12a
0078
(K12a
0078
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 9 10 12 4 6 1 7 11
Solving Sequence
7,11
12
4,8
9 1 3 2 10 6 5
c
11
c
7
c
8
c
12
c
3
c
1
c
10
c
6
c
5
c
2
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
81
− u
80
+ ··· + b − 1, u
81
+ u
80
+ ··· + a + 4u, u
82
+ 2u
81
+ ··· + u + 1i
I
u
2
= h−u
5
+ u
3
− u
2
+ b − u, −u
7
+ u
5
− u
4
− u
3
+ a − 1, u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 90 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
= h−u
81
−u
80
+· · · + b− 1, u
81
+u
80
+· · · + a+ 4u, u
82
+2u
81
+· · · + u+ 1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
4
=
−u
81
− u
80
+ ··· + 5u
2
− 4u
u
81
+ u
80
+ ··· + u + 1
a
8
=
u
−u
3
+ u
a
9
=
−u
14
+ 3u
12
− 6u
10
+ 7u
8
− 6u
6
+ 4u
4
− 2u
2
+ 1
−u
14
+ 2u
12
− 3u
10
+ 2u
8
+ u
2
a
1
=
−u
2
+ 1
−u
2
a
3
=
−u
81
− 2u
80
+ ··· − 4u − 1
−u
81
− 2u
80
+ ··· + 3u
2
− 1
a
2
=
u
81
+ u
80
+ ··· + 3u + 1
−u
76
+ 14u
74
+ ··· + 4u
3
− 4u
2
a
10
=
u
4
− u
2
+ 1
u
4
a
6
=
−u
9
+ 2u
7
− 3u
5
+ 2u
3
− u
−u
9
+ u
7
− u
5
+ u
a
5
=
u
19
− 4u
17
+ 10u
15
− 16u
13
+ 19u
11
− 18u
9
+ 14u
7
− 10u
5
+ 5u
3
− 2u
u
19
− 3u
17
+ 6u
15
− 7u
13
+ 5u
11
− 3u
9
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 3u
81
+ 2u
80
+ ··· + 19u + 9
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
82
+ 35u
81
+ ··· + 87u + 1
c
2
, c
4
u
82
− 9u
81
+ ··· − 15u + 1
c
3
, c
8
u
82
+ u
81
+ ··· − 384u + 256
c
5
, c
6
, c
9
u
82
− 2u
81
+ ··· + 231u + 49
c
7
, c
11
u
82
+ 2u
81
+ ··· + u + 1
c
10
, c
12
u
82
− 30u
81
+ ··· + u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
82
+ 33y
81
+ ··· − 3371y + 1
c
2
, c
4
y
82
− 35y
81
+ ··· − 87y + 1
c
3
, c
8
y
82
− 51y
81
+ ··· − 1949696y + 65536
c
5
, c
6
, c
9
y
82
− 86y
81
+ ··· + 637y + 2401
c
7
, c
11
y
82
− 30y
81
+ ··· + y + 1
c
10
, c
12
y
82
+ 46y
81
+ ··· + 25y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.742770 + 0.681590I
a = 0.244580 + 1.360110I
b = −0.293780 + 1.217510I
−3.63738 + 0.78744I 0
u = −0.742770 − 0.681590I
a = 0.244580 − 1.360110I
b = −0.293780 − 1.217510I
−3.63738 − 0.78744I 0
u = −0.983711 + 0.113504I
a = 1.135410 + 0.395982I
b = 1.160820 + 0.257051I
5.61858 − 1.16403I 0
u = −0.983711 − 0.113504I
a = 1.135410 − 0.395982I
b = 1.160820 − 0.257051I
5.61858 + 1.16403I 0
u = −0.972534 + 0.184730I
a = −1.180950 − 0.509477I
b = −1.235990 − 0.259714I
4.40261 − 6.54773I 0
u = −0.972534 − 0.184730I
a = −1.180950 + 0.509477I
b = −1.235990 + 0.259714I
4.40261 + 6.54773I 0
u = −0.555615 + 0.812609I
a = 0.41623 + 2.64855I
b = −2.05957 + 2.46409I
6.23213 + 10.42700I 0
u = −0.555615 − 0.812609I
a = 0.41623 − 2.64855I
b = −2.05957 − 2.46409I
6.23213 − 10.42700I 0
u = 0.673619 + 0.707958I
a = −0.84455 + 1.16088I
b = 0.674960 + 1.170380I
0.191354 − 0.953328I 0
u = 0.673619 − 0.707958I
a = −0.84455 − 1.16088I
b = 0.674960 − 1.170380I
0.191354 + 0.953328I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.779687 + 0.667468I
a = 0.42764 − 2.37824I
b = −2.31222 − 1.69191I
−4.10775 + 1.34673I 0
u = 0.779687 − 0.667468I
a = 0.42764 + 2.37824I
b = −2.31222 + 1.69191I
−4.10775 − 1.34673I 0
u = −0.540301 + 0.807682I
a = −0.36241 − 2.48744I
b = 2.13262 − 2.27043I
8.19364 + 4.30348I 0
u = −0.540301 − 0.807682I
a = −0.36241 + 2.48744I
b = 2.13262 + 2.27043I
8.19364 − 4.30348I 0
u = −0.825888 + 0.624112I
a = −0.354205 − 0.713190I
b = −0.024309 − 0.522887I
−1.70128 − 2.34818I 0
u = −0.825888 − 0.624112I
a = −0.354205 + 0.713190I
b = −0.024309 + 0.522887I
−1.70128 + 2.34818I 0
u = 0.722897 + 0.746924I
a = 1.37403 − 1.51592I
b = −0.61425 − 1.93274I
−1.47202 − 5.71308I 0
u = 0.722897 − 0.746924I
a = 1.37403 + 1.51592I
b = −0.61425 + 1.93274I
−1.47202 + 5.71308I 0
u = 0.540413 + 0.787249I
a = −0.992598 − 0.687172I
b = −0.759757 + 0.395132I
2.76485 − 3.98227I 0
u = 0.540413 − 0.787249I
a = −0.992598 + 0.687172I
b = −0.759757 − 0.395132I
2.76485 + 3.98227I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.936730 + 0.474455I
a = 0.714457 + 0.315117I
b = 1.46416 − 1.29373I
2.91414 − 0.93006I 0
u = 0.936730 − 0.474455I
a = 0.714457 − 0.315117I
b = 1.46416 + 1.29373I
2.91414 + 0.93006I 0
u = 0.608329 + 0.721946I
a = −0.607735 + 0.576661I
b = 0.243718 + 0.648491I
0.322998 − 0.904010I 10.47601 + 0.I
u = 0.608329 − 0.721946I
a = −0.607735 − 0.576661I
b = 0.243718 − 0.648491I
0.322998 + 0.904010I 10.47601 + 0.I
u = −0.532481 + 0.772000I
a = −0.21660 + 2.33589I
b = −2.78716 + 1.94540I
1.25959 + 1.45124I 7.57778 + 0.I
u = −0.532481 − 0.772000I
a = −0.21660 − 2.33589I
b = −2.78716 − 1.94540I
1.25959 − 1.45124I 7.57778 + 0.I
u = −0.489775 + 0.784882I
a = −0.27408 − 1.70120I
b = 2.05575 − 1.44650I
8.50290 − 1.07286I 10.21024 + 0.I
u = −0.489775 − 0.784882I
a = −0.27408 + 1.70120I
b = 2.05575 + 1.44650I
8.50290 + 1.07286I 10.21024 + 0.I
u = 0.512685 + 0.769270I
a = 0.731052 + 0.931298I
b = 0.784174 − 0.176818I
2.94847 + 0.98484I 6.99928 − 2.55231I
u = 0.512685 − 0.769270I
a = 0.731052 − 0.931298I
b = 0.784174 + 0.176818I
2.94847 − 0.98484I 6.99928 + 2.55231I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.08016
a = 0.460931
b = 0.497360
5.68315 0
u = −0.891351 + 0.616034I
a = −0.632357 − 0.558408I
b = −0.062681 − 0.410371I
−1.49964 − 2.51086I 0
u = −0.891351 − 0.616034I
a = −0.632357 + 0.558408I
b = −0.062681 + 0.410371I
−1.49964 + 2.51086I 0
u = −0.469791 + 0.773485I
a = 0.305390 + 1.361410I
b = −1.91125 + 1.15480I
6.75343 − 7.22433I 8.05200 + 5.28342I
u = −0.469791 − 0.773485I
a = 0.305390 − 1.361410I
b = −1.91125 − 1.15480I
6.75343 + 7.22433I 8.05200 − 5.28342I
u = 0.963981 + 0.534698I
a = −1.006010 − 0.207935I
b = −1.58881 + 1.45848I
3.33664 + 4.49587I 0
u = 0.963981 − 0.534698I
a = −1.006010 + 0.207935I
b = −1.58881 − 1.45848I
3.33664 − 4.49587I 0
u = −0.831377 + 0.727164I
a = −0.694470 + 1.035420I
b = −0.983570 + 0.376220I
−3.10733 − 4.66813I 0
u = −0.831377 − 0.727164I
a = −0.694470 − 1.035420I
b = −0.983570 − 0.376220I
−3.10733 + 4.66813I 0
u = 0.869045 + 0.089866I
a = −0.106416 + 0.510526I
b = 0.181534 − 0.847373I
1.05325 + 1.74872I 11.28230 − 4.54859I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.869045 − 0.089866I
a = −0.106416 − 0.510526I
b = 0.181534 + 0.847373I
1.05325 − 1.74872I 11.28230 + 4.54859I
u = 0.914722 + 0.658273I
a = 2.36183 − 0.57009I
b = 1.43535 − 3.27787I
−3.69309 + 3.79631I 0
u = 0.914722 − 0.658273I
a = 2.36183 + 0.57009I
b = 1.43535 + 3.27787I
−3.69309 − 3.79631I 0
u = 1.13174
a = −2.77777
b = −0.171583
6.95973 0
u = −0.879897 + 0.717160I
a = 1.072790 − 0.475215I
b = 0.915535 + 0.286937I
−2.95968 − 0.83056I 0
u = −0.879897 − 0.717160I
a = 1.072790 + 0.475215I
b = 0.915535 − 0.286937I
−2.95968 + 0.83056I 0
u = −1.136470 + 0.010119I
a = 0.107222 + 0.732042I
b = 0.127712 + 0.826867I
8.58956 − 2.55565I 0
u = −1.136470 − 0.010119I
a = 0.107222 − 0.732042I
b = 0.127712 − 0.826867I
8.58956 + 2.55565I 0
u = 1.146400 + 0.030933I
a = −2.25094 + 0.29979I
b = −0.481995 − 0.450536I
12.2980 + 9.0487I 0
u = 1.146400 − 0.030933I
a = −2.25094 − 0.29979I
b = −0.481995 + 0.450536I
12.2980 − 9.0487I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.149210 + 0.018459I
a = 2.36564 − 0.19446I
b = 0.488971 + 0.274865I
14.1727 + 2.8208I 0
u = 1.149210 − 0.018459I
a = 2.36564 + 0.19446I
b = 0.488971 − 0.274865I
14.1727 − 2.8208I 0
u = −0.938888 + 0.666226I
a = 1.169250 + 0.543123I
b = 0.099267 + 0.943058I
−3.04447 − 6.00273I 0
u = −0.938888 − 0.666226I
a = 1.169250 − 0.543123I
b = 0.099267 − 0.943058I
−3.04447 + 6.00273I 0
u = 0.974939 + 0.669400I
a = −1.18700 + 0.86217I
b = −0.31958 + 2.22104I
1.07667 + 6.24979I 0
u = 0.974939 − 0.669400I
a = −1.18700 − 0.86217I
b = −0.31958 − 2.22104I
1.07667 − 6.24979I 0
u = 0.963754 + 0.699919I
a = 1.32913 − 1.40918I
b = −0.10864 − 2.78263I
−0.74834 + 11.21750I 0
u = 0.963754 − 0.699919I
a = 1.32913 + 1.40918I
b = −0.10864 + 2.78263I
−0.74834 − 11.21750I 0
u = −0.794366
a = −1.92015
b = −1.54546
−0.199376 16.2640
u = 1.021690 + 0.661610I
a = −0.354244 + 0.502031I
b = 0.100352 + 0.975761I
1.54134 + 6.23006I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.021690 − 0.661610I
a = −0.354244 − 0.502031I
b = 0.100352 − 0.975761I
1.54134 − 6.23006I 0
u = −1.066280 + 0.628903I
a = 1.342030 − 0.354279I
b = 1.54130 + 2.68662I
8.48924 + 1.95535I 0
u = −1.066280 − 0.628903I
a = 1.342030 + 0.354279I
b = 1.54130 − 2.68662I
8.48924 − 1.95535I 0
u = 1.057180 + 0.645866I
a = −0.744289 − 0.464674I
b = −1.45280 + 0.13979I
4.53494 + 4.36156I 0
u = 1.057180 − 0.645866I
a = −0.744289 + 0.464674I
b = −1.45280 − 0.13979I
4.53494 − 4.36156I 0
u = −1.054170 + 0.653560I
a = 2.57977 − 0.47238I
b = 2.22655 + 3.81336I
2.78674 − 6.84253I 0
u = −1.054170 − 0.653560I
a = 2.57977 + 0.47238I
b = 2.22655 − 3.81336I
2.78674 + 6.84253I 0
u = −1.067760 + 0.639442I
a = −1.68971 + 0.24602I
b = −1.64820 − 3.06661I
10.19720 − 4.27807I 0
u = −1.067760 − 0.639442I
a = −1.68971 − 0.24602I
b = −1.64820 + 3.06661I
10.19720 + 4.27807I 0
u = 1.057590 + 0.659971I
a = 0.615096 + 0.674075I
b = 1.45428 + 0.26538I
4.29069 + 9.43826I 0
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.057590 − 0.659971I
a = 0.615096 − 0.674075I
b = 1.45428 − 0.26538I
4.29069 − 9.43826I 0
u = −1.065160 + 0.665399I
a = −2.46286 − 0.25088I
b = −1.53551 − 3.87903I
9.75434 − 9.83173I 0
u = −1.065160 − 0.665399I
a = −2.46286 + 0.25088I
b = −1.53551 + 3.87903I
9.75434 + 9.83173I 0
u = −1.062690 + 0.672745I
a = 2.60161 + 0.42830I
b = 1.41154 + 3.99113I
7.7452 − 15.9995I 0
u = −1.062690 − 0.672745I
a = 2.60161 − 0.42830I
b = 1.41154 − 3.99113I
7.7452 + 15.9995I 0
u = 0.296289 + 0.524344I
a = −0.50945 + 1.36283I
b = 0.648894 + 0.183040I
1.79627 − 0.48586I 8.09606 − 0.14290I
u = 0.296289 − 0.524344I
a = −0.50945 − 1.36283I
b = 0.648894 − 0.183040I
1.79627 + 0.48586I 8.09606 + 0.14290I
u = 0.160959 + 0.532274I
a = 0.73342 − 1.50239I
b = −0.576401 − 0.234087I
1.01882 + 4.40596I 5.40774 − 5.91692I
u = 0.160959 − 0.532274I
a = 0.73342 + 1.50239I
b = −0.576401 + 0.234087I
1.01882 − 4.40596I 5.40774 + 5.91692I
u = 0.498541
a = −0.505794
b = 0.337787
0.680463 14.8240
12
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.121084 + 0.275781I
a = 0.71570 − 1.87090I
b = −0.450058 − 0.371328I
−1.65227 − 0.65424I −2.63477 + 1.82437I
u = −0.121084 − 0.275781I
a = 0.71570 + 1.87090I
b = −0.450058 + 0.371328I
−1.65227 + 0.65424I −2.63477 − 1.82437I
13
II. I
u
2
= h−u
5
+ u
3
− u
2
+ b − u, −u
7
+ u
5
− u
4
− u
3
+ a − 1, u
8
− u
7
− u
6
+
2u
5
+ u
4
− 2u
3
+ 2u − 1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
4
=
u
7
− u
5
+ u
4
+ u
3
+ 1
u
5
− u
3
+ u
2
+ u
a
8
=
u
−u
3
+ u
a
9
=
u
−u
3
+ u
a
1
=
−u
2
+ 1
−u
2
a
3
=
u
7
− u
5
+ u
4
+ u
3
+ 1
u
5
− u
3
+ u
2
+ u
a
2
=
u
7
− u
5
+ u
4
+ u
3
− u
2
+ 2
u
5
− u
3
+ u
a
10
=
u
4
− u
2
+ 1
u
4
a
6
=
u
6
− u
4
+ 2u
2
− 1
−u
7
+ u
6
+ 2u
5
− u
4
− 2u
3
+ 2u
2
+ 2u − 1
a
5
=
u
2
− 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
7
+ u
6
− 5u
5
+ 5u
3
− u
2
− 4u + 5
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
8
c
3
, c
8
u
8
c
4
(u + 1)
8
c
5
, c
6
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
c
7
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1
c
9
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
c
10
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
c
11
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
c
12
u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
8
c
3
, c
8
y
8
c
5
, c
6
, c
9
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
c
7
, c
11
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
c
10
, c
12
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.570868 + 0.730671I
a = 0.325934 + 0.693334I
b = 0.972127 + 0.565636I
−0.604279 − 1.131230I 1.47926 + 0.84929I
u = 0.570868 − 0.730671I
a = 0.325934 − 0.693334I
b = 0.972127 − 0.565636I
−0.604279 + 1.131230I 1.47926 − 0.84929I
u = −0.855237 + 0.665892I
a = −1.03462 − 0.99451I
b = 0.39611 − 1.88650I
−3.80435 − 2.57849I 2.50535 + 3.23297I
u = −0.855237 − 0.665892I
a = −1.03462 + 0.99451I
b = 0.39611 + 1.88650I
−3.80435 + 2.57849I 2.50535 − 3.23297I
u = −1.09818
a = 0.801005
b = −0.165005
4.85780 7.45240
u = 1.031810 + 0.655470I
a = −0.842429 − 0.289836I
b = −0.699541 + 1.033710I
0.73474 + 6.44354I 3.27544 − 5.90525I
u = 1.031810 − 0.655470I
a = −0.842429 + 0.289836I
b = −0.699541 − 1.033710I
0.73474 − 6.44354I 3.27544 + 5.90525I
u = 0.603304
a = 1.30123
b = 0.827616
−0.799899 3.02750
17
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
82
+ 35u
81
+ ··· + 87u + 1)
c
2
((u − 1)
8
)(u
82
− 9u
81
+ ··· − 15u + 1)
c
3
, c
8
u
8
(u
82
+ u
81
+ ··· − 384u + 256)
c
4
((u + 1)
8
)(u
82
− 9u
81
+ ··· − 15u + 1)
c
5
, c
6
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)(u
82
− 2u
81
+ ··· + 231u + 49)
c
7
(u
8
+ u
7
+ ··· − 2u − 1)(u
82
+ 2u
81
+ ··· + u + 1)
c
9
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
82
− 2u
81
+ ··· + 231u + 49)
c
10
(u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
· (u
82
− 30u
81
+ ··· + u + 1)
c
11
(u
8
− u
7
+ ··· + 2u − 1)(u
82
+ 2u
81
+ ··· + u + 1)
c
12
(u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
· (u
82
− 30u
81
+ ··· + u + 1)
18
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
8
)(y
82
+ 33y
81
+ ··· − 3371y + 1)
c
2
, c
4
((y −1)
8
)(y
82
− 35y
81
+ ··· − 87y + 1)
c
3
, c
8
y
8
(y
82
− 51y
81
+ ··· − 1949696y + 65536)
c
5
, c
6
, c
9
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
82
− 86y
81
+ ··· + 637y + 2401)
c
7
, c
11
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
82
− 30y
81
+ ··· + y + 1)
c
10
, c
12
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
82
+ 46y
81
+ ··· + 25y + 1)
19
