12a
0160
(K12a
0160
)
A knot diagram
1
Linearized knot diagam
3 5 9 2 11 12 1 10 4 8 6 7
Solving Sequence
6,12
7 1 8 11
2,5
3 4 10 9
c
6
c
12
c
7
c
11
c
5
c
2
c
4
c
10
c
9
c
1
, c
3
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
51
+ u
50
+ ··· + b − u, −u
51
− u
50
+ ··· + a − 1, u
52
+ 2u
51
+ ··· + 2u + 1i
I
u
2
= hb + u + 2, a − u − 1, u
2
− u − 1i
* 2 irreducible components of dim
C
= 0, with total 54 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
51
+u
50
+· · · + b− u, −u
51
−u
50
+· · · + a− 1, u
52
+2u
51
+· · · + 2u+ 1i
(i) Arc colorings
a
6
=
1
0
a
12
=
0
u
a
7
=
1
−u
2
a
1
=
u
−u
3
+ u
a
8
=
−u
2
+ 1
u
4
− 2u
2
a
11
=
−u
u
a
2
=
u
51
+ u
50
+ ··· + 7u + 1
−u
51
− u
50
+ ··· − 4u
2
+ u
a
5
=
−u
2
+ 1
u
2
a
3
=
u
50
+ u
49
+ ··· + 18u
2
+ 5u
u
51
− 32u
49
+ ··· + 2u + 1
a
4
=
−2u
51
− u
50
+ ··· − 7u − 1
3u
51
+ 2u
50
+ ··· + u + 1
a
10
=
u
7
− 4u
5
+ 4u
3
− 2u
−u
9
+ 5u
7
− 7u
5
+ 2u
3
+ u
a
9
=
−u
12
+ 7u
10
− 17u
8
+ 18u
6
− 10u
4
+ u
2
+ 1
u
14
− 8u
12
+ 23u
10
− 28u
8
+ 12u
6
+ 2u
4
− 3u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 8u
51
+ 9u
50
+ ··· − 9u + 11
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
52
+ 29u
51
+ ··· + 29u + 1
c
2
, c
4
u
52
− 3u
51
+ ··· − 9u + 1
c
3
, c
9
u
52
− u
51
+ ··· − 4u + 4
c
5
, c
6
, c
7
c
11
, c
12
u
52
− 2u
51
+ ··· − 2u + 1
c
8
, c
10
u
52
− 15u
51
+ ··· − 248u + 16
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
52
− 9y
51
+ ··· − 593y + 1
c
2
, c
4
y
52
− 29y
51
+ ··· − 29y + 1
c
3
, c
9
y
52
− 15y
51
+ ··· − 248y + 16
c
5
, c
6
, c
7
c
11
, c
12
y
52
− 66y
51
+ ··· + 6y + 1
c
8
, c
10
y
52
+ 41y
51
+ ··· − 9504y + 256
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.002200 + 0.101685I
a = 0.620325 + 0.568001I
b = −0.338736 − 0.201694I
5.42587 + 0.91536I 15.8250 + 0.I
u = 1.002200 − 0.101685I
a = 0.620325 − 0.568001I
b = −0.338736 + 0.201694I
5.42587 − 0.91536I 15.8250 + 0.I
u = 0.900408 + 0.415514I
a = −0.686316 + 0.426160I
b = −1.32817 − 1.98079I
−2.89703 + 10.97490I 6.84879 − 9.10557I
u = 0.900408 − 0.415514I
a = −0.686316 − 0.426160I
b = −1.32817 + 1.98079I
−2.89703 − 10.97490I 6.84879 + 9.10557I
u = 0.993272 + 0.210152I
a = 0.306112 − 0.310737I
b = −1.09437 − 1.05834I
4.36145 + 5.53576I 12.8247 − 7.7042I
u = 0.993272 − 0.210152I
a = 0.306112 + 0.310737I
b = −1.09437 + 1.05834I
4.36145 − 5.53576I 12.8247 + 7.7042I
u = 0.887221 + 0.376671I
a = 0.216936 + 0.254530I
b = 0.512607 − 0.524064I
0.28073 + 6.02352I 10.12786 − 6.24694I
u = 0.887221 − 0.376671I
a = 0.216936 − 0.254530I
b = 0.512607 + 0.524064I
0.28073 − 6.02352I 10.12786 + 6.24694I
u = −0.853207 + 0.384206I
a = 0.903606 + 0.104965I
b = 1.14253 − 1.97773I
−3.75080 − 4.74137I 5.53664 + 4.96484I
u = −0.853207 − 0.384206I
a = 0.903606 − 0.104965I
b = 1.14253 + 1.97773I
−3.75080 + 4.74137I 5.53664 − 4.96484I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.825680 + 0.382316I
a = −0.111743 − 0.969230I
b = 1.57224 + 1.67207I
−3.92292 + 1.91465I 5.33907 − 3.86780I
u = 0.825680 − 0.382316I
a = −0.111743 + 0.969230I
b = 1.57224 − 1.67207I
−3.92292 − 1.91465I 5.33907 + 3.86780I
u = −0.768059 + 0.439344I
a = −0.195399 − 0.893583I
b = −1.17619 + 1.70576I
−3.69972 + 3.78870I 5.61035 − 2.00055I
u = −0.768059 − 0.439344I
a = −0.195399 + 0.893583I
b = −1.17619 − 1.70576I
−3.69972 − 3.78870I 5.61035 + 2.00055I
u = −0.869138 + 0.090566I
a = −0.624942 − 0.906814I
b = 0.124670 − 0.964759I
1.28982 − 1.61982I 10.04112 + 4.38556I
u = −0.869138 − 0.090566I
a = −0.624942 + 0.906814I
b = 0.124670 + 0.964759I
1.28982 + 1.61982I 10.04112 − 4.38556I
u = −0.778648 + 0.361263I
a = −0.381137 + 0.106397I
b = −0.541900 − 0.279638I
−0.394378 − 0.492969I 9.04055 + 1.45710I
u = −0.778648 − 0.361263I
a = −0.381137 − 0.106397I
b = −0.541900 + 0.279638I
−0.394378 + 0.492969I 9.04055 − 1.45710I
u = 0.803120
a = −0.675162
b = 2.05557
0.0369555 14.8580
u = −0.062250 + 0.639708I
a = −3.04229 + 0.33326I
b = 0.028753 − 0.273837I
−5.82794 − 7.41916I 2.02402 + 6.23213I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.062250 − 0.639708I
a = −3.04229 − 0.33326I
b = 0.028753 + 0.273837I
−5.82794 + 7.41916I 2.02402 − 6.23213I
u = 0.013286 + 0.599769I
a = 3.16045 + 0.60539I
b = 0.050264 − 0.426280I
−6.37556 + 1.41175I 0.512224 − 0.772694I
u = 0.013286 − 0.599769I
a = 3.16045 − 0.60539I
b = 0.050264 + 0.426280I
−6.37556 − 1.41175I 0.512224 + 0.772694I
u = −0.051648 + 0.589699I
a = −0.118187 + 0.761845I
b = 0.019107 + 0.393978I
−2.57484 − 2.75018I 4.73313 + 3.20106I
u = −0.051648 − 0.589699I
a = −0.118187 − 0.761845I
b = 0.019107 − 0.393978I
−2.57484 + 2.75018I 4.73313 − 3.20106I
u = −0.429505 + 0.363992I
a = −0.283846 + 0.092161I
b = −0.501430 + 0.658806I
0.985801 + 0.373941I 10.64681 + 0.40187I
u = −0.429505 − 0.363992I
a = −0.283846 − 0.092161I
b = −0.501430 − 0.658806I
0.985801 − 0.373941I 10.64681 − 0.40187I
u = −0.260485 + 0.467225I
a = −1.65751 + 0.54880I
b = −0.417669 − 0.022465I
0.44749 − 3.27943I 7.33672 + 8.28421I
u = −0.260485 − 0.467225I
a = −1.65751 − 0.54880I
b = −0.417669 + 0.022465I
0.44749 + 3.27943I 7.33672 − 8.28421I
u = −0.449135
a = −0.461577
b = −0.372331
0.706606 14.1070
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.58988
a = 2.20327
b = −3.13699
7.80979 0
u = 1.62815 + 0.10007I
a = 2.77026 + 1.09581I
b = −3.78872 − 1.31091I
4.49371 − 1.83490I 0
u = 1.62815 − 0.10007I
a = 2.77026 − 1.09581I
b = −3.78872 + 1.31091I
4.49371 + 1.83490I 0
u = 1.65492 + 0.07803I
a = 0.986803 − 0.272171I
b = −1.64509 + 0.11104I
8.08427 + 2.03863I 0
u = 1.65492 − 0.07803I
a = 0.986803 + 0.272171I
b = −1.64509 − 0.11104I
8.08427 − 2.03863I 0
u = −1.66220 + 0.09221I
a = −3.12446 + 1.04099I
b = 4.21717 − 1.24397I
4.73033 − 3.67102I 0
u = −1.66220 − 0.09221I
a = −3.12446 − 1.04099I
b = 4.21717 + 1.24397I
4.73033 + 3.67102I 0
u = −1.67125
a = −3.25510
b = 4.37303
8.84251 0
u = 1.67019 + 0.09664I
a = −2.99043 − 2.84568I
b = 4.30768 + 4.47913I
5.04044 + 6.55656I 0
u = 1.67019 − 0.09664I
a = −2.99043 + 2.84568I
b = 4.30768 − 4.47913I
5.04044 − 6.55656I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.68080 + 0.01746I
a = 0.01213 − 2.15798I
b = −0.39566 + 3.61473I
10.32590 + 1.99807I 0
u = 1.68080 − 0.01746I
a = 0.01213 + 2.15798I
b = −0.39566 − 3.61473I
10.32590 − 1.99807I 0
u = −1.68130 + 0.09775I
a = −0.765995 − 0.522974I
b = 1.297310 + 0.421881I
9.25802 − 7.84981I 0
u = −1.68130 − 0.09775I
a = −0.765995 + 0.522974I
b = 1.297310 − 0.421881I
9.25802 + 7.84981I 0
u = −1.68310 + 0.11088I
a = 3.15694 − 2.48000I
b = −4.50686 + 3.86266I
6.1040 − 13.0204I 0
u = −1.68310 − 0.11088I
a = 3.15694 + 2.48000I
b = −4.50686 − 3.86266I
6.1040 + 13.0204I 0
u = −1.70835 + 0.02404I
a = 0.042019 + 0.436305I
b = 0.256089 − 1.030400I
15.0483 − 1.4053I 0
u = −1.70835 − 0.02404I
a = 0.042019 − 0.436305I
b = 0.256089 + 1.030400I
15.0483 + 1.4053I 0
u = −1.70818 + 0.04745I
a = 1.55667 − 1.83823I
b = −1.98030 + 3.09035I
13.9392 − 6.5232I 0
u = −1.70818 − 0.04745I
a = 1.55667 + 1.83823I
b = −1.98030 − 3.09035I
13.9392 + 6.5232I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.123626 + 0.224585I
a = 0.84430 + 2.89252I
b = 0.727029 − 0.150879I
−1.62775 + 0.54260I −3.36212 − 1.40035I
u = 0.123626 − 0.224585I
a = 0.84430 − 2.89252I
b = 0.727029 + 0.150879I
−1.62775 − 0.54260I −3.36212 + 1.40035I
10
II. I
u
2
= hb + u + 2, a − u − 1, u
2
− u − 1i
(i) Arc colorings
a
6
=
1
0
a
12
=
0
u
a
7
=
1
−u − 1
a
1
=
u
−u − 1
a
8
=
−u
u
a
11
=
−u
u
a
2
=
u + 1
−u − 2
a
5
=
−u
u + 1
a
3
=
1
−1
a
4
=
1
−1
a
10
=
−u
u
a
9
=
−u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
c
3
, c
8
, c
9
c
10
u
2
c
4
(u + 1)
2
c
5
, c
6
, c
7
u
2
− u − 1
c
11
, c
12
u
2
+ u − 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y − 1)
2
c
3
, c
8
, c
9
c
10
y
2
c
5
, c
6
, c
7
c
11
, c
12
y
2
− 3y + 1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 0.381966
b = −1.38197
−0.657974 3.00000
u = 1.61803
a = 2.61803
b = −3.61803
7.23771 3.00000
14
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
2
)(u
52
+ 29u
51
+ ··· + 29u + 1)
c
2
((u − 1)
2
)(u
52
− 3u
51
+ ··· − 9u + 1)
c
3
, c
9
u
2
(u
52
− u
51
+ ··· − 4u + 4)
c
4
((u + 1)
2
)(u
52
− 3u
51
+ ··· − 9u + 1)
c
5
, c
6
, c
7
(u
2
− u − 1)(u
52
− 2u
51
+ ··· − 2u + 1)
c
8
, c
10
u
2
(u
52
− 15u
51
+ ··· − 248u + 16)
c
11
, c
12
(u
2
+ u − 1)(u
52
− 2u
51
+ ··· − 2u + 1)
15
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y − 1)
2
)(y
52
− 9y
51
+ ··· − 593y + 1)
c
2
, c
4
((y − 1)
2
)(y
52
− 29y
51
+ ··· − 29y + 1)
c
3
, c
9
y
2
(y
52
− 15y
51
+ ··· − 248y + 16)
c
5
, c
6
, c
7
c
11
, c
12
(y
2
− 3y + 1)(y
52
− 66y
51
+ ··· + 6y + 1)
c
8
, c
10
y
2
(y
52
+ 41y
51
+ ··· − 9504y + 256)
16
