12n
0564
(K12n
0564
)
A knot diagram
1
Linearized knot diagam
3 6 8 12 11 2 4 3 6 5 10 9
Solving Sequence
5,11 3,6
2 1 10 12 4 9 8 7
c
5
c
2
c
1
c
10
c
11
c
4
c
9
c
8
c
7
c
3
, c
6
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
16
− 3u
15
+ ··· + b + 3, 3u
16
+ 7u
15
+ ··· + 2a − 7, u
17
+ 3u
16
+ ··· − 5u − 2i
I
u
2
= hu
5
a + u
5
− 2u
3
a + u
2
a − u
3
+ 2au + b + u + 1, 2u
4
a + 2u
5
− u
3
a − u
4
− 2u
2
a − 2u
3
+ a
2
+ 2au + u
2
− u,
u
6
− u
5
− u
4
+ 2u
3
− u + 1i
I
u
3
= hu
6
− 2u
4
+ u
3
+ u
2
+ b − u + 1, u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ u
3
+ u
2
+ a + u − 2, u
10
− 3u
8
+ 4u
6
− u
4
− u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 39 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
16
−3u
15
+· · ·+b+3, 3u
16
+7u
15
+· · ·+2a−7, u
17
+3u
16
+· · ·−5u−2i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
3
=
−
3
2
u
16
−
7
2
u
15
+ ··· +
11
2
u +
7
2
u
16
+ 3u
15
+ ··· − 5u − 3
a
6
=
1
−u
2
a
2
=
−
5
2
u
16
−
11
2
u
15
+ ··· +
17
2
u +
9
2
2u
16
+ 5u
15
+ ··· − 8u − 5
a
1
=
u
11
− 2u
9
+ 2u
7
+ u
3
−u
13
+ 3u
11
− 5u
9
+ 4u
7
− 2u
5
− u
3
+ u
a
10
=
−u
u
a
12
=
u
3
−u
3
+ u
a
4
=
u
6
− u
4
+ 1
−u
6
+ 2u
4
− u
2
a
9
=
−u
3
u
5
− u
3
+ u
a
8
=
1
2
u
16
+
3
2
u
15
+ ··· −
5
2
u −
3
2
u
16
+ 2u
15
+ ··· − 2u − 1
a
7
=
−
1
2
u
16
−
1
2
u
15
+ ··· +
1
2
u −
1
2
u
16
+ 2u
15
+ ··· − 2u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
16
− 4u
15
+ 6u
14
+ 22u
13
+ 2u
12
− 44u
11
− 38u
10
+ 30u
9
+
64u
8
+ 26u
7
− 36u
6
− 48u
5
− 16u
4
+ 18u
3
+ 12u
2
+ 8u + 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
17
+ 30u
16
+ ··· − 8u − 1
c
2
, c
3
, c
6
c
7
, c
8
u
17
+ 15u
15
+ ··· + 2u − 1
c
4
, c
9
u
17
+ 9u
16
+ ··· − 77u − 26
c
5
, c
10
u
17
+ 3u
16
+ ··· − 5u − 2
c
11
u
17
− 9u
16
+ ··· + 5u − 4
c
12
u
17
+ u
16
+ ··· + 953u − 416
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
17
− 98y
16
+ ··· + 64y −1
c
2
, c
3
, c
6
c
7
, c
8
y
17
+ 30y
16
+ ··· − 8y −1
c
4
, c
9
y
17
+ 11y
16
+ ··· + 3589y −676
c
5
, c
10
y
17
− 9y
16
+ ··· + 5y −4
c
11
y
17
− y
16
+ ··· + 193y −16
c
12
y
17
+ 83y
16
+ ··· + 3035633y −173056
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.270803 + 0.900641I
a = −0.709209 − 0.184245I
b = −0.73238 − 2.39254I
−15.8747 + 5.8792I 1.30073 − 1.97139I
u = −0.270803 − 0.900641I
a = −0.709209 + 0.184245I
b = −0.73238 + 2.39254I
−15.8747 − 5.8792I 1.30073 + 1.97139I
u = −0.810610 + 0.750728I
a = −1.317050 + 0.001505I
b = 0.058875 + 0.461436I
−19.2384 − 2.7966I 0.86008 + 2.64584I
u = −0.810610 − 0.750728I
a = −1.317050 − 0.001505I
b = 0.058875 − 0.461436I
−19.2384 + 2.7966I 0.86008 − 2.64584I
u = −0.776721 + 0.418654I
a = 0.541225 + 0.547668I
b = −0.518708 − 0.343588I
−0.92435 − 1.82362I 3.05958 + 5.69158I
u = −0.776721 − 0.418654I
a = 0.541225 − 0.547668I
b = −0.518708 + 0.343588I
−0.92435 + 1.82362I 3.05958 − 5.69158I
u = 1.158560 + 0.445647I
a = −0.511951 + 0.740081I
b = 0.883342 + 0.177811I
4.23482 + 2.69674I 10.58660 + 0.80633I
u = 1.158560 − 0.445647I
a = −0.511951 − 0.740081I
b = 0.883342 − 0.177811I
4.23482 − 2.69674I 10.58660 − 0.80633I
u = −1.172560 + 0.461545I
a = −1.225960 + 0.029576I
b = 1.068970 − 0.795251I
4.11934 − 5.61810I 10.07697 + 8.39649I
u = −1.172560 − 0.461545I
a = −1.225960 − 0.029576I
b = 1.068970 + 0.795251I
4.11934 + 5.61810I 10.07697 − 8.39649I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.277260 + 0.264559I
a = 0.07926 − 2.43683I
b = −1.56412 + 1.91811I
−10.78020 − 2.11031I 5.71043 + 0.08866I
u = 1.277260 − 0.264559I
a = 0.07926 + 2.43683I
b = −1.56412 − 1.91811I
−10.78020 + 2.11031I 5.71043 − 0.08866I
u = −0.050223 + 0.684600I
a = 0.511482 − 0.219170I
b = 0.581679 + 0.400436I
0.95257 + 1.33285I 6.82195 − 5.20077I
u = −0.050223 − 0.684600I
a = 0.511482 + 0.219170I
b = 0.581679 − 0.400436I
0.95257 − 1.33285I 6.82195 + 5.20077I
u = 0.682195
a = 0.679402
b = 0.0468590
0.845049 13.0200
u = −1.196000 + 0.589565I
a = 2.54251 − 1.12472I
b = −0.80109 + 2.93271I
−13.0820 − 11.3274I 4.07376 + 5.51607I
u = −1.196000 − 0.589565I
a = 2.54251 + 1.12472I
b = −0.80109 − 2.93271I
−13.0820 + 11.3274I 4.07376 − 5.51607I
6
II. I
u
2
= hu
5
a + u
5
− 2u
3
a + u
2
a − u
3
+ 2au + b + u + 1, 2u
4
a + 2u
5
+ · · · +
a
2
− u, u
6
− u
5
− u
4
+ 2u
3
− u + 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
3
=
a
−u
5
a − u
5
+ 2u
3
a − u
2
a + u
3
− 2au − u − 1
a
6
=
1
−u
2
a
2
=
u
5
a + u
5
− 2u
3
a − u
3
+ 2au + a + u + 1
−u
5
a + u
4
a − 2u
5
+ 2u
3
a + u
4
− 2u
2
a + 2u
3
− 2au − 2u
2
+ a − u
a
1
=
2u
3
+ 1
−2u
3
+ 2u
a
10
=
−u
u
a
12
=
u
3
−u
3
+ u
a
4
=
u
5
− 2u
3
+ u
−u
5
+ u
4
+ 2u
3
− u
2
− u + 1
a
9
=
−u
3
u
5
− u
3
+ u
a
8
=
u
5
a − u
3
a + 2u
4
− u
3
+ au − 2u
2
+ a + 2u + 1
−u
5
a + u
4
a − u
5
+ u
3
a − 2u
2
a + 2u
3
− u
2
+ a − 2u + 1
a
7
=
−u
5
a − u
5
+ 2u
3
a + 2u
4
− u
2
a + u
3
− 2au − 2u
2
+ a + u + 1
u
5
a − 2u
3
a − u
4
+ u
2
a + 2au − a − u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
4
+ 4u
2
− 4u + 2
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
12
+ 15u
11
+ ··· + 1324u + 289
c
2
, c
3
, c
6
c
7
, c
8
u
12
− u
11
+ ··· − 28u + 17
c
4
, c
9
, c
11
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
2
c
5
, c
10
, c
12
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
12
− 13y
11
+ ··· + 423772y + 83521
c
2
, c
3
, c
6
c
7
, c
8
y
12
+ 15y
11
+ ··· + 1324y + 289
c
4
, c
9
, c
11
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
c
5
, c
10
, c
12
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.002190 + 0.295542I
a = 0.912198 − 0.739675I
b = −0.414477 − 0.040596I
−1.39926 − 0.92430I 7.71672 + 0.79423I
u = −1.002190 + 0.295542I
a = 1.20218 + 2.00149I
b = −1.84373 − 0.52857I
−1.39926 − 0.92430I 7.71672 + 0.79423I
u = −1.002190 − 0.295542I
a = 0.912198 + 0.739675I
b = −0.414477 + 0.040596I
−1.39926 + 0.92430I 7.71672 − 0.79423I
u = −1.002190 − 0.295542I
a = 1.20218 − 2.00149I
b = −1.84373 + 0.52857I
−1.39926 + 0.92430I 7.71672 − 0.79423I
u = 0.428243 + 0.664531I
a = −0.486207 + 0.830594I
b = 0.015656 − 0.966738I
−5.18047 − 0.92430I 0.283283 + 0.794226I
u = 0.428243 + 0.664531I
a = −0.860954 − 0.361329I
b = −1.09861 + 1.55912I
−5.18047 − 0.92430I 0.283283 + 0.794226I
u = 0.428243 − 0.664531I
a = −0.486207 − 0.830594I
b = 0.015656 + 0.966738I
−5.18047 + 0.92430I 0.283283 − 0.794226I
u = 0.428243 − 0.664531I
a = −0.860954 + 0.361329I
b = −1.09861 − 1.55912I
−5.18047 + 0.92430I 0.283283 − 0.794226I
u = 1.073950 + 0.558752I
a = −1.37466 − 0.68992I
b = 0.524769 + 0.629076I
−3.28987 + 5.69302I 4.00000 − 5.51057I
u = 1.073950 + 0.558752I
a = 2.60745 − 0.30647I
b = −1.68361 − 1.82922I
−3.28987 + 5.69302I 4.00000 − 5.51057I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.073950 − 0.558752I
a = −1.37466 + 0.68992I
b = 0.524769 − 0.629076I
−3.28987 − 5.69302I 4.00000 + 5.51057I
u = 1.073950 − 0.558752I
a = 2.60745 + 0.30647I
b = −1.68361 + 1.82922I
−3.28987 − 5.69302I 4.00000 + 5.51057I
11
III. I
u
3
=
hu
6
−2u
4
+u
3
+u
2
+b−u+1, u
8
+u
7
+· · ·+a−2, u
10
−3u
8
+4u
6
−u
4
−u
2
+1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
3
=
−u
8
− u
7
+ 3u
6
+ 2u
5
− 3u
4
− u
3
− u
2
− u + 2
−u
6
+ 2u
4
− u
3
− u
2
+ u − 1
a
6
=
1
−u
2
a
2
=
u
9
− u
8
− 3u
7
+ 3u
6
+ 3u
5
− 3u
4
+ u
3
− u
2
− 2u + 2
−u
9
+ 3u
7
− u
6
− 3u
5
+ 2u
4
− u
3
− u
2
+ 2u − 1
a
1
=
u
9
− 2u
7
+ u
5
+ 2u
3
− u
−u
9
+ 3u
7
− 3u
5
+ u
a
10
=
−u
u
a
12
=
u
3
−u
3
+ u
a
4
=
u
6
− u
4
+ 1
−u
6
+ 2u
4
− u
2
a
9
=
−u
3
u
5
− u
3
+ u
a
8
=
−u
8
+ u
7
+ 2u
6
− 2u
5
− 2u
4
+ u
3
− u
2
+ u + 1
−u
9
+ u
8
+ 2u
7
− 2u
6
− u
5
+ 2u
4
− 2u
3
+ u
a
7
=
−u
8
+ u
7
+ 2u
6
− 2u
5
− 2u
4
+ 2u
3
− u
2
+ u + 1
−u
9
+ u
8
+ 2u
7
− 2u
6
− 2u
5
+ 2u
4
− u
3
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
8
+ 8u
6
− 8u
4
− 4u
2
+ 4
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
10
c
2
, c
3
, c
6
c
7
, c
8
(u
2
+ 1)
5
c
4
, c
9
u
10
+ 5u
8
+ 8u
6
+ 3u
4
− u
2
+ 1
c
5
, c
10
u
10
− 3u
8
+ 4u
6
− u
4
− u
2
+ 1
c
11
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
c
12
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y −1)
10
c
2
, c
3
, c
6
c
7
, c
8
(y + 1)
10
c
4
, c
9
(y
5
+ 5y
4
+ 8y
3
+ 3y
2
− y + 1)
2
c
5
, c
10
(y
5
− 3y
4
+ 4y
3
− y
2
− y + 1)
2
c
11
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
c
12
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.822375 + 0.339110I
a = 1.88547 + 1.25135I
b = −1.75626 − 0.65077I
−2.96077 − 1.53058I 0.51511 + 4.43065I
u = −0.822375 − 0.339110I
a = 1.88547 − 1.25135I
b = −1.75626 + 0.65077I
−2.96077 + 1.53058I 0.51511 − 4.43065I
u = 0.822375 + 0.339110I
a = −0.32986 − 1.50891I
b = −0.656443 + 0.030936I
−2.96077 + 1.53058I 0.51511 − 4.43065I
u = 0.822375 − 0.339110I
a = −0.32986 + 1.50891I
b = −0.656443 − 0.030936I
−2.96077 − 1.53058I 0.51511 + 4.43065I
u = 0.766826I
a = 0.821196 + 0.370286I
b = 0.482881 + 1.217740I
−0.888787 1.48110
u = − 0.766826I
a = 0.821196 − 0.370286I
b = 0.482881 − 1.217740I
−0.888787 1.48110
u = −1.200150 + 0.455697I
a = −1.56305 + 1.07974I
b = 0.74575 − 2.04068I
2.58269 − 4.40083I 4.74431 + 3.49859I
u = −1.200150 − 0.455697I
a = −1.56305 − 1.07974I
b = 0.74575 + 2.04068I
2.58269 + 4.40083I 4.74431 − 3.49859I
u = 1.200150 + 0.455697I
a = 0.186244 + 1.292420I
b = 1.18408 − 0.79689I
2.58269 + 4.40083I 4.74431 − 3.49859I
u = 1.200150 − 0.455697I
a = 0.186244 − 1.292420I
b = 1.18408 + 0.79689I
2.58269 − 4.40083I 4.74431 + 3.49859I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
10
)(u
12
+ 15u
11
+ ··· + 1324u + 289)
· (u
17
+ 30u
16
+ ··· − 8u − 1)
c
2
, c
3
, c
6
c
7
, c
8
((u
2
+ 1)
5
)(u
12
− u
11
+ ··· − 28u + 17)(u
17
+ 15u
15
+ ··· + 2u − 1)
c
4
, c
9
((u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
2
)(u
10
+ 5u
8
+ 8u
6
+ 3u
4
− u
2
+ 1)
· (u
17
+ 9u
16
+ ··· − 77u − 26)
c
5
, c
10
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
(u
10
− 3u
8
+ 4u
6
− u
4
− u
2
+ 1)
· (u
17
+ 3u
16
+ ··· − 5u − 2)
c
11
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
2
· (u
17
− 9u
16
+ ··· + 5u − 4)
c
12
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
· (u
17
+ u
16
+ ··· + 953u − 416)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
10
)(y
12
− 13y
11
+ ··· + 423772y + 83521)
· (y
17
− 98y
16
+ ··· + 64y −1)
c
2
, c
3
, c
6
c
7
, c
8
((y + 1)
10
)(y
12
+ 15y
11
+ ··· + 1324y + 289)
· (y
17
+ 30y
16
+ ··· − 8y −1)
c
4
, c
9
(y
5
+ 5y
4
+ 8y
3
+ 3y
2
− y + 1)
2
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
· (y
17
+ 11y
16
+ ··· + 3589y −676)
c
5
, c
10
(y
5
− 3y
4
+ 4y
3
− y
2
− y + 1)
2
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
· (y
17
− 9y
16
+ ··· + 5y −4)
c
11
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
· (y
17
− y
16
+ ··· + 193y −16)
c
12
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
· (y
17
+ 83y
16
+ ··· + 3035633y −173056)
17
