12n
0368
(K12n
0368
)
A knot diagram
1
Linearized knot diagam
3 6 7 10 2 9 3 12 4 6 8 11
Solving Sequence
8,12 3,9
7 4 6 2 5 11 1 10
c
8
c
7
c
3
c
6
c
2
c
5
c
11
c
12
c
10
c
1
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h4.99290 × 10
29
u
27
− 9.04354 × 10
29
u
26
+ ··· + 1.48123 × 10
30
b + 2.22464 × 10
30
,
1.79527 × 10
31
u
27
− 3.40599 × 10
31
u
26
+ ··· + 4.44370 × 10
30
a + 1.93917 × 10
32
, u
28
− 2u
27
+ ··· + 15u − 1i
I
u
2
= h−u
15
− u
14
+ ··· + b − 1, −6u
15
+ 2u
14
+ ··· + a − 10,
u
16
− 5u
14
+ u
13
+ 13u
12
− 3u
11
− 23u
10
+ 6u
9
+ 29u
8
− 8u
7
− 26u
6
+ 8u
5
+ 16u
4
− 4u
3
− 6u
2
+ u + 1i
I
u
3
= hb − 1, a − 1, u + 1i
* 3 irreducible components of dim
C
= 0, with total 45 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
=
h4.99×10
29
u
27
−9.04×10
29
u
26
+· · ·+1.48×10
30
b+2.22×10
30
, 1.80×10
31
u
27
−
3.41 × 10
31
u
26
+ · · · + 4.44 × 10
30
a + 1.94 × 10
32
, u
28
− 2u
27
+ · · · + 15u − 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
3
=
−4.04003u
27
+ 7.66476u
26
+ ··· + 200.372u − 43.6386
−0.337077u
27
+ 0.610541u
26
+ ··· + 8.04513u − 1.50188
a
9
=
1
u
2
a
7
=
4.84634u
27
− 9.14511u
26
+ ··· − 218.138u + 47.6168
0.162770u
27
− 0.345209u
26
+ ··· − 15.7091u + 2.22482
a
4
=
−9.48708u
27
+ 18.0943u
26
+ ··· + 467.082u − 96.8502
−0.312623u
27
+ 0.470169u
26
+ ··· + 9.63585u − 2.96138
a
6
=
4.97301u
27
− 9.41740u
26
+ ··· − 230.480u + 49.2941
0.173164u
27
− 0.347370u
26
+ ··· − 15.2980u + 2.20586
a
2
=
−12.5790u
27
+ 23.8637u
26
+ ··· + 606.925u − 129.004
−0.600077u
27
+ 1.10181u
26
+ ··· + 27.4907u − 4.83602
a
5
=
−16.7676u
27
+ 31.8085u
26
+ ··· + 818.466u − 172.294
−0.699475u
27
+ 1.15355u
26
+ ··· + 25.6721u − 5.70735
a
11
=
u
u
a
1
=
−u
3
−u
3
+ u
a
10
=
15.6689u
27
− 29.9384u
26
+ ··· − 773.348u + 161.864
0.340769u
27
− 0.720300u
26
+ ··· − 15.5800u + 4.83370
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2.36235u
27
− 4.45969u
26
+ ··· − 84.8018u + 5.95686
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
28
+ 56u
27
+ ··· + 733262u + 28561
c
2
, c
5
u
28
+ 2u
27
+ ··· − 1450u − 169
c
3
, c
7
u
28
− 4u
27
+ ··· + 156u + 23
c
4
, c
9
u
28
− 24u
26
+ ··· + 46u − 43
c
6
u
28
− 3u
27
+ ··· − u − 1
c
8
, c
11
u
28
+ 2u
27
+ ··· − 15u − 1
c
10
u
28
− 63u
26
+ ··· + 129664u + 33653
c
12
u
28
+ 26u
27
+ ··· + 87u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
28
− 188y
27
+ ··· − 348035375138y + 815730721
c
2
, c
5
y
28
− 56y
27
+ ··· − 733262y + 28561
c
3
, c
7
y
28
+ 32y
27
+ ··· + 1516y + 529
c
4
, c
9
y
28
− 48y
27
+ ··· − 11662y + 1849
c
6
y
28
− 3y
27
+ ··· − 13y + 1
c
8
, c
11
y
28
− 26y
27
+ ··· − 87y + 1
c
10
y
28
− 126y
27
+ ··· − 33547178288y + 1132524409
c
12
y
28
− 38y
27
+ ··· − 11231y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.906896 + 0.646886I
a = −0.018709 − 0.380835I
b = −0.638662 + 0.021294I
1.71311 + 2.54690I −1.43844 − 1.11774I
u = −0.906896 − 0.646886I
a = −0.018709 + 0.380835I
b = −0.638662 − 0.021294I
1.71311 − 2.54690I −1.43844 + 1.11774I
u = 0.967377 + 0.607852I
a = 0.156659 − 0.808047I
b = 0.094276 − 0.264187I
−1.84113 − 4.28473I −9.39382 + 5.33049I
u = 0.967377 − 0.607852I
a = 0.156659 + 0.808047I
b = 0.094276 + 0.264187I
−1.84113 + 4.28473I −9.39382 − 5.33049I
u = −1.213110 + 0.327569I
a = 0.57852 − 1.90768I
b = −0.26231 − 1.86751I
−4.18195 + 4.33996I −13.7175 − 7.6147I
u = −1.213110 − 0.327569I
a = 0.57852 + 1.90768I
b = −0.26231 + 1.86751I
−4.18195 − 4.33996I −13.7175 + 7.6147I
u = 0.614251 + 0.338068I
a = −0.531012 + 0.325089I
b = 0.489062 − 0.020644I
−0.982240 + 0.119994I −8.17175 + 0.02561I
u = 0.614251 − 0.338068I
a = −0.531012 − 0.325089I
b = 0.489062 + 0.020644I
−0.982240 − 0.119994I −8.17175 − 0.02561I
u = 0.696063
a = −0.481673
b = 0.373251
−0.946260 −10.4550
u = 1.257950 + 0.387635I
a = −0.74898 − 1.66086I
b = 0.123685 − 1.210180I
−4.49564 − 1.87179I −14.6317 − 1.5633I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.257950 − 0.387635I
a = −0.74898 + 1.66086I
b = 0.123685 + 1.210180I
−4.49564 + 1.87179I −14.6317 + 1.5633I
u = −0.07059 + 1.45889I
a = −0.0475163 − 0.0400773I
b = −0.21400 − 1.78782I
−18.0323 + 4.3258I −11.73246 − 2.08575I
u = −0.07059 − 1.45889I
a = −0.0475163 + 0.0400773I
b = −0.21400 + 1.78782I
−18.0323 − 4.3258I −11.73246 + 2.08575I
u = 1.50606
a = −1.25220
b = −3.31724
−15.4303 −22.9460
u = −0.036535 + 0.484008I
a = −0.642458 + 0.305211I
b = −0.141755 + 0.946555I
−0.78712 − 1.33307I −6.90000 + 4.82501I
u = −0.036535 − 0.484008I
a = −0.642458 − 0.305211I
b = −0.141755 − 0.946555I
−0.78712 + 1.33307I −6.90000 − 4.82501I
u = 1.57314 + 0.15250I
a = 0.34179 + 1.42667I
b = −0.20624 + 2.13211I
−10.36200 + 1.52321I −13.77194 − 1.67469I
u = 1.57314 − 0.15250I
a = 0.34179 − 1.42667I
b = −0.20624 − 2.13211I
−10.36200 − 1.52321I −13.77194 + 1.67469I
u = −1.61185
a = −0.871275
b = 0.575915
−16.6517 −17.2310
u = −1.61164 + 0.12556I
a = −0.01556 + 1.51530I
b = 0.42383 + 1.60874I
−11.04510 + 5.13574I −13.9862 − 3.3545I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.61164 − 0.12556I
a = −0.01556 − 1.51530I
b = 0.42383 − 1.60874I
−11.04510 − 5.13574I −13.9862 + 3.3545I
u = 0.055371 + 0.305690I
a = −1.91148 + 0.93139I
b = 0.441705 − 1.203630I
−4.59867 − 3.49978I −13.2785 + 5.9062I
u = 0.055371 − 0.305690I
a = −1.91148 − 0.93139I
b = 0.441705 + 1.203630I
−4.59867 + 3.49978I −13.2785 − 5.9062I
u = 1.56528 + 0.69723I
a = 0.61493 + 1.30760I
b = −0.73222 + 1.97314I
16.4052 − 11.9046I −13.05540 + 4.81403I
u = 1.56528 − 0.69723I
a = 0.61493 − 1.30760I
b = −0.73222 − 1.97314I
16.4052 + 11.9046I −13.05540 − 4.81403I
u = −1.53853 + 0.79216I
a = −0.793628 + 0.857092I
b = 0.34136 + 1.61067I
17.0288 + 3.5616I −13.06254 + 0.I
u = −1.53853 − 0.79216I
a = −0.793628 − 0.857092I
b = 0.34136 − 1.61067I
17.0288 − 3.5616I −13.06254 + 0.I
u = 0.0975704
a = −30.3600
b = −1.06939
−10.1503 0.912190
7
II.
I
u
2
= h−u
15
−u
14
+· · ·+b−1, −6u
15
+2u
14
+· · ·+a−10, u
16
−5u
14
+· · ·+u+1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
3
=
6u
15
− 2u
14
+ ··· − 15u + 10
u
15
+ u
14
+ ··· − 4u + 1
a
9
=
1
u
2
a
7
=
5u
15
− 4u
14
+ ··· − 10u + 13
3u
15
− 2u
14
+ ··· − 2u + 3
a
4
=
13u
15
− 5u
14
+ ··· − 30u + 25
u
14
− u
13
+ ··· − 5u + 1
a
6
=
6u
15
− 4u
14
+ ··· − 11u + 12
3u
15
− 2u
14
+ ··· − 3u + 3
a
2
=
16u
15
− 7u
14
+ ··· − 35u + 29
3u
15
− u
14
+ ··· − 4u + 5
a
5
=
−18u
15
+ 9u
14
+ ··· + 38u − 40
−u
15
+ 5u
13
+ ··· + 5u − 4
a
11
=
u
u
a
1
=
−u
3
−u
3
+ u
a
10
=
−21u
15
+ 11u
14
+ ··· + 40u − 40
−5u
15
+ 3u
14
+ ··· + 5u − 9
(ii) Obstruction class = 1
(iii) Cusp Shapes = −13u
15
+ 4u
14
+ 60u
13
− 31u
12
− 141u
11
+ 76u
10
+ 230u
9
−
132u
8
− 260u
7
+ 157u
6
+ 200u
5
− 136u
4
− 96u
3
+ 62u
2
+ 25u − 31
8
(iv) u-Polynomials at the component
9
Crossings u-Polynomials at each crossing
c
1
u
16
− 16u
15
+ ··· − 8u + 1
c
2
u
16
+ 4u
15
+ ··· + 4u + 1
c
3
u
16
+ 4u
14
+ ··· − 6u + 1
c
4
u
16
− 10u
14
+ ··· − 2u − 1
c
5
u
16
− 4u
15
+ ··· − 4u + 1
c
6
u
16
− 3u
15
+ ··· − 3u − 1
c
7
u
16
+ 4u
14
+ ··· + 6u + 1
c
8
u
16
− 5u
14
+ ··· + u + 1
c
9
u
16
− 10u
14
+ ··· + 2u − 1
c
10
u
16
+ 2u
15
+ ··· + 7u
2
− 1
c
11
u
16
− 5u
14
+ ··· − u + 1
c
12
u
16
+ 10u
15
+ ··· + 13u + 1
10
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
16
− 52y
15
+ ··· + 20y + 1
c
2
, c
5
y
16
− 16y
15
+ ··· − 8y + 1
c
3
, c
7
y
16
+ 8y
15
+ ··· − 10y + 1
c
4
, c
9
y
16
− 20y
15
+ ··· + 4y + 1
c
6
y
16
+ y
15
+ ··· − 15y + 1
c
8
, c
11
y
16
− 10y
15
+ ··· − 13y + 1
c
10
y
16
− 38y
15
+ ··· − 14y + 1
c
12
y
16
+ 2y
15
+ ··· − 17y + 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.772761 + 0.712653I
a = 0.214642 + 0.139490I
b = −0.174212 + 0.983327I
−3.18731 − 4.89171I −13.3342 + 7.1832I
u = 0.772761 − 0.712653I
a = 0.214642 − 0.139490I
b = −0.174212 − 0.983327I
−3.18731 + 4.89171I −13.3342 − 7.1832I
u = −1.026470 + 0.385848I
a = 0.97974 − 2.34875I
b = −0.64749 − 1.56601I
−6.20371 + 5.58512I −14.9139 − 6.6172I
u = −1.026470 − 0.385848I
a = 0.97974 + 2.34875I
b = −0.64749 + 1.56601I
−6.20371 − 5.58512I −14.9139 + 6.6172I
u = −0.868992 + 0.775777I
a = 0.422206 − 0.337071I
b = −0.267354 + 0.100433I
1.11054 + 2.92387I −11.66623 − 5.71389I
u = −0.868992 − 0.775777I
a = 0.422206 + 0.337071I
b = −0.267354 − 0.100433I
1.11054 − 2.92387I −11.66623 + 5.71389I
u = 1.153510 + 0.323030I
a = −0.52474 − 1.80473I
b = −0.125626 − 1.307330I
−4.07711 − 3.14561I −13.9405 + 2.6304I
u = 1.153510 − 0.323030I
a = −0.52474 + 1.80473I
b = −0.125626 + 1.307330I
−4.07711 + 3.14561I −13.9405 − 2.6304I
u = −0.730829 + 0.328251I
a = −0.476380 − 0.106565I
b = −0.722249 + 1.146800I
−5.07627 − 2.52540I −16.3893 + 0.5079I
u = −0.730829 − 0.328251I
a = −0.476380 + 0.106565I
b = −0.722249 − 1.146800I
−5.07627 + 2.52540I −16.3893 − 0.5079I
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.004110 + 0.721375I
a = −1.101500 − 0.625779I
b = 0.144368 − 1.128410I
−3.89724 − 0.61018I −12.26838 + 0.63265I
u = 1.004110 − 0.721375I
a = −1.101500 + 0.625779I
b = 0.144368 + 1.128410I
−3.89724 + 0.61018I −12.26838 − 0.63265I
u = 0.698430 + 0.203647I
a = −1.272980 + 0.463255I
b = 0.148546 + 0.648408I
−2.24714 + 0.87508I −15.0335 − 1.9463I
u = 0.698430 − 0.203647I
a = −1.272980 − 0.463255I
b = 0.148546 − 0.648408I
−2.24714 − 0.87508I −15.0335 + 1.9463I
u = −0.491820
a = 7.05656
b = 1.05815
−10.4382 −27.2890
u = −1.51322
a = 0.461470
b = 2.22988
−14.7824 −9.61910
14
III. I
u
3
= hb − 1, a − 1, u + 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
−1
a
3
=
1
1
a
9
=
1
1
a
7
=
2
1
a
4
=
3
2
a
6
=
1
0
a
2
=
2
1
a
5
=
−1
−1
a
11
=
−1
−1
a
1
=
1
0
a
10
=
−2
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −18
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
c
12
u + 1
c
4
, c
8
, c
9
c
10
, c
11
u − 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y −1
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 1.00000
b = 1.00000
−4.93480 −18.0000
18
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u + 1)(u
16
− 16u
15
+ ··· − 8u + 1)
· (u
28
+ 56u
27
+ ··· + 733262u + 28561)
c
2
(u + 1)(u
16
+ 4u
15
+ ··· + 4u + 1)(u
28
+ 2u
27
+ ··· − 1450u − 169)
c
3
(u + 1)(u
16
+ 4u
14
+ ··· − 6u + 1)(u
28
− 4u
27
+ ··· + 156u + 23)
c
4
(u − 1)(u
16
− 10u
14
+ ··· − 2u − 1)(u
28
− 24u
26
+ ··· + 46u − 43)
c
5
(u + 1)(u
16
− 4u
15
+ ··· − 4u + 1)(u
28
+ 2u
27
+ ··· − 1450u − 169)
c
6
(u + 1)(u
16
− 3u
15
+ ··· − 3u − 1)(u
28
− 3u
27
+ ··· − u − 1)
c
7
(u + 1)(u
16
+ 4u
14
+ ··· + 6u + 1)(u
28
− 4u
27
+ ··· + 156u + 23)
c
8
(u − 1)(u
16
− 5u
14
+ ··· + u + 1)(u
28
+ 2u
27
+ ··· − 15u − 1)
c
9
(u − 1)(u
16
− 10u
14
+ ··· + 2u − 1)(u
28
− 24u
26
+ ··· + 46u − 43)
c
10
(u − 1)(u
16
+ 2u
15
+ ··· + 7u
2
− 1)
· (u
28
− 63u
26
+ ··· + 129664u + 33653)
c
11
(u − 1)(u
16
− 5u
14
+ ··· − u + 1)(u
28
+ 2u
27
+ ··· − 15u − 1)
c
12
(u + 1)(u
16
+ 10u
15
+ ··· + 13u + 1)(u
28
+ 26u
27
+ ··· + 87u + 1)
19
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)(y
16
− 52y
15
+ ··· + 20y + 1)
· (y
28
− 188y
27
+ ··· − 348035375138y + 815730721)
c
2
, c
5
(y −1)(y
16
− 16y
15
+ ··· − 8y + 1)
· (y
28
− 56y
27
+ ··· − 733262y + 28561)
c
3
, c
7
(y −1)(y
16
+ 8y
15
+ ··· − 10y + 1)(y
28
+ 32y
27
+ ··· + 1516y + 529)
c
4
, c
9
(y −1)(y
16
− 20y
15
+ ··· + 4y + 1)(y
28
− 48y
27
+ ··· − 11662y + 1849)
c
6
(y −1)(y
16
+ y
15
+ ··· − 15y + 1)(y
28
− 3y
27
+ ··· − 13y + 1)
c
8
, c
11
(y −1)(y
16
− 10y
15
+ ··· − 13y + 1)(y
28
− 26y
27
+ ··· − 87y + 1)
c
10
(y −1)(y
16
− 38y
15
+ ··· − 14y + 1)
· (y
28
− 126y
27
+ ··· − 33547178288y + 1132524409)
c
12
(y −1)(y
16
+ 2y
15
+ ··· − 17y + 1)(y
28
− 38y
27
+ ··· − 11231y + 1)
20
