12n
0558
(K12n
0558
)
A knot diagram
1
Linearized knot diagam
3 7 9 8 11 2 6 4 12 6 9 10
Solving Sequence
3,9 4,11
12 10 1 8 5 6 7 2
c
3
c
11
c
9
c
12
c
8
c
4
c
5
c
7
c
2
c
1
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h6.38928 × 10
32
u
29
− 1.63982 × 10
33
u
28
+ ··· + 2.11658 × 10
34
b + 1.61565 × 10
34
,
− 5.75591 × 10
32
u
29
+ 4.74067 × 10
32
u
28
+ ··· + 6.34974 × 10
34
a + 7.64726 × 10
34
,
u
30
− 2u
29
+ ··· − 36u − 36i
I
u
2
= hu
4
− u
3
+ 3u
2
+ b − 3u, −u
4
+ u
3
− 4u
2
+ a + 3u − 3, u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1i
I
u
3
= hbau + b
2
+ 2ba − 2bu + b + a − 2u, a
2
− au + 1, u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 43 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
= h6.39 × 10
32
u
29
− 1.64 × 10
33
u
28
+ · · · + 2.12 × 10
34
b + 1.62 ×
10
34
, −5.76 × 10
32
u
29
+ 4.74 × 10
32
u
28
+ · · · + 6.35 × 10
34
a + 7.65 ×
10
34
, u
30
− 2u
29
+ · · · − 36u − 36i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
u
2
a
11
=
0.00906479u
29
− 0.00746591u
28
+ ··· − 3.45875u − 1.20434
−0.0301868u
29
+ 0.0774751u
28
+ ··· + 0.497856u − 0.763331
a
12
=
0.00906479u
29
− 0.00746591u
28
+ ··· − 3.45875u − 1.20434
−0.0230609u
29
+ 0.0603049u
28
+ ··· − 0.212369u − 1.14722
a
10
=
−0.0180236u
29
+ 0.0235211u
28
+ ··· − 1.46858u + 0.229478
−0.0262408u
29
+ 0.0578653u
28
+ ··· + 0.566767u − 0.217670
a
1
=
−0.00197135u
29
− 0.0109705u
28
+ ··· − 0.133902u + 1.62976
−0.0140099u
29
+ 0.0220006u
28
+ ··· + 1.02890u + 0.106895
a
8
=
u
u
3
+ u
a
5
=
u
2
+ 1
u
4
+ 2u
2
a
6
=
0.0118634u
29
− 0.0426200u
28
+ ··· − 1.74552u + 0.488618
0.00457667u
29
+ 0.00334632u
28
+ ··· − 2.20933u − 1.11355
a
7
=
−0.0141492u
29
+ 0.0501839u
28
+ ··· + 1.57769u − 1.21616
0.0122465u
29
− 0.0288317u
28
+ ··· + 0.716119u + 0.452662
a
2
=
0.0120385u
29
− 0.0329711u
28
+ ··· − 1.16280u + 1.52287
−0.0140099u
29
+ 0.0220006u
28
+ ··· + 1.02890u + 0.106895
(ii) Obstruction class = −1
(iii) Cusp Shapes = 0.239398u
29
− 0.548218u
28
+ ··· − 2.40002u − 3.00002
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
30
+ 18u
29
+ ··· − 108u + 81
c
2
, c
6
u
30
− 2u
29
+ ··· + 24u − 9
c
3
, c
4
, c
8
u
30
− 2u
29
+ ··· − 36u − 36
c
5
, c
10
u
30
+ u
29
+ ··· − 192u + 32
c
9
, c
11
, c
12
u
30
+ 10u
29
+ ··· − 6u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
30
− 6y
29
+ ··· − 343440y + 6561
c
2
, c
6
y
30
− 18y
29
+ ··· + 108y + 81
c
3
, c
4
, c
8
y
30
+ 10y
29
+ ··· + 6120y + 1296
c
5
, c
10
y
30
− 21y
29
+ ··· − 37376y + 1024
c
9
, c
11
, c
12
y
30
− 14y
29
+ ··· − 62y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.060949 + 0.993960I
a = 0.054675 − 1.026910I
b = 8.32351 − 0.35869I
3.35106 − 2.03998I −64.0149 − 21.4668I
u = 0.060949 − 0.993960I
a = 0.054675 + 1.026910I
b = 8.32351 + 0.35869I
3.35106 + 2.03998I −64.0149 + 21.4668I
u = −0.537847 + 0.922281I
a = −0.159880 + 0.394407I
b = 0.747743 + 0.961542I
1.90616 + 2.59959I −5.37556 − 3.84761I
u = −0.537847 − 0.922281I
a = −0.159880 − 0.394407I
b = 0.747743 − 0.961542I
1.90616 − 2.59959I −5.37556 + 3.84761I
u = 0.885353
a = 1.71518
b = 1.30868
−0.666718 −7.65480
u = −1.017350 + 0.486039I
a = −0.574022 − 1.162990I
b = −2.06665 − 1.74207I
−2.24459 + 3.47011I −6.14707 − 3.44778I
u = −1.017350 − 0.486039I
a = −0.574022 + 1.162990I
b = −2.06665 + 1.74207I
−2.24459 − 3.47011I −6.14707 + 3.44778I
u = −0.273950 + 0.781452I
a = 0.71806 + 1.92269I
b = 0.548292 − 0.259925I
9.49867 − 1.47383I −5.91236 − 0.53100I
u = −0.273950 − 0.781452I
a = 0.71806 − 1.92269I
b = 0.548292 + 0.259925I
9.49867 + 1.47383I −5.91236 + 0.53100I
u = −0.136443 + 1.288730I
a = −0.477581 − 0.116821I
b = 0.689136 + 1.080140I
1.36845 + 1.35871I −6.00000 − 0.19188I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.136443 − 1.288730I
a = −0.477581 + 0.116821I
b = 0.689136 − 1.080140I
1.36845 − 1.35871I −6.00000 + 0.19188I
u = 0.019571 + 0.697685I
a = 0.237922 − 0.197067I
b = −0.624234 + 1.163700I
1.57503 + 2.29578I −6.91536 − 5.32282I
u = 0.019571 − 0.697685I
a = 0.237922 + 0.197067I
b = −0.624234 − 1.163700I
1.57503 − 2.29578I −6.91536 + 5.32282I
u = −1.208520 + 0.721449I
a = 0.813857 − 0.414850I
b = −0.02930 − 2.67603I
−3.67883 + 0.34034I −3.88135 − 0.22027I
u = −1.208520 − 0.721449I
a = 0.813857 + 0.414850I
b = −0.02930 + 2.67603I
−3.67883 − 0.34034I −3.88135 + 0.22027I
u = 0.272145 + 0.413956I
a = 0.49020 − 1.91653I
b = 1.014450 − 0.329885I
1.70474 − 0.86259I 1.77057 + 2.23681I
u = 0.272145 − 0.413956I
a = 0.49020 + 1.91653I
b = 1.014450 + 0.329885I
1.70474 + 0.86259I 1.77057 − 2.23681I
u = 1.41053 + 0.55324I
a = −0.960628 − 0.426036I
b = −1.07262 − 3.12469I
−7.52040 + 5.61944I −5.75678 − 3.05482I
u = 1.41053 − 0.55324I
a = −0.960628 + 0.426036I
b = −1.07262 + 3.12469I
−7.52040 − 5.61944I −5.75678 + 3.05482I
u = −0.10183 + 1.58047I
a = 0.100802 + 0.775964I
b = 0.045827 − 1.200310I
13.09890 + 3.14293I 6.86746 − 0.28273I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.10183 − 1.58047I
a = 0.100802 − 0.775964I
b = 0.045827 + 1.200310I
13.09890 − 3.14293I 6.86746 + 0.28273I
u = −0.89832 + 1.30629I
a = 0.032327 + 0.946721I
b = 3.00551 + 0.63169I
−1.74378 + 7.39574I −1.50771 − 3.90519I
u = −0.89832 − 1.30629I
a = 0.032327 − 0.946721I
b = 3.00551 − 0.63169I
−1.74378 − 7.39574I −1.50771 + 3.90519I
u = 1.20555 + 1.04353I
a = −0.758006 − 0.545395I
b = 1.26924 − 3.19920I
−8.33843 − 5.76803I −6.15654 + 3.24176I
u = 1.20555 − 1.04353I
a = −0.758006 + 0.545395I
b = 1.26924 + 3.19920I
−8.33843 + 5.76803I −6.15654 − 3.24176I
u = −0.379780
a = 0.318553
b = −0.215645
−0.719557 −14.3210
u = 0.83269 + 1.39448I
a = −0.055727 + 1.016580I
b = −3.22456 + 0.35008I
−4.6721 − 13.5988I −3.02274 + 6.83229I
u = 0.83269 − 1.39448I
a = −0.055727 − 1.016580I
b = −3.22456 − 0.35008I
−4.6721 + 13.5988I −3.02274 − 6.83229I
u = 1.12003 + 1.24900I
a = 0.104473 + 0.946436I
b = −3.17286 + 1.59367I
−7.72406 − 2.77206I −6.00000 + 1.57550I
u = 1.12003 − 1.24900I
a = 0.104473 − 0.946436I
b = −3.17286 − 1.59367I
−7.72406 + 2.77206I −6.00000 − 1.57550I
7
II. I
u
2
=
hu
4
−u
3
+3u
2
+b−3u, −u
4
+u
3
−4u
2
+a+3u−3, u
5
−u
4
+4u
3
−3u
2
+3u−1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
u
2
a
11
=
u
4
− u
3
+ 4u
2
− 3u + 3
−u
4
+ u
3
− 3u
2
+ 3u
a
12
=
u
4
− u
3
+ 4u
2
− 3u + 3
−u
4
+ u
3
− 3u
2
+ 2u
a
10
=
u
4
− u
3
+ 4u
2
− 3u + 3
−u
4
+ u
3
− 3u
2
+ 3u
a
1
=
0
−u
a
8
=
u
u
3
+ u
a
5
=
u
2
+ 1
u
4
+ 2u
2
a
6
=
u
2
+ 1
u
4
+ 2u
2
a
7
=
u
3
+ 2u
u
4
− u
3
+ 3u
2
− 2u + 1
a
2
=
u
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
4
+ 5u
3
− 12u
2
+ 16u − 8
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1
c
2
u
5
− u
4
+ u
2
+ u − 1
c
5
, c
10
u
5
c
6
u
5
+ u
4
− u
2
+ u + 1
c
7
, c
8
u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1
c
9
(u + 1)
5
c
11
, c
12
(u − 1)
5
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1
c
2
, c
6
y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1
c
5
, c
10
y
5
c
9
, c
11
, c
12
(y −1)
5
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.233677 + 0.885557I
a = 0.278580 − 1.055720I
b = 1.99181 + 1.46959I
3.46474 − 2.21397I 0.36497 + 8.87119I
u = 0.233677 − 0.885557I
a = 0.278580 + 1.055720I
b = 1.99181 − 1.46959I
3.46474 + 2.21397I 0.36497 − 8.87119I
u = 0.416284
a = 2.40221
b = 0.771083
0.762751 −3.17840
u = 0.05818 + 1.69128I
a = 0.020316 − 0.590570I
b = 0.122644 + 0.787371I
12.60320 − 3.33174I −7.77577 + 5.09400I
u = 0.05818 − 1.69128I
a = 0.020316 + 0.590570I
b = 0.122644 − 0.787371I
12.60320 + 3.33174I −7.77577 − 5.09400I
11
III. I
u
3
= hbau + b
2
+ 2ba − 2bu + b + a − 2u, a
2
− au + 1, u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−1
a
11
=
a
b
a
12
=
a
b + a
a
10
=
−a − u
bau − a
a
1
=
−a − u
bau − u
a
8
=
u
0
a
5
=
0
−1
a
6
=
au − 1
ba − 1
a
7
=
2bau − b − a
ba − u − 1
a
2
=
−bau − a
bau − u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4bau + 4u
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)
4
c
2
, c
6
(u
4
− u
2
+ 1)
2
c
3
, c
4
, c
8
(u
2
+ 1)
4
c
5
, c
10
(u
4
+ 3u
2
+ 1)
2
c
7
(u
2
+ u + 1)
4
c
9
(u
2
− u − 1)
4
c
11
, c
12
(u
2
+ u − 1)
4
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
(y
2
+ y + 1)
4
c
2
, c
6
(y
2
− y + 1)
4
c
3
, c
4
, c
8
(y + 1)
8
c
5
, c
10
(y
2
+ 3y + 1)
4
c
9
, c
11
, c
12
(y
2
− 3y + 1)
4
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = − 0.618034I
b = −0.809017 + 0.216775I
2.63189 − 2.02988I 2.00000 + 3.46410I
u = 1.000000I
a = − 0.618034I
b = −0.80902 + 3.01929I
2.63189 + 2.02988I 2.00000 − 3.46410I
u = 1.000000I
a = 1.61803I
b = 0.309017 − 1.153270I
10.52760 + 2.02988I 2.00000 − 3.46410I
u = 1.000000I
a = 1.61803I
b = 0.309017 − 0.082801I
10.52760 − 2.02988I 2.00000 + 3.46410I
u = − 1.000000I
a = 0.618034I
b = −0.809017 − 0.216775I
2.63189 + 2.02988I 2.00000 − 3.46410I
u = − 1.000000I
a = 0.618034I
b = −0.80902 − 3.01929I
2.63189 − 2.02988I 2.00000 + 3.46410I
u = − 1.000000I
a = − 1.61803I
b = 0.309017 + 1.153270I
10.52760 − 2.02988I 2.00000 + 3.46410I
u = − 1.000000I
a = − 1.61803I
b = 0.309017 + 0.082801I
10.52760 + 2.02988I 2.00000 − 3.46410I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)
4
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)
· (u
30
+ 18u
29
+ ··· − 108u + 81)
c
2
((u
4
− u
2
+ 1)
2
)(u
5
− u
4
+ u
2
+ u − 1)(u
30
− 2u
29
+ ··· + 24u − 9)
c
3
, c
4
((u
2
+ 1)
4
)(u
5
− u
4
+ ··· + 3u − 1)(u
30
− 2u
29
+ ··· − 36u − 36)
c
5
, c
10
u
5
(u
4
+ 3u
2
+ 1)
2
(u
30
+ u
29
+ ··· − 192u + 32)
c
6
((u
4
− u
2
+ 1)
2
)(u
5
+ u
4
− u
2
+ u + 1)(u
30
− 2u
29
+ ··· + 24u − 9)
c
7
(u
2
+ u + 1)
4
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)
· (u
30
+ 18u
29
+ ··· − 108u + 81)
c
8
((u
2
+ 1)
4
)(u
5
+ u
4
+ ··· + 3u + 1)(u
30
− 2u
29
+ ··· − 36u − 36)
c
9
((u + 1)
5
)(u
2
− u − 1)
4
(u
30
+ 10u
29
+ ··· − 6u − 1)
c
11
, c
12
((u − 1)
5
)(u
2
+ u − 1)
4
(u
30
+ 10u
29
+ ··· − 6u − 1)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
(y
2
+ y + 1)
4
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)
· (y
30
− 6y
29
+ ··· − 343440y + 6561)
c
2
, c
6
(y
2
− y + 1)
4
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)
· (y
30
− 18y
29
+ ··· + 108y + 81)
c
3
, c
4
, c
8
(y + 1)
8
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)
· (y
30
+ 10y
29
+ ··· + 6120y + 1296)
c
5
, c
10
y
5
(y
2
+ 3y + 1)
4
(y
30
− 21y
29
+ ··· − 37376y + 1024)
c
9
, c
11
, c
12
((y −1)
5
)(y
2
− 3y + 1)
4
(y
30
− 14y
29
+ ··· − 62y + 1)
17
