12n
0591
(K12n
0591
)
A knot diagram
1
Linearized knot diagam
3 7 11 8 10 2 5 12 3 5 8 9
Solving Sequence
8,12
9 1
3,11
4 5 7 2 6 10
c
8
c
12
c
11
c
3
c
4
c
7
c
2
c
6
c
10
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
6
+ 3u
5
+ 3u
4
− 17u
3
+ 12u
2
+ b + 4u − 2, 2u
7
− 9u
6
+ 2u
5
+ 44u
4
− 70u
3
+ 21u
2
+ a + 16u − 4,
u
8
− 4u
7
− u
6
+ 22u
5
− 25u
4
− 4u
3
+ 12u
2
+ u − 1i
I
u
2
= hu
4
+ 2u
3
− u
2
+ b − u, u
4
+ 3u
3
+ u
2
+ a − 2u − 1, u
5
+ 3u
4
− 3u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 13 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
6
+ 3u
5
+ 3u
4
− 17u
3
+ 12u
2
+ b + 4u − 2, 2u
7
− 9u
6
+ · · · + a −
4, u
8
− 4u
7
+ · · · + u − 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
9
=
1
u
2
a
1
=
−u
−u
3
+ u
a
3
=
−2u
7
+ 9u
6
− 2u
5
− 44u
4
+ 70u
3
− 21u
2
− 16u + 4
u
6
− 3u
5
− 3u
4
+ 17u
3
− 12u
2
− 4u + 2
a
11
=
u
u
a
4
=
−3u
7
+ 12u
6
+ u
5
− 61u
4
+ 82u
3
− 17u
2
− 18u + 4
−u
7
+ 4u
6
− 20u
4
+ 29u
3
− 8u
2
− 6u + 2
a
5
=
−2u
7
+ 8u
6
+ u
5
− 41u
4
+ 53u
3
− 9u
2
− 12u + 2
−u
7
+ 4u
6
− 20u
4
+ 29u
3
− 8u
2
− 6u + 2
a
7
=
−u
6
+ 2u
5
+ 4u
4
− 11u
3
+ 6u
2
+ 1
−u
7
+ u
6
+ 7u
5
− 8u
4
− 10u
3
+ 12u
2
+ u − 1
a
2
=
u
7
− u
6
− 8u
5
+ 9u
4
+ 15u
3
− 19u
2
− u + 1
3u
7
− 6u
6
− 16u
5
+ 37u
4
+ u
3
− 25u
2
+ 2
a
6
=
3u
7
− 11u
6
− 6u
5
+ 58u
4
− 56u
3
− u
2
+ 12u − 1
2u
7
− 8u
6
− 5u
5
+ 42u
4
− 35u
3
− 4u
2
+ 6u − 1
a
10
=
−u
7
+ 2u
6
+ 5u
5
− 12u
4
+ u
3
+ 6u
2
+ u
−2u
7
+ 4u
6
+ 10u
5
− 24u
4
+ 2u
3
+ 13u
2
+ u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −8u
7
+ 32u
6
+ 4u
5
− 163u
4
+ 213u
3
− 40u
2
− 51u − 4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
8
+ 7u
7
+ 35u
6
+ 244u
5
+ 493u
4
+ 885u
3
+ 772u
2
+ 608u + 64
c
2
, c
6
u
8
− 9u
7
+ 37u
6
− 88u
5
+ 125u
4
− 101u
3
+ 34u
2
+ 8u − 8
c
3
u
8
+ 2u
7
+ 4u
6
− 6u
5
+ 10u
4
− 3u
3
+ 5u
2
− u − 1
c
4
, c
7
u
8
− 3u
7
+ 9u
6
− 2u
5
− 14u
3
− 19u
2
− 8u − 1
c
5
, c
9
, c
10
u
8
+ 2u
7
+ 10u
6
+ 51u
5
+ 8u
4
+ 28u
3
+ 10u
2
+ 2u + 1
c
8
, c
11
, c
12
u
8
+ 4u
7
− u
6
− 22u
5
− 25u
4
+ 4u
3
+ 12u
2
− u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
8
+ 21y
7
+ ··· − 270848y + 4096
c
2
, c
6
y
8
− 7y
7
+ 35y
6
− 244y
5
+ 493y
4
− 885y
3
+ 772y
2
− 608y + 64
c
3
y
8
+ 4y
7
+ 60y
6
+ 66y
5
+ 106y
4
+ 71y
3
− y
2
− 11y + 1
c
4
, c
7
y
8
+ 9y
7
+ 69y
6
− 126y
5
− 448y
4
− 246y
3
+ 137y
2
− 26y + 1
c
5
, c
9
, c
10
y
8
+ 16y
7
− 88y
6
− 2533y
5
− 2598y
4
− 808y
3
+ 4y
2
+ 16y + 1
c
8
, c
11
, c
12
y
8
− 18y
7
+ 127y
6
− 442y
5
+ 783y
4
− 658y
3
+ 202y
2
− 25y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.357740 + 0.195354I
a = −0.300439 + 0.153458I
b = −0.766915 + 0.837154I
−3.40670 + 1.82723I −16.1033 − 2.4545I
u = 1.357740 − 0.195354I
a = −0.300439 − 0.153458I
b = −0.766915 − 0.837154I
−3.40670 − 1.82723I −16.1033 + 2.4545I
u = −0.432193 + 0.048912I
a = 0.26933 + 2.64283I
b = 0.144443 + 0.793261I
2.34111 − 2.75408I −10.98687 + 7.43235I
u = −0.432193 − 0.048912I
a = 0.26933 − 2.64283I
b = 0.144443 − 0.793261I
2.34111 + 2.75408I −10.98687 − 7.43235I
u = 0.276903
a = −0.812540
b = 0.311154
−0.562481 −17.6050
u = 2.08809 + 0.20687I
a = 0.80818 + 1.18869I
b = 1.72548 + 2.10674I
3.25293 − 6.36321I −12.47350 + 2.40837I
u = 2.08809 − 0.20687I
a = 0.80818 − 1.18869I
b = 1.72548 − 2.10674I
3.25293 + 6.36321I −12.47350 − 2.40837I
u = −2.30418
a = −0.741615
b = −0.517168
−18.6166 −11.2680
5
II.
I
u
2
= hu
4
+ 2u
3
− u
2
+ b − u, u
4
+ 3u
3
+ u
2
+ a − 2u − 1, u
5
+ 3u
4
− 3u
2
+ u − 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
9
=
1
u
2
a
1
=
−u
−u
3
+ u
a
3
=
−u
4
− 3u
3
− u
2
+ 2u + 1
−u
4
− 2u
3
+ u
2
+ u
a
11
=
u
u
a
4
=
−2u
3
− 3u
2
+ 3u
−u
3
− u
2
+ 2u − 1
a
5
=
−u
3
− 2u
2
+ u + 1
−u
3
− u
2
+ 2u − 1
a
7
=
u
4
+ 2u
3
− 2u
2
− 2u + 2
−u
3
− 2u
2
+ 2u
a
2
=
−2u
2
− 3u + 3
−u
4
− 3u
3
+ 2u + 1
a
6
=
−u
4
− 3u
3
+ 4u − 1
u
4
+ u
3
− u
2
+ u − 2
a
10
=
−u
4
− 3u
3
+ 4u
u
2
+ u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
4
− 3u
3
− 4u
2
− u − 8
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
5
− 7u
4
+ 15u
3
− 20u
2
+ 9u − 1
c
2
u
5
− u
4
− 3u
3
+ 3u − 1
c
3
u
5
− u
4
− 3u
3
− 2u
2
− 3u − 1
c
4
u
5
− 2u
4
− u
3
+ 4u
2
− 6u + 3
c
5
, c
9
u
5
− 3u
4
+ u
3
+ u
2
− 4u + 1
c
6
u
5
+ u
4
− 3u
3
+ 3u + 1
c
7
u
5
+ 2u
4
− u
3
− 4u
2
− 6u − 3
c
8
u
5
+ 3u
4
− 3u
2
+ u − 1
c
10
u
5
+ 3u
4
+ u
3
− u
2
− 4u − 1
c
11
, c
12
u
5
− 3u
4
+ 3u
2
+ u + 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
5
− 19y
4
− 37y
3
− 144y
2
+ 41y −1
c
2
, c
6
y
5
− 7y
4
+ 15y
3
− 20y
2
+ 9y −1
c
3
y
5
− 7y
4
− y
3
+ 12y
2
+ 5y −1
c
4
, c
7
y
5
− 6y
4
+ 5y
3
+ 8y
2
+ 12y −9
c
5
, c
9
, c
10
y
5
− 7y
4
− y
3
− 3y
2
+ 14y −1
c
8
, c
11
, c
12
y
5
− 9y
4
+ 20y
3
− 3y
2
− 5y −1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.896190
a = −0.815182
b = −0.385277
−2.59633 −14.9130
u = 0.116133 + 0.503198I
a = 1.68814 + 1.26672I
b = 0.005908 + 0.890217I
2.78994 + 2.01434I −6.94110 − 0.59350I
u = 0.116133 − 0.503198I
a = 1.68814 − 1.26672I
b = 0.005908 − 0.890217I
2.78994 − 2.01434I −6.94110 + 0.59350I
u = −1.78655
a = 1.15452
b = 2.62235
−12.8780 −12.0610
u = −2.34191
a = −0.715612
b = −1.24889
−19.7144 −19.1430
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
5
− 7u
4
+ 15u
3
− 20u
2
+ 9u − 1)
· (u
8
+ 7u
7
+ 35u
6
+ 244u
5
+ 493u
4
+ 885u
3
+ 772u
2
+ 608u + 64)
c
2
(u
5
− u
4
− 3u
3
+ 3u − 1)
· (u
8
− 9u
7
+ 37u
6
− 88u
5
+ 125u
4
− 101u
3
+ 34u
2
+ 8u − 8)
c
3
(u
5
− u
4
− 3u
3
− 2u
2
− 3u − 1)
· (u
8
+ 2u
7
+ 4u
6
− 6u
5
+ 10u
4
− 3u
3
+ 5u
2
− u − 1)
c
4
(u
5
− 2u
4
− u
3
+ 4u
2
− 6u + 3)
· (u
8
− 3u
7
+ 9u
6
− 2u
5
− 14u
3
− 19u
2
− 8u − 1)
c
5
, c
9
(u
5
− 3u
4
+ u
3
+ u
2
− 4u + 1)
· (u
8
+ 2u
7
+ 10u
6
+ 51u
5
+ 8u
4
+ 28u
3
+ 10u
2
+ 2u + 1)
c
6
(u
5
+ u
4
− 3u
3
+ 3u + 1)
· (u
8
− 9u
7
+ 37u
6
− 88u
5
+ 125u
4
− 101u
3
+ 34u
2
+ 8u − 8)
c
7
(u
5
+ 2u
4
− u
3
− 4u
2
− 6u − 3)
· (u
8
− 3u
7
+ 9u
6
− 2u
5
− 14u
3
− 19u
2
− 8u − 1)
c
8
(u
5
+ 3u
4
− 3u
2
+ u − 1)
· (u
8
+ 4u
7
− u
6
− 22u
5
− 25u
4
+ 4u
3
+ 12u
2
− u − 1)
c
10
(u
5
+ 3u
4
+ u
3
− u
2
− 4u − 1)
· (u
8
+ 2u
7
+ 10u
6
+ 51u
5
+ 8u
4
+ 28u
3
+ 10u
2
+ 2u + 1)
c
11
, c
12
(u
5
− 3u
4
+ 3u
2
+ u + 1)
· (u
8
+ 4u
7
− u
6
− 22u
5
− 25u
4
+ 4u
3
+ 12u
2
− u − 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
5
− 19y
4
− 37y
3
− 144y
2
+ 41y −1)
· (y
8
+ 21y
7
+ ··· − 270848y + 4096)
c
2
, c
6
(y
5
− 7y
4
+ 15y
3
− 20y
2
+ 9y −1)
· (y
8
− 7y
7
+ 35y
6
− 244y
5
+ 493y
4
− 885y
3
+ 772y
2
− 608y + 64)
c
3
(y
5
− 7y
4
− y
3
+ 12y
2
+ 5y −1)
· (y
8
+ 4y
7
+ 60y
6
+ 66y
5
+ 106y
4
+ 71y
3
− y
2
− 11y + 1)
c
4
, c
7
(y
5
− 6y
4
+ 5y
3
+ 8y
2
+ 12y −9)
· (y
8
+ 9y
7
+ 69y
6
− 126y
5
− 448y
4
− 246y
3
+ 137y
2
− 26y + 1)
c
5
, c
9
, c
10
(y
5
− 7y
4
− y
3
− 3y
2
+ 14y −1)
· (y
8
+ 16y
7
− 88y
6
− 2533y
5
− 2598y
4
− 808y
3
+ 4y
2
+ 16y + 1)
c
8
, c
11
, c
12
(y
5
− 9y
4
+ 20y
3
− 3y
2
− 5y −1)
· (y
8
− 18y
7
+ 127y
6
− 442y
5
+ 783y
4
− 658y
3
+ 202y
2
− 25y + 1)
11
