12a
0371
(K12a
0371
)
A knot diagram
1
Linearized knot diagam
3 6 9 10 2 11 12 1 5 4 7 8
Solving Sequence
7,12
8 1
3,9
4 11 6 2 5 10
c
7
c
12
c
8
c
3
c
11
c
6
c
2
c
5
c
10
c
1
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h6.22403 × 10
15
u
53
− 2.16644 × 10
16
u
52
+ ··· + 1.15315 × 10
16
b + 1.86130 × 10
15
,
− 3.84141 × 10
15
u
53
− 5.38285 × 10
15
u
52
+ ··· + 3.45944 × 10
16
a + 4.72334 × 10
16
, u
54
− 2u
53
+ ··· − 9u + 3i
I
u
2
= hb + 1, a − 1, u
2
− u − 1i
I
u
3
= h−au + b − u − 1, a
2
+ 2a + 3, u
2
+ u − 1i
* 3 irreducible components of dim
C
= 0, with total 60 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
= h6.22×10
15
u
53
−2.17×10
16
u
52
+· · ·+1.15×10
16
b+1.86×10
15
, −3.84×
10
15
u
53
−5.38×10
15
u
52
+· · ·+3.46×10
16
a+4.72×10
16
, u
54
−2u
53
+· · ·−9u+3i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
8
=
1
−u
2
a
1
=
u
−u
3
+ u
a
3
=
0.111041u
53
+ 0.155599u
52
+ ··· + 4.11220u − 1.36535
−0.539742u
53
+ 1.87871u
52
+ ··· + 3.32643u − 0.161411
a
9
=
−u
2
+ 1
u
4
− 2u
2
a
4
=
0.607017u
53
+ 0.0379752u
52
+ ··· + 3.56687u − 1.29746
−1.30686u
53
+ 2.07232u
52
+ ··· − 0.567204u + 1.29954
a
11
=
−u
u
a
6
=
−u
2
+ 1
u
2
a
2
=
0.874650u
53
+ 0.540026u
52
+ ··· + 6.16049u − 2.08423
−1.85817u
53
+ 1.24119u
52
+ ··· − 7.27374u + 3.92680
a
5
=
0.818828u
53
− 1.31508u
52
+ ··· − 12.3614u + 1.51037
−1.22880u
53
+ 1.80213u
52
+ ··· + 3.71901u − 0.853328
a
10
=
−0.287418u
53
− 0.715155u
52
+ ··· − 7.57410u + 3.47537
0.496261u
53
+ 1.55979u
52
+ ··· + 11.6363u − 3.00772
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
1362927332552725
5765740254433699
u
53
−
23381515261364111
5765740254433699
u
52
+···−
315387112075661492
5765740254433699
u+
81812943695667795
5765740254433699
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
54
+ 25u
53
+ ··· + 4038u + 121
c
2
, c
5
u
54
+ 3u
53
+ ··· − 12u + 11
c
3
u
54
− u
53
+ ··· − 1064u + 212
c
4
, c
9
, c
10
u
54
+ u
53
+ ··· − 36u
2
+ 4
c
6
, c
7
, c
8
c
11
, c
12
u
54
+ 2u
53
+ ··· + 9u + 3
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
54
+ 15y
53
+ ··· − 5943246y + 14641
c
2
, c
5
y
54
− 25y
53
+ ··· − 4038y + 121
c
3
y
54
− 11y
53
+ ··· − 679264y + 44944
c
4
, c
9
, c
10
y
54
+ 49y
53
+ ··· − 288y + 16
c
6
, c
7
, c
8
c
11
, c
12
y
54
− 72y
53
+ ··· + 15y + 9
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.033200 + 0.205031I
a = −0.154928 − 0.608627I
b = 0.043531 − 1.156360I
2.48098 − 2.65644I 0
u = −1.033200 − 0.205031I
a = −0.154928 + 0.608627I
b = 0.043531 + 1.156360I
2.48098 + 2.65644I 0
u = −0.909027 + 0.201496I
a = 1.133080 − 0.239512I
b = −1.44506 + 0.44518I
−4.31420 − 2.29321I 6.00000 + 3.97838I
u = −0.909027 − 0.201496I
a = 1.133080 + 0.239512I
b = −1.44506 − 0.44518I
−4.31420 + 2.29321I 6.00000 − 3.97838I
u = −1.002340 + 0.391017I
a = 0.373522 − 0.490684I
b = 0.159427 − 1.337780I
−0.92608 − 10.90870I 0
u = −1.002340 − 0.391017I
a = 0.373522 + 0.490684I
b = 0.159427 + 1.337780I
−0.92608 + 10.90870I 0
u = −1.035390 + 0.306908I
a = −0.826876 + 0.402579I
b = 0.073409 + 0.249362I
1.28679 − 5.35006I 0
u = −1.035390 − 0.306908I
a = −0.826876 − 0.402579I
b = 0.073409 − 0.249362I
1.28679 + 5.35006I 0
u = 1.035020 + 0.327197I
a = −0.175793 − 0.504243I
b = −0.136856 − 1.285110I
4.34470 + 6.91339I 0
u = 1.035020 − 0.327197I
a = −0.175793 + 0.504243I
b = −0.136856 + 1.285110I
4.34470 − 6.91339I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.084440 + 0.195251I
a = 0.775059 + 0.381941I
b = −0.098656 + 0.410716I
5.85353 + 1.52576I 0
u = 1.084440 − 0.195251I
a = 0.775059 − 0.381941I
b = −0.098656 − 0.410716I
5.85353 − 1.52576I 0
u = −1.120580 + 0.004894I
a = −0.589724 + 0.389273I
b = 0.128535 + 0.742200I
3.05827 + 2.30358I 0
u = −1.120580 − 0.004894I
a = −0.589724 − 0.389273I
b = 0.128535 − 0.742200I
3.05827 − 2.30358I 0
u = 0.849728 + 0.155769I
a = 0.39360 − 1.41513I
b = 0.39372 − 1.40393I
−4.89459 + 1.45597I 7.23813 − 5.57336I
u = 0.849728 − 0.155769I
a = 0.39360 + 1.41513I
b = 0.39372 + 1.40393I
−4.89459 − 1.45597I 7.23813 + 5.57336I
u = 0.824308
a = −1.10776
b = 1.34525
−0.0557748 15.4170
u = 0.611997 + 0.492090I
a = −0.641324 − 0.317061I
b = 0.772587 + 0.690967I
−3.28887 − 3.64964I 5.12459 + 1.94620I
u = 0.611997 − 0.492090I
a = −0.641324 + 0.317061I
b = 0.772587 − 0.690967I
−3.28887 + 3.64964I 5.12459 − 1.94620I
u = 0.184324 + 0.638362I
a = 1.16880 + 1.08855I
b = 0.356445 − 0.466281I
−4.58312 + 7.42689I 2.31046 − 6.99724I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.184324 − 0.638362I
a = 1.16880 − 1.08855I
b = 0.356445 + 0.466281I
−4.58312 − 7.42689I 2.31046 + 6.99724I
u = 0.454241 + 0.453212I
a = 0.661574 + 0.833835I
b = 0.301434 − 0.167958I
−2.14560 + 0.88769I 6.18591 − 4.35554I
u = 0.454241 − 0.453212I
a = 0.661574 − 0.833835I
b = 0.301434 + 0.167958I
−2.14560 − 0.88769I 6.18591 + 4.35554I
u = −0.455135 + 0.442932I
a = 0.490955 − 0.173110I
b = −0.587686 + 0.546971I
1.046370 + 0.494844I 10.64023 + 0.18722I
u = −0.455135 − 0.442932I
a = 0.490955 + 0.173110I
b = −0.587686 − 0.546971I
1.046370 − 0.494844I 10.64023 − 0.18722I
u = −0.248292 + 0.564754I
a = −0.98947 + 1.14172I
b = −0.374122 − 0.374831I
0.36246 − 3.88016I 7.34275 + 7.06872I
u = −0.248292 − 0.564754I
a = −0.98947 − 1.14172I
b = −0.374122 + 0.374831I
0.36246 + 3.88016I 7.34275 − 7.06872I
u = 0.260818 + 0.544636I
a = −0.240106 − 0.253128I
b = 0.306471 + 0.627052I
−2.73742 + 2.46050I 4.87493 − 3.38491I
u = 0.260818 − 0.544636I
a = −0.240106 + 0.253128I
b = 0.306471 − 0.627052I
−2.73742 − 2.46050I 4.87493 + 3.38491I
u = −1.54880 + 0.05145I
a = −0.928695 − 0.013670I
b = 0.951552 + 0.345464I
3.71593 + 1.81884I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.54880 − 0.05145I
a = −0.928695 + 0.013670I
b = 0.951552 − 0.345464I
3.71593 − 1.81884I 0
u = −0.421997
a = −0.252936
b = −0.323293
0.611710 16.5140
u = 1.59012
a = 1.01605
b = −1.17062
7.70118 0
u = 0.209460 + 0.318057I
a = 0.26966 + 1.81111I
b = 0.598382 − 0.217505I
−1.44891 + 0.87205I −0.45042 − 2.57700I
u = 0.209460 − 0.318057I
a = 0.26966 − 1.81111I
b = 0.598382 + 0.217505I
−1.44891 − 0.87205I −0.45042 + 2.57700I
u = 0.052280 + 0.369852I
a = −2.16277 + 2.09624I
b = −0.774037 − 0.521212I
−7.25657 + 0.32248I −3.59043 − 0.24507I
u = 0.052280 − 0.369852I
a = −2.16277 − 2.09624I
b = −0.774037 + 0.521212I
−7.25657 − 0.32248I −3.59043 + 0.24507I
u = −1.68847 + 0.03150I
a = −0.90904 − 3.14347I
b = 1.74981 + 5.17567I
4.15513 − 2.12237I 0
u = −1.68847 − 0.03150I
a = −0.90904 + 3.14347I
b = 1.74981 − 5.17567I
4.15513 + 2.12237I 0
u = −1.69091
a = −1.02001
b = 0.854840
8.98922 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.69828 + 0.04672I
a = 1.022170 + 0.018668I
b = −0.834861 + 0.045952I
4.95768 + 3.23508I 0
u = 1.69828 − 0.04672I
a = 1.022170 − 0.018668I
b = −0.834861 − 0.045952I
4.95768 − 3.23508I 0
u = 1.71944 + 0.10562I
a = −0.42831 − 2.69040I
b = 0.36112 + 4.43030I
8.6699 + 12.9219I 0
u = 1.71944 − 0.10562I
a = −0.42831 + 2.69040I
b = 0.36112 − 4.43030I
8.6699 − 12.9219I 0
u = 1.72963 + 0.05599I
a = 0.14187 − 2.61907I
b = −0.54631 + 4.33951I
12.36250 + 3.74731I 0
u = 1.72963 − 0.05599I
a = 0.14187 + 2.61907I
b = −0.54631 − 4.33951I
12.36250 − 3.74731I 0
u = 1.72946 + 0.07949I
a = 0.62410 + 1.44096I
b = −1.29088 − 2.56923I
11.12150 + 6.92662I 0
u = 1.72946 − 0.07949I
a = 0.62410 − 1.44096I
b = −1.29088 + 2.56923I
11.12150 − 6.92662I 0
u = −1.72986 + 0.08588I
a = 0.19966 − 2.64199I
b = 0.00573 + 4.36280I
14.1686 − 8.6037I 0
u = −1.72986 − 0.08588I
a = 0.19966 + 2.64199I
b = 0.00573 − 4.36280I
14.1686 + 8.6037I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.73989 + 0.05063I
a = −0.61741 + 1.65827I
b = 1.29019 − 2.88401I
15.9719 − 2.5529I 0
u = −1.73989 − 0.05063I
a = −0.61741 − 1.65827I
b = 1.29019 + 2.88401I
15.9719 + 2.5529I 0
u = 1.74111 + 0.01390I
a = 0.59275 + 1.97467I
b = −1.25696 − 3.35550I
13.30770 − 2.12200I 0
u = 1.74111 − 0.01390I
a = 0.59275 − 1.97467I
b = −1.25696 + 3.35550I
13.30770 + 2.12200I 0
10
II. I
u
2
= hb + 1, a − 1, u
2
− u − 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
8
=
1
−u − 1
a
1
=
u
−u − 1
a
3
=
1
−1
a
9
=
−u
u
a
4
=
1
−1
a
11
=
−u
u
a
6
=
−u
u + 1
a
2
=
u + 1
−u − 2
a
5
=
1
−1
a
10
=
−u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
c
3
, c
4
, c
9
c
10
u
2
c
5
(u + 1)
2
c
6
, c
7
, c
8
u
2
− u − 1
c
11
, c
12
u
2
+ u − 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
2
c
3
, c
4
, c
9
c
10
y
2
c
6
, c
7
, c
8
c
11
, c
12
y
2
− 3y + 1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 1.00000
b = −1.00000
−0.657974 2.00000
u = 1.61803
a = 1.00000
b = −1.00000
7.23771 2.00000
14
III. I
u
3
= h−au + b − u − 1, a
2
+ 2a + 3, u
2
+ u − 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
8
=
1
u − 1
a
1
=
u
−u + 1
a
3
=
a
au + u + 1
a
9
=
u
−u
a
4
=
au + a + u
1
a
11
=
−u
u
a
6
=
u
−u + 1
a
2
=
a + u
au + 2
a
5
=
−a
−au − u − 1
a
10
=
a + u + 3
−a + u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
4
c
2
(u + 1)
4
c
3
, c
4
, c
9
c
10
(u
2
+ 2)
2
c
6
, c
7
, c
8
(u
2
+ u − 1)
2
c
11
, c
12
(u
2
− u − 1)
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
4
c
3
, c
4
, c
9
c
10
(y + 2)
4
c
6
, c
7
, c
8
c
11
, c
12
(y
2
− 3y + 1)
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −1.00000 + 1.41421I
b = 1.000000 + 0.874032I
−5.59278 4.00000
u = 0.618034
a = −1.00000 − 1.41421I
b = 1.000000 − 0.874032I
−5.59278 4.00000
u = −1.61803
a = −1.00000 + 1.41421I
b = 1.00000 − 2.28825I
2.30291 4.00000
u = −1.61803
a = −1.00000 − 1.41421I
b = 1.00000 + 2.28825I
2.30291 4.00000
18
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
54
+ 25u
53
+ ··· + 4038u + 121)
c
2
((u − 1)
2
)(u + 1)
4
(u
54
+ 3u
53
+ ··· − 12u + 11)
c
3
u
2
(u
2
+ 2)
2
(u
54
− u
53
+ ··· − 1064u + 212)
c
4
, c
9
, c
10
u
2
(u
2
+ 2)
2
(u
54
+ u
53
+ ··· − 36u
2
+ 4)
c
5
((u − 1)
4
)(u + 1)
2
(u
54
+ 3u
53
+ ··· − 12u + 11)
c
6
, c
7
, c
8
(u
2
− u − 1)(u
2
+ u − 1)
2
(u
54
+ 2u
53
+ ··· + 9u + 3)
c
11
, c
12
((u
2
− u − 1)
2
)(u
2
+ u − 1)(u
54
+ 2u
53
+ ··· + 9u + 3)
19
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
54
+ 15y
53
+ ··· − 5943246y + 14641)
c
2
, c
5
((y −1)
6
)(y
54
− 25y
53
+ ··· − 4038y + 121)
c
3
y
2
(y + 2)
4
(y
54
− 11y
53
+ ··· − 679264y + 44944)
c
4
, c
9
, c
10
y
2
(y + 2)
4
(y
54
+ 49y
53
+ ··· − 288y + 16)
c
6
, c
7
, c
8
c
11
, c
12
((y
2
− 3y + 1)
3
)(y
54
− 72y
53
+ ··· + 15y + 9)
20
