12a
0772
(K12a
0772
)
A knot diagram
1
Linearized knot diagam
3 8 10 12 11 9 2 6 7 1 5 4
Solving Sequence
4,12
5 1 11
6,9
7 8 10 3 2
c
4
c
12
c
11
c
5
c
6
c
8
c
10
c
3
c
2
c
1
, c
7
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
67
+ 2u
66
+ ··· + b + 1, u
67
+ 2u
66
+ ··· + a + 2, u
68
+ 2u
67
+ ··· + 6u + 1i
I
u
2
= hb, u
3
+ a + 2u, u
4
− u
3
+ 3u
2
− 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 72 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
= hu
67
+2u
66
+· · ·+b +1, u
67
+2u
66
+· · ·+a +2, u
68
+2u
67
+· · ·+6u +1i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
u
a
5
=
1
−u
2
a
1
=
u
u
a
11
=
−u
u
3
+ u
a
6
=
u
2
+ 1
−u
4
− 2u
2
a
9
=
−u
67
− 2u
66
+ ··· − 6u − 2
−u
67
− 2u
66
+ ··· − 4u − 1
a
7
=
u
67
+ u
66
+ ··· + 2u + 2
−u
41
− 23u
39
+ ··· − 2u
2
− u
a
8
=
−u
67
− 3u
66
+ ··· − 8u − 3
−2u
67
− 4u
66
+ ··· − 10u − 2
a
10
=
u
5
+ 2u
3
− u
u
5
+ 3u
3
+ u
a
3
=
−u
10
− 5u
8
− 6u
6
+ u
4
+ u
2
+ 1
−u
10
− 6u
8
− 11u
6
− 6u
4
− u
2
a
2
=
u
19
+ 10u
17
+ 38u
15
+ 66u
13
+ 47u
11
+ 4u
9
− 8u
7
− 10u
5
− 3u
3
u
19
+ 11u
17
+ 48u
15
+ 105u
13
+ 121u
11
+ 75u
9
+ 30u
7
+ 8u
5
+ u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = u
67
+ 2u
66
+ ··· − u − 9
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
68
+ 27u
67
+ ··· + 1344u + 256
c
2
, c
7
u
68
+ u
67
+ ··· − 40u − 16
c
3
u
68
+ 2u
67
+ ··· + 5322u + 1049
c
4
, c
5
, c
11
c
12
u
68
+ 2u
67
+ ··· + 6u + 1
c
6
, c
8
, c
9
u
68
− 5u
67
+ ··· + 6u − 1
c
10
u
68
− 18u
67
+ ··· − 1008u + 49
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
68
+ 21y
67
+ ··· − 1232896y + 65536
c
2
, c
7
y
68
− 27y
67
+ ··· − 1344y + 256
c
3
y
68
− 18y
67
+ ··· − 32379118y + 1100401
c
4
, c
5
, c
11
c
12
y
68
+ 78y
67
+ ··· − 6y + 1
c
6
, c
8
, c
9
y
68
− 59y
67
+ ··· − 12y + 1
c
10
y
68
− 6y
67
+ ··· − 75166y + 2401
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.197347 + 0.873475I
a = −1.89430 + 0.94588I
b = −1.68578 − 0.87347I
−6.76573 + 5.38396I −12.62780 + 0.I
u = −0.197347 − 0.873475I
a = −1.89430 − 0.94588I
b = −1.68578 + 0.87347I
−6.76573 − 5.38396I −12.62780 + 0.I
u = −0.400919 + 0.776860I
a = −1.74029 + 2.14685I
b = −2.54116 − 0.78686I
−10.07420 − 3.27035I −14.2388 + 4.5595I
u = −0.400919 − 0.776860I
a = −1.74029 − 2.14685I
b = −2.54116 + 0.78686I
−10.07420 + 3.27035I −14.2388 − 4.5595I
u = −0.516725 + 0.700827I
a = −0.69973 + 2.63609I
b = −2.80366 − 0.05209I
−4.70584 − 11.85590I −9.20800 + 9.67141I
u = −0.516725 − 0.700827I
a = −0.69973 − 2.63609I
b = −2.80366 + 0.05209I
−4.70584 + 11.85590I −9.20800 − 9.67141I
u = −0.495876 + 0.679831I
a = 0.682404 − 0.604436I
b = 1.120320 + 0.179470I
0.39345 − 7.62900I −5.28637 + 9.02700I
u = −0.495876 − 0.679831I
a = 0.682404 + 0.604436I
b = 1.120320 − 0.179470I
0.39345 + 7.62900I −5.28637 − 9.02700I
u = 0.474084 + 0.680555I
a = 0.96634 + 3.09061I
b = 3.15400 − 0.19444I
−2.72085 + 5.66166I −8.20435 − 6.61637I
u = 0.474084 − 0.680555I
a = 0.96634 − 3.09061I
b = 3.15400 + 0.19444I
−2.72085 − 5.66166I −8.20435 + 6.61637I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.450615 + 0.661293I
a = 0.423151 − 0.930817I
b = 0.078954 + 0.208633I
−1.99409 − 3.06832I −9.13804 + 6.00596I
u = −0.450615 − 0.661293I
a = 0.423151 + 0.930817I
b = 0.078954 − 0.208633I
−1.99409 + 3.06832I −9.13804 − 6.00596I
u = 0.481088 + 0.620510I
a = −0.751512 − 0.690976I
b = −1.050000 + 0.375121I
1.64508 + 2.45629I −1.94498 − 3.97918I
u = 0.481088 − 0.620510I
a = −0.751512 + 0.690976I
b = −1.050000 − 0.375121I
1.64508 − 2.45629I −1.94498 + 3.97918I
u = −0.170138 + 0.764448I
a = 0.198071 − 0.774648I
b = 0.091517 + 0.187313I
−1.56053 + 1.82446I −9.18170 − 2.54653I
u = −0.170138 − 0.764448I
a = 0.198071 + 0.774648I
b = 0.091517 − 0.187313I
−1.56053 − 1.82446I −9.18170 + 2.54653I
u = 0.532445 + 0.564325I
a = −0.388234 − 1.011420I
b = 0.050264 + 0.201909I
−2.03493 − 0.49893I −8.07852 − 0.03154I
u = 0.532445 − 0.564325I
a = −0.388234 + 1.011420I
b = 0.050264 − 0.201909I
−2.03493 + 0.49893I −8.07852 + 0.03154I
u = 0.243977 + 0.727217I
a = 2.80371 + 1.23958I
b = 1.88635 − 1.53609I
−4.18607 − 0.03385I −11.57789 − 1.81848I
u = 0.243977 − 0.727217I
a = 2.80371 − 1.23958I
b = 1.88635 + 1.53609I
−4.18607 + 0.03385I −11.57789 + 1.81848I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.311198 + 0.686872I
a = 0.980284 − 0.595708I
b = 0.701988 − 0.150651I
−2.94398 − 2.12230I −12.85059 + 5.39939I
u = −0.311198 − 0.686872I
a = 0.980284 + 0.595708I
b = 0.701988 + 0.150651I
−2.94398 + 2.12230I −12.85059 − 5.39939I
u = 0.563412 + 0.362435I
a = −0.247629 − 1.102900I
b = 0.277678 + 0.121770I
−1.44819 + 4.26469I −5.83422 − 6.54784I
u = 0.563412 − 0.362435I
a = −0.247629 + 1.102900I
b = 0.277678 − 0.121770I
−1.44819 − 4.26469I −5.83422 + 6.54784I
u = −0.606358 + 0.193080I
a = −1.39324 − 1.03510I
b = 2.26203 + 0.05493I
−3.21668 + 8.03552I −5.98685 − 4.67899I
u = −0.606358 − 0.193080I
a = −1.39324 + 1.03510I
b = 2.26203 − 0.05493I
−3.21668 − 8.03552I −5.98685 + 4.67899I
u = 0.330146 + 0.517285I
a = −0.660081 − 0.413783I
b = −0.217944 + 0.480205I
−0.032689 + 1.215400I −0.57880 − 5.39464I
u = 0.330146 − 0.517285I
a = −0.660081 + 0.413783I
b = −0.217944 − 0.480205I
−0.032689 − 1.215400I −0.57880 + 5.39464I
u = −0.561161 + 0.207787I
a = 0.337609 + 0.749165I
b = −0.707722 + 0.316264I
1.76608 + 3.99561I −1.44534 − 3.66904I
u = −0.561161 − 0.207787I
a = 0.337609 − 0.749165I
b = −0.707722 − 0.316264I
1.76608 − 3.99561I −1.44534 + 3.66904I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.518120 + 0.292169I
a = −0.539528 + 0.663157I
b = 0.669029 + 0.534018I
2.59858 + 1.02255I 0.98197 − 3.38285I
u = 0.518120 − 0.292169I
a = −0.539528 − 0.663157I
b = 0.669029 − 0.534018I
2.59858 − 1.02255I 0.98197 + 3.38285I
u = −0.576388
a = −1.92938
b = 2.19889
−7.75399 −9.67190
u = 0.07796 + 1.43530I
a = −0.162067 + 1.034400I
b = −0.840024 + 0.237993I
−7.09488 + 6.44284I 0
u = 0.07796 − 1.43530I
a = −0.162067 − 1.034400I
b = −0.840024 − 0.237993I
−7.09488 − 6.44284I 0
u = 0.524636 + 0.189192I
a = 1.87813 − 1.35011I
b = −2.32094 − 0.02678I
−1.30607 − 2.19733I −4.11257 + 1.04166I
u = 0.524636 − 0.189192I
a = 1.87813 + 1.35011I
b = −2.32094 + 0.02678I
−1.30607 + 2.19733I −4.11257 − 1.04166I
u = 0.03050 + 1.44934I
a = 0.083282 − 1.286440I
b = −0.069756 − 0.938775I
−2.76626 + 2.66512I 0
u = 0.03050 − 1.44934I
a = 0.083282 + 1.286440I
b = −0.069756 + 0.938775I
−2.76626 − 2.66512I 0
u = −0.01471 + 1.47586I
a = 0.02141 + 1.58635I
b = 0.899720 + 0.769271I
−6.14370 − 1.26145I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.01471 − 1.47586I
a = 0.02141 − 1.58635I
b = 0.899720 − 0.769271I
−6.14370 + 1.26145I 0
u = −0.456710 + 0.227741I
a = 0.035275 − 1.215450I
b = −0.378539 − 0.029211I
−0.733539 − 0.161688I −4.49004 − 0.59711I
u = −0.456710 − 0.227741I
a = 0.035275 + 1.215450I
b = −0.378539 + 0.029211I
−0.733539 + 0.161688I −4.49004 + 0.59711I
u = 0.14519 + 1.54639I
a = 0.253403 + 0.483127I
b = −0.280463 − 0.206557I
−9.06713 + 1.92504I 0
u = 0.14519 − 1.54639I
a = 0.253403 − 0.483127I
b = −0.280463 + 0.206557I
−9.06713 − 1.92504I 0
u = 0.08880 + 1.56603I
a = 0.845921 − 0.221814I
b = 0.418511 − 0.710414I
−7.19692 + 2.68331I 0
u = 0.08880 − 1.56603I
a = 0.845921 + 0.221814I
b = 0.418511 + 0.710414I
−7.19692 − 2.68331I 0
u = 0.13443 + 1.57872I
a = 1.53855 + 0.40023I
b = 1.353120 − 0.222566I
−5.78611 + 4.68931I 0
u = 0.13443 − 1.57872I
a = 1.53855 − 0.40023I
b = 1.353120 + 0.222566I
−5.78611 − 4.68931I 0
u = −0.12937 + 1.59418I
a = −0.328067 + 0.224406I
b = 0.094922 − 0.452255I
−9.65837 − 5.20890I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.12937 − 1.59418I
a = −0.328067 − 0.224406I
b = 0.094922 + 0.452255I
−9.65837 + 5.20890I 0
u = −0.08976 + 1.60098I
a = −1.30221 + 0.61992I
b = −0.936662 + 0.307947I
−10.77820 − 3.62355I 0
u = −0.08976 − 1.60098I
a = −1.30221 − 0.61992I
b = −0.936662 − 0.307947I
−10.77820 + 3.62355I 0
u = −0.14415 + 1.59782I
a = −1.49950 + 0.50175I
b = −1.43108 − 0.02226I
−7.32065 − 10.00200I 0
u = −0.14415 − 1.59782I
a = −1.49950 − 0.50175I
b = −1.43108 + 0.02226I
−7.32065 + 10.00200I 0
u = 0.13700 + 1.59875I
a = −3.57982 − 2.12253I
b = −3.78928 + 0.27230I
−10.45510 + 7.92717I 0
u = 0.13700 − 1.59875I
a = −3.57982 + 2.12253I
b = −3.78928 − 0.27230I
−10.45510 − 7.92717I 0
u = −0.06158 + 1.60581I
a = −0.295079 − 0.181302I
b = −0.063867 − 0.737957I
−9.61163 + 0.87781I 0
u = −0.06158 − 1.60581I
a = −0.295079 + 0.181302I
b = −0.063867 + 0.737957I
−9.61163 − 0.87781I 0
u = 0.07627 + 1.60624I
a = −3.85696 + 1.01925I
b = −2.50796 + 2.49681I
−12.16480 + 1.21091I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.07627 − 1.60624I
a = −3.85696 − 1.01925I
b = −2.50796 − 2.49681I
−12.16480 − 1.21091I 0
u = −0.15212 + 1.60475I
a = 2.90531 − 1.99256I
b = 3.25598 + 0.08173I
−12.5124 − 14.3522I 0
u = −0.15212 − 1.60475I
a = 2.90531 + 1.99256I
b = 3.25598 − 0.08173I
−12.5124 + 14.3522I 0
u = −0.11057 + 1.62515I
a = 3.52667 − 0.54454I
b = 3.06754 + 1.31578I
−18.2927 − 5.1877I 0
u = −0.11057 − 1.62515I
a = 3.52667 + 0.54454I
b = 3.06754 − 1.31578I
−18.2927 + 5.1877I 0
u = −0.05163 + 1.62908I
a = 2.77109 + 0.87233I
b = 1.87630 + 1.80364I
−15.2957 + 4.4711I 0
u = −0.05163 − 1.62908I
a = 2.77109 − 0.87233I
b = 1.87630 − 1.80364I
−15.2957 − 4.4711I 0
u = −0.297877
a = −0.895338
b = −0.465644
−1.07159 −8.30070
11
II. I
u
2
= hb, u
3
+ a + 2u, u
4
− u
3
+ 3u
2
− 2u + 1i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
u
a
5
=
1
−u
2
a
1
=
u
u
a
11
=
−u
u
3
+ u
a
6
=
u
2
+ 1
−u
3
+ u
2
− 2u + 1
a
9
=
−u
3
− 2u
0
a
7
=
−u
3
+ u
2
− 2u + 1
−u
3
+ u
2
− 2u + 1
a
8
=
−u
3
+ u
2
− 2u + 1
−u
3
+ u
2
− 2u + 1
a
10
=
−u
2
− 1
u
3
− u
2
+ 2u − 1
a
3
=
u
u
a
2
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
3
+ 3u
2
− 10u − 4
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
u
4
c
3
u
4
− u
3
+ u
2
+ 1
c
4
, c
5
u
4
− u
3
+ 3u
2
− 2u + 1
c
6
(u − 1)
4
c
8
, c
9
(u + 1)
4
c
10
u
4
+ u
3
+ u
2
+ 1
c
11
, c
12
u
4
+ u
3
+ 3u
2
+ 2u + 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
7
y
4
c
3
, c
10
y
4
+ y
3
+ 3y
2
+ 2y + 1
c
4
, c
5
, c
11
c
12
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
c
6
, c
8
, c
9
(y −1)
4
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.395123 + 0.506844I
a = −0.547424 − 1.120870I
b = 0
−1.43393 + 1.41510I −7.52507 − 4.18840I
u = 0.395123 − 0.506844I
a = −0.547424 + 1.120870I
b = 0
−1.43393 − 1.41510I −7.52507 + 4.18840I
u = 0.10488 + 1.55249I
a = 0.547424 + 0.585652I
b = 0
−8.43568 + 3.16396I −9.97493 − 3.47609I
u = 0.10488 − 1.55249I
a = 0.547424 − 0.585652I
b = 0
−8.43568 − 3.16396I −9.97493 + 3.47609I
15
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
4
(u
68
+ 27u
67
+ ··· + 1344u + 256)
c
2
, c
7
u
4
(u
68
+ u
67
+ ··· − 40u − 16)
c
3
(u
4
− u
3
+ u
2
+ 1)(u
68
+ 2u
67
+ ··· + 5322u + 1049)
c
4
, c
5
(u
4
− u
3
+ 3u
2
− 2u + 1)(u
68
+ 2u
67
+ ··· + 6u + 1)
c
6
((u − 1)
4
)(u
68
− 5u
67
+ ··· + 6u − 1)
c
8
, c
9
((u + 1)
4
)(u
68
− 5u
67
+ ··· + 6u − 1)
c
10
(u
4
+ u
3
+ u
2
+ 1)(u
68
− 18u
67
+ ··· − 1008u + 49)
c
11
, c
12
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
68
+ 2u
67
+ ··· + 6u + 1)
16
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
4
(y
68
+ 21y
67
+ ··· − 1232896y + 65536)
c
2
, c
7
y
4
(y
68
− 27y
67
+ ··· − 1344y + 256)
c
3
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
68
− 18y
67
+ ··· − 3.23791 × 10
7
y + 1100401)
c
4
, c
5
, c
11
c
12
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)(y
68
+ 78y
67
+ ··· − 6y + 1)
c
6
, c
8
, c
9
((y −1)
4
)(y
68
− 59y
67
+ ··· − 12y + 1)
c
10
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
68
− 6y
67
+ ··· − 75166y + 2401)
17
