12n
0162
(K12n
0162
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 11 9 4 5 12 6 10 8
Solving Sequence
6,10
11 12
2,5
3 1 4 9 7 8
c
10
c
11
c
5
c
2
c
1
c
4
c
9
c
6
c
8
c
3
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h2u
38
− 4u
37
+ ··· + b − 2, −2u
38
+ 2u
37
+ ··· + a + 3, u
39
− 2u
38
+ ··· + 2u + 1i
I
u
2
= h−u
7
− u
5
− u
3
+ u
2
+ b, −u
6
− u
4
− 2u
2
+ a − 1, u
9
+ u
8
+ 2u
7
+ u
6
+ 3u
5
+ u
4
+ 2u
3
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 48 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
= h2u
38
−4u
37
+· · ·+b−2, −2u
38
+2u
37
+· · ·+a+3, u
39
−2u
38
+· · ·+2u+1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
12
=
u
2
+ 1
−u
2
a
2
=
2u
38
− 2u
37
+ ··· + 3u − 3
−2u
38
+ 4u
37
+ ··· + 6u + 2
a
5
=
−u
u
3
+ u
a
3
=
3u
38
− 3u
37
+ ··· + 3u − 3
−3u
38
+ 6u
37
+ ··· + 9u + 3
a
1
=
−u
20
− 3u
18
− 7u
16
− 10u
14
− 10u
12
− 7u
10
− u
8
+ 2u
6
+ 3u
4
+ u
2
− 1
u
22
+ 4u
20
+ ··· + 2u
4
+ u
2
a
4
=
u
38
− u
37
+ ··· + 3u − 2
−u
38
+ 2u
37
+ ··· + 4u + 1
a
9
=
u
4
+ u
2
+ 1
−u
4
a
7
=
u
9
+ 2u
7
+ 3u
5
+ 2u
3
+ u
−u
9
− u
7
− u
5
+ u
a
8
=
−u
8
− u
6
− u
4
+ 1
u
10
+ 2u
8
+ 3u
6
+ 2u
4
+ u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −4u
38
+ 6u
37
−28u
36
+ 32u
35
−120u
34
+ 122u
33
−369u
32
+ 323u
31
−893u
30
+ 680u
29
−
1762u
28
+ 1139u
27
− 2917u
26
+ 1542u
25
− 4078u
24
+ 1638u
23
− 4860u
22
+ 1250u
21
−
4902u
20
+ 407u
19
−4141u
18
−578u
17
−2860u
16
−1302u
15
−1518u
14
−1484u
13
−540u
12
−
1166u
11
− 62u
10
− 622u
9
+ 52u
8
− 188u
7
+ 8u
6
+ 28u
5
− 32u
4
+ 47u
3
− 17u
2
+ 8u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
39
+ 4u
38
+ ··· + 6u + 1
c
2
, c
4
u
39
− 10u
38
+ ··· − 10u + 1
c
3
, c
7
u
39
+ u
38
+ ··· + 1024u + 512
c
5
, c
10
u
39
− 2u
38
+ ··· + 2u + 1
c
6
u
39
+ 10u
38
+ ··· + 1722u + 193
c
8
u
39
− 2u
38
+ ··· + 2u + 1
c
9
, c
11
u
39
+ 12u
38
+ ··· + 18u − 1
c
12
u
39
+ 8u
38
+ ··· + 1136340u − 591991
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
39
+ 72y
38
+ ··· + 42y −1
c
2
, c
4
y
39
− 4y
38
+ ··· + 6y −1
c
3
, c
7
y
39
+ 57y
38
+ ··· − 3145728y −262144
c
5
, c
10
y
39
+ 12y
38
+ ··· + 18y −1
c
6
y
39
− 20y
38
+ ··· + 882042y −37249
c
8
y
39
− 52y
38
+ ··· + 18y −1
c
9
, c
11
y
39
+ 32y
38
+ ··· + 438y −1
c
12
y
39
− 112y
38
+ ··· + 32115490504994y −350453344081
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.120246 + 1.009060I
a = 0.245784 − 1.091900I
b = 0.063842 + 0.428041I
−2.16299 + 2.79564I −3.22234 − 5.02358I
u = 0.120246 − 1.009060I
a = 0.245784 + 1.091900I
b = 0.063842 − 0.428041I
−2.16299 − 2.79564I −3.22234 + 5.02358I
u = −0.784016 + 0.696729I
a = 1.54879 + 0.28318I
b = −1.30776 + 1.31337I
3.90920 + 2.54093I 3.58364 − 2.83796I
u = −0.784016 − 0.696729I
a = 1.54879 − 0.28318I
b = −1.30776 − 1.31337I
3.90920 − 2.54093I 3.58364 + 2.83796I
u = 0.596432 + 0.887614I
a = −0.515948 − 0.665357I
b = −0.671166 + 1.160530I
0.19918 + 2.33431I 0.66043 − 2.69942I
u = 0.596432 − 0.887614I
a = −0.515948 + 0.665357I
b = −0.671166 − 1.160530I
0.19918 − 2.33431I 0.66043 + 2.69942I
u = −0.293261 + 1.042380I
a = 0.412665 + 0.456169I
b = 0.948855 + 0.872815I
7.61588 + 0.58808I −2.37285 + 1.24200I
u = −0.293261 − 1.042380I
a = 0.412665 − 0.456169I
b = 0.948855 − 0.872815I
7.61588 − 0.58808I −2.37285 − 1.24200I
u = −0.248492 + 1.061350I
a = −1.15816 − 0.97210I
b = −0.264108 + 0.521722I
7.32167 − 7.23317I −2.97850 + 5.46930I
u = −0.248492 − 1.061350I
a = −1.15816 + 0.97210I
b = −0.264108 − 0.521722I
7.32167 + 7.23317I −2.97850 − 5.46930I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.727925 + 0.815935I
a = 1.03158 − 1.88331I
b = −2.28549 + 0.08879I
1.36861 + 0.67210I −0.227831 − 0.290291I
u = 0.727925 − 0.815935I
a = 1.03158 + 1.88331I
b = −2.28549 − 0.08879I
1.36861 − 0.67210I −0.227831 + 0.290291I
u = 0.219912 + 0.863651I
a = −0.840864 − 0.340640I
b = −0.408735 + 0.084317I
−0.71566 + 1.83000I −2.05496 − 5.14676I
u = 0.219912 − 0.863651I
a = −0.840864 + 0.340640I
b = −0.408735 − 0.084317I
−0.71566 − 1.83000I −2.05496 + 5.14676I
u = −0.699877 + 0.873292I
a = −0.469502 + 0.267136I
b = −0.70229 − 1.61147I
−0.05262 − 2.68764I 1.74155 + 3.66223I
u = −0.699877 − 0.873292I
a = −0.469502 − 0.267136I
b = −0.70229 + 1.61147I
−0.05262 + 2.68764I 1.74155 − 3.66223I
u = −0.801951 + 0.791302I
a = −1.14040 − 0.96544I
b = 1.86997 + 0.11290I
5.59974 − 0.03074I 4.36542 + 0.32402I
u = −0.801951 − 0.791302I
a = −1.14040 + 0.96544I
b = 1.86997 − 0.11290I
5.59974 + 0.03074I 4.36542 − 0.32402I
u = 0.864121 + 0.732996I
a = −1.98825 + 1.82338I
b = 3.31808 + 0.25253I
14.7085 − 6.7067I 2.84933 + 2.28339I
u = 0.864121 − 0.732996I
a = −1.98825 − 1.82338I
b = 3.31808 − 0.25253I
14.7085 + 6.7067I 2.84933 − 2.28339I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.060322 + 0.863529I
a = 1.31866 + 0.55049I
b = −0.236777 − 1.257920I
−3.43952 − 0.92375I −7.85826 − 1.02249I
u = −0.060322 − 0.863529I
a = 1.31866 − 0.55049I
b = −0.236777 + 1.257920I
−3.43952 + 0.92375I −7.85826 + 1.02249I
u = 0.863742 + 0.759791I
a = 1.91967 + 0.29557I
b = −1.83974 − 1.49581I
15.2033 + 1.5546I 3.37626 − 1.94136I
u = 0.863742 − 0.759791I
a = 1.91967 − 0.29557I
b = −1.83974 + 1.49581I
15.2033 − 1.5546I 3.37626 + 1.94136I
u = 0.718483 + 0.920822I
a = −1.98095 + 0.84172I
b = 2.64227 + 1.17341I
1.04684 + 4.85277I −1.08190 − 5.10627I
u = 0.718483 − 0.920822I
a = −1.98095 − 0.84172I
b = 2.64227 − 1.17341I
1.04684 − 4.85277I −1.08190 + 5.10627I
u = −0.756351 + 0.955946I
a = 1.29073 + 0.97706I
b = −1.70884 + 0.75010I
5.09426 − 5.82741I 3.18362 + 5.23319I
u = −0.756351 − 0.955946I
a = 1.29073 − 0.97706I
b = −1.70884 − 0.75010I
5.09426 + 5.82741I 3.18362 − 5.23319I
u = −0.711811 + 0.997242I
a = −0.37938 − 1.46724I
b = 2.21221 + 0.85051I
3.00218 − 8.19358I 1.84671 + 8.19526I
u = −0.711811 − 0.997242I
a = −0.37938 + 1.46724I
b = 2.21221 − 0.85051I
3.00218 + 8.19358I 1.84671 − 8.19526I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.777121 + 1.000210I
a = −0.10510 + 1.98160I
b = 1.90838 − 0.99134I
14.4580 + 4.5482I 2.23048 − 2.92809I
u = 0.777121 − 1.000210I
a = −0.10510 − 1.98160I
b = 1.90838 + 0.99134I
14.4580 − 4.5482I 2.23048 + 2.92809I
u = 0.764056 + 1.013960I
a = 2.01030 − 1.76059I
b = −3.73540 − 0.89810I
13.8390 + 12.7651I 1.42561 − 7.10495I
u = 0.764056 − 1.013960I
a = 2.01030 + 1.76059I
b = −3.73540 + 0.89810I
13.8390 − 12.7651I 1.42561 + 7.10495I
u = −0.727545 + 0.029883I
a = 0.918981 + 0.363391I
b = −0.008973 + 1.054810I
10.90740 − 4.01643I 3.18279 + 2.27518I
u = −0.727545 − 0.029883I
a = 0.918981 − 0.363391I
b = −0.008973 − 1.054810I
10.90740 + 4.01643I 3.18279 − 2.27518I
u = 0.541767 + 0.138295I
a = −0.429515 + 0.451386I
b = 0.417940 + 0.509275I
1.42481 + 0.81540I 4.93808 − 2.26255I
u = 0.541767 − 0.138295I
a = −0.429515 − 0.451386I
b = 0.417940 − 0.509275I
1.42481 − 0.81540I 4.93808 + 2.26255I
u = −0.220361
a = −3.37818
b = −0.424531
−1.26360 −9.17460
8
II. I
u
2
= h−u
7
− u
5
− u
3
+ u
2
+ b, −u
6
− u
4
− 2u
2
+ a − 1, u
9
+ u
8
+ 2u
7
+
u
6
+ 3u
5
+ u
4
+ 2u
3
+ u − 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
12
=
u
2
+ 1
−u
2
a
2
=
u
6
+ u
4
+ 2u
2
+ 1
u
7
+ u
5
+ u
3
− u
2
a
5
=
−u
u
3
+ u
a
3
=
u
6
+ u
4
+ 2u
2
− u + 1
u
7
+ u
5
+ 2u
3
− u
2
+ u
a
1
=
u
−u
3
− u
a
4
=
u
6
+ u
4
+ 2u
2
− u + 1
u
7
+ u
5
+ 2u
3
− u
2
+ u
a
9
=
u
4
+ u
2
+ 1
−u
4
a
7
=
−u
8
− u
6
− u
4
+ 1
u
8
+ u
7
+ u
6
+ 2u
5
+ u
4
+ 2u
3
+ 2u − 1
a
8
=
−u
8
− u
6
− u
4
+ 1
u
8
+ u
7
+ u
6
+ 2u
5
+ u
4
+ 2u
3
+ 2u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
7
+ 4u
6
+ 5u
5
+ 5u
4
+ 10u
3
+ 5u
2
+ u − 1
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
9
c
3
, c
7
u
9
c
4
(u + 1)
9
c
5
u
9
− u
8
+ 2u
7
− u
6
+ 3u
5
− u
4
+ 2u
3
+ u + 1
c
6
u
9
+ 5u
8
+ 12u
7
+ 15u
6
+ 9u
5
− u
4
− 4u
3
− 2u
2
+ u + 1
c
8
, c
12
u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1
c
9
u
9
− 3u
8
+ 8u
7
− 13u
6
+ 17u
5
− 17u
4
+ 12u
3
− 6u
2
+ u + 1
c
10
u
9
+ u
8
+ 2u
7
+ u
6
+ 3u
5
+ u
4
+ 2u
3
+ u − 1
c
11
u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
9
c
3
, c
7
y
9
c
5
, c
10
y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1
c
6
y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y −1
c
8
, c
12
y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1
c
9
, c
11
y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.140343 + 0.966856I
a = −0.630598 + 0.707882I
b = 0.392669 − 0.901894I
−3.42837 + 2.09337I −7.72019 − 4.44592I
u = 0.140343 − 0.966856I
a = −0.630598 − 0.707882I
b = 0.392669 + 0.901894I
−3.42837 − 2.09337I −7.72019 + 4.44592I
u = 0.628449 + 0.875112I
a = 0.481040 + 0.507127I
b = 0.79657 − 1.60206I
−1.02799 + 2.45442I −7.83797 − 2.47153I
u = 0.628449 − 0.875112I
a = 0.481040 − 0.507127I
b = 0.79657 + 1.60206I
−1.02799 − 2.45442I −7.83797 + 2.47153I
u = −0.796005 + 0.733148I
a = −0.552775 − 1.001020I
b = 1.094590 − 0.173964I
2.72642 + 1.33617I 1.031098 − 0.174453I
u = −0.796005 − 0.733148I
a = −0.552775 + 1.001020I
b = 1.094590 + 0.173964I
2.72642 − 1.33617I 1.031098 + 0.174453I
u = −0.728966 + 0.986295I
a = 0.896321 + 0.526299I
b = −1.74212 + 0.33916I
1.95319 − 7.08493I −0.87316 + 5.18429I
u = −0.728966 − 0.986295I
a = 0.896321 − 0.526299I
b = −1.74212 − 0.33916I
1.95319 + 7.08493I −0.87316 − 5.18429I
u = 0.512358
a = 1.61202
b = −0.0834351
−0.446489 2.80040
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
39
+ 4u
38
+ ··· + 6u + 1)
c
2
((u − 1)
9
)(u
39
− 10u
38
+ ··· − 10u + 1)
c
3
, c
7
u
9
(u
39
+ u
38
+ ··· + 1024u + 512)
c
4
((u + 1)
9
)(u
39
− 10u
38
+ ··· − 10u + 1)
c
5
(u
9
− u
8
+ ··· + u + 1)(u
39
− 2u
38
+ ··· + 2u + 1)
c
6
(u
9
+ 5u
8
+ 12u
7
+ 15u
6
+ 9u
5
− u
4
− 4u
3
− 2u
2
+ u + 1)
· (u
39
+ 10u
38
+ ··· + 1722u + 193)
c
8
(u
9
+ u
8
+ ··· − u − 1)(u
39
− 2u
38
+ ··· + 2u + 1)
c
9
(u
9
− 3u
8
+ 8u
7
− 13u
6
+ 17u
5
− 17u
4
+ 12u
3
− 6u
2
+ u + 1)
· (u
39
+ 12u
38
+ ··· + 18u − 1)
c
10
(u
9
+ u
8
+ ··· + u − 1)(u
39
− 2u
38
+ ··· + 2u + 1)
c
11
(u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1)
· (u
39
+ 12u
38
+ ··· + 18u − 1)
c
12
(u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1)
· (u
39
+ 8u
38
+ ··· + 1136340u − 591991)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
9
)(y
39
+ 72y
38
+ ··· + 42y −1)
c
2
, c
4
((y −1)
9
)(y
39
− 4y
38
+ ··· + 6y −1)
c
3
, c
7
y
9
(y
39
+ 57y
38
+ ··· − 3145728y −262144)
c
5
, c
10
(y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1)
· (y
39
+ 12y
38
+ ··· + 18y −1)
c
6
(y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y −1)
· (y
39
− 20y
38
+ ··· + 882042y −37249)
c
8
(y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1)
· (y
39
− 52y
38
+ ··· + 18y −1)
c
9
, c
11
(y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1)
· (y
39
+ 32y
38
+ ··· + 438y −1)
c
12
(y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1)
· (y
39
− 112y
38
+ ··· + 32115490504994y −350453344081)
14
