.36903 − 5.36546I −17.8756 + 8.3270I
u = 1.282720 − 0.266061I
a = −0.417258 − 1.231710I
b = −0.313111 + 0.558407I
−2.36903 + 5.36546I −17.8756 − 8.3270I
u = −0.123057 + 1.316600I
a = 0.002520 − 1.240400I
b = −0.32423 + 2.21788I
10.67480 + 4.86745I −9.82219 − 2.45720I
u = −0.123057 − 1.316600I
a = 0.002520 + 1.240400I
b = −0.32423 − 2.21788I
10.67480 − 4.86745I −9.82219 + 2.45720I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.315130 + 0.199500I
a = 0.931929 + 0.744429I
b = 0.399732 − 1.228200I
−2.04826 + 3.83523I −16.1350 − 2.1387I
u = −1.315130 − 0.199500I
a = 0.931929 − 0.744429I
b = 0.399732 + 1.228200I
−2.04826 − 3.83523I −16.1350 + 2.1387I
u = −0.410599 + 0.509120I
a = 0.56444 + 1.46880I
b = 0.821859 + 0.235411I
2.03875 + 2.53994I −11.82047 − 4.16914I
u = −0.410599 − 0.509120I
a = 0.56444 − 1.46880I
b = 0.821859 − 0.235411I
2.03875 − 2.53994I −11.82047 + 4.16914I
u = 1.49784
a = 0.0823949
b = −1.05480
−11.6286 −23.4910
u = 0.394751
a = 1.18127
b = 1.60965
−7.19054 −5.70560
u = −1.41645 + 0.78202I
a = −0.817155 − 0.477348I
b = −0.71160 + 2.00454I
6.79929 + 2.45061I −11.06872 − 1.28046I
u = −1.41645 − 0.78202I
a = −0.817155 + 0.477348I
b = −0.71160 − 2.00454I
6.79929 − 2.45061I −11.06872 + 1.28046I
u = 1.50198 + 0.60802I
a = 0.868783 − 0.609431I
b = 1.06909 + 1.77146I
5.60719 − 11.65040I −12.86830 + 5.47222I
u = 1.50198 − 0.60802I
a = 0.868783 + 0.609431I
b = 1.06909 − 1.77146I
5.60719 + 11.65040I −12.86830 − 5.47222I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.313467
a = −0.425309
b = −0.324845
−0.515739 −19.2450
u = −0.232568 + 0.185578I
a = 2.34163 + 3.73562I
b = 0.349022 + 0.617701I
2.00166 + 2.59049I −13.9321 − 5.9466I
u = −0.232568 − 0.185578I
a = 2.34163 − 3.73562I
b = 0.349022 − 0.617701I
2.00166 − 2.59049I −13.9321 + 5.9466I
u = −1.78276
a = −0.490994
b = −0.811626
−16.2087 −2.53170
7
II.
I
u
2
= hu
11
− 7u
9
+ · · · + b + 3u, u
9
− u
8
+ · · · + a + 1, u
12
− 8u
10
+ · · · − 2u − 1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
6
=
−u
−u
3
+ u
a
7
=
u
3
− 2u
−u
3
+ u
a
3
=
−u
9
+ u
8
+ 6u
7
− 6u
6
− 11u
5
+ 11u
4
+ 5u
3
− 6u
2
+ 3u − 1
−u
11
+ 7u
9
− 17u
7
− u
6
+ 15u
5
+ 3u
4
− 2u
2
− 3u
a
12
=
−u
2
+ 1
−u
4
+ 2u
2
a
9
=
−u
11
+ u
10
+ ··· + u + 3
u
10
− 6u
8
+ 11u
6
+ u
5
− 4u
4
− 2u
3
− 5u
2
a
4
=
u
11
− 8u
9
+ u
8
+ 24u
7
− 5u
6
− 31u
5
+ 8u
4
+ 12u
3
− 3u
2
+ 4u − 2
−u
11
+ 7u
9
− 17u
7
− u
6
+ 15u
5
+ 3u
4
− 2u
2
− 2u
a
8
=
−u
11
+ u
10
+ ··· + u + 3
−u
6
+ 4u
4
− 4u
2
− u − 1
a
2
=
−u
9
+ u
8
+ 6u
7
− 6u
6
− 11u
5
+ 11u
4
+ 5u
3
− 5u
2
+ 3u − 3
u
10
− 7u
8
+ 17u
6
+ u
5
− 16u
4
− 3u
3
+ 2u
2
+ 2u + 2
a
1
=
u
11
− 8u
9
+ u
8
+ 24u
7
− 5u
6
− 32u
5
+ 8u
4
+ 16u
3
− 3u
2
− 3
u
11
+ 2u
10
+ ··· + 7u + 3
(ii) Obstruction class = 1
(iii) Cusp Shapes
= −6u
11
+ 4u
10
+ 43u
9
− 25u
8
− 108u
7
+ 42u
6
+ 108u
5
+ u
4
− 21u
3
− 40u
2
− 14u − 8
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
12
− 12u
11
+ ··· − 15u + 1
c
2
u
12
− 6u
10
+ u
9
+ 15u
8
− 3u
7
− 21u
6
+ 4u
5
+ 17u
4
− 4u
3
− 7u
2
+ u + 1
c
3
u
12
+ 3u
11
+ u
10
− 5u
9
− 8u
8
− 5u
7
+ 2u
6
+ 5u
5
+ 5u
4
+ 3u
3
− 1
c
4
u
12
+ u
11
− u
10
− 3u
8
− 2u
7
− u
6
+ u
5
+ 3u
4
+ u
3
+ 2u
2
− 1
c
5
u
12
− 8u
10
+ 24u
8
− u
7
− 32u
6
+ 4u
5
+ 15u
4
− 5u
3
+ 2u
2
+ 2u − 1
c
6
u
12
− 8u
10
+ ··· + 2u − 1
c
7
u
12
− 6u
10
− u
9
+ 15u
8
+ 3u
7
− 21u
6
− 4u
5
+ 17u
4
+ 4u
3
− 7u
2
− u + 1
c
8
u
12
− u
11
− u
10
− 3u
8
+ 2u
7
− u
6
− u
5
+ 3u
4
− u
3
+ 2u
2
− 1
c
9
u
12
− 2u
10
− u
9
− 3u
8
− u
7
+ u
6
+ 2u
5
+ 3u
4
+ u
2
− u − 1
c
10
, c
11
u
12
− 8u
10
+ 24u
8
+ u
7
− 32u
6
− 4u
5
+ 15u
4
+ 5u
3
+ 2u
2
− 2u − 1
c
12
u
12
− 4u
11
+ ··· + 7u + 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
12
− 12y
11
+ ··· − 43y + 1
c
2
, c
7
y
12
− 12y
11
+ ··· − 15y + 1
c
3
y
12
− 7y
11
+ ··· − 10y
2
+ 1
c
4
, c
8
y
12
− 3y
11
+ ··· − 4y + 1
c
5
, c
10
, c
11
y
12
− 16y
11
+ ··· − 8y + 1
c
6
y
12
− 16y
11
+ ··· − 20y + 1
c
9
y
12
− 4y
11
+ ··· − 3y + 1
c
12
y
12
− 24y
11
+ ··· + 17y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.17998
a = 0.800947
b = 1.82750
−9.65747 −15.7210
u = −1.23606
a = −1.56622
b = −1.10566
−6.93408 −26.3380
u = −1.324120 + 0.237549I
a = 0.849347 + 0.965329I
b = 0.107633 − 0.994842I
−1.41781 + 4.88882I −10.78753 − 6.22489I
u = −1.324120 − 0.237549I
a = 0.849347 − 0.965329I
b = 0.107633 + 0.994842I
−1.41781 − 4.88882I −10.78753 + 6.22489I
u = 0.579754
a = 0.128754
b = −1.45303
−7.57612 −27.2480
u = −0.136756 + 0.512426I
a = 1.53089 + 2.63864I
b = 0.199486 − 0.931452I
2.56609 − 2.20336I −4.12994 + 0.41603I
u = −0.136756 − 0.512426I
a = 1.53089 − 2.63864I
b = 0.199486 + 0.931452I
2.56609 + 2.20336I −4.12994 − 0.41603I
u = 1.46831 + 0.31043I
a = −0.963926 + 0.306069I
b = −0.527150 − 0.827184I
−2.88723 − 0.94663I −11.62610 + 0.41133I
u = 1.46831 − 0.31043I
a = −0.963926 − 0.306069I
b = −0.527150 + 0.827184I
−2.88723 + 0.94663I −11.62610 − 0.41133I
u = 1.58481
a = 0.328758
b = −0.536398
−11.0733 −8.22380
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.371528
a = −2.93289
b = 0.802551
−4.02071 −7.78700
u = −1.75183
a = 0.408041
b = 0.905098
−16.4780 −32.5950
12
III. I
u
3
= hb, a − 1, u − 1i
(i) Arc colorings
a
5
=
0
1
a
10
=
1
0
a
11
=
1
1
a
6
=
−1
0
a
7
=
−1
0
a
3
=
1
0
a
12
=
0
1
a
9
=
1
0
a
4
=
1
1
a
8
=
1
−1
a
2
=
2
−1
a
1
=
1
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −18
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
12
u + 1
c
2
, c
4
, c
5
c
7
, c
8
, c
10
c
11
u − 1
c
6
, c
9
u
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
7
c
8
, c
10
, c
11
c
12
y −1
c
6
, c
9
y
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 0
−4.93480 −18.0000
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u + 1)(u
12
− 12u
11
+ ··· − 15u + 1)(u
24
+ 4u
23
+ ··· + 1789u + 1)
c
2
(u − 1)
· (u
12
− 6u
10
+ u
9
+ 15u
8
− 3u
7
− 21u
6
+ 4u
5
+ 17u
4
− 4u
3
− 7u
2
+ u + 1)
· (u
24
+ 2u
23
+ ··· + 47u + 1)
c
3
(u + 1)(u
12
+ 3u
11
+ ··· + 3u
3
− 1)
· (u
24
− 3u
23
+ ··· + 14u + 1)
c
4
(u − 1)(u
12
+ u
11
− u
10
− 3u
8
− 2u
7
− u
6
+ u
5
+ 3u
4
+ u
3
+ 2u
2
− 1)
· (u
24
+ 3u
23
+ ··· + 80u + 19)
c
5
(u − 1)(u
12
− 8u
10
+ ··· + 2u − 1)
· (u
24
− 10u
22
+ ··· + 6u + 1)
c
6
u(u
12
− 8u
10
+ ··· + 2u − 1)(u
24
+ 3u
23
+ ··· + 696u + 37)
c
7
(u − 1)
· (u
12
− 6u
10
− u
9
+ 15u
8
+ 3u
7
− 21u
6
− 4u
5
+ 17u
4
+ 4u
3
− 7u
2
− u + 1)
· (u
24
+ 2u
23
+ ··· + 47u + 1)
c
8
(u − 1)(u
12
− u
11
− u
10
− 3u
8
+ 2u
7
− u
6
− u
5
+ 3u
4
− u
3
+ 2u
2
− 1)
· (u
24
+ 3u
23
+ ··· + 80u + 19)
c
9
u(u
12
− 2u
10
− u
9
− 3u
8
− u
7
+ u
6
+ 2u
5
+ 3u
4
+ u
2
− u − 1)
· (u
24
− u
23
+ ··· + 40u − 8)
c
10
, c
11
(u − 1)(u
12
− 8u
10
+ ··· − 2u − 1)
· (u
24
− 10u
22
+ ··· + 6u + 1)
c
12
(u + 1)(u
12
− 4u
11
+ ··· + 7u + 1)(u
24
+ 2u
23
+ ··· + 5u − 1)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)(y
12
− 12y
11
+ ··· − 43y + 1)(y
24
+ 44y
23
+ ··· − 3160089y + 1)
c
2
, c
7
(y −1)(y
12
− 12y
11
+ ··· − 15y + 1)(y
24
− 4y
23
+ ··· − 1789y + 1)
c
3
(y −1)(y
12
− 7y
11
+ ··· − 10y
2
+ 1)(y
24
+ 13y
23
+ ··· − 214y + 1)
c
4
, c
8
(y −1)(y
12
− 3y
11
+ ··· − 4y + 1)(y
24
+ 29y
23
+ ··· − 5754y + 361)
c
5
, c
10
, c
11
(y −1)(y
12
− 16y
11
+ ··· − 8y + 1)(y
24
− 20y
23
+ ··· − 26y + 1)
c
6
y(y
12
− 16y
11
+ ··· − 20y + 1)(y
24
+ 47y
23
+ ··· − 161850y + 1369)
c
9
y(y
12
− 4y
11
+ ··· − 3y + 1)(y
24
+ 23y
23
+ ··· − 4064y + 64)
c
12
(y −1)(y
12
− 24y
11
+ ··· + 17y + 1)(y
24
− 60y
23
+ ··· − 93y + 1)
18
