12a
0056
(K12a
0056
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 10 4 1 12 11 6 9 8
Solving Sequence
5,10
6
3,11
2 1 4 7 9 12 8
c
5
c
10
c
2
c
1
c
4
c
6
c
9
c
11
c
8
c
3
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
48
− u
47
+ ··· + b + 1, u
45
− 4u
43
+ ··· + a − 5u, u
49
+ 2u
48
+ ··· + u − 1i
I
u
2
= hb + 1, u
4
− u
2
+ a + u + 2, u
5
− u
4
+ u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 54 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
48
−u
47
+· · ·+b+1, u
45
−4u
43
+· · ·+a−5u, u
49
+2u
48
+· · ·+u−1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
u
2
a
3
=
−u
45
+ 4u
43
+ ··· − 22u
3
+ 5u
u
48
+ u
47
+ ··· + 4u
2
− 1
a
11
=
−u
−u
3
+ u
a
2
=
u
48
+ u
47
+ ··· + 5u − 1
u
48
+ u
47
+ ··· + 4u
2
− 1
a
1
=
u
9
+ 3u
5
+ u
u
11
− u
9
+ 4u
7
− 3u
5
+ 3u
3
− u
a
4
=
2u
48
+ 2u
47
+ ··· + 6u − 1
u
48
+ u
47
+ ··· + u − 1
a
7
=
u
11
+ 4u
7
+ 3u
3
u
13
− u
11
+ 5u
9
− 4u
7
+ 6u
5
− 3u
3
+ u
a
9
=
u
3
u
5
− u
3
+ u
a
12
=
−u
5
− u
−u
7
+ u
5
− 2u
3
+ u
a
8
=
u
7
+ 2u
3
u
9
− u
7
+ 3u
5
− 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6u
48
+ 5u
47
+ ··· + 9u − 20
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
49
+ 20u
48
+ ··· + 79u + 1
c
2
, c
4
u
49
− 6u
48
+ ··· − u + 1
c
3
, c
6
u
49
− u
48
+ ··· + 64u + 32
c
5
, c
10
u
49
− 2u
48
+ ··· + u + 1
c
7
, c
8
, c
9
c
11
, c
12
u
49
+ 8u
48
+ ··· + 13u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
49
+ 24y
48
+ ··· + 2991y −1
c
2
, c
4
y
49
− 20y
48
+ ··· + 79y −1
c
3
, c
6
y
49
+ 33y
48
+ ··· − 9728y −1024
c
5
, c
10
y
49
− 8y
48
+ ··· + 13y −1
c
7
, c
8
, c
9
c
11
, c
12
y
49
+ 68y
48
+ ··· − 67y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.760736 + 0.719246I
a = 1.12247 + 1.09140I
b = −0.784658 − 0.558197I
3.14913 + 0.38825I −7.13985 + 0.I
u = −0.760736 − 0.719246I
a = 1.12247 − 1.09140I
b = −0.784658 + 0.558197I
3.14913 − 0.38825I −7.13985 + 0.I
u = 0.796734 + 0.681722I
a = −0.291180 + 1.036890I
b = −1.281700 + 0.033312I
1.51702 − 2.54299I −6.23264 + 3.96997I
u = 0.796734 − 0.681722I
a = −0.291180 − 1.036890I
b = −1.281700 − 0.033312I
1.51702 + 2.54299I −6.23264 − 3.96997I
u = −0.892137 + 0.327379I
a = 2.17207 + 1.31935I
b = 1.005090 − 0.623169I
0.43580 + 6.91892I −11.9536 − 9.3736I
u = −0.892137 − 0.327379I
a = 2.17207 − 1.31935I
b = 1.005090 + 0.623169I
0.43580 − 6.91892I −11.9536 + 9.3736I
u = −0.844082 + 0.428322I
a = −0.187763 + 0.837953I
b = 0.617924 + 0.579853I
1.59253 + 2.02971I −8.12894 − 3.92342I
u = −0.844082 − 0.428322I
a = −0.187763 − 0.837953I
b = 0.617924 − 0.579853I
1.59253 − 2.02971I −8.12894 + 3.92342I
u = 0.693592 + 0.795699I
a = −0.527393 + 1.026490I
b = 1.070340 − 0.723216I
7.01029 + 5.24368I −5.42899 − 3.02603I
u = 0.693592 − 0.795699I
a = −0.527393 − 1.026490I
b = 1.070340 + 0.723216I
7.01029 − 5.24368I −5.42899 + 3.02603I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.842140 + 0.687789I
a = −0.05555 − 2.42181I
b = −0.868271 + 0.553100I
2.88211 + 4.83632I −8.23420 − 6.42667I
u = −0.842140 − 0.687789I
a = −0.05555 + 2.42181I
b = −0.868271 − 0.553100I
2.88211 − 4.83632I −8.23420 + 6.42667I
u = 0.749509 + 0.792916I
a = −0.060126 − 1.378320I
b = 0.588767 + 0.880625I
8.45422 − 0.68674I −3.54074 + 1.96318I
u = 0.749509 − 0.792916I
a = −0.060126 + 1.378320I
b = 0.588767 − 0.880625I
8.45422 + 0.68674I −3.54074 − 1.96318I
u = −0.737633 + 0.516972I
a = 0.414474 + 0.300895I
b = 0.346209 + 0.168150I
1.34830 + 2.00478I −4.48597 − 4.55079I
u = −0.737633 − 0.516972I
a = 0.414474 − 0.300895I
b = 0.346209 − 0.168150I
1.34830 − 2.00478I −4.48597 + 4.55079I
u = 0.880418 + 0.058419I
a = 1.79760 − 0.81935I
b = 0.877222 − 0.561628I
−0.98474 + 2.22735I −14.5110 − 2.6928I
u = 0.880418 − 0.058419I
a = 1.79760 + 0.81935I
b = 0.877222 + 0.561628I
−0.98474 − 2.22735I −14.5110 + 2.6928I
u = 0.892884 + 0.720049I
a = −1.122640 + 0.355505I
b = 0.517458 − 0.877459I
7.97137 − 4.88104I −4.58899 + 4.14916I
u = 0.892884 − 0.720049I
a = −1.122640 − 0.355505I
b = 0.517458 + 0.877459I
7.97137 + 4.88104I −4.58899 − 4.14916I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.922095 + 0.682387I
a = 0.94618 − 2.27074I
b = 1.097470 + 0.693580I
6.24237 − 10.68800I −7.37912 + 8.93754I
u = 0.922095 − 0.682387I
a = 0.94618 + 2.27074I
b = 1.097470 − 0.693580I
6.24237 + 10.68800I −7.37912 − 8.93754I
u = 0.740442 + 0.287353I
a = −1.50962 + 2.57763I
b = −0.946359 − 0.333388I
−2.14769 − 2.41886I −15.9646 + 6.9978I
u = 0.740442 − 0.287353I
a = −1.50962 − 2.57763I
b = −0.946359 + 0.333388I
−2.14769 + 2.41886I −15.9646 − 6.9978I
u = −0.359412 + 0.611149I
a = −0.165040 + 1.109630I
b = 0.758551 − 0.696515I
3.14714 + 1.64287I −3.84623 − 3.06683I
u = −0.359412 − 0.611149I
a = −0.165040 − 1.109630I
b = 0.758551 + 0.696515I
3.14714 − 1.64287I −3.84623 + 3.06683I
u = 0.931772 + 0.895572I
a = 0.237488 − 0.406711I
b = 0.694049 − 0.014591I
9.96355 − 3.30520I 0
u = 0.931772 − 0.895572I
a = 0.237488 + 0.406711I
b = 0.694049 + 0.014591I
9.96355 + 3.30520I 0
u = −0.682651 + 0.184302I
a = −2.50646 − 0.45724I
b = −1.096840 − 0.143397I
−2.69048 + 0.60292I −15.3689 − 9.6396I
u = −0.682651 − 0.184302I
a = −2.50646 + 0.45724I
b = −1.096840 + 0.143397I
−2.69048 − 0.60292I −15.3689 + 9.6396I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.948044 + 0.926579I
a = 0.195056 − 0.777042I
b = −1.395190 − 0.006072I
11.83110 + 3.40658I 0
u = −0.948044 − 0.926579I
a = 0.195056 + 0.777042I
b = −1.395190 + 0.006072I
11.83110 − 3.40658I 0
u = 0.942403 + 0.933530I
a = 1.03952 − 1.20944I
b = −0.855209 + 0.703197I
13.54070 − 0.72694I 0
u = 0.942403 − 0.933530I
a = 1.03952 + 1.20944I
b = −0.855209 − 0.703197I
13.54070 + 0.72694I 0
u = −0.927792 + 0.948411I
a = −0.633748 − 1.030830I
b = 1.155820 + 0.749430I
17.4053 − 6.0517I 0
u = −0.927792 − 0.948411I
a = −0.633748 + 1.030830I
b = 1.155820 − 0.749430I
17.4053 + 6.0517I 0
u = 0.956706 + 0.925364I
a = 0.22666 + 2.17249I
b = −0.870319 − 0.699281I
13.4933 − 6.1049I 0
u = 0.956706 − 0.925364I
a = 0.22666 − 2.17249I
b = −0.870319 + 0.699281I
13.4933 + 6.1049I 0
u = −0.939680 + 0.946771I
a = −0.07120 + 1.53829I
b = 0.540687 − 1.020080I
19.3115 + 0.3590I 0
u = −0.939680 − 0.946771I
a = −0.07120 − 1.53829I
b = 0.540687 + 1.020080I
19.3115 − 0.3590I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.975390 + 0.920268I
a = 0.26091 + 2.26210I
b = 1.161320 − 0.741973I
17.2463 + 12.9142I 0
u = −0.975390 − 0.920268I
a = 0.26091 − 2.26210I
b = 1.161320 + 0.741973I
17.2463 − 12.9142I 0
u = −0.968824 + 0.929241I
a = −1.16534 − 0.91357I
b = 0.526652 + 1.020740I
19.2138 + 6.5292I 0
u = −0.968824 − 0.929241I
a = −1.16534 + 0.91357I
b = 0.526652 − 1.020740I
19.2138 − 6.5292I 0
u = −0.222542 + 0.611946I
a = −0.356092 − 1.021050I
b = 0.934438 + 0.678142I
2.61636 − 3.65615I −4.75819 + 3.34819I
u = −0.222542 − 0.611946I
a = −0.356092 + 1.021050I
b = 0.934438 − 0.678142I
2.61636 + 3.65615I −4.75819 − 3.34819I
u = 0.560438
a = 0.882793
b = −0.134956
−0.730326 −14.0240
u = 0.314290 + 0.268544I
a = 1.79832 − 0.55498I
b = −0.725982 + 0.146511I
−0.980654 + 0.106245I −9.70626 + 1.04735I
u = 0.314290 − 0.268544I
a = 1.79832 + 0.55498I
b = −0.725982 − 0.146511I
−0.980654 − 0.106245I −9.70626 − 1.04735I
9
II. I
u
2
= hb + 1, u
4
− u
2
+ a + u + 2, u
5
− u
4
+ u
2
+ u − 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
u
2
a
3
=
−u
4
+ u
2
− u − 2
−1
a
11
=
−u
−u
3
+ u
a
2
=
−u
4
+ u
2
− u − 3
−1
a
1
=
−1
0
a
4
=
−u
4
+ u
2
− u − 2
−1
a
7
=
1
u
2
a
9
=
u
3
u
4
− u
3
− u
2
+ 1
a
12
=
−u
4
+ u
2
− 1
u
4
a
8
=
−u
2
+ 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
3
+ 3u
2
− u − 14
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
5
c
3
, c
6
u
5
c
4
(u + 1)
5
c
5
u
5
− u
4
+ u
2
+ u − 1
c
7
, c
8
, c
9
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1
c
10
u
5
+ u
4
− u
2
+ u + 1
c
11
, c
12
u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
5
c
3
, c
6
y
5
c
5
, c
10
y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1
c
7
, c
8
, c
9
c
11
, c
12
y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.758138 + 0.584034I
a = −0.278580 − 1.055720I
b = −1.00000
0.17487 + 2.21397I −12.88087 − 4.04855I
u = −0.758138 − 0.584034I
a = −0.278580 + 1.055720I
b = −1.00000
0.17487 − 2.21397I −12.88087 + 4.04855I
u = 0.935538 + 0.903908I
a = −0.020316 + 0.590570I
b = −1.00000
9.31336 − 3.33174I −13.28666 + 2.53508I
u = 0.935538 − 0.903908I
a = −0.020316 − 0.590570I
b = −1.00000
9.31336 + 3.33174I −13.28666 − 2.53508I
u = 0.645200
a = −2.40221
b = −1.00000
−2.52712 −13.6650
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
49
+ 20u
48
+ ··· + 79u + 1)
c
2
((u − 1)
5
)(u
49
− 6u
48
+ ··· − u + 1)
c
3
, c
6
u
5
(u
49
− u
48
+ ··· + 64u + 32)
c
4
((u + 1)
5
)(u
49
− 6u
48
+ ··· − u + 1)
c
5
(u
5
− u
4
+ u
2
+ u − 1)(u
49
− 2u
48
+ ··· + u + 1)
c
7
, c
8
, c
9
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)(u
49
+ 8u
48
+ ··· + 13u + 1)
c
10
(u
5
+ u
4
− u
2
+ u + 1)(u
49
− 2u
48
+ ··· + u + 1)
c
11
, c
12
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)(u
49
+ 8u
48
+ ··· + 13u + 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
5
)(y
49
+ 24y
48
+ ··· + 2991y −1)
c
2
, c
4
((y −1)
5
)(y
49
− 20y
48
+ ··· + 79y −1)
c
3
, c
6
y
5
(y
49
+ 33y
48
+ ··· − 9728y −1024)
c
5
, c
10
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)(y
49
− 8y
48
+ ··· + 13y −1)
c
7
, c
8
, c
9
c
11
, c
12
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)(y
49
+ 68y
48
+ ··· − 67y −1)
15
