12n
0079
(K12n
0079
)
A knot diagram
1
Linearized knot diagam
3 5 6 2 10 4 5 12 11 6 9 8
Solving Sequence
6,11
10
2,5
4 7 3 1 9 12 8
c
10
c
5
c
4
c
6
c
3
c
1
c
9
c
11
c
8
c
2
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
15
− u
14
+ 2u
12
+ 4u
11
− 5u
10
+ u
9
+ 8u
8
+ 3u
7
− 7u
6
+ 3u
5
+ 9u
4
− 2u
3
− u
2
+ b + 2u + 1, a − u,
u
16
− 2u
15
+ u
14
+ 2u
13
+ 3u
12
− 10u
11
+ 6u
10
+ 8u
9
− u
8
− 14u
7
+ 10u
6
+ 9u
5
− 8u
4
− 2u
3
+ 4u
2
+ u − 1i
I
u
2
= h−u
4
− 2u
3
+ b + 2u − 1, a + u, u
5
+ u
4
− u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 21 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
15
− u
14
+ · · · + b + 1, a − u, u
16
− 2u
15
+ · · · + u − 1i
(i) Arc colorings
a
6
=
0
u
a
11
=
1
0
a
10
=
1
−u
2
a
2
=
u
−u
15
+ u
14
+ ··· − 2u − 1
a
5
=
u
−u
3
+ u
a
4
=
u
15
− u
14
+ ··· + u + 1
2u
14
− u
13
+ ··· − 2u − 2
a
7
=
−u
10
+ u
8
− 4u
6
+ 3u
4
− 3u
2
+ 1
u
12
− 2u
10
+ 4u
8
− 6u
6
+ 3u
4
− 2u
2
a
3
=
u
15
− u
14
+ ··· + u + 1
u
14
− u
13
+ ··· − 2u − 1
a
1
=
−u
8
+ u
6
− 3u
4
+ 2u
2
− 1
−u
8
− 2u
4
a
9
=
−u
2
+ 1
−u
2
a
12
=
u
4
− u
2
+ 1
u
4
a
8
=
−u
6
+ u
4
− 2u
2
+ 1
−u
6
− u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
15
− 5u
14
+ 4u
13
+ 4u
12
+ 4u
11
− 23u
10
+ 23u
9
+ 14u
8
− 9u
7
−
26u
6
+ 37u
5
+ 12u
4
− 22u
3
+ 7u
2
+ 13u − 4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
16
− 2u
15
+ ··· + 27u + 1
c
2
, c
4
u
16
− 6u
15
+ ··· + 7u − 1
c
3
, c
6
u
16
+ u
15
+ ··· + 192u + 32
c
5
, c
10
u
16
− 2u
15
+ ··· + u − 1
c
7
u
16
− 2u
15
+ ··· − 4251u − 809
c
8
, c
9
, c
11
c
12
u
16
+ 2u
15
+ ··· + 9u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
16
+ 46y
15
+ ··· − 955y + 1
c
2
, c
4
y
16
+ 2y
15
+ ··· − 27y + 1
c
3
, c
6
y
16
− 33y
15
+ ··· − 10752y + 1024
c
5
, c
10
y
16
− 2y
15
+ ··· − 9y + 1
c
7
y
16
+ 110y
15
+ ··· − 17236113y + 654481
c
8
, c
9
, c
11
c
12
y
16
+ 26y
15
+ ··· − 9y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.858338 + 0.530060I
a = 0.858338 + 0.530060I
b = 1.51033 − 0.15720I
1.84411 − 3.14735I −3.35371 + 5.50197I
u = 0.858338 − 0.530060I
a = 0.858338 − 0.530060I
b = 1.51033 + 0.15720I
1.84411 + 3.14735I −3.35371 − 5.50197I
u = 0.617248 + 0.695994I
a = 0.617248 + 0.695994I
b = 0.650070 − 0.294152I
2.74741 − 1.38572I −1.45957 + 2.65740I
u = 0.617248 − 0.695994I
a = 0.617248 − 0.695994I
b = 0.650070 + 0.294152I
2.74741 + 1.38572I −1.45957 − 2.65740I
u = −0.817802 + 0.908616I
a = −0.817802 + 0.908616I
b = −1.96334 − 1.19161I
10.71530 + 0.17194I −2.05934 − 0.49098I
u = −0.817802 − 0.908616I
a = −0.817802 − 0.908616I
b = −1.96334 + 1.19161I
10.71530 − 0.17194I −2.05934 + 0.49098I
u = −0.964243 + 0.807862I
a = −0.964243 + 0.807862I
b = −2.94387 − 0.69457I
10.19210 + 6.12692I −2.87109 − 4.85275I
u = −0.964243 − 0.807862I
a = −0.964243 − 0.807862I
b = −2.94387 + 0.69457I
10.19210 − 6.12692I −2.87109 + 4.85275I
u = −0.713406
a = −0.713406
b = −0.602633
−1.05859 −9.11520
u = −0.521527 + 0.359954I
a = −0.521527 + 0.359954I
b = 0.685034 + 0.457929I
−1.01816 + 1.20835I −7.94696 − 4.63242I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.521527 − 0.359954I
a = −0.521527 − 0.359954I
b = 0.685034 − 0.457929I
−1.01816 − 1.20835I −7.94696 + 4.63242I
u = 0.955354 + 0.984451I
a = 0.955354 + 0.984451I
b = 2.82102 − 2.06263I
−15.8844 + 1.1391I −2.89439 + 0.24176I
u = 0.955354 − 0.984451I
a = 0.955354 − 0.984451I
b = 2.82102 + 2.06263I
−15.8844 − 1.1391I −2.89439 − 0.24176I
u = 0.995518 + 0.955803I
a = 0.995518 + 0.955803I
b = 4.02659 − 1.00397I
−16.0250 − 8.2475I −3.10340 + 4.00403I
u = 0.995518 − 0.955803I
a = 0.995518 − 0.955803I
b = 4.02659 + 1.00397I
−16.0250 + 8.2475I −3.10340 − 4.00403I
u = 0.467631
a = 0.467631
b = −1.96901
−2.17859 2.49220
6
II. I
u
2
= h−u
4
− 2u
3
+ b + 2u − 1, a + u, u
5
+ u
4
− u
2
+ u + 1i
(i) Arc colorings
a
6
=
0
u
a
11
=
1
0
a
10
=
1
−u
2
a
2
=
−u
u
4
+ 2u
3
− 2u + 1
a
5
=
u
−u
3
+ u
a
4
=
0
u
4
+ u
3
− u + 1
a
7
=
0
u
a
3
=
0
u
4
+ u
3
− u + 1
a
1
=
−u
u
3
− u
a
9
=
−u
2
+ 1
−u
2
a
12
=
u
4
− u
2
+ 1
u
4
a
8
=
−u
3
−u
4
− u
3
+ u
2
− 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8u
4
− u
3
+ 5u
2
+ 7u − 18
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
5
c
3
, c
6
u
5
c
4
(u + 1)
5
c
5
u
5
− u
4
+ u
2
+ u − 1
c
7
, c
11
, c
12
u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1
c
8
, c
9
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1
c
10
u
5
+ u
4
− u
2
+ u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
5
c
3
, c
6
y
5
c
5
, c
10
y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1
c
7
, c
8
, c
9
c
11
, c
12
y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.758138 + 0.584034I
a = −0.758138 − 0.584034I
b = −1.92595 + 0.86150I
0.17487 − 2.21397I −5.34777 + 4.39723I
u = 0.758138 − 0.584034I
a = −0.758138 + 0.584034I
b = −1.92595 − 0.86150I
0.17487 + 2.21397I −5.34777 − 4.39723I
u = −0.935538 + 0.903908I
a = 0.935538 − 0.903908I
b = 2.96269 + 1.26507I
9.31336 + 3.33174I −2.87586 − 2.18947I
u = −0.935538 − 0.903908I
a = 0.935538 + 0.903908I
b = 2.96269 − 1.26507I
9.31336 − 3.33174I −2.87586 + 2.18947I
u = −0.645200
a = 0.645200
b = 1.92652
−2.52712 −21.5530
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
16
− 2u
15
+ ··· + 27u + 1)
c
2
((u − 1)
5
)(u
16
− 6u
15
+ ··· + 7u − 1)
c
3
, c
6
u
5
(u
16
+ u
15
+ ··· + 192u + 32)
c
4
((u + 1)
5
)(u
16
− 6u
15
+ ··· + 7u − 1)
c
5
(u
5
− u
4
+ u
2
+ u − 1)(u
16
− 2u
15
+ ··· + u − 1)
c
7
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)(u
16
− 2u
15
+ ··· − 4251u − 809)
c
8
, c
9
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)(u
16
+ 2u
15
+ ··· + 9u + 1)
c
10
(u
5
+ u
4
− u
2
+ u + 1)(u
16
− 2u
15
+ ··· + u − 1)
c
11
, c
12
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)(u
16
+ 2u
15
+ ··· + 9u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
5
)(y
16
+ 46y
15
+ ··· − 955y + 1)
c
2
, c
4
((y −1)
5
)(y
16
+ 2y
15
+ ··· − 27y + 1)
c
3
, c
6
y
5
(y
16
− 33y
15
+ ··· − 10752y + 1024)
c
5
, c
10
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)(y
16
− 2y
15
+ ··· − 9y + 1)
c
7
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)
· (y
16
+ 110y
15
+ ··· − 17236113y + 654481)
c
8
, c
9
, c
11
c
12
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)(y
16
+ 26y
15
+ ··· − 9y + 1)
12
