12n
0573
(K12n
0573
)
A knot diagram
1
Linearized knot diagam
3 7 10 9 8 2 12 3 4 5 1 8
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
= hu
16
− u
15
+ ··· + b + 1, −u
18
+ 3u
17
+ ··· + 2a − 6, u
19
− 3u
18
+ ··· + 6u − 2i
I
u
2
= h24u
8
a + 207u
8
+ ··· + 31a − 237,
2u
8
a + u
8
+ 6u
6
a + 5u
6
+ 6u
4
a − u
5
+ 8u
4
− 2u
2
a − 2u
3
+ a
2
+ au + 2u
2
− 4a − 3,
u
9
+ u
8
+ 4u
7
+ 3u
6
+ 5u
5
+ 3u
4
− 3u − 1i
I
u
3
= hb − 1, −2u
3
− 3u
2
+ 3a − 3u − 3, u
4
+ 3u
2
+ 3i
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 46 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
16
−u
15
+· · ·+b+1, −u
18
+3u
17
+· · ·+2a−6, u
19
−3u
18
+· · ·+6u−2i
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
a
4
=
1
−u
2
a
7
=
1
2
u
18
−
3
2
u
17
+ ··· −
7
2
u + 3
−u
16
+ u
15
+ ··· + u − 1
a
2
=
−
3
2
u
18
+
5
2
u
17
+ ··· +
7
2
u − 1
u
18
− 2u
17
+ ··· − 2u + 1
a
1
=
−
1
2
u
18
+
1
2
u
17
+ ··· −
5
2
u
2
+
3
2
u
u
18
− 2u
17
+ ··· − 2u + 1
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
=
3
2
u
18
−
5
2
u
17
+ ··· −
5
2
u + 2
−u
18
+ 2u
17
+ ··· + 3u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
17
+ 6u
16
− 22u
15
+ 44u
14
− 88u
13
+ 124u
12
− 164u
11
+
158u
10
− 134u
9
+ 68u
8
− 10u
7
− 26u
6
+ 42u
5
− 14u
4
+ 2u
3
+ 18u
2
− 20u + 8
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
11
u
19
+ 3u
18
+ ··· + 7u + 1
c
2
, c
6
, c
7
c
12
u
19
− u
18
+ ··· − u − 1
c
3
, c
4
, c
9
u
19
− 3u
18
+ ··· + 6u − 2
c
5
u
19
+ 21u
18
+ ··· + 2406u + 562
c
8
, c
10
u
19
+ 3u
18
+ ··· + 14u − 10
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
11
y
19
+ 37y
18
+ ··· + 7y −1
c
2
, c
6
, c
7
c
12
y
19
− 3y
18
+ ··· + 7y −1
c
3
, c
4
, c
9
y
19
+ 15y
18
+ ··· − 16y −4
c
5
y
19
− 45y
18
+ ··· − 2597328y −315844
c
8
, c
10
y
19
− 21y
18
+ ··· − 384y −100
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.304317 + 0.981930I
a = −0.73640 + 1.23198I
b = 0.534624 − 0.866654I
0.619243 + 0.825287I 1.58746 − 1.31207I
u = −0.304317 − 0.981930I
a = −0.73640 − 1.23198I
b = 0.534624 + 0.866654I
0.619243 − 0.825287I 1.58746 + 1.31207I
u = 0.929404 + 0.054061I
a = −1.85707 + 2.86205I
b = 1.16834 − 0.97470I
11.9496 + 7.7615I 2.99197 − 4.29762I
u = 0.929404 − 0.054061I
a = −1.85707 − 2.86205I
b = 1.16834 + 0.97470I
11.9496 − 7.7615I 2.99197 + 4.29762I
u = 0.744027
a = 0.660757
b = −0.577590
2.21133 4.57840
u = −0.689684 + 0.229296I
a = 0.50396 + 2.44275I
b = −0.768036 − 0.810356I
2.79530 − 4.62119I 3.18170 + 6.56238I
u = −0.689684 − 0.229296I
a = 0.50396 − 2.44275I
b = −0.768036 + 0.810356I
2.79530 + 4.62119I 3.18170 − 6.56238I
u = 0.315935 + 1.282700I
a = −0.038879 + 0.454563I
b = 0.606078 + 0.079526I
−1.79110 + 3.82280I −0.01419 − 2.05902I
u = 0.315935 − 1.282700I
a = −0.038879 − 0.454563I
b = 0.606078 − 0.079526I
−1.79110 − 3.82280I −0.01419 + 2.05902I
u = 0.473566 + 1.246700I
a = 1.69399 + 1.30200I
b = −1.12434 − 1.01004I
8.26951 − 2.76755I 0.165642 + 1.152780I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.473566 − 1.246700I
a = 1.69399 − 1.30200I
b = −1.12434 + 1.01004I
8.26951 + 2.76755I 0.165642 − 1.152780I
u = −0.000906 + 1.344200I
a = −0.628189 − 0.110568I
b = −0.631711 + 0.410114I
−5.42437 + 1.46948I −4.90135 − 4.71907I
u = −0.000906 − 1.344200I
a = −0.628189 + 0.110568I
b = −0.631711 − 0.410114I
−5.42437 − 1.46948I −4.90135 + 4.71907I
u = −0.250312 + 1.349130I
a = 0.61482 − 1.58026I
b = 0.876351 + 0.702623I
−2.18836 − 7.94720I −2.76731 + 8.17106I
u = −0.250312 − 1.349130I
a = 0.61482 + 1.58026I
b = 0.876351 − 0.702623I
−2.18836 + 7.94720I −2.76731 − 8.17106I
u = 0.435648 + 1.328780I
a = 0.25932 − 2.69648I
b = −1.19156 + 0.93252I
7.6280 + 12.6384I −0.71689 − 6.92034I
u = 0.435648 − 1.328780I
a = 0.25932 + 2.69648I
b = −1.19156 − 0.93252I
7.6280 − 12.6384I −0.71689 + 6.92034I
u = 0.218652 + 0.470395I
a = 0.358071 + 0.971620I
b = 0.319050 − 0.558488I
0.065587 + 1.130710I 1.18374 − 5.82659I
u = 0.218652 − 0.470395I
a = 0.358071 − 0.971620I
b = 0.319050 + 0.558488I
0.065587 − 1.130710I 1.18374 + 5.82659I
6
II. I
u
2
= h24u
8
a + 207u
8
+ · · · + 31a − 237, 2u
8
a + u
8
+ · · · − 4a − 3, u
9
+
u
8
+ 4u
7
+ 3u
6
+ 5u
5
+ 3u
4
− 3u − 1i
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
a
4
=
1
−u
2
a
7
=
a
−0.0892193au
8
− 0.769517u
8
+ ··· − 0.115242a + 0.881041
a
2
=
−0.769517au
8
− 0.762082u
8
+ ··· + 0.881041a + 1.97398
0.163569au
8
− 0.0892193u
8
+ ··· + 0.0446097a − 1.11524
a
1
=
−0.605948au
8
− 0.851301u
8
+ ··· + 0.925651a + 0.858736
0.163569au
8
− 0.0892193u
8
+ ··· + 0.0446097a − 1.11524
a
6
=
−2u
8
− u
7
− 6u
6
− 2u
5
− 6u
4
+ 2u + 2
u
8
+ u
7
+ 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
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
=
−0.769517au
8
− 0.762082u
8
+ ··· + 0.881041a − 0.0260223
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
8
− 4u
7
− 12u
6
− 8u
5
− 8u
4
− 4u
3
+ 8u
2
+ 4u + 6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
11
u
18
+ 5u
17
+ ··· + 4u + 1
c
2
, c
6
, c
7
c
12
u
18
− u
17
+ ··· + 2u − 1
c
3
, c
4
, c
9
(u
9
+ u
8
+ 4u
7
+ 3u
6
+ 5u
5
+ 3u
4
− 3u − 1)
2
c
5
(u
9
− 7u
8
+ 6u
7
+ 37u
6
− 21u
5
− 89u
4
− 66u
3
− 54u
2
− 39u − 7)
2
c
8
, c
10
(u
9
− u
8
− 6u
7
+ 5u
6
+ 11u
5
− 7u
4
− 6u
3
+ 4u
2
− u − 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
11
y
18
+ 15y
17
+ ··· − 52y + 1
c
2
, c
6
, c
7
c
12
y
18
− 5y
17
+ ··· − 4y + 1
c
3
, c
4
, c
9
(y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ y
5
− 31y
4
− 24y
3
+ 6y
2
+ 9y −1)
2
c
5
(y
9
− 37y
8
+ ··· + 765y −49)
2
c
8
, c
10
(y
9
− 13y
8
+ 68y
7
− 183y
6
+ 269y
5
− 211y
4
+ 80y
3
− 18y
2
+ 9y −1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.940385
a = −1.67785 + 2.94580I
b = 0.88600 − 1.16403I
12.9028 4.12280
u = −0.940385
a = −1.67785 − 2.94580I
b = 0.88600 + 1.16403I
12.9028 4.12280
u = −0.105528 + 1.193370I
a = −0.612327 − 0.108328I
b = −1.214940 + 0.117733I
−6.13776 − 1.55423I −5.05960 + 4.30527I
u = −0.105528 + 1.193370I
a = −0.85424 − 2.40749I
b = 0.870781 + 0.348555I
−6.13776 − 1.55423I −5.05960 + 4.30527I
u = −0.105528 − 1.193370I
a = −0.612327 + 0.108328I
b = −1.214940 − 0.117733I
−6.13776 + 1.55423I −5.05960 − 4.30527I
u = −0.105528 − 1.193370I
a = −0.85424 + 2.40749I
b = 0.870781 − 0.348555I
−6.13776 + 1.55423I −5.05960 − 4.30527I
u = 0.743788
a = 0.661558 + 0.082738I
b = −0.577633 − 0.031295I
2.21133 4.57530
u = 0.743788
a = 0.661558 − 0.082738I
b = −0.577633 + 0.031295I
2.21133 4.57530
u = 0.328404 + 1.225450I
a = 0.279234 − 0.828501I
b = 0.067133 + 0.481523I
−1.53180 + 3.86354I 0.03791 − 4.00946I
u = 0.328404 + 1.225450I
a = −0.455774 + 1.279350I
b = 1.048570 − 0.263166I
−1.53180 + 3.86354I 0.03791 − 4.00946I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.328404 − 1.225450I
a = 0.279234 + 0.828501I
b = 0.067133 − 0.481523I
−1.53180 − 3.86354I 0.03791 + 4.00946I
u = 0.328404 − 1.225450I
a = −0.455774 − 1.279350I
b = 1.048570 + 0.263166I
−1.53180 − 3.86354I 0.03791 + 4.00946I
u = −0.460882 + 1.295330I
a = 1.69755 − 1.44384I
b = −0.82021 + 1.17863I
8.87899 − 4.99486I 0.86627 + 2.90812I
u = −0.460882 + 1.295330I
a = 0.22926 + 2.59994I
b = −0.94094 − 1.12597I
8.87899 − 4.99486I 0.86627 + 2.90812I
u = −0.460882 − 1.295330I
a = 1.69755 + 1.44384I
b = −0.82021 − 1.17863I
8.87899 + 4.99486I 0.86627 − 2.90812I
u = −0.460882 − 1.295330I
a = 0.22926 − 2.59994I
b = −0.94094 + 1.12597I
8.87899 + 4.99486I 0.86627 − 2.90812I
u = −0.327390
a = −0.523848
b = 1.13069
−2.72863 5.61280
u = −0.327390
a = 4.98902
b = −0.768210
−2.72863 5.61280
11
III. I
u
3
= 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
=
−
5
3
u
3
− u
2
− 3u − 1
u
3
+ u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
− 12
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
c
11
(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
12
(u + 1)
4
13
(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
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.340625 + 1.271230I
a = −1.23394 + 1.06269I
b = 1.00000
−3.28987 + 4.05977I −6.00000 − 3.46410I
u = 0.340625 − 1.271230I
a = −1.23394 − 1.06269I
b = 1.00000
−3.28987 − 4.05977I −6.00000 + 3.46410I
u = −0.340625 + 1.271230I
a = 0.233945 − 0.669365I
b = 1.00000
−3.28987 − 4.05977I −6.00000 + 3.46410I
u = −0.340625 − 1.271230I
a = 0.233945 + 0.669365I
b = 1.00000
−3.28987 + 4.05977I −6.00000 − 3.46410I
15
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
+ 3u + 1
−u
3
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
− 4
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
11
c
12
(u − 1)
4
c
2
, c
7
(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
17
(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
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.786151
a = 2.40419
b = −1.00000
0.657974 −1.52790
u = −0.786151
a = 0.831883
b = −1.00000
0.657974 −1.52790
u = 1.272020I
a = −0.618030 + 1.272020I
b = −1.00000
−7.23771 −10.4720
u = − 1.272020I
a = −0.618030 − 1.272020I
b = −1.00000
−7.23771 −10.4720
19
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 = −12
20
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
c
11
u − 1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
c
6
, c
12
u + 1
21
(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
22
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = 1.00000
−3.28987 −12.0000
23
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
11
((u − 1)
9
)(u
18
+ 5u
17
+ ··· + 4u + 1)(u
19
+ 3u
18
+ ··· + 7u + 1)
c
2
, c
7
((u − 1)
5
)(u + 1)
4
(u
18
− u
17
+ ··· + 2u − 1)(u
19
− u
18
+ ··· − u − 1)
c
3
, c
4
, c
9
u(u
4
+ u
2
− 1)(u
4
+ 3u
2
+ 3)(u
9
+ u
8
+ ··· − 3u − 1)
2
· (u
19
− 3u
18
+ ··· + 6u − 2)
c
5
u(u
4
− 3u
2
+ 3)(u
4
− u
2
− 1)
· (u
9
− 7u
8
+ 6u
7
+ 37u
6
− 21u
5
− 89u
4
− 66u
3
− 54u
2
− 39u − 7)
2
· (u
19
+ 21u
18
+ ··· + 2406u + 562)
c
6
, c
12
((u − 1)
4
)(u + 1)
5
(u
18
− u
17
+ ··· + 2u − 1)(u
19
− u
18
+ ··· − u − 1)
c
8
, c
10
u(u
4
− 3u
2
+ 3)(u
4
− u
2
− 1)
· (u
9
− u
8
− 6u
7
+ 5u
6
+ 11u
5
− 7u
4
− 6u
3
+ 4u
2
− u − 1)
2
· (u
19
+ 3u
18
+ ··· + 14u − 10)
24
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
11
((y − 1)
9
)(y
18
+ 15y
17
+ ··· − 52y + 1)(y
19
+ 37y
18
+ ··· + 7y − 1)
c
2
, c
6
, c
7
c
12
((y − 1)
9
)(y
18
− 5y
17
+ ··· − 4y + 1)(y
19
− 3y
18
+ ··· + 7y − 1)
c
3
, c
4
, c
9
y(y
2
+ y − 1)
2
(y
2
+ 3y + 3)
2
· (y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ y
5
− 31y
4
− 24y
3
+ 6y
2
+ 9y − 1)
2
· (y
19
+ 15y
18
+ ··· − 16y − 4)
c
5
y(y
2
− 3y + 3)
2
(y
2
− y − 1)
2
(y
9
− 37y
8
+ ··· + 765y − 49)
2
· (y
19
− 45y
18
+ ··· − 2597328y − 315844)
c
8
, c
10
y(y
2
− 3y + 3)
2
(y
2
− y − 1)
2
· (y
9
− 13y
8
+ 68y
7
− 183y
6
+ 269y
5
− 211y
4
+ 80y
3
− 18y
2
+ 9y − 1)
2
· (y
19
− 21y
18
+ ··· − 384y − 100)
25
