12n
0004
(K12n
0004
)
A knot diagram
1
Linearized knot diagam
3 5 6 8 2 9 10 4 12 8 9 11
Solving Sequence
5,8
4
9,11
12 10 7 6 3 2 1
c
4
c
8
c
11
c
10
c
7
c
6
c
3
c
2
c
1
c
5
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h2.68313 × 10
45
u
49
+ 5.65169 × 10
45
u
48
+ ··· + 2.22303 × 10
45
b + 2.08265 × 10
45
,
5.37170 × 10
43
u
49
+ 1.30102 × 10
44
u
48
+ ··· + 2.22303 × 10
45
a − 4.96236 × 10
44
, u
50
+ 2u
49
+ ··· + u + 1i
I
u
2
= hu
3
+ b + u + 1, a, u
4
+ u
2
+ u + 1i
I
u
3
= hu
5
− u
4
+ 2u
3
− 2u
2
+ b + 2u − 2, a, u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1i
* 3 irreducible components of dim
C
= 0, with total 60 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
= h2.68 × 10
45
u
49
+ 5.65 × 10
45
u
48
+ · · · + 2.22 × 10
45
b + 2.08 × 10
45
, 5.37 ×
10
43
u
49
+1.30×10
44
u
48
+· · ·+2.22×10
45
a−4.96×10
44
, u
50
+2u
49
+· · ·+u+1i
(i) Arc colorings
a
5
=
1
0
a
8
=
0
u
a
4
=
1
u
2
a
9
=
u
u
3
+ u
a
11
=
−0.0241638u
49
− 0.0585243u
48
+ ··· + 2.46234u + 0.223225
−1.20697u
49
− 2.54233u
48
+ ··· − 1.10388u − 0.936850
a
12
=
u
−1.23912u
49
− 2.70146u
48
+ ··· − 2.60058u − 1.17027
a
10
=
−0.0241638u
49
− 0.0585243u
48
+ ··· + 2.46234u + 0.223225
−1.15066u
49
− 2.32468u
48
+ ··· − 1.06952u − 0.926653
a
7
=
−0.172888u
49
− 0.633762u
48
+ ··· − 0.157137u − 0.0708064
−0.00568877u
49
− 0.0730463u
48
+ ··· − 0.998737u + 0.163739
a
6
=
−0.296253u
49
− 0.972434u
48
+ ··· − 0.415283u − 0.214269
−0.0837652u
49
− 0.360352u
48
+ ··· − 1.04157u + 0.112220
a
3
=
−0.0373866u
49
+ 0.100580u
48
+ ··· − 0.326440u + 1.04404
0.373816u
49
+ 0.661193u
48
+ ··· − 0.0581820u − 0.269424
a
2
=
−0.411203u
49
− 0.560613u
48
+ ··· − 0.268258u + 1.31346
0.373816u
49
+ 0.661193u
48
+ ··· − 0.0581820u − 0.269424
a
1
=
−0.114683u
49
− 0.372829u
48
+ ··· + 1.30247u + 0.0534395
0.181570u
49
+ 0.599605u
48
+ ··· + 1.71776u + 0.267709
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4.70229u
49
+ 5.71053u
48
+ ··· − 8.33362u − 6.55580
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
50
+ 22u
49
+ ··· + 5u + 1
c
2
, c
5
u
50
+ 2u
49
+ ··· + 5u + 1
c
3
u
50
− 2u
49
+ ··· − 48u + 36
c
4
, c
8
u
50
− 2u
49
+ ··· − u + 1
c
6
u
50
− 10u
49
+ ··· − 6028015u + 3579401
c
7
, c
10
u
50
+ 5u
49
+ ··· + 5120u + 1024
c
9
, c
11
u
50
− 11u
49
+ ··· − 10u + 1
c
12
u
50
+ 9u
49
+ ··· − 10u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
50
+ 14y
49
+ ··· + 105y + 1
c
2
, c
5
y
50
+ 22y
49
+ ··· + 5y + 1
c
3
y
50
+ 6y
49
+ ··· + 28872y + 1296
c
4
, c
8
y
50
+ 10y
49
+ ··· + 5y + 1
c
6
y
50
+ 74y
49
+ ··· + 905556314476485y + 12812111518801
c
7
, c
10
y
50
− 63y
49
+ ··· − 14155776y + 1048576
c
9
, c
11
y
50
− 9y
49
+ ··· + 10y + 1
c
12
y
50
+ 75y
49
+ ··· + 10y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.585003 + 0.735265I
a = 1.57008 + 0.40175I
b = 0.294139 + 0.670355I
0.18981 + 6.85618I −0.26689 − 9.43060I
u = 0.585003 − 0.735265I
a = 1.57008 − 0.40175I
b = 0.294139 − 0.670355I
0.18981 − 6.85618I −0.26689 + 9.43060I
u = 0.892429 + 0.266725I
a = 0.668641 − 0.033988I
b = 0.763934 + 0.111453I
−0.49506 − 3.19030I 2.46244 + 4.05593I
u = 0.892429 − 0.266725I
a = 0.668641 + 0.033988I
b = 0.763934 − 0.111453I
−0.49506 + 3.19030I 2.46244 − 4.05593I
u = −0.627517 + 0.673413I
a = −1.357940 + 0.261790I
b = −0.510233 + 0.568117I
1.65860 − 2.16501I 3.61128 + 3.98050I
u = −0.627517 − 0.673413I
a = −1.357940 − 0.261790I
b = −0.510233 − 0.568117I
1.65860 + 2.16501I 3.61128 − 3.98050I
u = −0.412013 + 0.747742I
a = −0.058407 + 0.811246I
b = 0.131377 + 0.894939I
1.19055 − 1.89480I 3.58493 + 4.65187I
u = −0.412013 − 0.747742I
a = −0.058407 − 0.811246I
b = 0.131377 − 0.894939I
1.19055 + 1.89480I 3.58493 − 4.65187I
u = −0.713195 + 0.463752I
a = −0.882675 + 0.101195I
b = −0.711974 + 0.297517I
1.34929 − 0.89664I 5.51965 + 2.35436I
u = −0.713195 − 0.463752I
a = −0.882675 − 0.101195I
b = −0.711974 − 0.297517I
1.34929 + 0.89664I 5.51965 − 2.35436I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.442833 + 1.118810I
a = 0.228660 + 0.614406I
b = 0.496365 + 0.310678I
−0.96543 − 3.58766I 2.90819 + 4.89226I
u = −0.442833 − 1.118810I
a = 0.228660 − 0.614406I
b = 0.496365 − 0.310678I
−0.96543 + 3.58766I 2.90819 − 4.89226I
u = 0.415615 + 0.648154I
a = 1.13644 + 0.88310I
b = 0.232356 + 0.006577I
−2.15443 + 1.20962I −5.26784 − 4.21990I
u = 0.415615 − 0.648154I
a = 1.13644 − 0.88310I
b = 0.232356 − 0.006577I
−2.15443 − 1.20962I −5.26784 + 4.21990I
u = −0.239224 + 0.729553I
a = −0.92587 + 1.52769I
b = 0.184105 − 0.655650I
−3.56813 − 4.51753I −7.12654 + 7.85182I
u = −0.239224 − 0.729553I
a = −0.92587 − 1.52769I
b = 0.184105 + 0.655650I
−3.56813 + 4.51753I −7.12654 − 7.85182I
u = 0.474690 + 0.586628I
a = 0.317436 + 0.867818I
b = 0.336934 + 1.108570I
0.36504 − 2.86959I 1.52034 + 1.42682I
u = 0.474690 − 0.586628I
a = 0.317436 − 0.867818I
b = 0.336934 − 1.108570I
0.36504 + 2.86959I 1.52034 − 1.42682I
u = 0.276652 + 1.215690I
a = −0.130051 + 0.523865I
b = −0.256894 + 0.216714I
−5.21381 + 0.38052I 0
u = 0.276652 − 1.215690I
a = −0.130051 − 0.523865I
b = −0.256894 − 0.216714I
−5.21381 − 0.38052I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.932827 + 0.874426I
a = −1.22918 − 1.02414I
b = −1.88245 + 0.61714I
8.53540 − 6.83590I 0
u = −0.932827 − 0.874426I
a = −1.22918 + 1.02414I
b = −1.88245 − 0.61714I
8.53540 + 6.83590I 0
u = −0.084592 + 0.713079I
a = −0.33510 + 1.70480I
b = 0.060912 − 1.093010I
−4.25283 + 1.34403I −9.08170 − 1.74638I
u = −0.084592 − 0.713079I
a = −0.33510 − 1.70480I
b = 0.060912 + 1.093010I
−4.25283 − 1.34403I −9.08170 + 1.74638I
u = −0.986608 + 0.831816I
a = −0.974415 − 0.890813I
b = −1.69006 + 0.29018I
4.09805 + 0.13420I 0
u = −0.986608 − 0.831816I
a = −0.974415 + 0.890813I
b = −1.69006 − 0.29018I
4.09805 − 0.13420I 0
u = 0.953965 + 0.875078I
a = 1.13306 − 1.04945I
b = 1.91375 + 0.48552I
10.16820 + 1.33885I 0
u = 0.953965 − 0.875078I
a = 1.13306 + 1.04945I
b = 1.91375 − 0.48552I
10.16820 − 1.33885I 0
u = 0.503491 + 1.215080I
a = −0.292017 + 0.535955I
b = −0.543268 + 0.125989I
−3.59441 + 8.42989I 0
u = 0.503491 − 1.215080I
a = −0.292017 − 0.535955I
b = −0.543268 − 0.125989I
−3.59441 − 8.42989I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.864809 + 0.998500I
a = 1.021580 + 0.934002I
b = 2.26238 + 0.01234I
8.12452 + 0.19479I 0
u = −0.864809 − 0.998500I
a = 1.021580 − 0.934002I
b = 2.26238 − 0.01234I
8.12452 − 0.19479I 0
u = 1.004750 + 0.876459I
a = 0.905947 − 1.070140I
b = 1.91696 + 0.17358I
9.78917 − 2.20715I 0
u = 1.004750 − 0.876459I
a = 0.905947 + 1.070140I
b = 1.91696 − 0.17358I
9.78917 + 2.20715I 0
u = 0.879539 + 1.011550I
a = −1.081270 + 0.875542I
b = −2.29261 − 0.17584I
9.71806 + 5.41424I 0
u = 0.879539 − 1.011550I
a = −1.081270 − 0.875542I
b = −2.29261 + 0.17584I
9.71806 − 5.41424I 0
u = −1.024250 + 0.878258I
a = −0.822290 − 1.069240I
b = −1.90015 + 0.05959I
7.85339 + 7.67379I 0
u = −1.024250 − 0.878258I
a = −0.822290 + 1.069240I
b = −1.90015 − 0.05959I
7.85339 − 7.67379I 0
u = −0.877514 + 1.052570I
a = 1.041610 + 0.710120I
b = 1.99699 − 0.41591I
3.38860 − 6.97331I 0
u = −0.877514 − 1.052570I
a = 1.041610 − 0.710120I
b = 1.99699 + 0.41591I
3.38860 + 6.97331I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.241768 + 0.574851I
a = 0.656765 + 1.098310I
b = 0.388520 − 0.721943I
−1.91283 + 0.92101I −3.25247 − 0.86020I
u = 0.241768 − 0.574851I
a = 0.656765 − 1.098310I
b = 0.388520 + 0.721943I
−1.91283 − 0.92101I −3.25247 + 0.86020I
u = 0.907639 + 1.041190I
a = −1.179790 + 0.720010I
b = −2.25110 − 0.59042I
9.24238 + 9.20482I 0
u = 0.907639 − 1.041190I
a = −1.179790 − 0.720010I
b = −2.25110 + 0.59042I
9.24238 − 9.20482I 0
u = −0.916608 + 1.051050I
a = 1.204760 + 0.661896I
b = 2.21127 − 0.72670I
7.2749 − 14.7585I 0
u = −0.916608 − 1.051050I
a = 1.204760 − 0.661896I
b = 2.21127 + 0.72670I
7.2749 + 14.7585I 0
u = 0.418143 + 0.325356I
a = 0.746640 + 0.640935I
b = 2.00114 + 0.03022I
−1.36557 + 1.46875I −6.84223 − 10.34978I
u = 0.418143 − 0.325356I
a = 0.746640 − 0.640935I
b = 2.00114 − 0.03022I
−1.36557 − 1.46875I −6.84223 + 10.34978I
u = −0.431697 + 0.145919I
a = −0.862616 + 0.330550I
b = −3.15240 − 0.05844I
−1.85076 + 2.37111I 9.8016 + 19.1551I
u = −0.431697 − 0.145919I
a = −0.862616 − 0.330550I
b = −3.15240 + 0.05844I
−1.85076 − 2.37111I 9.8016 − 19.1551I
9
II. I
u
2
= hu
3
+ b + u + 1, a, u
4
+ u
2
+ u + 1i
(i) Arc colorings
a
5
=
1
0
a
8
=
0
u
a
4
=
1
u
2
a
9
=
u
u
3
+ u
a
11
=
0
−u
3
− u − 1
a
12
=
−u
−2u
3
− 2u − 1
a
10
=
0
−u
3
− u − 1
a
7
=
0
u
a
6
=
u
3
−u
2
a
3
=
u
3
+ u
2
+ 1
−u
a
2
=
u
3
+ u
2
+ u + 1
−u
a
1
=
−u
−u
3
− u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5u
3
− 3u
2
+ u + 1
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
4
− 2u
3
+ 3u
2
− u + 1
c
2
, c
4
u
4
+ u
2
+ u + 1
c
3
u
4
+ 3u
3
+ 4u
2
+ 3u + 2
c
5
, c
8
u
4
+ u
2
− u + 1
c
7
, c
10
u
4
c
9
(u − 1)
4
c
11
, c
12
(u + 1)
4
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
4
+ 2y
3
+ 7y
2
+ 5y + 1
c
2
, c
4
, c
5
c
8
y
4
+ 2y
3
+ 3y
2
+ y + 1
c
3
y
4
− y
3
+ 2y
2
+ 7y + 4
c
7
, c
10
y
4
c
9
, c
11
, c
12
(y −1)
4
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.547424 + 0.585652I
a = 0
b = −0.851808 − 0.911292I
−0.66484 − 1.39709I 2.57868 + 4.13745I
u = −0.547424 − 0.585652I
a = 0
b = −0.851808 + 0.911292I
−0.66484 + 1.39709I 2.57868 − 4.13745I
u = 0.547424 + 1.120870I
a = 0
b = 0.351808 − 0.720342I
−4.26996 + 7.64338I −5.07868 − 4.56334I
u = 0.547424 − 1.120870I
a = 0
b = 0.351808 + 0.720342I
−4.26996 − 7.64338I −5.07868 + 4.56334I
13
III.
I
u
3
= hu
5
− u
4
+ 2u
3
− 2u
2
+ b + 2u − 2, a, u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
8
=
0
u
a
4
=
1
u
2
a
9
=
u
u
3
+ u
a
11
=
0
−u
5
+ u
4
− 2u
3
+ 2u
2
− 2u + 2
a
12
=
−u
−u
5
+ u
4
− 3u
3
+ 2u
2
− 3u + 2
a
10
=
0
−u
5
+ u
4
− 2u
3
+ 2u
2
− 2u + 2
a
7
=
0
u
a
6
=
u
3
u
5
+ u
3
+ u
a
3
=
−u
5
+ u
4
− 2u
3
+ 2u
2
− 2u + 2
−u
5
− 2u
3
+ u
2
− u + 1
a
2
=
u
4
+ u
2
− u + 1
−u
5
− 2u
3
+ u
2
− u + 1
a
1
=
−u
−u
3
− u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
5
− u
4
+ 8u
3
− u
2
+ 7u − 8
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
6
− 3u
5
+ 4u
4
− 2u
3
+ 1
c
2
, c
4
u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1
c
3
(u
3
− u
2
+ 1)
2
c
5
, c
8
u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1
c
7
, c
10
u
6
c
9
(u − 1)
6
c
11
, c
12
(u + 1)
6
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1
c
2
, c
4
, c
5
c
8
y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1
c
3
(y
3
− y
2
+ 2y −1)
2
c
7
, c
10
y
6
c
9
, c
11
, c
12
(y −1)
6
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.498832 + 1.001300I
a = 0
b = −0.398606 − 0.800120I
−1.91067 − 2.82812I −1.88527 + 2.08748I
u = −0.498832 − 1.001300I
a = 0
b = −0.398606 + 0.800120I
−1.91067 + 2.82812I −1.88527 − 2.08748I
u = 0.284920 + 1.115140I
a = 0
b = 0.215080 − 0.841795I
−6.04826 −10.27439 + 0.99756I
u = 0.284920 − 1.115140I
a = 0
b = 0.215080 + 0.841795I
−6.04826 −10.27439 − 0.99756I
u = 0.713912 + 0.305839I
a = 0
b = 1.183530 − 0.507021I
−1.91067 − 2.82812I −2.34034 + 5.36114I
u = 0.713912 − 0.305839I
a = 0
b = 1.183530 + 0.507021I
−1.91067 + 2.82812I −2.34034 − 5.36114I
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
4
− 2u
3
+ 3u
2
− u + 1)(u
6
− 3u
5
+ 4u
4
− 2u
3
+ 1)
· (u
50
+ 22u
49
+ ··· + 5u + 1)
c
2
(u
4
+ u
2
+ u + 1)(u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1)
· (u
50
+ 2u
49
+ ··· + 5u + 1)
c
3
((u
3
− u
2
+ 1)
2
)(u
4
+ 3u
3
+ ··· + 3u + 2)(u
50
− 2u
49
+ ··· − 48u + 36)
c
4
(u
4
+ u
2
+ u + 1)(u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1)
· (u
50
− 2u
49
+ ··· − u + 1)
c
5
(u
4
+ u
2
− u + 1)(u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1)
· (u
50
+ 2u
49
+ ··· + 5u + 1)
c
6
(u
4
− 2u
3
+ 3u
2
− u + 1)(u
6
− 3u
5
+ 4u
4
− 2u
3
+ 1)
· (u
50
− 10u
49
+ ··· − 6028015u + 3579401)
c
7
, c
10
u
10
(u
50
+ 5u
49
+ ··· + 5120u + 1024)
c
8
(u
4
+ u
2
− u + 1)(u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1)
· (u
50
− 2u
49
+ ··· − u + 1)
c
9
((u − 1)
10
)(u
50
− 11u
49
+ ··· − 10u + 1)
c
11
((u + 1)
10
)(u
50
− 11u
49
+ ··· − 10u + 1)
c
12
((u + 1)
10
)(u
50
+ 9u
49
+ ··· − 10u + 1)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
4
+ 2y
3
+ 7y
2
+ 5y + 1)(y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1)
· (y
50
+ 14y
49
+ ··· + 105y + 1)
c
2
, c
5
(y
4
+ 2y
3
+ 3y
2
+ y + 1)(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
· (y
50
+ 22y
49
+ ··· + 5y + 1)
c
3
(y
3
− y
2
+ 2y −1)
2
(y
4
− y
3
+ 2y
2
+ 7y + 4)
· (y
50
+ 6y
49
+ ··· + 28872y + 1296)
c
4
, c
8
(y
4
+ 2y
3
+ 3y
2
+ y + 1)(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
· (y
50
+ 10y
49
+ ··· + 5y + 1)
c
6
(y
4
+ 2y
3
+ 7y
2
+ 5y + 1)(y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1)
· (y
50
+ 74y
49
+ ··· + 905556314476485y + 12812111518801)
c
7
, c
10
y
10
(y
50
− 63y
49
+ ··· − 1.41558 × 10
7
y + 1048576)
c
9
, c
11
((y −1)
10
)(y
50
− 9y
49
+ ··· + 10y + 1)
c
12
((y −1)
10
)(y
50
+ 75y
49
+ ··· + 10y + 1)
19
