12a
0981
(K12a
0981
)
A knot diagram
1
Linearized knot diagam
4 6 11 9 2 10 12 1 5 3 7 8
Solving Sequence
2,5
6
3,9
10 7 11 4 1 8 12
c
5
c
2
c
9
c
6
c
10
c
4
c
1
c
8
c
12
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−1747u
50
+ 78946u
49
+ ··· + 1990656b − 43767094,
− 14055938u
50
+ 67741913u
49
+ ··· + 279355392a − 5726762096,
u
51
− 6u
50
+ ··· + 8662u − 842i
I
u
2
= h−a
2
+ b − a, a
3
+ 2a
2
+ a + 1, u + 1i
I
u
3
= hb
4
a
2
+ 2b
3
a − 2b
2
a
2
− b
2
a + b
2
− 2ba + a
2
− b + a − 1, u + 1i
I
v
1
= ha, b
6
− 2b
4
− b
3
+ b
2
+ b − 1, v − 1i
I
v
2
= ha, b + 1, v − 1i
* 4 irreducible components of dim
C
= 0, with total 61 representations.
* 1 irreducible components of dim
C
= 1
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−1747u
50
+7.89×10
4
u
49
+· · ·+1.99×10
6
b−4.38×10
7
, −1.41×10
7
u
50
+
6.77 × 10
7
u
49
+ · · · + 2.79 × 10
8
a − 5.73 × 10
9
, u
51
− 6u
50
+ · · · + 8662u − 842i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
3
+ u
a
9
=
0.0503156u
50
− 0.242494u
49
+ ··· − 256.094u + 20.4999
0.000877600u
50
− 0.0396583u
49
+ ··· − 185.410u + 21.9863
a
10
=
0.0511932u
50
− 0.282152u
49
+ ··· − 441.504u + 42.4862
0.000877600u
50
− 0.0396583u
49
+ ··· − 185.410u + 21.9863
a
7
=
0.0185165u
50
− 0.0729252u
49
+ ··· − 12.3427u + 3.52550
0.0182020u
50
− 0.0910577u
49
+ ··· − 146.302u + 15.3639
a
11
=
−0.0685537u
50
+ 0.309278u
49
+ ··· + 380.668u − 44.0036
0.00243739u
50
− 0.00644160u
49
+ ··· − 7.88344u + 1.49815
a
4
=
0.0211970u
50
− 0.106355u
49
+ ··· − 161.802u + 15.3309
−0.00435384u
50
+ 0.0207994u
49
+ ··· + 26.1973u − 2.31048
a
1
=
−0.00426257u
50
+ 0.0286204u
49
+ ··· + 70.7619u − 7.40462
0.0330833u
50
− 0.152421u
49
+ ··· − 177.572u + 18.3423
a
8
=
0.114056u
50
− 0.513210u
49
+ ··· − 473.287u + 40.8488
0.0646460u
50
− 0.330940u
49
+ ··· − 491.932u + 52.8520
a
12
=
0.0705646u
50
− 0.342297u
49
+ ··· − 486.734u + 50.5146
0.0477518u
50
− 0.225647u
49
+ ··· − 248.299u + 25.3200
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
383093
1492992
u
50
+
3585271
2985984
u
49
+ ··· +
86360671
55296
u −
228139061
1492992
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
16(16u
51
+ 84u
49
+ ··· − 138159u − 46107)
c
2
, c
5
u
51
− 6u
50
+ ··· + 8662u − 842
c
3
, c
10
9(9u
51
− 18u
50
+ ··· + 3u − 1)
c
4
, c
9
9(9u
51
+ 18u
50
+ ··· − u − 1)
c
6
16(16u
51
− 16u
50
+ ··· − 333u + 261)
c
7
, c
8
, c
11
c
12
u
51
− 4u
50
+ ··· − 186u − 46
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
256(256y
51
+ 2688y
50
+ ··· − 3.63961 × 10
9
y − 2.12586 × 10
9
)
c
2
, c
5
y
51
− 34y
50
+ ··· + 30048920y − 708964
c
3
, c
10
81(81y
51
− 2376y
50
+ ··· + 27y − 1)
c
4
, c
9
81(81y
51
− 3024y
50
+ ··· + 11y − 1)
c
6
256(256y
51
− 3200y
50
+ ··· − 3345273y − 68121)
c
7
, c
8
, c
11
c
12
y
51
− 60y
50
+ ··· + 35976y − 2116
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.871937 + 0.599275I
a = −1.43039 − 0.76680I
b = 1.075950 − 0.106403I
1.61712 − 0.65965I 7.95896 − 0.68823I
u = 0.871937 − 0.599275I
a = −1.43039 + 0.76680I
b = 1.075950 + 0.106403I
1.61712 + 0.65965I 7.95896 + 0.68823I
u = −0.954975 + 0.488693I
a = 1.183620 + 0.097225I
b = 0.055327 − 0.421750I
4.92043 − 1.04343I − 60.500387 + 0.10I
u = −0.954975 − 0.488693I
a = 1.183620 − 0.097225I
b = 0.055327 + 0.421750I
4.92043 + 1.04343I − 60.500387 + 0.10I
u = 0.288710 + 0.865056I
a = 1.49117 + 0.44766I
b = −1.271070 − 0.230386I
6.15912 + 2.38142I 10.95071 − 1.99222I
u = 0.288710 − 0.865056I
a = 1.49117 − 0.44766I
b = −1.271070 + 0.230386I
6.15912 − 2.38142I 10.95071 + 1.99222I
u = 0.190005 + 0.889876I
a = −1.57561 − 0.59687I
b = 1.44689 + 0.33886I
15.4058 + 4.1385I 11.16132 − 1.35851I
u = 0.190005 − 0.889876I
a = −1.57561 + 0.59687I
b = 1.44689 − 0.33886I
15.4058 − 4.1385I 11.16132 + 1.35851I
u = −0.864821
a = −0.975469
b = −0.447949
−1.39737 −9.01210
u = 0.063386 + 1.145060I
a = −1.72917 + 0.07049I
b = 1.39200 − 0.36995I
12.5583 − 9.7853I 8.08488 + 5.85275I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.063386 − 1.145060I
a = −1.72917 − 0.07049I
b = 1.39200 + 0.36995I
12.5583 + 9.7853I 8.08488 − 5.85275I
u = 1.071180 + 0.478231I
a = 1.11714 + 1.01410I
b = −1.126820 + 0.397205I
0.37782 − 4.14573I 0. + 5.58002I
u = 1.071180 − 0.478231I
a = 1.11714 − 1.01410I
b = −1.126820 − 0.397205I
0.37782 + 4.14573I 0. − 5.58002I
u = −1.20895
a = 0.563495
b = 0.856182
0.370489 11.3980
u = 1.22486
a = 1.64302
b = 0.210756
6.17402 −3.35810
u = −0.133856 + 0.749902I
a = −0.218006 + 0.669940I
b = −0.279882 − 0.872377I
7.32400 + 5.35250I 5.78289 − 4.81378I
u = −0.133856 − 0.749902I
a = −0.218006 − 0.669940I
b = −0.279882 + 0.872377I
7.32400 − 5.35250I 5.78289 + 4.81378I
u = 1.151360 + 0.486900I
a = −1.031110 − 0.933045I
b = 1.33347 − 0.56033I
3.45393 − 7.28871I 0
u = 1.151360 − 0.486900I
a = −1.031110 + 0.933045I
b = 1.33347 + 0.56033I
3.45393 + 7.28871I 0
u = 0.116607 + 1.248630I
a = 1.55658 − 0.00466I
b = −1.258200 + 0.254835I
3.85227 − 6.56215I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.116607 − 1.248630I
a = 1.55658 + 0.00466I
b = −1.258200 − 0.254835I
3.85227 + 6.56215I 0
u = 0.721707
a = 3.42336
b = −0.761960
8.07048 15.1420
u = 1.185540 + 0.490657I
a = 1.035740 + 0.915650I
b = −1.51238 + 0.64262I
12.3415 − 9.1062I 0
u = 1.185540 − 0.490657I
a = 1.035740 − 0.915650I
b = −1.51238 − 0.64262I
12.3415 + 9.1062I 0
u = −0.222036 + 0.676897I
a = 0.395217 − 0.289769I
b = 0.196870 + 0.589331I
−0.49730 + 3.53349I 2.22017 − 7.44112I
u = −0.222036 − 0.676897I
a = 0.395217 + 0.289769I
b = 0.196870 − 0.589331I
−0.49730 − 3.53349I 2.22017 + 7.44112I
u = −0.489136 + 0.507443I
a = −0.863036 − 0.069456I
b = −0.124638 − 0.153724I
−1.58698 + 0.44085I −4.04045 − 0.10164I
u = −0.489136 − 0.507443I
a = −0.863036 + 0.069456I
b = −0.124638 + 0.153724I
−1.58698 − 0.44085I −4.04045 + 0.10164I
u = 1.284590 + 0.367030I
a = −0.415487 − 0.264559I
b = 0.160260 − 1.287920I
3.01762 − 9.37305I 0
u = 1.284590 − 0.367030I
a = −0.415487 + 0.264559I
b = 0.160260 + 1.287920I
3.01762 + 9.37305I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.301300 + 0.341701I
a = 0.201290 + 0.276131I
b = −0.027671 + 1.082990I
−5.08700 − 7.24377I 0
u = 1.301300 − 0.341701I
a = 0.201290 − 0.276131I
b = −0.027671 − 1.082990I
−5.08700 + 7.24377I 0
u = 1.337140 + 0.307386I
a = 0.068553 − 0.241472I
b = −0.132614 − 0.848734I
−6.92934 − 3.57492I 0
u = 1.337140 − 0.307386I
a = 0.068553 + 0.241472I
b = −0.132614 + 0.848734I
−6.92934 + 3.57492I 0
u = 1.40404 + 0.19556I
a = −0.500238 + 0.206535I
b = 0.226409 + 0.417307I
−3.34999 − 0.41537I 0
u = 1.40404 − 0.19556I
a = −0.500238 − 0.206535I
b = 0.226409 − 0.417307I
−3.34999 + 0.41537I 0
u = 0.52748 + 1.35639I
a = −1.345560 − 0.200993I
b = 1.163450 − 0.055837I
1.80797 − 1.23291I 0
u = 0.52748 − 1.35639I
a = −1.345560 + 0.200993I
b = 1.163450 + 0.055837I
1.80797 + 1.23291I 0
u = −1.38742 + 0.54293I
a = 1.15060 − 0.96574I
b = −1.44935 − 0.55020I
8.0411 + 15.7082I 0
u = −1.38742 − 0.54293I
a = 1.15060 + 0.96574I
b = −1.44935 + 0.55020I
8.0411 − 15.7082I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.40409 + 0.56693I
a = −1.121190 + 0.837048I
b = 1.34279 + 0.51016I
−0.81260 + 12.82570I 0
u = −1.40409 − 0.56693I
a = −1.121190 − 0.837048I
b = 1.34279 − 0.51016I
−0.81260 − 12.82570I 0
u = −1.54951 + 0.14699I
a = 0.172341 − 0.256428I
b = −1.170700 + 0.028166I
9.83449 − 0.04248I 0
u = −1.54951 − 0.14699I
a = 0.172341 + 0.256428I
b = −1.170700 − 0.028166I
9.83449 + 0.04248I 0
u = −1.43969 + 0.61276I
a = 1.067200 − 0.671303I
b = −1.214210 − 0.430814I
−3.57762 + 8.19895I 0
u = −1.43969 − 0.61276I
a = 1.067200 + 0.671303I
b = −1.214210 + 0.430814I
−3.57762 − 8.19895I 0
u = 1.37780 + 0.82539I
a = 0.906207 + 0.483019I
b = −1.213230 − 0.143156I
8.68259 + 3.09879I 0
u = 1.37780 − 0.82539I
a = 0.906207 − 0.483019I
b = −1.213230 + 0.143156I
8.68259 − 3.09879I 0
u = −1.60562 + 0.66191I
a = −0.900138 + 0.467041I
b = 1.127080 + 0.236987I
−0.72424 + 3.17343I 0
u = −1.60562 − 0.66191I
a = −0.900138 − 0.467041I
b = 1.127080 − 0.236987I
−0.72424 − 3.17343I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.157699
a = −4.50862
b = 0.663524
0.907699 11.5760
10
II. I
u
2
= h−a
2
+ b − a, a
3
+ 2a
2
+ a + 1, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
5
=
1
0
a
6
=
1
1
a
3
=
1
0
a
9
=
a
a
2
+ a
a
10
=
a
2
+ 2a
a
2
+ a
a
7
=
−a
−a
2
− a
a
11
=
a
a
2
+ a
a
4
=
−a
2
− a
−a
a
1
=
a
a
2
+ a
a
8
=
a
a
2
+ a
a
12
=
a
a
2
+ a
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
9
, c
10
u
3
− u − 1
c
2
, c
5
(u + 1)
3
c
6
u
3
− 2u
2
+ u − 1
c
7
, c
8
, c
11
c
12
u
3
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
9
, c
10
y
3
− 2y
2
+ y − 1
c
2
, c
5
(y − 1)
3
c
6
y
3
− 2y
2
− 3y − 1
c
7
, c
8
, c
11
c
12
y
3
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −0.122561 + 0.744862I
b = −0.662359 + 0.562280I
−1.64493 −6.00000
u = −1.00000
a = −0.122561 − 0.744862I
b = −0.662359 − 0.562280I
−1.64493 −6.00000
u = −1.00000
a = −1.75488
b = 1.32472
−1.64493 −6.00000
14
III. I
u
3
= hb
4
a
2
+ 2b
3
a − 2b
2
a
2
− b
2
a + b
2
− 2ba + a
2
− b + a − 1, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
5
=
1
0
a
6
=
1
1
a
3
=
1
0
a
9
=
a
b
a
10
=
b + a
b
a
7
=
ba + a
2
+ 1
ba + 1
a
11
=
a
b
a
4
=
ba + 1
b
2
a
1
=
−b
2
a
2
− 2ba − 1
−b
3
a − b
2
− 1
a
8
=
−b
3
a
2
− a
3
b
2
− b
2
a
2
− 2b
2
a + a
3
− ba + a
2
− b + a
−b
3
a
2
− b
3
a − 2b
2
a + a
2
b − b
2
+ ba − b + a
a
12
=
b
3
a
2
+ a
3
b
2
+ 2b
2
a − a
3
+ b
b
3
a
2
+ 2b
2
a − a
2
b + 2b − a
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4
(iv) u-Polynomials at the component : It cannot be defined for a positive
dimension component.
(v) Riley Polynomials at the component : It cannot be defined for a positive
dimension component.
15
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
7.23771 4.00000
16
IV. I
v
1
= ha, b
6
− 2b
4
− b
3
+ b
2
+ b − 1, v − 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
1
0
a
6
=
1
0
a
3
=
1
0
a
9
=
0
b
a
10
=
b
b
a
7
=
−b
2
+ 1
−b
2
a
11
=
0
b
a
4
=
1
b
2
a
1
=
−b
2
+ 1
−b
4
a
8
=
−b
5
+ 2b
3
− b
−b
5
− b
4
+ b
3
+ b
2
a
12
=
b
5
− 2b
3
+ b
b
5
− b
3
+ b
(ii) Obstruction class = −1
(iii) Cusp Shapes = 10
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
6
+ 4u
5
+ 6u
4
+ 7u
3
+ 7u
2
+ 3u + 1
c
2
, c
5
u
6
c
3
, c
4
, c
6
c
9
, c
10
u
6
− 2u
4
− u
3
+ u
2
+ u − 1
c
7
, c
8
, c
11
c
12
(u
2
+ u − 1)
3
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
6
− 4y
5
− 6y
4
+ 13y
3
+ 19y
2
+ 5y + 1
c
2
, c
5
y
6
c
3
, c
4
, c
6
c
9
, c
10
y
6
− 4y
5
+ 6y
4
− 7y
3
+ 7y
2
− 3y + 1
c
7
, c
8
, c
11
c
12
(y
2
− 3y + 1)
3
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = −0.726823 + 0.764732I
8.88264 10.0000
v = 1.00000
a = 0
b = −0.726823 − 0.764732I
8.88264 10.0000
v = 1.00000
a = 0
b = −1.22636
0.986960 10.0000
v = 1.00000
a = 0
b = 0.613180 + 0.357727I
0.986960 10.0000
v = 1.00000
a = 0
b = 0.613180 − 0.357727I
0.986960 10.0000
v = 1.00000
a = 0
b = 1.45365
8.88264 10.0000
20
V. I
v
2
= ha, b + 1, v − 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
1
0
a
6
=
1
0
a
3
=
1
0
a
9
=
0
−1
a
10
=
−1
−1
a
7
=
0
−1
a
11
=
0
−1
a
4
=
1
1
a
1
=
0
−1
a
8
=
0
−1
a
12
=
0
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
21
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
9
, c
10
u − 1
c
2
, c
5
, c
7
c
8
, c
11
, c
12
u
c
3
, c
4
, c
6
u + 1
22
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
6
, c
9
, c
10
y − 1
c
2
, c
5
, c
7
c
8
, c
11
, c
12
y
23
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
2
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = −1.00000
0 0
24
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
16(u − 1)(u
3
− u − 1)(u
6
+ 4u
5
+ 6u
4
+ 7u
3
+ 7u
2
+ 3u + 1)
· (16u
51
+ 84u
49
+ ··· − 138159u − 46107)
c
2
, c
5
u
7
(u + 1)
3
(u
51
− 6u
50
+ ··· + 8662u − 842)
c
3
9(u + 1)(u
3
− u − 1)(u
6
− 2u
4
− u
3
+ u
2
+ u − 1)
· (9u
51
− 18u
50
+ ··· + 3u − 1)
c
4
9(u + 1)(u
3
− u − 1)(u
6
− 2u
4
+ ··· + u − 1)(9u
51
+ 18u
50
+ ··· − u − 1)
c
6
16(u + 1)(u
3
− 2u
2
+ u − 1)(u
6
− 2u
4
− u
3
+ u
2
+ u − 1)
· (16u
51
− 16u
50
+ ··· − 333u + 261)
c
7
, c
8
, c
11
c
12
u
4
(u
2
+ u − 1)
3
(u
51
− 4u
50
+ ··· − 186u − 46)
c
9
9(u − 1)(u
3
− u − 1)(u
6
− 2u
4
+ ··· + u − 1)(9u
51
+ 18u
50
+ ··· − u − 1)
c
10
9(u − 1)(u
3
− u − 1)(u
6
− 2u
4
− u
3
+ u
2
+ u − 1)
· (9u
51
− 18u
50
+ ··· + 3u − 1)
25
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
256(y − 1)(y
3
− 2y
2
+ y − 1)(y
6
− 4y
5
+ ··· + 5y + 1)
· (256y
51
+ 2688y
50
+ ··· − 3639614229y − 2125855449)
c
2
, c
5
y
7
(y − 1)
3
(y
51
− 34y
50
+ ··· + 3.00489 × 10
7
y − 708964)
c
3
, c
10
81(y − 1)(y
3
− 2y
2
+ y − 1)(y
6
− 4y
5
+ 6y
4
− 7y
3
+ 7y
2
− 3y + 1)
· (81y
51
− 2376y
50
+ ··· + 27y − 1)
c
4
, c
9
81(y − 1)(y
3
− 2y
2
+ y − 1)(y
6
− 4y
5
+ 6y
4
− 7y
3
+ 7y
2
− 3y + 1)
· (81y
51
− 3024y
50
+ ··· + 11y − 1)
c
6
256(y − 1)(y
3
− 2y
2
− 3y − 1)(y
6
− 4y
5
+ ··· − 3y + 1)
· (256y
51
− 3200y
50
+ ··· − 3345273y − 68121)
c
7
, c
8
, c
11
c
12
y
4
(y
2
− 3y + 1)
3
(y
51
− 60y
50
+ ··· + 35976y − 2116)
26
