12a
0690
(K12a
0690
)
A knot diagram
1
Linearized knot diagam
3 7 12 11 9 8 2 6 1 5 4 10
Solving Sequence
2,8
7 3 1 6 9 10 5 11 4 12
c
7
c
2
c
1
c
6
c
8
c
9
c
5
c
10
c
4
c
12
c
3
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
44
+ u
43
+ ··· + 3u
2
+ 1i
* 1 irreducible components of dim
C
= 0, with total 44 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
44
+ u
43
+ · · · + 3u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
8
=
1
0
a
7
=
1
u
2
a
3
=
u
u
3
+ u
a
1
=
u
3
u
5
+ u
3
+ u
a
6
=
u
2
+ 1
u
2
a
9
=
u
4
+ u
2
+ 1
u
4
a
10
=
−u
12
− u
10
− 3u
8
− 2u
6
+ u
2
+ 1
−u
14
− 2u
12
− 5u
10
− 6u
8
− 6u
6
− 2u
4
− u
2
a
5
=
u
6
+ u
4
+ 2u
2
+ 1
u
6
+ u
2
a
11
=
u
26
+ 3u
24
+ ··· + 3u
2
+ 1
u
26
+ 2u
24
+ ··· − u
6
− u
2
a
4
=
u
41
+ 4u
39
+ ··· − 2u
3
+ u
u
43
+ 5u
41
+ ··· + 3u
3
+ u
a
12
=
u
21
+ 2u
19
+ 7u
17
+ 10u
15
+ 14u
13
+ 12u
11
+ 5u
9
− 2u
7
− 5u
5
− 2u
3
− u
u
23
+ 3u
21
+ ··· + 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
42
− 4u
41
+ ··· − 12u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
8
u
44
+ 9u
43
+ ··· + 6u + 1
c
2
, c
7
u
44
+ u
43
+ ··· + 3u
2
+ 1
c
3
, c
4
, c
10
c
11
u
44
− u
43
+ ··· + 2u + 1
c
9
, c
12
u
44
+ 9u
43
+ ··· − 8u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
8
y
44
+ 53y
43
+ ··· + 38y + 1
c
2
, c
7
y
44
+ 9y
43
+ ··· + 6y + 1
c
3
, c
4
, c
10
c
11
y
44
+ 49y
43
+ ··· + 6y + 1
c
9
, c
12
y
44
+ 17y
43
+ ··· − 10y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.655099 + 0.751397I
10.47210 + 2.42057I 5.00512 − 3.50428I
u = 0.655099 − 0.751397I
10.47210 − 2.42057I 5.00512 + 3.50428I
u = 0.457290 + 0.880807I
−1.50378 + 2.68816I −5.95162 − 3.22752I
u = 0.457290 − 0.880807I
−1.50378 − 2.68816I −5.95162 + 3.22752I
u = −0.495314 + 0.905747I
−0.91674 − 6.42453I −3.67057 + 9.87108I
u = −0.495314 − 0.905747I
−0.91674 + 6.42453I −3.67057 − 9.87108I
u = −0.379628 + 0.881690I
4.48646 − 0.62493I −2.71495 + 3.55329I
u = −0.379628 − 0.881690I
4.48646 + 0.62493I −2.71495 − 3.55329I
u = 0.520660 + 0.926121I
6.20074 + 8.92581I −0.06506 − 8.28337I
u = 0.520660 − 0.926121I
6.20074 − 8.92581I −0.06506 + 8.28337I
u = −0.075736 + 0.916828I
2.88611 − 4.18901I −5.84139 + 3.97616I
u = −0.075736 − 0.916828I
2.88611 + 4.18901I −5.84139 − 3.97616I
u = −0.558233 + 0.724774I
2.78376 − 2.11026I 4.66535 + 4.95546I
u = −0.558233 − 0.724774I
2.78376 + 2.11026I 4.66535 − 4.95546I
u = 0.027533 + 0.895789I
−3.76094 + 1.84552I −10.24133 − 4.26971I
u = 0.027533 − 0.895789I
−3.76094 − 1.84552I −10.24133 + 4.26971I
u = 0.672800 + 0.488934I
7.59840 − 4.48618I 3.71469 + 2.30966I
u = 0.672800 − 0.488934I
7.59840 + 4.48618I 3.71469 − 2.30966I
u = −0.608827 + 0.473439I
0.43015 + 2.25708I 0.48990 − 3.95946I
u = −0.608827 − 0.473439I
0.43015 − 2.25708I 0.48990 + 3.95946I
u = 0.855723 + 0.916416I
11.73180 + 3.17974I 1.84300 − 2.50377I
u = 0.855723 − 0.916416I
11.73180 − 3.17974I 1.84300 + 2.50377I
u = −0.888935 + 0.892951I
6.98756 − 1.09197I −1.86937 + 2.51379I
u = −0.888935 − 0.892951I
6.98756 + 1.09197I −1.86937 − 2.51379I
u = 0.902652 + 0.887567I
8.07252 − 2.80136I 0. + 3.37397I
u = 0.902652 − 0.887567I
8.07252 + 2.80136I 0. − 3.37397I
u = −0.913567 + 0.886340I
15.5582 + 5.3967I 3.71419 − 2.08485I
u = −0.913567 − 0.886340I
15.5582 − 5.3967I 3.71419 + 2.08485I
u = 0.885790 + 0.925568I
11.28050 + 3.27174I 4.41927 − 2.54057I
u = 0.885790 − 0.925568I
11.28050 − 3.27174I 4.41927 + 2.54057I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.864280 + 0.949514I
6.80746 − 5.39327I −2.17460 + 2.29017I
u = −0.864280 − 0.949514I
6.80746 + 5.39327I −2.17460 − 2.29017I
u = 0.868528 + 0.961101I
7.83679 + 9.34211I 0. − 8.07187I
u = 0.868528 − 0.961101I
7.83679 − 9.34211I 0. + 8.07187I
u = −0.903374 + 0.934414I
−19.6382 − 3.3289I 5.91897 + 2.37042I
u = −0.903374 − 0.934414I
−19.6382 + 3.3289I 5.91897 − 2.37042I
u = −0.873373 + 0.968938I
15.2918 − 11.9872I 3.20276 + 6.75389I
u = −0.873373 − 0.968938I
15.2918 + 11.9872I 3.20276 − 6.75389I
u = 0.326627 + 0.532554I
−0.166341 + 0.920466I −3.92535 − 6.51014I
u = 0.326627 − 0.532554I
−0.166341 − 0.920466I −3.92535 + 6.51014I
u = −0.558336 + 0.208664I
6.42918 − 2.63981I 3.65305 + 2.69802I
u = −0.558336 − 0.208664I
6.42918 + 2.63981I 3.65305 − 2.69802I
u = 0.446901 + 0.368826I
−0.171380 + 0.972064I −1.26603 − 5.13929I
u = 0.446901 − 0.368826I
−0.171380 − 0.972064I −1.26603 + 5.13929I
6
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
8
u
44
+ 9u
43
+ ··· + 6u + 1
c
2
, c
7
u
44
+ u
43
+ ··· + 3u
2
+ 1
c
3
, c
4
, c
10
c
11
u
44
− u
43
+ ··· + 2u + 1
c
9
, c
12
u
44
+ 9u
43
+ ··· − 8u + 1
7
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
8
y
44
+ 53y
43
+ ··· + 38y + 1
c
2
, c
7
y
44
+ 9y
43
+ ··· + 6y + 1
c
3
, c
4
, c
10
c
11
y
44
+ 49y
43
+ ··· + 6y + 1
c
9
, c
12
y
44
+ 17y
43
+ ··· − 10y + 1
8
