12n
0163
(K12n
0163
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 12 11 3 5 7 6 10 9
Solving Sequence
6,11
7 10 12
3,5
2 1 4 9 8
c
6
c
10
c
11
c
5
c
2
c
1
c
4
c
9
c
8
c
3
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
31
− u
30
+ ··· + b − 2u, −u
31
+ u
30
+ ··· + a + 5u, u
32
− 2u
31
+ ··· + 5u − 1i
I
u
2
= hu
6
− 2u
4
− u
3
+ u
2
+ b + u + 1, u
7
+ u
6
− 2u
5
− 2u
4
+ u
3
+ u
2
+ a + u + 1,
u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1i
* 2 irreducible components of dim
C
= 0, with total 41 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
= hu
31
−u
30
+· · ·+b−2u, −u
31
+u
30
+· · ·+a+5u, u
32
−2u
31
+· · ·+5u−1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
u
2
a
10
=
u
u
a
12
=
−u
3
−u
3
+ u
a
3
=
u
31
− u
30
+ ··· + 8u
2
− 5u
−u
31
+ u
30
+ ··· − 2u
2
+ 2u
a
5
=
u
6
− u
4
+ 1
u
6
− 2u
4
+ u
2
a
2
=
−u
29
+ 8u
27
+ ··· + 4u
2
− 2u
−u
31
+ u
30
+ ··· − u
2
+ 2u
a
1
=
−u
11
+ 2u
9
− 2u
7
− u
3
−u
13
+ 3u
11
− 5u
9
+ 4u
7
− 2u
5
− u
3
+ u
a
4
=
u
31
− u
30
+ ··· + u + 1
u
31
− u
30
+ ··· + 6u
3
− u
a
9
=
u
3
u
5
− u
3
+ u
a
8
=
u
17
− 4u
15
+ 7u
13
− 4u
11
− 3u
9
+ 6u
7
− 2u
5
+ u
u
17
− 5u
15
+ 11u
13
− 12u
11
+ 5u
9
+ 2u
7
− 2u
5
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −10u
31
+ 11u
30
+ 82u
29
− 109u
28
− 303u
27
+ 492u
26
+ 589u
25
−
1289u
24
− 454u
23
+ 2036u
22
− 559u
21
− 1667u
20
+ 1827u
19
− 152u
18
− 1853u
17
+
1907u
16
+ 315u
15
− 1818u
14
+ 1090u
13
+ 295u
12
− 980u
11
+ 734u
10
+ 52u
9
− 568u
8
+
382u
7
+ 24u
6
− 173u
5
+ 145u
4
− 51u
3
− 48u
2
+ 51u − 23
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
32
+ 50u
31
+ ··· + 37u + 1
c
2
, c
4
u
32
− 10u
31
+ ··· + 7u − 1
c
3
, c
7
u
32
− u
31
+ ··· + 1024u + 512
c
5
, c
9
u
32
+ 6u
31
+ ··· + 49u + 5
c
6
, c
10
u
32
+ 2u
31
+ ··· − 5u − 1
c
8
u
32
+ 2u
31
+ ··· − 3u − 1
c
11
u
32
+ 18u
31
+ ··· + 9u + 1
c
12
u
32
− 6u
31
+ ··· + 1421u + 145
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
32
− 126y
31
+ ··· − 181y + 1
c
2
, c
4
y
32
− 50y
31
+ ··· − 37y + 1
c
3
, c
7
y
32
− 57y
31
+ ··· + 1310720y + 262144
c
5
, c
9
y
32
+ 30y
31
+ ··· − 461y + 25
c
6
, c
10
y
32
− 18y
31
+ ··· − 9y + 1
c
8
y
32
− 66y
31
+ ··· − 9y + 1
c
11
y
32
− 6y
31
+ ··· − 17y + 1
c
12
y
32
− 30y
31
+ ··· + 1291979y + 21025
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.920983 + 0.401471I
a = 0.917136 − 1.039840I
b = 0.415154 + 0.155618I
−2.04683 − 3.28761I −12.15173 + 6.50570I
u = 0.920983 − 0.401471I
a = 0.917136 + 1.039840I
b = 0.415154 − 0.155618I
−2.04683 + 3.28761I −12.15173 − 6.50570I
u = −0.907439 + 0.255427I
a = 1.69960 + 1.28460I
b = 1.49817 + 0.96455I
−3.08320 + 1.04878I −12.81708 − 5.09104I
u = −0.907439 − 0.255427I
a = 1.69960 − 1.28460I
b = 1.49817 − 0.96455I
−3.08320 − 1.04878I −12.81708 + 5.09104I
u = −0.777091 + 0.477881I
a = −0.536566 − 0.666041I
b = −0.649263 − 0.366597I
1.34788 + 1.99721I −0.92513 − 4.43380I
u = −0.777091 − 0.477881I
a = −0.536566 + 0.666041I
b = −0.649263 + 0.366597I
1.34788 − 1.99721I −0.92513 + 4.43380I
u = 0.953782 + 0.580631I
a = −0.13862 + 2.54860I
b = −0.84091 + 1.57585I
−11.82200 − 5.87879I −11.28951 + 5.22144I
u = 0.953782 − 0.580631I
a = −0.13862 − 2.54860I
b = −0.84091 − 1.57585I
−11.82200 + 5.87879I −11.28951 − 5.22144I
u = 0.124644 + 0.870094I
a = −0.488908 − 1.119880I
b = 2.86386 + 0.17983I
−16.5159 + 6.3822I −11.71583 − 2.55779I
u = 0.124644 − 0.870094I
a = −0.488908 + 1.119880I
b = 2.86386 − 0.17983I
−16.5159 − 6.3822I −11.71583 + 2.55779I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.534278 + 0.665517I
a = −1.317440 − 0.135023I
b = −1.75596 − 0.78631I
−10.62080 + 1.07852I −9.17455 + 0.26224I
u = 0.534278 − 0.665517I
a = −1.317440 + 0.135023I
b = −1.75596 + 0.78631I
−10.62080 − 1.07852I −9.17455 − 0.26224I
u = −1.15702
a = −2.82340
b = −1.47424
−16.1082 −16.3550
u = 0.030254 + 0.815370I
a = 0.892190 + 0.010385I
b = −1.84976 − 0.76361I
−5.35010 + 1.73289I −12.16292 − 1.23498I
u = 0.030254 − 0.815370I
a = 0.892190 − 0.010385I
b = −1.84976 + 0.76361I
−5.35010 − 1.73289I −12.16292 + 1.23498I
u = 1.183180 + 0.412649I
a = 0.200378 − 0.628582I
b = 0.533514 + 0.079289I
−4.65799 − 1.98947I −10.09461 + 1.11362I
u = 1.183180 − 0.412649I
a = 0.200378 + 0.628582I
b = 0.533514 − 0.079289I
−4.65799 + 1.98947I −10.09461 − 1.11362I
u = −0.104577 + 0.739226I
a = −0.423697 + 0.357022I
b = 0.508988 + 0.480725I
−1.00221 − 1.97931I −5.05108 + 2.63229I
u = −0.104577 − 0.739226I
a = −0.423697 − 0.357022I
b = 0.508988 − 0.480725I
−1.00221 + 1.97931I −5.05108 − 2.63229I
u = −1.179880 + 0.490061I
a = 1.118620 − 0.044684I
b = 1.040940 − 0.674261I
−4.09969 + 6.54761I −8.24294 − 5.21749I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.179880 − 0.490061I
a = 1.118620 + 0.044684I
b = 1.040940 + 0.674261I
−4.09969 − 6.54761I −8.24294 + 5.21749I
u = −1.222640 + 0.444091I
a = −2.70913 − 0.19363I
b = −2.46800 + 1.73197I
−9.06220 + 2.72579I −15.5836 − 2.2463I
u = −1.222640 − 0.444091I
a = −2.70913 + 0.19363I
b = −2.46800 − 1.73197I
−9.06220 − 2.72579I −15.5836 + 2.2463I
u = 1.218380 + 0.471883I
a = −1.38525 + 1.82670I
b = −2.45044 − 0.06343I
−8.86209 − 6.36925I −15.2047 + 4.5478I
u = 1.218380 − 0.471883I
a = −1.38525 − 1.82670I
b = −2.45044 + 0.06343I
−8.86209 + 6.36925I −15.2047 − 4.5478I
u = −1.257560 + 0.385869I
a = 2.56331 + 1.01533I
b = 2.30376 − 1.55155I
18.6867 − 2.0727I −15.8105 − 0.3085I
u = −1.257560 − 0.385869I
a = 2.56331 − 1.01533I
b = 2.30376 + 1.55155I
18.6867 + 2.0727I −15.8105 + 0.3085I
u = 1.223360 + 0.521615I
a = 2.66517 − 2.37727I
b = 4.04494 − 0.12748I
19.6654 − 11.4323I −14.6387 + 5.6746I
u = 1.223360 − 0.521615I
a = 2.66517 + 2.37727I
b = 4.04494 + 0.12748I
19.6654 + 11.4323I −14.6387 − 5.6746I
u = 0.653875
a = −0.683891
b = 0.233434
−0.923628 −10.7930
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.511886 + 0.195210I
a = −0.803157 + 0.226384I
b = 0.425410 − 0.116435I
−0.941625 + 0.020840I −9.56315 + 0.03156I
u = 0.511886 − 0.195210I
a = −0.803157 − 0.226384I
b = 0.425410 + 0.116435I
−0.941625 − 0.020840I −9.56315 − 0.03156I
8
II. I
u
2
= hu
6
− 2u
4
− u
3
+ u
2
+ b + u + 1, u
7
+ u
6
− 2u
5
− 2u
4
+ u
3
+ u
2
+
a + u + 1, u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
u
2
a
10
=
u
u
a
12
=
−u
3
−u
3
+ u
a
3
=
−u
7
− u
6
+ 2u
5
+ 2u
4
− u
3
− u
2
− u − 1
−u
6
+ 2u
4
+ u
3
− u
2
− u − 1
a
5
=
u
6
− u
4
+ 1
u
6
− 2u
4
+ u
2
a
2
=
−u
7
− 2u
6
+ 2u
5
+ 3u
4
− u
3
− u
2
− u − 2
−2u
6
+ 4u
4
+ u
3
− 2u
2
− u − 1
a
1
=
−u
6
+ u
4
− 1
−u
6
+ 2u
4
− u
2
a
4
=
−u
7
− u
6
+ 2u
5
+ 2u
4
− u
3
− u
2
− u − 1
−u
6
+ 2u
4
+ u
3
− u
2
− u − 1
a
9
=
u
3
u
5
− u
3
+ u
a
8
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
8
− 2u
7
− u
6
+ 4u
5
+ 3u
4
− 6u
3
− u
2
− u − 10
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
9
c
3
, c
7
u
9
c
4
(u + 1)
9
c
5
u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1
c
6
u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1
c
8
, c
12
u
9
− u
8
+ 2u
7
− u
6
+ 3u
5
− u
4
+ 2u
3
+ u + 1
c
9
u
9
− 3u
8
+ 8u
7
− 13u
6
+ 17u
5
− 17u
4
+ 12u
3
− 6u
2
+ u + 1
c
10
u
9
− u
8
− 2u
7
+ 3u
6
+ u
5
− 3u
4
+ 2u
3
− u + 1
c
11
u
9
+ 5u
8
+ 12u
7
+ 15u
6
+ 9u
5
− u
4
− 4u
3
− 2u
2
+ u + 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
9
c
3
, c
7
y
9
c
5
, c
9
y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1
c
6
, c
10
y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1
c
8
, c
12
y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1
c
11
y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y −1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.772920 + 0.510351I
a = −0.147032 − 1.012940I
b = −0.848670 − 0.225310I
0.13850 + 2.09337I −9.40455 − 4.13635I
u = −0.772920 − 0.510351I
a = −0.147032 + 1.012940I
b = −0.848670 + 0.225310I
0.13850 − 2.09337I −9.40455 + 4.13635I
u = 0.825933
a = −1.95176
b = −1.33142
−2.84338 −12.5800
u = 1.173910 + 0.391555I
a = 0.679689 + 0.626017I
b = 0.25695 + 1.39155I
−6.01628 − 1.33617I −15.1179 + 0.3856I
u = 1.173910 − 0.391555I
a = 0.679689 − 0.626017I
b = 0.25695 − 1.39155I
−6.01628 + 1.33617I −15.1179 − 0.3856I
u = −0.141484 + 0.739668I
a = −0.541407 + 0.753907I
b = 0.443165 − 0.284059I
−2.26187 − 2.45442I −10.97405 + 3.19656I
u = −0.141484 − 0.739668I
a = −0.541407 − 0.753907I
b = 0.443165 + 0.284059I
−2.26187 + 2.45442I −10.97405 − 3.19656I
u = −1.172470 + 0.500383I
a = 0.484630 + 0.655708I
b = 1.314260 + 0.168567I
−5.24306 + 7.08493I −14.2133 − 6.7157I
u = −1.172470 − 0.500383I
a = 0.484630 − 0.655708I
b = 1.314260 − 0.168567I
−5.24306 − 7.08493I −14.2133 + 6.7157I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
32
+ 50u
31
+ ··· + 37u + 1)
c
2
((u − 1)
9
)(u
32
− 10u
31
+ ··· + 7u − 1)
c
3
, c
7
u
9
(u
32
− u
31
+ ··· + 1024u + 512)
c
4
((u + 1)
9
)(u
32
− 10u
31
+ ··· + 7u − 1)
c
5
(u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1)
· (u
32
+ 6u
31
+ ··· + 49u + 5)
c
6
(u
9
+ u
8
+ ··· − u − 1)(u
32
+ 2u
31
+ ··· − 5u − 1)
c
8
(u
9
− u
8
+ ··· + u + 1)(u
32
+ 2u
31
+ ··· − 3u − 1)
c
9
(u
9
− 3u
8
+ 8u
7
− 13u
6
+ 17u
5
− 17u
4
+ 12u
3
− 6u
2
+ u + 1)
· (u
32
+ 6u
31
+ ··· + 49u + 5)
c
10
(u
9
− u
8
+ ··· − u + 1)(u
32
+ 2u
31
+ ··· − 5u − 1)
c
11
(u
9
+ 5u
8
+ 12u
7
+ 15u
6
+ 9u
5
− u
4
− 4u
3
− 2u
2
+ u + 1)
· (u
32
+ 18u
31
+ ··· + 9u + 1)
c
12
(u
9
− u
8
+ 2u
7
− u
6
+ 3u
5
− u
4
+ 2u
3
+ u + 1)
· (u
32
− 6u
31
+ ··· + 1421u + 145)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
9
)(y
32
− 126y
31
+ ··· − 181y + 1)
c
2
, c
4
((y −1)
9
)(y
32
− 50y
31
+ ··· − 37y + 1)
c
3
, c
7
y
9
(y
32
− 57y
31
+ ··· + 1310720y + 262144)
c
5
, c
9
(y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1)
· (y
32
+ 30y
31
+ ··· − 461y + 25)
c
6
, c
10
(y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1)
· (y
32
− 18y
31
+ ··· − 9y + 1)
c
8
(y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1)
· (y
32
− 66y
31
+ ··· − 9y + 1)
c
11
(y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y −1)
· (y
32
− 6y
31
+ ··· − 17y + 1)
c
12
(y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1)
· (y
32
− 30y
31
+ ··· + 1291979y + 21025)
14
