12n
0043
(K12n
0043
)
A knot diagram
1
Linearized knot diagam
3 5 6 10 2 10 11 12 5 1 8 7
Solving Sequence
2,5
3
6,10
7 1 11 4 9 12 8
c
2
c
5
c
6
c
1
c
10
c
4
c
9
c
12
c
8
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h15u
49
− 122u
48
+ ··· + 16b − 5, 6u
49
− 51u
48
+ ··· + 8a − 7, u
50
− 6u
49
+ ··· − 3u + 1i
I
u
2
= hb
5
− b
4
u − b
4
+ 2b
3
u + b
2
− bu − b + u, a, u
2
+ u + 1i
* 2 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
=
h15u
49
−122u
48
+· · ·+16b−5, 6u
49
−51u
48
+· · ·+8a−7, u
50
−6u
49
+· · ·−3u+1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
−u
2
a
6
=
u
u
a
10
=
−
3
4
u
49
+
51
8
u
48
+ ··· −
3
2
u +
7
8
−0.937500u
49
+ 7.62500u
48
+ ··· − 1.68750u + 0.312500
a
7
=
−u
2
− 1
−0.0625000u
48
+ 0.312500u
47
+ ··· + 2.12500u − 0.0625000
a
1
=
u
2
+ 1
−u
4
a
11
=
−2.81250u
49
+ 17.4375u
48
+ ··· − 6.56250u + 2.75000
−3.75000u
49
+ 23.4375u
48
+ ··· − 6.50000u + 1.18750
a
4
=
u
4
+ u
2
+ 1
u
4
a
9
=
−
3
4
u
49
+
51
8
u
48
+ ··· −
3
2
u +
7
8
−1.81250u
49
+ 14.8750u
48
+ ··· − 8.06250u + 2.18750
a
12
=
1
16
u
49
−
3
8
u
48
+ ··· −
29
16
u +
15
16
5
8
u
49
−
19
8
u
48
+ ··· −
9
8
u +
7
8
a
8
=
1
2
u
49
−
53
16
u
48
+ ··· +
25
4
u −
13
16
−0.187500u
49
− 0.687500u
48
+ ··· + 4.43750u − 1.75000
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
235
16
u
49
+
1341
16
u
48
+ ··· −
101
16
u −
1
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
50
+ 30u
49
+ ··· + 9u + 1
c
2
, c
5
u
50
+ 6u
49
+ ··· + 3u + 1
c
3
u
50
− 6u
49
+ ··· − 5u + 2
c
4
, c
9
u
50
− u
49
+ ··· + 1024u + 1024
c
6
u
50
+ 3u
49
+ ··· + 9u
2
+ 1
c
7
, c
8
, c
11
u
50
− 3u
49
+ ··· + 9u
2
+ 1
c
10
u
50
− 13u
49
+ ··· − 146u − 7
c
12
u
50
+ 9u
49
+ ··· + 227u + 32
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
50
− 14y
49
+ ··· + 9y + 1
c
2
, c
5
y
50
+ 30y
49
+ ··· + 9y + 1
c
3
y
50
− 58y
49
+ ··· + 63y + 4
c
4
, c
9
y
50
− 55y
49
+ ··· − 12582912y + 1048576
c
6
y
50
− 61y
49
+ ··· + 18y + 1
c
7
, c
8
, c
11
y
50
− 45y
49
+ ··· + 18y + 1
c
10
y
50
− y
49
+ ··· − 19146y + 49
c
12
y
50
+ 7y
49
+ ··· + 13303y + 1024
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.973341 + 0.149524I
a = −1.68794 − 0.12351I
b = 0.066174 − 0.254632I
−1.48997 − 8.27576I 2.25055 + 4.63368I
u = 0.973341 − 0.149524I
a = −1.68794 + 0.12351I
b = 0.066174 + 0.254632I
−1.48997 + 8.27576I 2.25055 − 4.63368I
u = 0.973094 + 0.107852I
a = 1.69811 + 0.08847I
b = −0.056748 + 0.182641I
−6.65156 − 4.37591I −2.22838 + 3.56988I
u = 0.973094 − 0.107852I
a = 1.69811 − 0.08847I
b = −0.056748 − 0.182641I
−6.65156 + 4.37591I −2.22838 − 3.56988I
u = −0.677365 + 0.699027I
a = −0.322440 − 0.902269I
b = −0.498316 − 0.530339I
4.64833 + 0.74609I 3.29001 + 0.I
u = −0.677365 − 0.699027I
a = −0.322440 + 0.902269I
b = −0.498316 + 0.530339I
4.64833 − 0.74609I 3.29001 + 0.I
u = 0.957021 + 0.043611I
a = −1.69470 − 0.03418I
b = 0.0746266 − 0.0721643I
−4.57488 − 0.26600I − 60.10 − 1.059498I
u = 0.957021 − 0.043611I
a = −1.69470 + 0.03418I
b = 0.0746266 + 0.0721643I
−4.57488 + 0.26600I − 60.10 + 1.059498I
u = 0.236301 + 0.909169I
a = −0.572242 + 0.629406I
b = 0.72257 + 1.84188I
5.03527 + 5.79377I −0.60515 − 2.37720I
u = 0.236301 − 0.909169I
a = −0.572242 − 0.629406I
b = 0.72257 − 1.84188I
5.03527 − 5.79377I −0.60515 + 2.37720I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.161591 + 0.918734I
a = 0.626446 − 0.670903I
b = −0.48442 − 1.64182I
−0.70904 + 2.61756I −4.03413 − 3.04976I
u = 0.161591 − 0.918734I
a = 0.626446 + 0.670903I
b = −0.48442 + 1.64182I
−0.70904 − 2.61756I −4.03413 + 3.04976I
u = −0.502925 + 0.747274I
a = −0.018799 + 0.638607I
b = 0.247387 + 0.438501I
0.02112 − 1.45050I −2.86693 + 5.13818I
u = −0.502925 − 0.747274I
a = −0.018799 − 0.638607I
b = 0.247387 − 0.438501I
0.02112 + 1.45050I −2.86693 − 5.13818I
u = −0.609640 + 0.928743I
a = −0.602842 − 0.340623I
b = −0.607870 − 0.045415I
−0.64445 − 3.09089I 0
u = −0.609640 − 0.928743I
a = −0.602842 + 0.340623I
b = −0.607870 + 0.045415I
−0.64445 + 3.09089I 0
u = −0.115719 + 1.106410I
a = −0.816238 + 0.672347I
b = −0.366736 + 1.302780I
−0.489731 + 0.237177I 0
u = −0.115719 − 1.106410I
a = −0.816238 − 0.672347I
b = −0.366736 − 1.302780I
−0.489731 − 0.237177I 0
u = −0.686560 + 0.915627I
a = 0.693027 + 0.536546I
b = 0.731923 + 0.193983I
4.03197 − 5.98158I 0
u = −0.686560 − 0.915627I
a = 0.693027 − 0.536546I
b = 0.731923 − 0.193983I
4.03197 + 5.98158I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.243021 + 1.135140I
a = 0.892386 − 0.542794I
b = 0.566389 − 1.069200I
−3.49782 − 3.09462I 0
u = −0.243021 − 1.135140I
a = 0.892386 + 0.542794I
b = 0.566389 + 1.069200I
−3.49782 + 3.09462I 0
u = −0.538980 + 1.033550I
a = 0.798066 + 0.043574I
b = 0.699077 − 0.292935I
2.94116 − 0.76505I 0
u = −0.538980 − 1.033550I
a = 0.798066 − 0.043574I
b = 0.699077 + 0.292935I
2.94116 + 0.76505I 0
u = 0.022608 + 0.789395I
a = −0.701358 + 0.689418I
b = 0.406014 + 1.098740I
−0.100668 − 0.941081I −0.92280 + 4.24243I
u = 0.022608 − 0.789395I
a = −0.701358 − 0.689418I
b = 0.406014 − 1.098740I
−0.100668 + 0.941081I −0.92280 − 4.24243I
u = −0.311220 + 1.172780I
a = −0.971626 + 0.461124I
b = −0.720591 + 0.967453I
1.10315 − 6.57415I 0
u = −0.311220 − 1.172780I
a = −0.971626 − 0.461124I
b = −0.720591 − 0.967453I
1.10315 + 6.57415I 0
u = 0.778276
a = 1.59204
b = −0.332020
2.53130 4.52920
u = 0.215175 + 0.693272I
a = 0.787829 − 0.545376I
b = −0.975052 − 0.975693I
5.65570 − 3.38770I 2.66671 + 5.35428I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215175 − 0.693272I
a = 0.787829 + 0.545376I
b = −0.975052 + 0.975693I
5.65570 + 3.38770I 2.66671 − 5.35428I
u = 0.476948 + 1.225580I
a = −0.211170 + 1.154370I
b = −0.37574 + 2.70281I
−0.98640 + 4.59430I 0
u = 0.476948 − 1.225580I
a = −0.211170 − 1.154370I
b = −0.37574 − 2.70281I
−0.98640 − 4.59430I 0
u = 0.510209 + 1.283800I
a = 0.195449 − 1.284720I
b = 0.45463 − 2.70776I
−8.38002 + 5.49910I 0
u = 0.510209 − 1.283800I
a = 0.195449 + 1.284720I
b = 0.45463 + 2.70776I
−8.38002 − 5.49910I 0
u = −0.565504 + 0.246324I
a = 0.06152 − 1.46066I
b = −0.182316 − 0.730281I
4.93756 − 3.55579I 5.87055 + 4.25691I
u = −0.565504 − 0.246324I
a = 0.06152 + 1.46066I
b = −0.182316 + 0.730281I
4.93756 + 3.55579I 5.87055 − 4.25691I
u = 0.562121 + 1.266420I
a = 0.086370 − 1.295980I
b = 0.45752 − 2.75201I
−4.9120 + 13.8047I 0
u = 0.562121 − 1.266420I
a = 0.086370 + 1.295980I
b = 0.45752 + 2.75201I
−4.9120 − 13.8047I 0
u = 0.383593 + 1.332350I
a = 0.449050 − 1.266840I
b = 0.50380 − 2.53593I
−6.24884 − 3.57394I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.383593 − 1.332350I
a = 0.449050 + 1.266840I
b = 0.50380 + 2.53593I
−6.24884 + 3.57394I 0
u = 0.456861 + 1.311070I
a = 0.310827 − 1.289690I
b = 0.47402 − 2.64241I
−8.79559 + 4.74233I 0
u = 0.456861 − 1.311070I
a = 0.310827 + 1.289690I
b = 0.47402 + 2.64241I
−8.79559 − 4.74233I 0
u = 0.543755 + 1.278370I
a = −0.130234 + 1.302520I
b = −0.45971 + 2.73698I
−10.2434 + 9.8226I 0
u = 0.543755 − 1.278370I
a = −0.130234 − 1.302520I
b = −0.45971 − 2.73698I
−10.2434 − 9.8226I 0
u = 0.416925 + 1.327490I
a = −0.391023 + 1.285420I
b = −0.49447 + 2.58673I
−11.20260 + 0.48443I 0
u = 0.416925 − 1.327490I
a = −0.391023 − 1.285420I
b = −0.49447 − 2.58673I
−11.20260 − 0.48443I 0
u = −0.200172 + 0.291553I
a = −0.68081 + 1.49034I
b = 0.248384 + 0.528654I
0.027965 − 1.099060I 0.35978 + 6.01400I
u = −0.200172 − 0.291553I
a = −0.68081 − 1.49034I
b = 0.248384 − 0.528654I
0.027965 + 1.099060I 0.35978 − 6.01400I
u = 0.344852
a = 1.81264
b = −0.529057
2.85126 2.12860
9
II. I
u
2
= hb
5
− b
4
u − b
4
+ 2b
3
u + b
2
− bu − b + u, a, u
2
+ u + 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
u + 1
a
6
=
u
u
a
10
=
0
b
a
7
=
u
b
2
u + u
a
1
=
−u
−u
a
11
=
bu + b
bu + 2b
a
4
=
0
u
a
9
=
0
b
a
12
=
−b
2
− u
−b
4
− b
2
− u
a
8
=
−b
4
u − b
4
+ b
2
+ u
−b
4
u − 2b
4
+ b
2
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −b
4
u − b
4
+ 4b
3
− 3b
2
u − 3b
2
+ 5bu + b + 4u + 6
10
(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
9
u
10
c
6
, c
10
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
c
7
, c
8
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
c
11
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
c
12
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
11
(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
9
y
10
c
6
, c
10
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
c
7
, c
8
, c
11
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
c
12
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0
b = −0.881753 + 0.117510I
0.32910 − 3.56046I 2.49844 + 7.77102I
u = −0.500000 + 0.866025I
a = 0
b = 0.542643 − 0.704866I
0.329100 − 0.499304I −0.01046 − 1.42329I
u = −0.500000 + 0.866025I
a = 0
b = 0.383413 + 0.664091I
2.40108 − 2.02988I 0.33682 + 4.42764I
u = −0.500000 + 0.866025I
a = 0
b = −0.811514 + 0.994721I
5.87256 − 6.43072I 6.88365 + 7.29164I
u = −0.500000 + 0.866025I
a = 0
b = 1.267210 − 0.205431I
5.87256 + 2.37095I 4.29156 + 0.98555I
u = −0.500000 − 0.866025I
a = 0
b = −0.881753 − 0.117510I
0.32910 + 3.56046I 2.49844 − 7.77102I
u = −0.500000 − 0.866025I
a = 0
b = 0.542643 + 0.704866I
0.329100 + 0.499304I −0.01046 + 1.42329I
u = −0.500000 − 0.866025I
a = 0
b = 0.383413 − 0.664091I
2.40108 + 2.02988I 0.33682 − 4.42764I
u = −0.500000 − 0.866025I
a = 0
b = −0.811514 − 0.994721I
5.87256 + 6.43072I 6.88365 − 7.29164I
u = −0.500000 − 0.866025I
a = 0
b = 1.267210 + 0.205431I
5.87256 − 2.37095I 4.29156 − 0.98555I
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
5
)(u
50
+ 30u
49
+ ··· + 9u + 1)
c
2
((u
2
+ u + 1)
5
)(u
50
+ 6u
49
+ ··· + 3u + 1)
c
3
((u
2
− u + 1)
5
)(u
50
− 6u
49
+ ··· − 5u + 2)
c
4
, c
9
u
10
(u
50
− u
49
+ ··· + 1024u + 1024)
c
5
((u
2
− u + 1)
5
)(u
50
+ 6u
49
+ ··· + 3u + 1)
c
6
((u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
)(u
50
+ 3u
49
+ ··· + 9u
2
+ 1)
c
7
, c
8
((u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
)(u
50
− 3u
49
+ ··· + 9u
2
+ 1)
c
10
((u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
)(u
50
− 13u
49
+ ··· − 146u − 7)
c
11
((u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
)(u
50
− 3u
49
+ ··· + 9u
2
+ 1)
c
12
((u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
)(u
50
+ 9u
49
+ ··· + 227u + 32)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
5
)(y
50
− 14y
49
+ ··· + 9y + 1)
c
2
, c
5
((y
2
+ y + 1)
5
)(y
50
+ 30y
49
+ ··· + 9y + 1)
c
3
((y
2
+ y + 1)
5
)(y
50
− 58y
49
+ ··· + 63y + 4)
c
4
, c
9
y
10
(y
50
− 55y
49
+ ··· − 1.25829 × 10
7
y + 1048576)
c
6
((y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
)(y
50
− 61y
49
+ ··· + 18y + 1)
c
7
, c
8
, c
11
((y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
)(y
50
− 45y
49
+ ··· + 18y + 1)
c
10
((y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
)(y
50
− y
49
+ ··· − 19146y + 49)
c
12
((y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
)(y
50
+ 7y
49
+ ··· + 13303y + 1024)
15
