12n
0572
(K12n
0572
)
A knot diagram
1
Linearized knot diagam
3 7 10 9 8 2 12 3 4 5 7 11
Solving Sequence
3,10 4,7
2 1 6 9 5 11 8 12
c
3
c
2
c
1
c
6
c
9
c
4
c
10
c
8
c
12
c
5
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h2u
30
+ 2u
29
+ ··· + 4b + 2, −2u
30
− u
29
+ ··· + 4a − 6, u
31
+ 2u
30
+ ··· + 2u + 2i
I
u
2
= hb − 1, 2u
3
− 3u
2
+ 3a + 3u − 3, u
4
+ 3u
2
+ 3i
I
u
3
= h−a
2
u
2
+ u
2
a − 2au + b + 2a − 2u, −2a
2
u
2
+ a
3
+ 2u
2
a − 2a
2
− 3au + 5a − u + 1, u
3
− u
2
+ 2u − 1i
I
u
4
= hb + 1, −u
2
+ a + u − 1, u
4
+ u
2
− 1i
I
v
1
= ha, b − 1, v −1i
* 5 irreducible components of dim
C
= 0, with total 49 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
= h2u
30
+2u
29
+· · ·+4b+2, −2u
30
−u
29
+· · ·+4a−6, u
31
+2u
30
+· · ·+2u+2i
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
a
4
=
1
−u
2
a
7
=
1
2
u
30
+
1
4
u
29
+ ··· + 2u +
3
2
−
1
2
u
30
−
1
2
u
29
+ ··· − u −
1
2
a
2
=
−
1
4
u
27
− 3u
25
+ ··· +
1
2
u + 1
1
4
u
27
+
11
4
u
25
+ ··· −
1
2
u
2
+
1
2
u
a
1
=
−
1
4
u
25
−
5
2
u
23
+ ··· + u + 1
1
4
u
27
+
11
4
u
25
+ ··· −
1
2
u
2
+
1
2
u
a
6
=
−u
10
− 5u
8
− 8u
6
− 3u
4
+ 3u
2
+ 1
u
10
+ 4u
8
+ 5u
6
− 3u
2
a
9
=
−u
u
3
+ u
a
5
=
u
2
+ 1
−u
4
− 2u
2
a
11
=
u
5
+ 2u
3
+ u
−u
7
− 3u
5
− 2u
3
+ u
a
8
=
−u
3
− 2u
u
3
+ u
a
12
=
1
2
u
30
+ u
29
+ ··· +
3
2
u +
5
2
−
1
4
u
28
−
1
4
u
27
+ ··· +
3
2
u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 2u
30
+ 4u
29
+ 30u
28
+ 48u
27
+ 192u
26
+ 248u
25
+ 682u
24
+ 702u
23
+ 1432u
22
+ 1112u
21
+
1656u
20
+ 764u
19
+ 556u
18
−384u
17
−1002u
16
−1040u
15
−1088u
14
−344u
13
+ 216u
12
+
568u
11
+ 830u
10
+ 454u
9
+ 206u
8
−108u
7
−284u
6
−180u
5
−122u
4
−6u
3
+ 30u
2
+ 4u + 12
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
31
+ 44u
30
+ ··· + 119u + 1
c
2
, c
6
u
31
− 2u
30
+ ··· + 3u − 1
c
3
, c
4
, c
9
u
31
+ 2u
30
+ ··· + 2u + 2
c
5
u
31
+ 7u
30
+ ··· − 88514u + 28438
c
7
, c
11
u
31
+ 2u
30
+ ··· − u − 1
c
8
, c
10
u
31
− 2u
30
+ ··· − 88u + 16
c
12
u
31
− 4u
30
+ ··· + 39u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
31
− 104y
30
+ ··· + 10991y −1
c
2
, c
6
y
31
− 44y
30
+ ··· + 119y −1
c
3
, c
4
, c
9
y
31
+ 26y
30
+ ··· + 8y −4
c
5
y
31
+ 55y
30
+ ··· + 4648364048y −808719844
c
7
, c
11
y
31
− 4y
30
+ ··· + 39y −1
c
8
, c
10
y
31
− 14y
30
+ ··· + 448y −256
c
12
y
31
+ 56y
30
+ ··· + 479y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.422005 + 0.970388I
a = 0.780023 − 0.122330I
b = −1.63363 − 0.20014I
−8.77509 − 4.02062I 3.17327 + 1.29124I
u = 0.422005 − 0.970388I
a = 0.780023 + 0.122330I
b = −1.63363 + 0.20014I
−8.77509 + 4.02062I 3.17327 − 1.29124I
u = −0.438452 + 0.819465I
a = 0.612911 + 0.875111I
b = −1.66514 − 0.05832I
−9.30026 − 3.06037I 2.53545 + 3.57264I
u = −0.438452 − 0.819465I
a = 0.612911 − 0.875111I
b = −1.66514 + 0.05832I
−9.30026 + 3.06037I 2.53545 − 3.57264I
u = −0.151148 + 1.120390I
a = −0.774438 + 0.936580I
b = 0.430006 − 0.577254I
−1.40117 + 1.17674I 6.48562 − 3.33148I
u = −0.151148 − 1.120390I
a = −0.774438 − 0.936580I
b = 0.430006 + 0.577254I
−1.40117 − 1.17674I 6.48562 + 3.33148I
u = 0.822269 + 0.212623I
a = −1.60479 + 1.78111I
b = 1.64313 − 0.27195I
−6.42043 + 8.50954I 6.05681 − 5.42341I
u = 0.822269 − 0.212623I
a = −1.60479 − 1.78111I
b = 1.64313 + 0.27195I
−6.42043 − 8.50954I 6.05681 + 5.42341I
u = −0.769464 + 0.275211I
a = −1.58833 − 1.20476I
b = 1.68783 + 0.04109I
−7.55262 − 1.26507I 4.68154 + 1.47900I
u = −0.769464 − 0.275211I
a = −1.58833 + 1.20476I
b = 1.68783 − 0.04109I
−7.55262 + 1.26507I 4.68154 − 1.47900I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.801134
a = 1.76838
b = −1.08175
2.34779 4.38740
u = −0.777905
a = −0.722824
b = −0.0906547
5.57117 16.4750
u = −0.709050 + 0.173396I
a = 0.38023 + 2.22913I
b = −0.614830 − 0.764424I
1.18939 − 4.50976I 8.37255 + 6.79420I
u = −0.709050 − 0.173396I
a = 0.38023 − 2.22913I
b = −0.614830 + 0.764424I
1.18939 + 4.50976I 8.37255 − 6.79420I
u = −0.332218 + 1.265660I
a = 0.704768 + 0.124425I
b = 0.095778 − 0.113924I
1.64712 − 4.00353I 11.89002 + 3.58978I
u = −0.332218 − 1.265660I
a = 0.704768 − 0.124425I
b = 0.095778 + 0.113924I
1.64712 + 4.00353I 11.89002 − 3.58978I
u = 0.355714 + 1.276500I
a = −0.665256 + 0.850651I
b = 1.105160 + 0.056084I
−1.62439 + 4.16232I 0.18076 − 3.51312I
u = 0.355714 − 1.276500I
a = −0.665256 − 0.850651I
b = 1.105160 − 0.056084I
−1.62439 − 4.16232I 0.18076 + 3.51312I
u = −0.054498 + 1.374780I
a = −0.492915 − 0.029751I
b = −0.992043 + 0.551725I
−6.79962 + 0.91660I −1.89239 − 1.83526I
u = −0.054498 − 1.374780I
a = −0.492915 + 0.029751I
b = −0.992043 − 0.551725I
−6.79962 − 0.91660I −1.89239 + 1.83526I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.292248 + 1.364020I
a = 0.75119 − 1.48738I
b = 0.689770 + 0.858986I
−3.67517 − 8.15180I 3.05070 + 7.43504I
u = −0.292248 − 1.364020I
a = 0.75119 + 1.48738I
b = 0.689770 − 0.858986I
−3.67517 + 8.15180I 3.05070 − 7.43504I
u = 0.34385 + 1.39505I
a = −0.22148 − 2.07364I
b = −1.66852 + 0.31356I
−11.5166 + 12.7194I 2.07160 − 6.66192I
u = 0.34385 − 1.39505I
a = −0.22148 + 2.07364I
b = −1.66852 − 0.31356I
−11.5166 − 12.7194I 2.07160 + 6.66192I
u = −0.30540 + 1.41307I
a = −0.24753 + 1.59262I
b = −1.74843 − 0.08810I
−12.92960 − 5.15155I 0.52455 + 2.55322I
u = −0.30540 − 1.41307I
a = −0.24753 − 1.59262I
b = −1.74843 + 0.08810I
−12.92960 + 5.15155I 0.52455 − 2.55322I
u = −0.047932 + 0.529217I
a = −0.316164 + 0.652908I
b = 0.693120 − 0.435687I
−1.07809 + 1.45065I 1.68946 − 3.85910I
u = −0.047932 − 0.529217I
a = −0.316164 − 0.652908I
b = 0.693120 + 0.435687I
−1.07809 − 1.45065I 1.68946 + 3.85910I
u = −0.02824 + 1.47075I
a = 1.096690 − 0.319605I
b = 1.77899 + 0.14272I
−16.7493 − 3.9918I −1.06457 + 2.30903I
u = −0.02824 − 1.47075I
a = 1.096690 + 0.319605I
b = 1.77899 − 0.14272I
−16.7493 + 3.9918I −1.06457 − 2.30903I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.346395
a = 2.12463
b = −0.429962
0.849256 13.6270
8
II. I
u
2
= hb − 1, 2u
3
− 3u
2
+ 3a + 3u − 3, u
4
+ 3u
2
+ 3i
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
a
4
=
1
−u
2
a
7
=
−
2
3
u
3
+ u
2
− u + 1
1
a
2
=
2
3
u
3
− u
2
+ u
−1
a
1
=
2
3
u
3
− u
2
+ u − 1
−1
a
6
=
1
0
a
9
=
−u
u
3
+ u
a
5
=
u
2
+ 1
u
2
+ 3
a
11
=
−u
3
− 2u
u
3
+ u
a
8
=
−u
3
− 2u
u
3
+ u
a
12
=
−
1
3
u
3
− u
2
− u − 1
u
3
+ u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
11
c
12
(u − 1)
4
c
3
, c
4
, c
9
u
4
+ 3u
2
+ 3
c
5
, c
8
, c
10
u
4
− 3u
2
+ 3
c
6
, c
7
(u + 1)
4
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
, c
12
(y −1)
4
c
3
, c
4
, c
9
(y
2
+ 3y + 3)
2
c
5
, c
8
, c
10
(y
2
− 3y + 3)
2
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.340625 + 1.271230I
a = 0.233945 + 0.669365I
b = 1.00000
4.05977I 6.00000 − 3.46410I
u = 0.340625 − 1.271230I
a = 0.233945 − 0.669365I
b = 1.00000
− 4.05977I 6.00000 + 3.46410I
u = −0.340625 + 1.271230I
a = −1.23394 − 1.06269I
b = 1.00000
− 4.05977I 6.00000 + 3.46410I
u = −0.340625 − 1.271230I
a = −1.23394 + 1.06269I
b = 1.00000
4.05977I 6.00000 − 3.46410I
12
III. I
u
3
=
h−a
2
u
2
+u
2
a−2au+b+2a−2u, −2a
2
u
2
+2u
2
a+· · ·+5a+1, u
3
−u
2
+2u−1i
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
a
4
=
1
−u
2
a
7
=
a
a
2
u
2
− u
2
a + 2au − 2a + 2u
a
2
=
a
2
u
2
+ 2au − a + 2u
a
2
u − u
2
a − a
2
+ 2au − 2u
2
+ 2u − 2
a
1
=
a
2
u
2
+ a
2
u − u
2
a − a
2
+ 4au − 2u
2
− a + 4u − 2
a
2
u − u
2
a − a
2
+ 2au − 2u
2
+ 2u − 2
a
6
=
u
2
+ 1
−u
2
+ u − 1
a
9
=
−u
u
2
− u + 1
a
5
=
u
2
+ 1
−u
2
+ u − 1
a
11
=
1
0
a
8
=
−u
2
− 1
u
2
− u + 1
a
12
=
a
2
u
2
+ 2au − a + 2u
a
2
u − u
2
a − a
2
+ 2au − 2u
2
+ 2u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
2
− 4u + 10
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
9
+ 6u
8
+ 15u
7
+ 21u
6
+ 19u
5
+ 12u
4
+ 7u
3
+ 5u
2
+ 2u + 1
c
2
, c
6
, c
7
c
11
u
9
− 3u
7
− u
6
+ 3u
5
+ 2u
4
− u
3
− u
2
+ 1
c
3
, c
4
, c
9
(u
3
− u
2
+ 2u − 1)
3
c
5
u
9
c
8
, c
10
(u
3
+ u
2
− 1)
3
c
12
u
9
− 6u
8
+ 15u
7
− 21u
6
+ 19u
5
− 12u
4
+ 7u
3
− 5u
2
+ 2u − 1
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
12
y
9
− 6y
8
+ 11y
7
− y
6
+ 11y
5
− 40y
4
− 37y
3
− 21y
2
− 6y −1
c
2
, c
6
, c
7
c
11
y
9
− 6y
8
+ 15y
7
− 21y
6
+ 19y
5
− 12y
4
+ 7y
3
− 5y
2
+ 2y −1
c
3
, c
4
, c
9
(y
3
+ 3y
2
+ 2y −1)
3
c
5
y
9
c
8
, c
10
(y
3
− y
2
+ 2y −1)
3
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = −1.110710 − 0.304480I
b = −1.324820 − 0.175904I
−3.02413 + 2.82812I 2.49024 − 2.97945I
u = 0.215080 + 1.307140I
a = −0.633796 − 0.350292I
b = 0.376870 + 0.700062I
−3.02413 + 2.82812I 2.49024 − 2.97945I
u = 0.215080 + 1.307140I
a = 0.41979 + 1.77933I
b = 0.947946 − 0.524157I
−3.02413 + 2.82812I 2.49024 − 2.97945I
u = 0.215080 − 1.307140I
a = −1.110710 + 0.304480I
b = −1.324820 + 0.175904I
−3.02413 − 2.82812I 2.49024 + 2.97945I
u = 0.215080 − 1.307140I
a = −0.633796 + 0.350292I
b = 0.376870 − 0.700062I
−3.02413 − 2.82812I 2.49024 + 2.97945I
u = 0.215080 − 1.307140I
a = 0.41979 − 1.77933I
b = 0.947946 + 0.524157I
−3.02413 − 2.82812I 2.49024 + 2.97945I
u = 0.569840
a = −0.101925
b = 1.26384
1.11345 9.01950
u = 0.569840
a = 1.37568 + 1.52573I
b = −0.631920 − 0.444935I
1.11345 9.01950
u = 0.569840
a = 1.37568 − 1.52573I
b = −0.631920 + 0.444935I
1.11345 9.01950
16
IV. I
u
4
= hb + 1, −u
2
+ a + u − 1, u
4
+ u
2
− 1i
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
a
4
=
1
−u
2
a
7
=
u
2
− u + 1
−1
a
2
=
u
2
− u + 2
−1
a
1
=
u
2
− u + 1
−1
a
6
=
−1
0
a
9
=
−u
u
3
+ u
a
5
=
u
2
+ 1
−u
2
− 1
a
11
=
u
3
+ 2u
−u
3
− u
a
8
=
−u
3
− 2u
u
3
+ u
a
12
=
u
3
+ u
2
+ u + 1
−u
3
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
+ 8
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
7
c
12
(u − 1)
4
c
2
, c
11
(u + 1)
4
c
3
, c
4
, c
9
u
4
+ u
2
− 1
c
5
, c
8
, c
10
u
4
− u
2
− 1
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
, c
12
(y −1)
4
c
3
, c
4
, c
9
(y
2
+ y −1)
2
c
5
, c
8
, c
10
(y
2
− y −1)
2
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.786151
a = 0.831883
b = −1.00000
3.94784 10.4720
u = −0.786151
a = 2.40419
b = −1.00000
3.94784 10.4720
u = 1.272020I
a = −0.618030 − 1.272020I
b = −1.00000
−3.94784 1.52790
u = − 1.272020I
a = −0.618030 + 1.272020I
b = −1.00000
−3.94784 1.52790
20
V. I
v
1
= ha, b − 1, v − 1i
(i) Arc colorings
a
3
=
1
0
a
10
=
1
0
a
4
=
1
0
a
7
=
0
1
a
2
=
1
−1
a
1
=
0
−1
a
6
=
1
0
a
9
=
1
0
a
5
=
1
0
a
11
=
1
0
a
8
=
1
0
a
12
=
1
−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
2
, c
11
c
12
u − 1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
c
6
, c
7
u + 1
22
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
, c
12
y −1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
y
23
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−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
((u − 1)
9
)(u
9
+ 6u
8
+ ··· + 2u + 1)
· (u
31
+ 44u
30
+ ··· + 119u + 1)
c
2
(u − 1)
5
(u + 1)
4
(u
9
− 3u
7
− u
6
+ 3u
5
+ 2u
4
− u
3
− u
2
+ 1)
· (u
31
− 2u
30
+ ··· + 3u − 1)
c
3
, c
4
, c
9
u(u
3
− u
2
+ 2u − 1)
3
(u
4
+ u
2
− 1)(u
4
+ 3u
2
+ 3)(u
31
+ 2u
30
+ ··· + 2u + 2)
c
5
u
10
(u
4
− 3u
2
+ 3)(u
4
− u
2
− 1)(u
31
+ 7u
30
+ ··· − 88514u + 28438)
c
6
(u − 1)
4
(u + 1)
5
(u
9
− 3u
7
− u
6
+ 3u
5
+ 2u
4
− u
3
− u
2
+ 1)
· (u
31
− 2u
30
+ ··· + 3u − 1)
c
7
(u − 1)
4
(u + 1)
5
(u
9
− 3u
7
− u
6
+ 3u
5
+ 2u
4
− u
3
− u
2
+ 1)
· (u
31
+ 2u
30
+ ··· − u − 1)
c
8
, c
10
u(u
3
+ u
2
− 1)
3
(u
4
− 3u
2
+ 3)(u
4
− u
2
− 1)(u
31
− 2u
30
+ ··· − 88u + 16)
c
11
(u − 1)
5
(u + 1)
4
(u
9
− 3u
7
− u
6
+ 3u
5
+ 2u
4
− u
3
− u
2
+ 1)
· (u
31
+ 2u
30
+ ··· − u − 1)
c
12
((u − 1)
9
)(u
9
− 6u
8
+ ··· + 2u − 1)
· (u
31
− 4u
30
+ ··· + 39u − 1)
25
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
9
)(y
9
− 6y
8
+ ··· − 6y −1)
· (y
31
− 104y
30
+ ··· + 10991y −1)
c
2
, c
6
((y −1)
9
)(y
9
− 6y
8
+ ··· + 2y −1)
· (y
31
− 44y
30
+ ··· + 119y −1)
c
3
, c
4
, c
9
y(y
2
+ y −1)
2
(y
2
+ 3y + 3)
2
(y
3
+ 3y
2
+ 2y −1)
3
· (y
31
+ 26y
30
+ ··· + 8y −4)
c
5
y
10
(y
2
− 3y + 3)
2
(y
2
− y −1)
2
· (y
31
+ 55y
30
+ ··· + 4648364048y −808719844)
c
7
, c
11
((y −1)
9
)(y
9
− 6y
8
+ ··· + 2y −1)
· (y
31
− 4y
30
+ ··· + 39y −1)
c
8
, c
10
y(y
2
− 3y + 3)
2
(y
2
− y −1)
2
(y
3
− y
2
+ 2y −1)
3
· (y
31
− 14y
30
+ ··· + 448y −256)
c
12
((y −1)
9
)(y
9
− 6y
8
+ ··· − 6y −1)
· (y
31
+ 56y
30
+ ··· + 479y −1)
26
