12n
0349
(K12n
0349
)
A knot diagram
1
Linearized knot diagam
3 5 8 10 2 11 3 4 1 7 10 4
Solving Sequence
4,8 9,12
1 10 5 3 2 6 7 11
c
8
c
12
c
9
c
4
c
3
c
2
c
5
c
7
c
11
c
1
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h5.03279 × 10
28
u
21
+ 1.41088 × 10
28
u
20
+ ··· + 3.11126 × 10
30
b − 1.47947 × 10
30
,
2.83952 × 10
29
u
21
+ 2.39177 × 10
30
u
20
+ ··· + 5.38247 × 10
32
a − 1.28920 × 10
33
,
u
22
− u
21
+ ··· + 570u − 173i
I
u
2
= h4u
15
+ u
14
+ ··· + b − 5, 5u
15
+ 2u
14
+ ··· + a − 5,
u
16
− 7u
14
− u
13
+ 22u
12
+ 6u
11
− 43u
10
− 15u
9
+ 58u
8
+ 19u
7
− 52u
6
− 13u
5
+ 29u
4
+ 5u
3
− 8u
2
− u + 1i
* 2 irreducible components of dim
C
= 0, with total 38 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
=
h5.03×10
28
u
21
+1.41×10
28
u
20
+· · ·+3.11×10
30
b−1.48×10
30
, 2.84×10
29
u
21
+
2.39 × 10
30
u
20
+ · · · + 5.38 × 10
32
a − 1.29 × 10
33
, u
22
− u
21
+ · · · + 570u − 173i
(i) Arc colorings
a
4
=
0
u
a
8
=
1
0
a
9
=
1
−u
2
a
12
=
−0.000527549u
21
− 0.00444362u
20
+ ··· − 2.94596u + 2.39519
−0.0161761u
21
− 0.00453474u
20
+ ··· + 4.84486u + 0.475523
a
1
=
−0.000527549u
21
− 0.00444362u
20
+ ··· − 2.94596u + 2.39519
−0.0112823u
21
− 0.00307489u
20
+ ··· + 2.10256u + 1.33554
a
10
=
−0.00747297u
21
− 0.00222999u
20
+ ··· + 1.32872u + 0.919896
−0.0287409u
21
+ 0.00951872u
20
+ ··· + 15.0584u − 4.13187
a
5
=
−0.0176760u
21
+ 0.00338113u
20
+ ··· + 7.60759u − 1.52246
−0.00312229u
21
+ 0.0168476u
20
+ ··· + 9.58458u − 3.73734
a
3
=
−u
u
a
2
=
0.0117801u
21
− 0.0136133u
20
+ ··· − 11.9200u + 5.73898
−0.0235899u
21
+ 0.00609484u
20
+ ··· + 11.0766u − 2.00826
a
6
=
0.0114740u
21
− 0.00633450u
20
+ ··· − 6.51280u + 1.90552
−0.0333527u
21
+ 0.0105246u
20
+ ··· + 16.7996u − 4.51821
a
7
=
−u
2
+ 1
u
2
a
11
=
−0.0267510u
21
− 0.00298230u
20
+ ··· + 5.83060u + 2.41175
−0.0193326u
21
+ 0.0165589u
20
+ ··· + 15.3927u − 4.75155
(ii) Obstruction class = −1
(iii) Cusp Shapes = 0.0344880u
21
− 0.0647662u
20
+ ··· − 54.7216u + 34.1950
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
22
+ 3u
21
+ ··· + 129u + 121
c
2
, c
5
u
22
+ 3u
21
+ ··· + 61u − 11
c
3
, c
7
, c
8
u
22
+ u
21
+ ··· − 570u − 173
c
4
u
22
+ 12u
21
+ ··· − 5056u − 1856
c
6
, c
10
u
22
− u
21
+ ··· + 387u − 119
c
9
u
22
− 3u
21
+ ··· − 17u + 1
c
11
u
22
+ u
21
+ ··· − 50523u + 14161
c
12
u
22
+ u
21
+ ··· + 8u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
22
+ 47y
21
+ ··· + 1408497y + 14641
c
2
, c
5
y
22
+ 3y
21
+ ··· + 129y + 121
c
3
, c
7
, c
8
y
22
− 33y
21
+ ··· − 222830y + 29929
c
4
y
22
− 48y
21
+ ··· − 17842176y + 3444736
c
6
, c
10
y
22
+ y
21
+ ··· − 50523y + 14161
c
9
y
22
+ 17y
21
+ ··· − 89y + 1
c
11
y
22
+ 57y
21
+ ··· − 20509146359y + 200533921
c
12
y
22
+ 49y
21
+ ··· − 52y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.888889 + 0.621508I
a = −0.785790 − 0.787936I
b = 1.41234 − 0.32273I
−2.09331 + 2.22663I 10.35334 − 5.55870I
u = 0.888889 − 0.621508I
a = −0.785790 + 0.787936I
b = 1.41234 + 0.32273I
−2.09331 − 2.22663I 10.35334 + 5.55870I
u = 1.125980 + 0.181359I
a = 0.194049 + 0.934438I
b = 0.533838 + 1.294670I
2.80761 − 4.32610I 7.93255 + 4.44451I
u = 1.125980 − 0.181359I
a = 0.194049 − 0.934438I
b = 0.533838 − 1.294670I
2.80761 + 4.32610I 7.93255 − 4.44451I
u = 0.468073 + 0.568730I
a = 0.033725 − 0.194825I
b = 0.67213 + 1.29726I
0.96204 − 3.13844I 5.33676 + 0.34451I
u = 0.468073 − 0.568730I
a = 0.033725 + 0.194825I
b = 0.67213 − 1.29726I
0.96204 + 3.13844I 5.33676 − 0.34451I
u = −1.219380 + 0.529046I
a = −1.29654 + 1.66639I
b = 0.021894 + 0.717525I
−9.56729 − 1.76902I 9.41797 + 0.67359I
u = −1.219380 − 0.529046I
a = −1.29654 − 1.66639I
b = 0.021894 − 0.717525I
−9.56729 + 1.76902I 9.41797 − 0.67359I
u = 1.138980 + 0.705814I
a = 0.762830 + 0.903956I
b = −1.373520 + 0.140422I
−5.89358 + 2.47710I 2.52368 − 2.07081I
u = 1.138980 − 0.705814I
a = 0.762830 − 0.903956I
b = −1.373520 − 0.140422I
−5.89358 − 2.47710I 2.52368 + 2.07081I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.390919 + 0.487203I
a = −0.142272 − 0.861639I
b = 0.433308 − 0.046954I
−1.67513 + 1.49906I 2.16861 − 5.00550I
u = 0.390919 − 0.487203I
a = −0.142272 + 0.861639I
b = 0.433308 + 0.046954I
−1.67513 − 1.49906I 2.16861 + 5.00550I
u = −0.486914
a = 0.684920
b = −0.232651
0.667389 15.1430
u = −1.51784 + 0.10872I
a = 0.152442 + 0.390876I
b = −0.0746395 + 0.0560374I
4.60852 − 3.54663I 3.76213 + 0.30741I
u = −1.51784 − 0.10872I
a = 0.152442 − 0.390876I
b = −0.0746395 − 0.0560374I
4.60852 + 3.54663I 3.76213 − 0.30741I
u = −1.16828 + 1.29949I
a = 0.771799 − 0.429092I
b = −0.50715 − 2.09278I
5.62161 − 3.04840I 8.24444 + 1.87699I
u = −1.16828 − 1.29949I
a = 0.771799 + 0.429092I
b = −0.50715 + 2.09278I
5.62161 + 3.04840I 8.24444 − 1.87699I
u = 1.88865 + 0.65161I
a = 0.557638 + 0.987237I
b = −1.32425 + 1.66024I
14.5063 + 11.3964I 7.17418 − 4.33997I
u = 1.88865 − 0.65161I
a = 0.557638 − 0.987237I
b = −1.32425 − 1.66024I
14.5063 − 11.3964I 7.17418 + 4.33997I
u = 2.18883
a = −0.226088
b = −2.17979
10.8028 8.45190
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −2.34694 + 0.29798I
a = 0.184555 − 0.987184I
b = −0.08772 − 2.56787I
16.2419 − 0.6601I 8.28889 + 0.I
u = −2.34694 − 0.29798I
a = 0.184555 + 0.987184I
b = −0.08772 + 2.56787I
16.2419 + 0.6601I 8.28889 + 0.I
7
II.
I
u
2
= h4u
15
+u
14
+· · ·+b −5, 5u
15
+2u
14
+· · ·+a −5, u
16
−7u
14
+· · ·−u +1i
(i) Arc colorings
a
4
=
0
u
a
8
=
1
0
a
9
=
1
−u
2
a
12
=
−5u
15
− 2u
14
+ ··· + 10u + 5
−4u
15
− u
14
+ ··· + 9u + 5
a
1
=
−5u
15
− 2u
14
+ ··· + 10u + 5
−2u
15
+ 14u
13
+ ··· + 6u + 3
a
10
=
−u
14
+ 6u
12
+ ··· − 8u
2
− u
−4u
15
− 2u
14
+ ··· + 7u + 6
a
5
=
2u
15
− 13u
13
+ ··· + 3u
2
− 2u
2u
15
+ u
14
+ ··· − 2u − 5
a
3
=
−u
u
a
2
=
−3u
15
− u
14
+ ··· + 5u + 3
−4u
15
− u
14
+ ··· + 11u + 5
a
6
=
5u
15
+ u
14
+ ··· − 12u − 5
4u
15
+ 2u
14
+ ··· − 10u − 6
a
7
=
−u
2
+ 1
u
2
a
11
=
−2u
15
− 2u
14
+ ··· + 3u + 2
−5u
15
− 2u
14
+ ··· + 9u + 7
(ii) Obstruction class = 1
(iii) Cusp Shapes = −14u
15
− 7u
14
+ 93u
13
+ 59u
12
− 272u
11
− 213u
10
+ 485u
9
+
439u
8
− 583u
7
− 544u
6
+ 451u
5
+ 409u
4
− 202u
3
− 183u
2
+ 22u + 36
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
16
− 14u
15
+ ··· − 18u + 1
c
2
u
16
+ 2u
15
+ ··· + 2u + 1
c
3
u
16
− 7u
14
+ ··· + u + 1
c
4
u
16
+ 2u
14
+ ··· − u + 1
c
5
u
16
− 2u
15
+ ··· − 2u + 1
c
6
u
16
+ 8u
14
+ ··· + 2u + 1
c
7
, c
8
u
16
− 7u
14
+ ··· − u + 1
c
9
u
16
− 4u
15
+ ··· + 4u
3
+ 1
c
10
u
16
+ 8u
14
+ ··· − 2u + 1
c
11
u
16
+ 16u
15
+ ··· + 18u + 1
c
12
u
16
+ 14u
14
+ ··· + 7u + 7
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
16
− 10y
15
+ ··· − 42y + 1
c
2
, c
5
y
16
+ 14y
15
+ ··· + 18y + 1
c
3
, c
7
, c
8
y
16
− 14y
15
+ ··· − 17y + 1
c
4
y
16
+ 4y
15
+ ··· + y + 1
c
6
, c
10
y
16
+ 16y
15
+ ··· + 18y + 1
c
9
y
16
− 4y
15
+ ··· − 12y
2
+ 1
c
11
y
16
− 16y
15
+ ··· − 18y + 1
c
12
y
16
+ 28y
15
+ ··· + 1505y + 49
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.895121 + 0.512839I
a = −0.764453 − 0.786216I
b = 1.48200 − 0.38704I
−2.57741 + 2.02646I −4.32270 − 0.49641I
u = 0.895121 − 0.512839I
a = −0.764453 + 0.786216I
b = 1.48200 + 0.38704I
−2.57741 − 2.02646I −4.32270 + 0.49641I
u = −0.991446 + 0.300154I
a = 0.72110 − 1.52416I
b = −1.200410 − 0.280363I
−5.52393 − 1.17654I 4.37675 − 1.28594I
u = −0.991446 − 0.300154I
a = 0.72110 + 1.52416I
b = −1.200410 + 0.280363I
−5.52393 + 1.17654I 4.37675 + 1.28594I
u = 1.042540 + 0.498206I
a = 1.57083 + 2.03165I
b = −0.562512 + 0.129135I
−10.21970 + 1.92477I −2.98411 − 3.89934I
u = 1.042540 − 0.498206I
a = 1.57083 − 2.03165I
b = −0.562512 − 0.129135I
−10.21970 − 1.92477I −2.98411 + 3.89934I
u = −0.947353 + 0.893061I
a = −0.995021 + 0.545877I
b = 1.157010 + 0.171797I
−5.04353 − 3.27906I 6.96610 + 5.21245I
u = −0.947353 − 0.893061I
a = −0.995021 − 0.545877I
b = 1.157010 − 0.171797I
−5.04353 + 3.27906I 6.96610 − 5.21245I
u = −0.535533 + 0.224627I
a = −0.50673 − 1.96194I
b = 0.091098 − 1.093350I
0.313495 − 1.189950I 7.79460 + 1.38246I
u = −0.535533 − 0.224627I
a = −0.50673 + 1.96194I
b = 0.091098 + 1.093350I
0.313495 + 1.189950I 7.79460 − 1.38246I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.42936 + 0.19543I
a = 0.667578 + 0.371646I
b = −0.005180 + 0.635975I
3.92703 − 0.68720I 9.16301 − 0.40551I
u = −1.42936 − 0.19543I
a = 0.667578 − 0.371646I
b = −0.005180 − 0.635975I
3.92703 + 0.68720I 9.16301 + 0.40551I
u = 1.46404 + 0.03561I
a = 0.004983 − 0.187762I
b = 0.403326 − 0.712059I
5.03010 + 4.12297I 11.6031 − 8.8115I
u = 1.46404 − 0.03561I
a = 0.004983 + 0.187762I
b = 0.403326 + 0.712059I
5.03010 − 4.12297I 11.6031 + 8.8115I
u = 0.501986 + 0.071058I
a = −0.69829 − 1.53708I
b = 0.634668 − 1.218490I
0.93447 + 4.27246I 4.40321 − 6.78367I
u = 0.501986 − 0.071058I
a = −0.69829 + 1.53708I
b = 0.634668 + 1.218490I
0.93447 − 4.27246I 4.40321 + 6.78367I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
16
− 14u
15
+ ··· − 18u + 1)(u
22
+ 3u
21
+ ··· + 129u + 121)
c
2
(u
16
+ 2u
15
+ ··· + 2u + 1)(u
22
+ 3u
21
+ ··· + 61u − 11)
c
3
(u
16
− 7u
14
+ ··· + u + 1)(u
22
+ u
21
+ ··· − 570u − 173)
c
4
(u
16
+ 2u
14
+ ··· − u + 1)(u
22
+ 12u
21
+ ··· − 5056u − 1856)
c
5
(u
16
− 2u
15
+ ··· − 2u + 1)(u
22
+ 3u
21
+ ··· + 61u − 11)
c
6
(u
16
+ 8u
14
+ ··· + 2u + 1)(u
22
− u
21
+ ··· + 387u − 119)
c
7
, c
8
(u
16
− 7u
14
+ ··· − u + 1)(u
22
+ u
21
+ ··· − 570u − 173)
c
9
(u
16
− 4u
15
+ ··· + 4u
3
+ 1)(u
22
− 3u
21
+ ··· − 17u + 1)
c
10
(u
16
+ 8u
14
+ ··· − 2u + 1)(u
22
− u
21
+ ··· + 387u − 119)
c
11
(u
16
+ 16u
15
+ ··· + 18u + 1)(u
22
+ u
21
+ ··· − 50523u + 14161)
c
12
(u
16
+ 14u
14
+ ··· + 7u + 7)(u
22
+ u
21
+ ··· + 8u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
16
− 10y
15
+ ··· − 42y + 1)(y
22
+ 47y
21
+ ··· + 1408497y + 14641)
c
2
, c
5
(y
16
+ 14y
15
+ ··· + 18y + 1)(y
22
+ 3y
21
+ ··· + 129y + 121)
c
3
, c
7
, c
8
(y
16
− 14y
15
+ ··· − 17y + 1)(y
22
− 33y
21
+ ··· − 222830y + 29929)
c
4
(y
16
+ 4y
15
+ ··· + y + 1)(y
22
− 48y
21
+ ··· − 1.78422 × 10
7
y + 3444736)
c
6
, c
10
(y
16
+ 16y
15
+ ··· + 18y + 1)(y
22
+ y
21
+ ··· − 50523y + 14161)
c
9
(y
16
− 4y
15
+ ··· − 12y
2
+ 1)(y
22
+ 17y
21
+ ··· − 89y + 1)
c
11
(y
16
− 16y
15
+ ··· − 18y + 1)
· (y
22
+ 57y
21
+ ··· − 20509146359y + 200533921)
c
12
(y
16
+ 28y
15
+ ··· + 1505y + 49)(y
22
+ 49y
21
+ ··· − 52y + 1)
14
