12n
0678
(K12n
0678
)
A knot diagram
1
Linearized knot diagam
4 5 7 2 10 3 12 11 6 7 8 10
Solving Sequence
7,12 4,8
3 6 11 10 1 5 2 9
c
7
c
3
c
6
c
11
c
10
c
12
c
5
c
2
c
9
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−171832171743933u
44
+ 565195223234702u
43
+ ··· + 319912709247818b − 45090425568559,
221527977237717u
44
− 489493690784658u
43
+ ··· + 319912709247818a + 1085130159951208,
u
45
− 3u
44
+ ··· − 8u + 1i
I
u
2
= h−au + b − u, u
2
a + a
2
+ au − u
2
+ 4a, u
3
+ u
2
+ 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 51 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−1.72×10
14
u
44
+5.65×10
14
u
43
+· · ·+3.20×10
14
b−4.51×10
13
, 2.22×
10
14
u
44
−4.89×10
14
u
43
+· · ·+3.20×10
14
a+1.09×10
15
, u
45
−3u
44
+· · ·−8u+1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
−0.692464u
44
+ 1.53009u
43
+ ··· − 21.6689u − 3.39196
0.537122u
44
− 1.76672u
43
+ ··· + 4.08754u + 0.140946
a
8
=
1
−u
2
a
3
=
−0.155342u
44
− 0.236632u
43
+ ··· − 17.5814u − 3.25101
0.537122u
44
− 1.76672u
43
+ ··· + 4.08754u + 0.140946
a
6
=
1.72119u
44
− 3.14555u
43
+ ··· + 18.4989u + 1.81534
−2.01801u
44
+ 5.92336u
43
+ ··· − 15.5848u + 1.72119
a
11
=
−u
u
3
+ u
a
10
=
−u
3
− 2u
u
3
+ u
a
1
=
u
7
+ 4u
5
+ 4u
3
−u
7
− 3u
5
− 2u
3
+ u
a
5
=
−0.470191u
44
+ 1.87005u
43
+ ··· + 7.18359u + 3.39261
−0.462878u
44
+ 1.23328u
43
+ ··· − 3.91246u + 0.140946
a
2
=
−1.21496u
44
+ 3.02326u
43
+ ··· − 22.9486u − 0.0362322
0.462878u
44
− 1.23328u
43
+ ··· + 3.91246u − 0.140946
a
9
=
u
2
+ 1
−u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
124420232195109
159956354623909
u
44
+
585183168556345
319912709247818
u
43
+ ··· −
6164660264148967
319912709247818
u −
3111644555281919
319912709247818
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
45
− 4u
44
+ ··· + 3u − 1
c
3
, c
6
u
45
+ 4u
44
+ ··· − u + 1
c
5
, c
9
u
45
+ 3u
44
+ ··· + 160u + 64
c
7
, c
8
, c
11
u
45
+ 3u
44
+ ··· − 8u − 1
c
10
u
45
− 3u
44
+ ··· − 542u − 97
c
12
u
45
+ 19u
44
+ ··· + 653792u − 13633
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
45
− 34y
44
+ ··· + 5y −1
c
3
, c
6
y
45
− 6y
44
+ ··· + 5y −1
c
5
, c
9
y
45
+ 35y
44
+ ··· + 58368y −4096
c
7
, c
8
, c
11
y
45
+ 37y
44
+ ··· + 120y −1
c
10
y
45
− 39y
44
+ ··· + 1081792y −9409
c
12
y
45
− 59y
44
+ ··· + 530299074808y −185858689
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.583612 + 0.749370I
a = −0.145378 − 1.180080I
b = 0.314212 + 0.693704I
−0.22746 − 3.52607I −0.53223 + 9.05892I
u = −0.583612 − 0.749370I
a = −0.145378 + 1.180080I
b = 0.314212 − 0.693704I
−0.22746 + 3.52607I −0.53223 − 9.05892I
u = 0.884188 + 0.167276I
a = −1.44217 + 3.00622I
b = 1.01437 − 1.10209I
4.46171 + 9.77082I −1.55814 − 6.22355I
u = 0.884188 − 0.167276I
a = −1.44217 − 3.00622I
b = 1.01437 + 1.10209I
4.46171 − 9.77082I −1.55814 + 6.22355I
u = −0.862556 + 0.189586I
a = −0.10994 + 1.47934I
b = −0.099189 − 0.621407I
1.49082 − 1.15898I 5.48142 + 4.79165I
u = −0.862556 − 0.189586I
a = −0.10994 − 1.47934I
b = −0.099189 + 0.621407I
1.49082 + 1.15898I 5.48142 − 4.79165I
u = 0.879134 + 0.073302I
a = 1.69262 − 2.92394I
b = −1.05605 + 1.10417I
8.52509 + 4.00354I 1.85103 − 2.75584I
u = 0.879134 − 0.073302I
a = 1.69262 + 2.92394I
b = −1.05605 − 1.10417I
8.52509 − 4.00354I 1.85103 + 2.75584I
u = 0.834305 + 0.026519I
a = −2.03456 − 2.80224I
b = 1.11558 + 1.09067I
4.18745 + 1.80014I 0.034492 − 1.194140I
u = 0.834305 − 0.026519I
a = −2.03456 + 2.80224I
b = 1.11558 − 1.09067I
4.18745 − 1.80014I 0.034492 + 1.194140I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.481295 + 1.085350I
a = −1.52416 − 1.39190I
b = 0.840283 + 1.091050I
1.64682 − 4.91710I −3.75732 + 2.86803I
u = 0.481295 − 1.085350I
a = −1.52416 + 1.39190I
b = 0.840283 − 1.091050I
1.64682 + 4.91710I −3.75732 − 2.86803I
u = −0.027576 + 1.213260I
a = 0.097830 + 0.955725I
b = 0.987113 − 0.071189I
−4.06865 − 0.00938I −7.42868 + 0.I
u = −0.027576 − 1.213260I
a = 0.097830 − 0.955725I
b = 0.987113 + 0.071189I
−4.06865 + 0.00938I −7.42868 + 0.I
u = −0.310904 + 1.181390I
a = −0.21426 − 1.54867I
b = 0.294215 + 0.670186I
−1.38332 − 2.86199I 0. + 3.33644I
u = −0.310904 − 1.181390I
a = −0.21426 + 1.54867I
b = 0.294215 − 0.670186I
−1.38332 + 2.86199I 0. − 3.33644I
u = 0.159859 + 1.260150I
a = 1.07101 − 1.51572I
b = −1.63864 + 0.16074I
−11.89460 + 2.24035I −8.35575 + 0.I
u = 0.159859 − 1.260150I
a = 1.07101 + 1.51572I
b = −1.63864 − 0.16074I
−11.89460 − 2.24035I −8.35575 + 0.I
u = 0.433209 + 1.205320I
a = 1.56995 + 1.36216I
b = −0.89067 − 1.16692I
5.04167 + 0.68814I 0
u = 0.433209 − 1.205320I
a = 1.56995 − 1.36216I
b = −0.89067 + 1.16692I
5.04167 − 0.68814I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.091883 + 1.292420I
a = 0.68882 + 2.04860I
b = 0.100880 − 0.832088I
−5.73773 − 2.01190I 0
u = −0.091883 − 1.292420I
a = 0.68882 − 2.04860I
b = 0.100880 + 0.832088I
−5.73773 + 2.01190I 0
u = 0.378074 + 1.248800I
a = −0.30888 + 2.43252I
b = 1.23412 − 0.93095I
0.40714 + 2.55603I 0
u = 0.378074 − 1.248800I
a = −0.30888 − 2.43252I
b = 1.23412 + 0.93095I
0.40714 − 2.55603I 0
u = −0.233067 + 1.286920I
a = −1.54727 − 1.27295I
b = −0.472121 + 0.195549I
−4.28958 − 3.06247I −27.0240 + 0.I
u = −0.233067 − 1.286920I
a = −1.54727 + 1.27295I
b = −0.472121 − 0.195549I
−4.28958 + 3.06247I −27.0240 + 0.I
u = 0.376537 + 1.291220I
a = −1.67227 − 1.26273I
b = 0.99238 + 1.22341I
0.08117 + 6.15166I 0
u = 0.376537 − 1.291220I
a = −1.67227 + 1.26273I
b = 0.99238 − 1.22341I
0.08117 − 6.15166I 0
u = −0.594574 + 0.259179I
a = −0.442778 + 1.247600I
b = −0.223270 − 0.529291I
1.36466 − 0.79548I 4.15835 + 2.62510I
u = −0.594574 − 0.259179I
a = −0.442778 − 1.247600I
b = −0.223270 + 0.529291I
1.36466 + 0.79548I 4.15835 − 2.62510I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.173829 + 1.352770I
a = −0.626070 + 0.431699I
b = −0.604107 − 0.273968I
−3.74207 − 3.36112I 0
u = −0.173829 − 1.352770I
a = −0.626070 − 0.431699I
b = −0.604107 + 0.273968I
−3.74207 + 3.36112I 0
u = −0.615344
a = −2.94504
b = −0.368354
−0.259777 −41.6450
u = 0.399717 + 1.327110I
a = 0.22643 − 2.45843I
b = −1.17155 + 1.01534I
4.14207 + 8.58803I 0
u = 0.399717 − 1.327110I
a = 0.22643 + 2.45843I
b = −1.17155 − 1.01534I
4.14207 − 8.58803I 0
u = −0.40967 + 1.36131I
a = 0.064099 + 1.291930I
b = −0.432795 − 0.645865I
−3.33493 − 5.79576I 0
u = −0.40967 − 1.36131I
a = 0.064099 − 1.291930I
b = −0.432795 + 0.645865I
−3.33493 + 5.79576I 0
u = 0.38534 + 1.38238I
a = −0.15294 + 2.46124I
b = 1.12554 − 1.06328I
−0.4279 + 14.3330I 0
u = 0.38534 − 1.38238I
a = −0.15294 − 2.46124I
b = 1.12554 + 1.06328I
−0.4279 − 14.3330I 0
u = −0.11768 + 1.50035I
a = −0.220688 − 0.588458I
b = 0.779647 + 0.534509I
−7.68575 − 5.74695I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.11768 − 1.50035I
a = −0.220688 + 0.588458I
b = 0.779647 − 0.534509I
−7.68575 + 5.74695I 0
u = 0.466619
a = 4.22060
b = −1.54831
−8.02557 −22.5480
u = −0.280147 + 0.187854I
a = 2.57065 + 1.29057I
b = 0.441345 − 0.382694I
−1.25055 − 0.68721I −6.23111 − 2.03128I
u = −0.280147 − 0.187854I
a = 2.57065 − 1.29057I
b = 0.441345 + 0.382694I
−1.25055 + 0.68721I −6.23111 + 2.03128I
u = 0.0963991
a = −5.35560
b = 0.614065
−0.870395 −12.0080
9
II. I
u
2
= h−au + b − u, u
2
a + a
2
+ au − u
2
+ 4a, u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
a
au + u
a
8
=
1
−u
2
a
3
=
au + a + u
au + u
a
6
=
u
2
− a + u + 1
−au − u − 1
a
11
=
−u
−u
2
− u − 1
a
10
=
u
2
+ 1
−u
2
− u − 1
a
1
=
−1
0
a
5
=
u
2
− a + u + 1
−au − u − 1
a
2
=
au + u
2
− a + 2u
−au − u − 1
a
9
=
u
2
+ 1
−u
2
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
2
a + 9u
2
− 3a + u − 1
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
(u
2
+ u − 1)
3
c
4
, c
6
(u
2
− u − 1)
3
c
5
, c
9
u
6
c
7
, c
8
(u
3
+ u
2
+ 2u + 1)
2
c
10
, c
12
(u
3
+ u
2
− 1)
2
c
11
(u
3
− u
2
+ 2u − 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
(y
2
− 3y + 1)
3
c
5
, c
9
y
6
c
7
, c
8
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
c
10
, c
12
(y
3
− y
2
+ 2y −1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.215080 + 1.307140I
a = −0.924253 + 0.460350I
b = −0.618034
−4.01109 − 2.82812I −8.01769 − 5.87116I
u = −0.215080 + 1.307140I
a = −1.19831 − 1.20521I
b = 1.61803
−11.90680 − 2.82812I −8.63833 + 7.89410I
u = −0.215080 − 1.307140I
a = −0.924253 − 0.460350I
b = −0.618034
−4.01109 + 2.82812I −8.01769 + 5.87116I
u = −0.215080 − 1.307140I
a = −1.19831 + 1.20521I
b = 1.61803
−11.90680 + 2.82812I −8.63833 − 7.89410I
u = −0.569840
a = 0.0845740
b = −0.618034
0.126494 1.18130
u = −0.569840
a = −3.83945
b = 1.61803
−7.76919 9.13080
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u
2
+ u − 1)
3
)(u
45
− 4u
44
+ ··· + 3u − 1)
c
3
((u
2
+ u − 1)
3
)(u
45
+ 4u
44
+ ··· − u + 1)
c
4
((u
2
− u − 1)
3
)(u
45
− 4u
44
+ ··· + 3u − 1)
c
5
, c
9
u
6
(u
45
+ 3u
44
+ ··· + 160u + 64)
c
6
((u
2
− u − 1)
3
)(u
45
+ 4u
44
+ ··· − u + 1)
c
7
, c
8
((u
3
+ u
2
+ 2u + 1)
2
)(u
45
+ 3u
44
+ ··· − 8u − 1)
c
10
((u
3
+ u
2
− 1)
2
)(u
45
− 3u
44
+ ··· − 542u − 97)
c
11
((u
3
− u
2
+ 2u − 1)
2
)(u
45
+ 3u
44
+ ··· − 8u − 1)
c
12
((u
3
+ u
2
− 1)
2
)(u
45
+ 19u
44
+ ··· + 653792u − 13633)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
((y
2
− 3y + 1)
3
)(y
45
− 34y
44
+ ··· + 5y −1)
c
3
, c
6
((y
2
− 3y + 1)
3
)(y
45
− 6y
44
+ ··· + 5y −1)
c
5
, c
9
y
6
(y
45
+ 35y
44
+ ··· + 58368y −4096)
c
7
, c
8
, c
11
((y
3
+ 3y
2
+ 2y −1)
2
)(y
45
+ 37y
44
+ ··· + 120y −1)
c
10
((y
3
− y
2
+ 2y −1)
2
)(y
45
− 39y
44
+ ··· + 1081792y −9409)
c
12
(y
3
− y
2
+ 2y −1)
2
· (y
45
− 59y
44
+ ··· + 530299074808y −185858689)
15
