12n
0044
(K12n
0044
)
A knot diagram
1
Linearized knot diagam
3 5 6 8 2 12 11 4 6 7 8 10
Solving Sequence
8,12
11 7
3,6
4 5 2 10 1 9
c
11
c
7
c
6
c
3
c
4
c
2
c
10
c
12
c
9
c
1
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h2u
34
+ 4u
33
+ ··· + 2b − 3u, −6u
34
− 12u
33
+ ··· + 2a − 5, u
35
+ 3u
34
+ ··· − u + 1i
I
u
2
= hu
3
a − au + b − a, −u
3
a − u
4
+ u
3
+ a
2
+ 2au + 2u
2
− 2u − 1, u
5
− u
4
− 2u
3
+ u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 45 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
34
+4u
33
+· · ·+2b−3u, −6u
34
−12u
33
+· · ·+2a−5, u
35
+3u
34
+· · ·−u+1i
(i) Arc colorings
a
8
=
0
u
a
12
=
1
0
a
11
=
1
−u
2
a
7
=
u
−u
3
+ u
a
3
=
3u
34
+ 6u
33
+ ··· −
3
2
u +
5
2
−u
34
− 2u
33
+ ··· −
7
2
u
2
+
3
2
u
a
6
=
−u
3
+ 2u
−u
3
+ u
a
4
=
u
34
+ 2u
33
+ ··· − u + 1
−2u
34
−
7
2
u
33
+ ··· + u −
1
2
a
5
=
u
34
+ 2u
33
+ ··· − u + 1
−4u
34
−
13
2
u
33
+ ··· + 3u −
3
2
a
2
=
u
34
+ 2u
33
+ ··· −
5
2
u +
1
2
1
2
u
32
+
1
2
u
31
+ ··· +
7
2
u
2
+
3
2
u
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
1
=
u
6
− 3u
4
+ 2u
2
+ 1
−u
8
+ 4u
6
− 4u
4
a
9
=
−u
10
+ 5u
8
− 8u
6
+ 3u
4
+ u
2
+ 1
−u
10
+ 4u
8
− 5u
6
+ 2u
4
− u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes =
13
2
u
34
+
21
2
u
33
+ ··· − 15u +
21
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
35
+ 24u
34
+ ··· − 4u − 1
c
2
, c
5
u
35
+ 6u
34
+ ··· − 8u − 1
c
3
u
35
− 6u
34
+ ··· + u − 2
c
4
, c
8
u
35
− u
34
+ ··· + 2048u + 1024
c
6
u
35
− 9u
34
+ ··· + 959u − 176
c
7
, c
10
, c
11
u
35
+ 3u
34
+ ··· − u + 1
c
9
u
35
− 3u
34
+ ··· + 109535u + 149381
c
12
u
35
+ 3u
34
+ ··· + 3u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
35
− 20y
34
+ ··· − 256y −1
c
2
, c
5
y
35
+ 24y
34
+ ··· − 4y −1
c
3
y
35
− 64y
34
+ ··· − 115y −4
c
4
, c
8
y
35
+ 55y
34
+ ··· − 7340032y −1048576
c
6
y
35
− 23y
34
+ ··· + 86145y −30976
c
7
, c
10
, c
11
y
35
− 35y
34
+ ··· − 13y −1
c
9
y
35
+ 109y
34
+ ··· − 963686176609y −22314683161
c
12
y
35
+ 49y
34
+ ··· − 13y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.567543 + 0.702485I
a = −1.79734 − 1.55953I
b = −2.27694 − 0.16811I
−13.52750 + 3.29825I −4.52460 − 0.12830I
u = 0.567543 − 0.702485I
a = −1.79734 + 1.55953I
b = −2.27694 + 0.16811I
−13.52750 − 3.29825I −4.52460 + 0.12830I
u = 0.480329 + 0.751506I
a = −2.12175 − 1.61713I
b = −2.31879 − 0.06298I
−13.2384 − 8.1531I −3.91050 + 5.47003I
u = 0.480329 − 0.751506I
a = −2.12175 + 1.61713I
b = −2.31879 + 0.06298I
−13.2384 + 8.1531I −3.91050 − 5.47003I
u = 0.509854 + 0.709144I
a = 1.95618 + 1.69499I
b = 2.34001 + 0.13274I
−9.08492 − 2.36143I −1.70250 + 2.73634I
u = 0.509854 − 0.709144I
a = 1.95618 − 1.69499I
b = 2.34001 − 0.13274I
−9.08492 + 2.36143I −1.70250 − 2.73634I
u = −1.217400 + 0.104228I
a = 0.556325 − 0.615249I
b = −0.001571 + 0.321064I
−3.05619 + 0.58484I −4.25548 + 0.I
u = −1.217400 − 0.104228I
a = 0.556325 + 0.615249I
b = −0.001571 − 0.321064I
−3.05619 − 0.58484I −4.25548 + 0.I
u = −0.293244 + 0.696609I
a = −0.846989 − 0.642400I
b = −0.423424 − 0.190758I
−2.39789 + 3.01714I −3.73777 − 3.74303I
u = −0.293244 − 0.696609I
a = −0.846989 + 0.642400I
b = −0.423424 + 0.190758I
−2.39789 − 3.01714I −3.73777 + 3.74303I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.588530 + 0.465780I
a = 0.008324 + 1.173240I
b = −0.042665 + 0.372046I
−3.52618 + 0.88882I −5.91074 − 2.69844I
u = −0.588530 − 0.465780I
a = 0.008324 − 1.173240I
b = −0.042665 − 0.372046I
−3.52618 − 0.88882I −5.91074 + 2.69844I
u = −1.31854
a = 0.766887
b = −0.555364
−3.00411 −1.27090
u = 1.351890 + 0.040890I
a = 0.157041 + 0.958259I
b = 0.25925 + 1.39116I
−3.48501 − 2.60863I −5.97774 + 4.12008I
u = 1.351890 − 0.040890I
a = 0.157041 − 0.958259I
b = 0.25925 − 1.39116I
−3.48501 + 2.60863I −5.97774 − 4.12008I
u = 1.389190 + 0.197736I
a = 0.262507 + 0.317399I
b = 0.407675 + 0.409042I
−5.20091 − 3.82410I −4.45749 + 4.83241I
u = 1.389190 − 0.197736I
a = 0.262507 − 0.317399I
b = 0.407675 − 0.409042I
−5.20091 + 3.82410I −4.45749 − 4.83241I
u = −1.41548 + 0.08723I
a = −1.030240 − 0.153922I
b = 1.236690 − 0.650078I
−5.56076 + 3.95289I −6.09734 + 0.I
u = −1.41548 − 0.08723I
a = −1.030240 + 0.153922I
b = 1.236690 + 0.650078I
−5.56076 − 3.95289I −6.09734 + 0.I
u = 1.40268 + 0.27759I
a = −0.526430 + 0.056708I
b = −0.706415 + 0.164348I
−7.78555 − 6.56645I −7.86488 + 0.I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.40268 − 0.27759I
a = −0.526430 − 0.056708I
b = −0.706415 − 0.164348I
−7.78555 + 6.56645I −7.86488 + 0.I
u = −0.234492 + 0.501034I
a = 0.744299 − 0.409065I
b = 0.189998 − 0.226205I
−0.008011 + 1.203150I −0.07375 − 5.95308I
u = −0.234492 − 0.501034I
a = 0.744299 + 0.409065I
b = 0.189998 + 0.226205I
−0.008011 − 1.203150I −0.07375 + 5.95308I
u = 1.49851 + 0.14408I
a = 0.114237 − 0.651402I
b = 0.105973 − 0.888851I
−10.28270 − 3.07368I 0
u = 1.49851 − 0.14408I
a = 0.114237 + 0.651402I
b = 0.105973 + 0.888851I
−10.28270 + 3.07368I 0
u = −1.50813 + 0.27067I
a = −0.11491 + 1.74410I
b = −2.45281 + 0.21864I
−19.6916 + 11.8912I 0
u = −1.50813 − 0.27067I
a = −0.11491 − 1.74410I
b = −2.45281 − 0.21864I
−19.6916 − 11.8912I 0
u = −1.51243 + 0.24615I
a = −0.07457 − 1.66019I
b = 2.49768 − 0.33604I
−15.6688 + 5.8511I 0
u = −1.51243 − 0.24615I
a = −0.07457 + 1.66019I
b = 2.49768 + 0.33604I
−15.6688 − 5.8511I 0
u = −1.53563 + 0.22688I
a = 0.106999 + 1.408270I
b = −2.40458 + 0.41233I
19.0477 + 0.0884I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.53563 − 0.22688I
a = 0.106999 − 1.408270I
b = −2.40458 − 0.41233I
19.0477 − 0.0884I 0
u = 0.022155 + 0.410874I
a = 1.75998 − 0.97960I
b = 0.132570 − 0.734165I
0.57528 + 1.38421I 4.86232 − 4.69401I
u = 0.022155 − 0.410874I
a = 1.75998 + 0.97960I
b = 0.132570 + 0.734165I
0.57528 − 1.38421I 4.86232 + 4.69401I
u = 0.242449 + 0.281603I
a = −1.53711 + 1.37950I
b = 0.735025 + 0.827616I
−0.19025 − 2.59587I 1.54589 + 1.28231I
u = 0.242449 − 0.281603I
a = −1.53711 − 1.37950I
b = 0.735025 − 0.827616I
−0.19025 + 2.59587I 1.54589 − 1.28231I
8
II. I
u
2
= hu
3
a − au + b − a, −u
3
a − u
4
+ u
3
+ a
2
+ 2au + 2u
2
− 2u − 1, u
5
−
u
4
− 2u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
8
=
0
u
a
12
=
1
0
a
11
=
1
−u
2
a
7
=
u
−u
3
+ u
a
3
=
a
−u
3
a + au + a
a
6
=
−u
3
+ 2u
−u
3
+ u
a
4
=
0
u
4
a − u
3
a − u
2
a
a
5
=
0
u
4
a − u
3
a − u
2
a
a
2
=
a
−u
3
a + u
3
+ au + a − u − 1
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
1
=
u
3
− 2u
u
3
− u
a
9
=
0
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
4
a − 4u
3
a − 2u
2
a + 4u
3
+ 4au + u
2
+ 3a − 9u − 7
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
5
c
2
(u
2
+ u + 1)
5
c
4
, c
8
u
10
c
6
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
c
7
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
c
9
, c
12
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
c
10
, c
11
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
(y
2
+ y + 1)
5
c
4
, c
8
y
10
c
6
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y − 1)
2
c
7
, c
10
, c
11
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y − 1)
2
c
9
, c
12
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y − 1)
2
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.21774
a = 0.314857 + 0.545349I
b = 0.500000 + 0.866025I
−2.40108 − 2.02988I −0.33682 + 2.50057I
u = −1.21774
a = 0.314857 − 0.545349I
b = 0.500000 − 0.866025I
−2.40108 + 2.02988I −0.33682 − 2.50057I
u = −0.309916 + 0.549911I
a = 1.394870 + 0.200669I
b = 0.500000 + 0.866025I
−0.32910 + 3.56046I 0.01046 − 8.35149I
u = −0.309916 + 0.549911I
a = −0.523653 − 1.308330I
b = 0.500000 − 0.866025I
−0.329100 − 0.499304I −2.49844 − 0.84282I
u = −0.309916 − 0.549911I
a = 1.394870 − 0.200669I
b = 0.500000 − 0.866025I
−0.32910 − 3.56046I 0.01046 + 8.35149I
u = −0.309916 − 0.549911I
a = −0.523653 + 1.308330I
b = 0.500000 + 0.866025I
−0.329100 + 0.499304I −2.49844 + 0.84282I
u = 1.41878 + 0.21917I
a = −0.850505 + 0.276175I
b = 0.500000 + 0.866025I
−5.87256 − 2.37095I −6.88365 + 0.36343I
u = 1.41878 + 0.21917I
a = 0.664427 + 0.598472I
b = 0.500000 − 0.866025I
−5.87256 − 6.43072I −4.29156 + 5.94266I
u = 1.41878 − 0.21917I
a = −0.850505 − 0.276175I
b = 0.500000 − 0.866025I
−5.87256 + 2.37095I −6.88365 − 0.36343I
u = 1.41878 − 0.21917I
a = 0.664427 − 0.598472I
b = 0.500000 + 0.866025I
−5.87256 + 6.43072I −4.29156 − 5.94266I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
5
)(u
35
+ 24u
34
+ ··· − 4u − 1)
c
2
((u
2
+ u + 1)
5
)(u
35
+ 6u
34
+ ··· − 8u − 1)
c
3
((u
2
− u + 1)
5
)(u
35
− 6u
34
+ ··· + u − 2)
c
4
, c
8
u
10
(u
35
− u
34
+ ··· + 2048u + 1024)
c
5
((u
2
− u + 1)
5
)(u
35
+ 6u
34
+ ··· − 8u − 1)
c
6
((u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
)(u
35
− 9u
34
+ ··· + 959u − 176)
c
7
((u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
)(u
35
+ 3u
34
+ ··· − u + 1)
c
9
((u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
)(u
35
− 3u
34
+ ··· + 109535u + 149381)
c
10
, c
11
((u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
)(u
35
+ 3u
34
+ ··· − u + 1)
c
12
((u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
)(u
35
+ 3u
34
+ ··· + 3u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
5
)(y
35
− 20y
34
+ ··· − 256y − 1)
c
2
, c
5
((y
2
+ y + 1)
5
)(y
35
+ 24y
34
+ ··· − 4y − 1)
c
3
((y
2
+ y + 1)
5
)(y
35
− 64y
34
+ ··· − 115y − 4)
c
4
, c
8
y
10
(y
35
+ 55y
34
+ ··· − 7340032y − 1048576)
c
6
((y
5
− y
4
+ 8y
3
− 3y
2
+ 3y − 1)
2
)(y
35
− 23y
34
+ ··· + 86145y − 30976)
c
7
, c
10
, c
11
((y
5
− 5y
4
+ 8y
3
− 3y
2
− y − 1)
2
)(y
35
− 35y
34
+ ··· − 13y − 1)
c
9
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y − 1)
2
· (y
35
+ 109y
34
+ ··· − 963686176609y − 22314683161)
c
12
((y
5
+ 3y
4
+ 4y
3
+ y
2
− y − 1)
2
)(y
35
+ 49y
34
+ ··· − 13y − 1)
14
