12n
0076
(K12n
0076
)
A knot diagram
1
Linearized knot diagam
3 5 6 2 10 4 5 12 11 6 8 9
Solving Sequence
5,10
6
3,11
2 1 4 7 8 9 12
c
5
c
10
c
2
c
1
c
4
c
6
c
7
c
9
c
12
c
3
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−1.99559 × 10
15
u
33
+ 5.12786 × 10
15
u
32
+ ··· + 3.29400 × 10
15
b + 2.31961 × 10
14
,
− 6.21494 × 10
15
u
33
+ 1.10612 × 10
16
u
32
+ ··· + 3.29400 × 10
15
a + 8.86419 × 10
15
, u
34
− 2u
33
+ ··· − u + 1i
I
u
2
= hb + 1, 2u
7
− u
6
− 3u
5
+ 3u
4
+ 4u
3
− 3u
2
+ a − 2u + 4, u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 42 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−2.00×10
15
u
33
+5.13×10
15
u
32
+· · ·+3.29×10
15
b+2.32×10
14
, −6.21×
10
15
u
33
+1.11×10
16
u
32
+· · ·+3.29×10
15
a+8.86×10
15
, u
34
−2u
33
+· · ·−u+1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
u
2
a
3
=
1.88675u
33
− 3.35800u
32
+ ··· − 0.414433u − 2.69102
0.605825u
33
− 1.55673u
32
+ ··· − 2.42216u − 0.0704193
a
11
=
−u
−u
3
+ u
a
2
=
2.49257u
33
− 4.91473u
32
+ ··· − 2.83659u − 2.76144
0.605825u
33
− 1.55673u
32
+ ··· − 2.42216u − 0.0704193
a
1
=
0.684721u
33
− 1.85986u
32
+ ··· − 3.17571u + 0.213786
−0.0613225u
33
+ 0.190476u
32
+ ··· + 0.906554u − 0.351909
a
4
=
2.09274u
33
− 3.89228u
32
+ ··· − 1.36534u − 2.34594
0.572218u
33
− 1.57386u
32
+ ··· − 2.75044u + 0.0518727
a
7
=
0.684721u
33
− 1.85986u
32
+ ··· − 3.17571u + 0.213786
0.395472u
33
− 0.984131u
32
+ ··· − 2.08170u + 0.842330
a
8
=
0.289249u
33
− 0.875732u
32
+ ··· − 1.09401u − 0.628544
0.395472u
33
− 0.984131u
32
+ ··· − 2.08170u + 0.842330
a
9
=
u
3
u
5
− u
3
+ u
a
12
=
0.466506u
33
− 1.30241u
32
+ ··· − 2.66126u + 0.0407295
−0.269323u
33
+ 0.553369u
32
+ ··· + 1.71300u − 0.617147
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
7661808169491132
3293995579890413
u
33
+
6218879522456028
3293995579890413
u
32
+ ···+
25977556548692283
3293995579890413
u −
23126112688430199
3293995579890413
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
34
+ 3u
33
+ ··· + 71u + 1
c
2
, c
4
u
34
− 9u
33
+ ··· − 15u + 1
c
3
, c
6
u
34
+ 3u
33
+ ··· + 2176u + 256
c
5
, c
10
u
34
− 2u
33
+ ··· − u + 1
c
7
u
34
− 6u
33
+ ··· + 1795665u + 338425
c
8
, c
11
, c
12
u
34
− 2u
33
+ ··· + 7u + 1
c
9
u
34
+ 6u
33
+ ··· + 11u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
34
+ 65y
33
+ ··· − 5331y + 1
c
2
, c
4
y
34
− 3y
33
+ ··· − 71y + 1
c
3
, c
6
y
34
− 51y
33
+ ··· − 1228800y + 65536
c
5
, c
10
y
34
− 6y
33
+ ··· − 11y + 1
c
7
y
34
+ 106y
33
+ ··· + 912331286925y + 114531480625
c
8
, c
11
, c
12
y
34
− 26y
33
+ ··· − 11y + 1
c
9
y
34
+ 46y
33
+ ··· + y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.635489 + 0.765565I
a = 0.023523 + 0.819812I
b = 0.143823 − 0.811670I
3.05379 − 1.22135I −2.44643 + 1.78317I
u = 0.635489 − 0.765565I
a = 0.023523 − 0.819812I
b = 0.143823 + 0.811670I
3.05379 + 1.22135I −2.44643 − 1.78317I
u = −0.729595 + 0.661430I
a = 0.034703 − 1.094050I
b = −0.136834 + 1.060550I
−0.23851 + 4.87038I −8.16084 − 6.79059I
u = −0.729595 − 0.661430I
a = 0.034703 + 1.094050I
b = −0.136834 − 1.060550I
−0.23851 − 4.87038I −8.16084 + 6.79059I
u = −0.479562 + 0.911974I
a = 0.046982 − 0.468540I
b = 0.371651 + 0.493500I
−1.11923 − 1.98539I −6.62012 + 2.37959I
u = −0.479562 − 0.911974I
a = 0.046982 + 0.468540I
b = 0.371651 − 0.493500I
−1.11923 + 1.98539I −6.62012 − 2.37959I
u = −0.766682 + 0.495753I
a = 1.50878 + 0.43111I
b = 0.146629 − 0.533111I
−0.524122 − 0.409066I −7.28048 − 0.84766I
u = −0.766682 − 0.495753I
a = 1.50878 − 0.43111I
b = 0.146629 + 0.533111I
−0.524122 + 0.409066I −7.28048 + 0.84766I
u = 0.971312 + 0.567163I
a = 1.103300 − 0.703542I
b = 0.496724 + 0.591318I
1.85693 − 3.80699I −4.56903 + 5.73620I
u = 0.971312 − 0.567163I
a = 1.103300 + 0.703542I
b = 0.496724 − 0.591318I
1.85693 + 3.80699I −4.56903 − 5.73620I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.17100
a = 0.851526
b = 0.548404
−7.23479 −9.88000
u = 0.718624 + 0.314023I
a = 0.116591 + 1.245780I
b = −1.100530 − 0.643963I
−4.65847 − 3.05078I −14.9930 + 6.5224I
u = 0.718624 − 0.314023I
a = 0.116591 − 1.245780I
b = −1.100530 + 0.643963I
−4.65847 + 3.05078I −14.9930 − 6.5224I
u = −1.123410 + 0.599536I
a = 0.797874 + 0.695393I
b = 0.704883 − 0.508995I
−3.27432 + 7.58793I −9.23689 − 7.74257I
u = −1.123410 − 0.599536I
a = 0.797874 − 0.695393I
b = 0.704883 + 0.508995I
−3.27432 − 7.58793I −9.23689 + 7.74257I
u = −0.721712
a = −0.340670
b = −1.45745
−6.03886 −17.6920
u = 0.900311 + 0.952119I
a = −0.732531 + 0.736305I
b = 0.96904 − 1.25880I
9.10120 − 4.37771I −7.49633 + 3.28771I
u = 0.900311 − 0.952119I
a = −0.732531 − 0.736305I
b = 0.96904 + 1.25880I
9.10120 + 4.37771I −7.49633 − 3.28771I
u = −0.905172 + 0.980418I
a = −0.763195 − 0.646287I
b = 1.05364 + 1.18320I
12.90170 − 0.67521I −4.51148 − 0.04928I
u = −0.905172 − 0.980418I
a = −0.763195 + 0.646287I
b = 1.05364 − 1.18320I
12.90170 + 0.67521I −4.51148 + 0.04928I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.995190 + 0.896687I
a = 0.62305 − 1.64005I
b = 1.05804 + 1.11835I
8.78395 − 2.42502I −7.89113 + 1.48359I
u = 0.995190 − 0.896687I
a = 0.62305 + 1.64005I
b = 1.05804 − 1.11835I
8.78395 + 2.42502I −7.89113 − 1.48359I
u = 0.901655 + 1.007390I
a = −0.755401 + 0.558173I
b = 1.09702 − 1.08890I
8.64740 + 5.63592I −8.00000 − 2.80908I
u = 0.901655 − 1.007390I
a = −0.755401 − 0.558173I
b = 1.09702 + 1.08890I
8.64740 − 5.63592I −8.00000 + 2.80908I
u = −0.530883 + 0.364299I
a = 0.43019 − 1.79437I
b = −0.779658 + 0.298976I
−1.01260 + 1.22984I −7.99935 − 4.73307I
u = −0.530883 − 0.364299I
a = 0.43019 + 1.79437I
b = −0.779658 − 0.298976I
−1.01260 − 1.22984I −7.99935 + 4.73307I
u = −1.012560 + 0.914952I
a = 0.49655 + 1.65048I
b = 1.15087 − 1.09510I
12.5409 + 7.6268I −5.13159 − 4.40800I
u = −1.012560 − 0.914952I
a = 0.49655 − 1.65048I
b = 1.15087 + 1.09510I
12.5409 − 7.6268I −5.13159 + 4.40800I
u = −0.620369
a = 1.18712
b = 0.117070
−0.969949 −9.86690
u = 1.031790 + 0.924118I
a = 0.39495 − 1.60999I
b = 1.21367 + 1.04076I
8.2072 − 12.7003I −8.60420 + 6.93082I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.031790 − 0.924118I
a = 0.39495 + 1.60999I
b = 1.21367 − 1.04076I
8.2072 + 12.7003I −8.60420 − 6.93082I
u = 0.246676 + 0.443752I
a = 4.30323 + 2.74241I
b = −0.948262 + 0.125356I
−3.28757 + 0.51694I −12.3807 + 13.4722I
u = 0.246676 − 0.443752I
a = 4.30323 − 2.74241I
b = −0.948262 − 0.125356I
−3.28757 − 0.51694I −12.3807 − 13.4722I
u = 0.464719
a = −2.95516
b = −1.08945
−2.17611 3.01310
8
II. I
u
2
= hb + 1, 2u
7
− u
6
− 3u
5
+ 3u
4
+ 4u
3
− 3u
2
+ a − 2u + 4, u
8
− u
7
−
u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
u
2
a
3
=
−2u
7
+ u
6
+ 3u
5
− 3u
4
− 4u
3
+ 3u
2
+ 2u − 4
−1
a
11
=
−u
−u
3
+ u
a
2
=
−2u
7
+ u
6
+ 3u
5
− 3u
4
− 4u
3
+ 3u
2
+ 2u − 5
−1
a
1
=
−1
0
a
4
=
−2u
7
+ u
6
+ 3u
5
− 3u
4
− 4u
3
+ 3u
2
+ 2u − 4
−1
a
7
=
1
u
2
a
8
=
−u
2
+ 1
u
2
a
9
=
u
3
u
5
− u
3
+ u
a
12
=
u
7
− 2u
5
+ 2u
3
− 2u
−u
7
+ u
5
− 2u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −10u
7
+ 2u
6
+ 16u
5
− 12u
4
− 19u
3
+ 9u
2
+ 8u − 27
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
8
c
3
, c
6
u
8
c
4
(u + 1)
8
c
5
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
c
7
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
c
8
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
c
9
u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1
c
10
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1
c
11
, c
12
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
8
c
3
, c
6
y
8
c
5
, c
10
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
c
7
, c
9
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
c
8
, c
11
, c
12
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.570868 + 0.730671I
a = 0.281371 + 1.128550I
b = −1.00000
−2.68559 + 1.13123I −9.56807 − 0.79885I
u = 0.570868 − 0.730671I
a = 0.281371 − 1.128550I
b = −1.00000
−2.68559 − 1.13123I −9.56807 + 0.79885I
u = −0.855237 + 0.665892I
a = −0.208670 − 0.825203I
b = −1.00000
0.51448 + 2.57849I −6.42531 − 3.25625I
u = −0.855237 − 0.665892I
a = −0.208670 + 0.825203I
b = −1.00000
0.51448 − 2.57849I −6.42531 + 3.25625I
u = −1.09818
a = −0.829189
b = −1.00000
−8.14766 −20.0060
u = 1.031810 + 0.655470I
a = −0.284386 + 0.605794I
b = −1.00000
−4.02461 − 6.44354I −11.71592 + 3.92092I
u = 1.031810 − 0.655470I
a = −0.284386 − 0.605794I
b = −1.00000
−4.02461 + 6.44354I −11.71592 − 3.92092I
u = 0.603304
a = −2.74744
b = −1.00000
−2.48997 −23.5750
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
34
+ 3u
33
+ ··· + 71u + 1)
c
2
((u − 1)
8
)(u
34
− 9u
33
+ ··· − 15u + 1)
c
3
, c
6
u
8
(u
34
+ 3u
33
+ ··· + 2176u + 256)
c
4
((u + 1)
8
)(u
34
− 9u
33
+ ··· − 15u + 1)
c
5
(u
8
− u
7
+ ··· + 2u − 1)(u
34
− 2u
33
+ ··· − u + 1)
c
7
(u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
· (u
34
− 6u
33
+ ··· + 1795665u + 338425)
c
8
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
34
− 2u
33
+ ··· + 7u + 1)
c
9
(u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
· (u
34
+ 6u
33
+ ··· + 11u + 1)
c
10
(u
8
+ u
7
+ ··· − 2u − 1)(u
34
− 2u
33
+ ··· − u + 1)
c
11
, c
12
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)(u
34
− 2u
33
+ ··· + 7u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
8
)(y
34
+ 65y
33
+ ··· − 5331y + 1)
c
2
, c
4
((y −1)
8
)(y
34
− 3y
33
+ ··· − 71y + 1)
c
3
, c
6
y
8
(y
34
− 51y
33
+ ··· − 1228800y + 65536)
c
5
, c
10
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
34
− 6y
33
+ ··· − 11y + 1)
c
7
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
34
+ 106y
33
+ ··· + 912331286925y + 114531480625)
c
8
, c
11
, c
12
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
34
− 26y
33
+ ··· − 11y + 1)
c
9
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
34
+ 46y
33
+ ··· + y + 1)
14
