12n
0575
(K12n
0575
)
A knot diagram
1
Linearized knot diagam
3 7 9 11 10 2 12 3 4 5 7 8
Solving Sequence
4,11 5,7
12 10 9 3 2 1 6 8
c
4
c
11
c
10
c
9
c
3
c
2
c
1
c
6
c
8
c
5
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 16u
4
+ 11u
3
+ 4u
2
+ b − 1,
u
9
+ 2u
8
+ 6u
7
+ 8u
6
+ 11u
5
+ 10u
4
+ 6u
3
+ 2u
2
+ 2a − u − 2,
u
10
+ 4u
9
+ 12u
8
+ 24u
7
+ 37u
6
+ 44u
5
+ 40u
4
+ 26u
3
+ 11u
2
− 2i
I
u
2
= hu
3
+ u
2
+ b + u + 2, u
3
+ 3a + 3u, u
4
+ 3u
2
+ 3i
I
u
3
= hu
3
− u
2
+ b + u, u
3
+ a + u, u
4
+ u
2
− 1i
I
v
1
= ha, b − 1, v + 1i
* 4 irreducible components of dim
C
= 0, with total 19 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
9
+ 3u
8
+ · · · + b − 1, u
9
+ 2u
8
+ · · · + 2a − 2, u
10
+ 4u
9
+ · · · + 11u
2
− 2i
(i) Arc colorings
a
4
=
1
0
a
11
=
0
u
a
5
=
1
u
2
a
7
=
−
1
2
u
9
− u
8
+ ··· +
1
2
u + 1
−u
9
− 3u
8
− 8u
7
− 13u
6
− 17u
5
− 16u
4
− 11u
3
− 4u
2
+ 1
a
12
=
1
2
u
9
+ 2u
8
+ ··· +
3
2
u − 2
u
7
+ 3u
6
+ 5u
5
+ 8u
4
+ 6u
3
+ 5u
2
+ 2u − 1
a
10
=
u
u
3
+ u
a
9
=
u
3
+ 2u
u
3
+ u
a
3
=
−u
6
− 3u
4
− 2u
2
+ 1
−u
6
− 2u
4
− u
2
a
2
=
1
2
u
9
+ 2u
8
+ ··· +
3
2
u − 1
u
7
+ 2u
6
+ 5u
5
+ 6u
4
+ 6u
3
+ 4u
2
+ u − 1
a
1
=
−
27
2
u
9
− 46u
8
+ ··· −
25
2
u + 24
−8u
9
− 32u
8
+ ··· − 11u + 19
a
6
=
u
2
+ 1
u
4
+ 2u
2
a
8
=
−u
9
− 4u
7
− 5u
5
+ 3u
−u
9
− 3u
7
− 3u
5
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
9
+ 8u
8
+ 22u
7
+ 40u
6
+ 52u
5
+ 48u
4
+ 28u
3
+ 4u
2
− 8u − 20
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
10
− 10u
9
+ ··· + 22u + 1
c
2
, c
6
, c
7
c
11
, c
12
u
10
− 2u
9
+ 7u
8
− 16u
7
+ 54u
6
+ 20u
5
− 38u
4
+ 9u
2
− 2u − 1
c
3
, c
8
, c
9
u
10
+ 4u
9
+ 4u
8
+ 6u
7
+ 47u
6
+ 34u
5
− 96u
4
− 96u
3
− 9u
2
− 28u − 10
c
4
, c
5
, c
10
u
10
− 4u
9
+ 12u
8
− 24u
7
+ 37u
6
− 44u
5
+ 40u
4
− 26u
3
+ 11u
2
− 2
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
10
+ 86y
9
+ ··· − 170y + 1
c
2
, c
6
, c
7
c
11
, c
12
y
10
+ 10y
9
+ ··· − 22y + 1
c
3
, c
8
, c
9
y
10
− 8y
9
+ ··· − 604y + 100
c
4
, c
5
, c
10
y
10
+ 8y
9
+ ··· − 44y + 4
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.011370 + 0.549500I
a = −1.84526 + 1.65245I
b = −1.36179 + 0.92941I
5.53688 + 3.23949I −14.4970 − 2.0733I
u = −1.011370 − 0.549500I
a = −1.84526 − 1.65245I
b = −1.36179 − 0.92941I
5.53688 − 3.23949I −14.4970 + 2.0733I
u = −0.815135
a = 0.753180
b = 1.02757
−5.74502 −15.7930
u = 0.055441 + 1.195260I
a = −0.294506 + 0.301650I
b = −0.641589 − 0.278823I
2.90689 − 1.22324I −8.89978 + 5.47255I
u = 0.055441 − 1.195260I
a = −0.294506 − 0.301650I
b = −0.641589 + 0.278823I
2.90689 + 1.22324I −8.89978 − 5.47255I
u = −0.362503 + 1.267330I
a = −0.160758 − 0.440175I
b = −0.898467 + 0.647980I
−1.81239 + 4.23636I −11.64407 − 4.22306I
u = −0.362503 − 1.267330I
a = −0.160758 + 0.440175I
b = −0.898467 − 0.647980I
−1.81239 − 4.23636I −11.64407 + 4.22306I
u = −0.42007 + 1.54013I
a = 1.45657 + 0.91885I
b = 3.27708 − 0.06119I
12.1053 + 8.5018I −12.65841 − 3.21110I
u = −0.42007 − 1.54013I
a = 1.45657 − 0.91885I
b = 3.27708 + 0.06119I
12.1053 − 8.5018I −12.65841 + 3.21110I
u = 0.292134
a = 0.934719
b = 0.221971
−0.474672 −20.8080
5
II. I
u
2
= hu
3
+ u
2
+ b + u + 2, u
3
+ 3a + 3u, u
4
+ 3u
2
+ 3i
(i) Arc colorings
a
4
=
1
0
a
11
=
0
u
a
5
=
1
u
2
a
7
=
−
1
3
u
3
− u
−u
3
− u
2
− u − 2
a
12
=
1
3
u
3
+ u
u
3
+ u
2
+ 2u + 2
a
10
=
u
u
3
+ u
a
9
=
u
3
+ 2u
u
3
+ u
a
3
=
u
2
+ 1
−u
2
− 3
a
2
=
1
3
u
3
+ u
2
+ u + 1
u
3
+ u − 1
a
1
=
1
3
u
3
+ u
u
3
+ u
2
+ u + 2
a
6
=
u
2
+ 1
−u
2
− 3
a
8
=
0
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
− 12
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
(u − 1)
4
c
3
, c
8
, c
9
u
4
− 3u
2
+ 3
c
4
, c
5
, c
10
u
4
+ 3u
2
+ 3
c
6
, c
11
, c
12
(u + 1)
4
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
, c
12
(y −1)
4
c
3
, c
8
, c
9
(y
2
− 3y + 3)
2
c
4
, c
5
, c
10
(y
2
+ 3y + 3)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.340625 + 1.271230I
a = 0.196660 − 0.733945I
b = 0.771230 − 0.525400I
−3.28987 − 4.05977I −18.0000 + 3.4641I
u = 0.340625 − 1.271230I
a = 0.196660 + 0.733945I
b = 0.771230 + 0.525400I
−3.28987 + 4.05977I −18.0000 − 3.4641I
u = −0.340625 + 1.271230I
a = −0.196660 − 0.733945I
b = −1.77123 + 1.20665I
−3.28987 + 4.05977I −18.0000 − 3.4641I
u = −0.340625 − 1.271230I
a = −0.196660 + 0.733945I
b = −1.77123 − 1.20665I
−3.28987 − 4.05977I −18.0000 + 3.4641I
9
III. I
u
3
= hu
3
− u
2
+ b + u, u
3
+ a + u, u
4
+ u
2
− 1i
(i) Arc colorings
a
4
=
1
0
a
11
=
0
u
a
5
=
1
u
2
a
7
=
−u
3
− u
−u
3
+ u
2
− u
a
12
=
−u
3
− u
−u
3
+ u
2
a
10
=
u
u
3
+ u
a
9
=
u
3
+ 2u
u
3
+ u
a
3
=
−u
2
− 1
−u
2
− 1
a
2
=
−u
3
− u
2
− u − 1
−u
3
− u − 1
a
1
=
−u
3
− u
−u
3
+ u
2
− u
a
6
=
u
2
+ 1
u
2
+ 1
a
8
=
0
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
− 20
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
11
c
12
(u − 1)
4
c
2
, c
7
(u + 1)
4
c
3
, c
8
, c
9
u
4
− u
2
− 1
c
4
, c
5
, c
10
u
4
+ u
2
− 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
, c
12
(y −1)
4
c
3
, c
8
, c
9
(y
2
− y −1)
2
c
4
, c
5
, c
10
(y
2
+ y −1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.786151
a = −1.27202
b = −0.653986
−7.23771 −22.4720
u = −0.786151
a = 1.27202
b = 1.89005
−7.23771 −22.4720
u = 1.272020I
a = 0.786151I
b = −1.61803 + 0.78615I
0.657974 −13.5280
u = − 1.272020I
a = − 0.786151I
b = −1.61803 − 0.78615I
0.657974 −13.5280
13
IV. I
v
1
= ha, b − 1, v + 1i
(i) Arc colorings
a
4
=
1
0
a
11
=
−1
0
a
5
=
1
0
a
7
=
0
1
a
12
=
−1
−1
a
10
=
−1
0
a
9
=
−1
0
a
3
=
1
0
a
2
=
1
−1
a
1
=
0
−1
a
6
=
1
0
a
8
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
u − 1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
c
6
, c
11
, c
12
u + 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
, c
12
y −1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
y
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = 1.00000
−3.28987 −12.0000
17
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
10
− 10u
9
+ ··· + 22u + 1)
c
2
, c
7
(u − 1)
5
(u + 1)
4
· (u
10
− 2u
9
+ 7u
8
− 16u
7
+ 54u
6
+ 20u
5
− 38u
4
+ 9u
2
− 2u − 1)
c
3
, c
8
, c
9
u(u
4
− 3u
2
+ 3)(u
4
− u
2
− 1)
· (u
10
+ 4u
9
+ 4u
8
+ 6u
7
+ 47u
6
+ 34u
5
− 96u
4
− 96u
3
− 9u
2
− 28u − 10)
c
4
, c
5
, c
10
u(u
4
+ u
2
− 1)(u
4
+ 3u
2
+ 3)
· (u
10
− 4u
9
+ 12u
8
− 24u
7
+ 37u
6
− 44u
5
+ 40u
4
− 26u
3
+ 11u
2
− 2)
c
6
, c
11
, c
12
(u − 1)
4
(u + 1)
5
· (u
10
− 2u
9
+ 7u
8
− 16u
7
+ 54u
6
+ 20u
5
− 38u
4
+ 9u
2
− 2u − 1)
18
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
9
)(y
10
+ 86y
9
+ ··· − 170y + 1)
c
2
, c
6
, c
7
c
11
, c
12
((y −1)
9
)(y
10
+ 10y
9
+ ··· − 22y + 1)
c
3
, c
8
, c
9
y(y
2
− 3y + 3)
2
(y
2
− y −1)
2
(y
10
− 8y
9
+ ··· − 604y + 100)
c
4
, c
5
, c
10
y(y
2
+ y −1)
2
(y
2
+ 3y + 3)
2
(y
10
+ 8y
9
+ ··· − 44y + 4)
19
