12a
0576
(K12a
0576
)
A knot diagram
1
Linearized knot diagam
3 7 9 10 11 2 12 1 4 5 6 8
Solving Sequence
2,6
7
3,12
8 1 9 11 5 10 4
c
6
c
2
c
7
c
1
c
8
c
11
c
5
c
10
c
4
c
3
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h11985u
24
+ 25005u
23
+ ··· + 72188b − 244730,
− 211795u
24
− 38567u
23
+ ··· + 505316a − 3814672, u
25
− u
24
+ ··· − 9u − 7i
I
u
2
= hu
3
+ b − u, a + u, u
9
− 3u
7
+ u
6
+ 3u
5
− 2u
4
− 3u
3
+ u
2
+ 2u − 1i
I
u
3
= hb − a + 1, a
2
− 2a − 2, u + 1i
I
u
4
= hb − 1, a, u − 1i
I
u
5
= hb + 1, a + 2, u − 1i
I
u
6
= hb − 1, a − 1, u − 1i
I
u
7
= hb, a − 1, u + 1i
I
v
1
= ha, b + 1, v + 1i
* 8 irreducible components of dim
C
= 0, with total 41 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
= h11985u
24
+ 25005u
23
+ · · · + 72188b − 244730, −2.12 × 10
5
u
24
−
3.86 × 10
4
u
23
+ · · · + 5.05 × 10
5
a − 3.81 × 10
6
, u
25
− u
24
+ · · · − 9u − 7i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
3
=
−u
−u
3
+ u
a
12
=
0.419134u
24
+ 0.0763225u
23
+ ··· − 0.423143u + 7.54908
−0.166025u
24
− 0.346387u
23
+ ··· + 2.60629u + 3.39018
a
8
=
0.484311u
24
− 0.650336u
23
+ ··· + 12.6564u − 0.752505
−0.101956u
24
− 0.126046u
23
+ ··· + 0.520322u − 3.17143
a
1
=
u
3
u
5
− u
3
+ u
a
9
=
−0.523558u
24
− 0.576639u
23
+ ··· + 3.23109u − 5.84862
0.409514u
24
+ 0.597509u
23
+ ··· − 4.14134u − 3.60527
a
11
=
0.585159u
24
+ 0.422710u
23
+ ··· − 3.02944u + 4.15891
−0.166025u
24
− 0.346387u
23
+ ··· + 2.60629u + 3.39018
a
5
=
−0.995781u
24
− 0.0732195u
23
+ ··· − 7.99477u + 3.18634
0.246842u
24
+ 0.673949u
23
+ ··· − 7.09679u + 2.13396
a
10
=
−0.479844u
24
− 0.692303u
23
+ ··· + 4.66701u − 9.17722
0.348285u
24
+ 0.294232u
23
+ ··· − 0.412423u − 4.84612
a
4
=
1.37385u
24
− 0.126816u
23
+ ··· + 11.2503u − 2.86998
0.198759u
24
− 1.06936u
23
+ ··· + 15.0646u + 1.75878
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
60407
36094
u
24
+
67585
36094
u
23
+ ··· −
588579
36094
u +
309390
18047
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
25
+ 7u
24
+ ··· + 739u + 49
c
2
, c
6
u
25
− u
24
+ ··· − 9u − 7
c
3
, c
4
, c
5
c
9
, c
10
, c
11
u
25
+ 2u
24
+ ··· + 2u + 2
c
7
, c
8
, c
12
u
25
+ u
24
+ ··· + 39u − 9
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
25
+ 29y
24
+ ··· + 138539y −2401
c
2
, c
6
y
25
− 7y
24
+ ··· + 739y −49
c
3
, c
4
, c
5
c
9
, c
10
, c
11
y
25
− 36y
24
+ ··· + 147y
2
− 4
c
7
, c
8
, c
12
y
25
− 31y
24
+ ··· + 1755y −81
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.896141 + 0.476708I
a = 0.264009 + 1.247940I
b = −0.518330 + 0.326163I
−0.17909 − 3.47166I 8.30266 + 8.94170I
u = 0.896141 − 0.476708I
a = 0.264009 − 1.247940I
b = −0.518330 − 0.326163I
−0.17909 + 3.47166I 8.30266 − 8.94170I
u = 0.712580 + 0.800068I
a = −0.424664 − 0.601315I
b = −0.730693 − 0.421612I
6.89461 + 0.59668I 14.2492 − 1.6095I
u = 0.712580 − 0.800068I
a = −0.424664 + 0.601315I
b = −0.730693 + 0.421612I
6.89461 − 0.59668I 14.2492 + 1.6095I
u = −0.871015 + 0.279677I
a = −0.337858 + 0.872649I
b = −0.111064 + 0.372277I
−1.37494 + 1.08288I 0.78629 − 1.57388I
u = −0.871015 − 0.279677I
a = −0.337858 − 0.872649I
b = −0.111064 − 0.372277I
−1.37494 − 1.08288I 0.78629 + 1.57388I
u = −1.09658
a = −1.64444
b = −1.74239
11.6378 5.99100
u = −0.874419 + 0.729333I
a = 0.344893 − 1.149310I
b = 0.073544 − 0.603993I
4.42212 + 2.78000I 9.55915 − 2.23614I
u = −0.874419 − 0.729333I
a = 0.344893 + 1.149310I
b = 0.073544 + 0.603993I
4.42212 − 2.78000I 9.55915 + 2.23614I
u = −0.582867 + 0.979338I
a = 0.091026 − 0.120602I
b = 1.354550 − 0.178353I
13.77140 − 2.72282I 16.3024 + 1.1923I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.582867 − 0.979338I
a = 0.091026 + 0.120602I
b = 1.354550 + 0.178353I
13.77140 + 2.72282I 16.3024 − 1.1923I
u = −0.947893 + 0.650696I
a = −0.21022 + 1.47307I
b = 1.250280 + 0.142199I
5.58852 + 5.07025I 11.97391 − 5.86575I
u = −0.947893 − 0.650696I
a = −0.21022 − 1.47307I
b = 1.250280 − 0.142199I
5.58852 − 5.07025I 11.97391 + 5.86575I
u = 0.544687 + 1.115460I
a = 0.171496 + 0.032811I
b = −1.82545 − 0.04307I
−13.8936 + 3.7790I 16.4732 − 0.8824I
u = 0.544687 − 1.115460I
a = 0.171496 − 0.032811I
b = −1.82545 + 0.04307I
−13.8936 − 3.7790I 16.4732 + 0.8824I
u = 1.006720 + 0.738523I
a = −0.027411 − 1.413620I
b = 0.601135 − 0.505413I
6.00828 − 6.40238I 12.19839 + 6.95987I
u = 1.006720 − 0.738523I
a = −0.027411 + 1.413620I
b = 0.601135 + 0.505413I
6.00828 + 6.40238I 12.19839 − 6.95987I
u = 1.000710 + 0.748113I
a = 0.18239 + 1.58644I
b = −1.80123 + 0.03491I
16.8362 − 5.8723I 12.48696 + 4.57611I
u = 1.000710 − 0.748113I
a = 0.18239 − 1.58644I
b = −1.80123 − 0.03491I
16.8362 + 5.8723I 12.48696 − 4.57611I
u = −1.124230 + 0.767226I
a = −0.28369 − 1.52440I
b = −1.290050 − 0.256043I
12.1250 + 9.0916I 14.3993 − 5.8572I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.124230 − 0.767226I
a = −0.28369 + 1.52440I
b = −1.290050 + 0.256043I
12.1250 − 9.0916I 14.3993 + 5.8572I
u = 1.20486 + 0.78616I
a = 0.47352 − 1.57215I
b = 1.81036 − 0.06682I
−15.9680 − 10.6007I 14.6369 + 4.9142I
u = 1.20486 − 0.78616I
a = 0.47352 + 1.57215I
b = 1.81036 + 0.06682I
−15.9680 + 10.6007I 14.6369 − 4.9142I
u = 0.539518
a = −0.734615
b = 0.400542
0.694972 14.9110
u = −0.373488
a = 4.17778
b = 1.71576
14.6126 18.3610
7
II. I
u
2
= hu
3
+ b − u, a + u, u
9
− 3u
7
+ u
6
+ 3u
5
− 2u
4
− 3u
3
+ u
2
+ 2u − 1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
3
=
−u
−u
3
+ u
a
12
=
−u
−u
3
+ u
a
8
=
u
2
+ 1
u
4
a
1
=
u
3
u
5
− u
3
+ u
a
9
=
−u
4
+ u
2
+ 1
−u
6
+ 2u
4
− u
2
a
11
=
u
3
− 2u
−u
3
+ u
a
5
=
−u
6
+ 3u
4
− 2u
2
+ 1
u
6
− 2u
4
+ u
2
a
10
=
u
7
+ u
6
− 2u
5
− 2u
4
+ u
3
+ u
2
− u − 1
−u
6
+ 2u
4
+ u
3
− u
2
− u + 1
a
4
=
u
7
− 2u
5
−u
6
+ 2u
4
+ u
3
− u
2
− u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 14
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
9
+ 6u
8
+ 15u
7
+ 25u
6
+ 35u
5
+ 36u
4
+ 27u
3
+ 17u
2
+ 6u + 1
c
2
, c
6
, c
7
c
8
, c
12
u
9
− 3u
7
+ u
6
+ 3u
5
− 2u
4
− 3u
3
+ u
2
+ 2u − 1
c
3
, c
4
, c
5
c
9
, c
10
, c
11
(u
3
− u
2
− 2u + 1)
3
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
9
− 6y
8
− 5y
7
+ 47y
6
+ 43y
5
− 88y
4
− 125y
3
− 37y
2
+ 2y − 1
c
2
, c
6
, c
7
c
8
, c
12
y
9
− 6y
8
+ 15y
7
− 25y
6
+ 35y
5
− 36y
4
+ 27y
3
− 17y
2
+ 6y − 1
c
3
, c
4
, c
5
c
9
, c
10
, c
11
(y
3
− 5y
2
+ 6y − 1)
3
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.689884 + 0.654080I
a = 0.689884 − 0.654080I
b = −1.24698
6.34475 14.0000
u = −0.689884 − 0.654080I
a = 0.689884 + 0.654080I
b = −1.24698
6.34475 14.0000
u = 0.743582 + 0.811631I
a = −0.743582 − 0.811631I
b = 1.80194
17.6243 14.0000
u = 0.743582 − 0.811631I
a = −0.743582 + 0.811631I
b = 1.80194
17.6243 14.0000
u = −1.17430
a = 1.17430
b = 0.445042
0.704972 14.0000
u = 1.37977
a = −1.37977
b = −1.24698
6.34475 14.0000
u = 0.587151 + 0.185036I
a = −0.587151 − 0.185036I
b = 0.445042
0.704972 14.0000
u = 0.587151 − 0.185036I
a = −0.587151 + 0.185036I
b = 0.445042
0.704972 14.0000
u = −1.48716
a = 1.48716
b = 1.80194
17.6243 14.0000
11
III. I
u
3
= hb − a + 1, a
2
− 2a − 2, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
6
=
1
0
a
7
=
1
1
a
3
=
1
0
a
12
=
a
a − 1
a
8
=
a + 1
a
a
1
=
−1
−1
a
9
=
a
a − 1
a
11
=
1
a − 1
a
5
=
a
3
a
10
=
−a − 1
−2a + 2
a
4
=
−a − 1
−3
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
12
(u − 1)
2
c
3
, c
4
, c
5
c
9
, c
10
, c
11
u
2
− 3
c
6
, c
7
, c
8
(u + 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
8
, c
12
(y − 1)
2
c
3
, c
4
, c
5
c
9
, c
10
, c
11
(y − 3)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −0.732051
b = −1.73205
13.1595 12.0000
u = −1.00000
a = 2.73205
b = 1.73205
13.1595 12.0000
15
IV. I
u
4
= hb − 1, a, u − 1i
(i) Arc colorings
a
2
=
0
1
a
6
=
1
0
a
7
=
1
1
a
3
=
−1
0
a
12
=
0
1
a
8
=
1
0
a
1
=
1
1
a
9
=
0
−1
a
11
=
−1
1
a
5
=
0
1
a
10
=
−1
0
a
4
=
−1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
7
c
8
u − 1
c
2
, c
9
, c
10
c
11
, c
12
u + 1
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y − 1
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0
b = 1.00000
3.28987 12.0000
19
V. I
u
5
= hb + 1, a + 2, u − 1i
(i) Arc colorings
a
2
=
0
1
a
6
=
1
0
a
7
=
1
1
a
3
=
−1
0
a
12
=
−2
−1
a
8
=
3
2
a
1
=
1
1
a
9
=
2
1
a
11
=
−1
−1
a
5
=
2
1
a
10
=
1
0
a
4
=
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
6
, c
7
c
8
, c
9
, c
10
c
11
u − 1
c
2
, c
3
, c
4
c
5
, c
12
u + 1
21
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y − 1
22
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −2.00000
b = −1.00000
3.28987 12.0000
23
VI. I
u
6
= hb − 1, a − 1, u − 1i
(i) Arc colorings
a
2
=
0
1
a
6
=
1
0
a
7
=
1
1
a
3
=
−1
0
a
12
=
1
1
a
8
=
1
1
a
1
=
1
1
a
9
=
1
1
a
11
=
0
1
a
5
=
1
1
a
10
=
−1
0
a
4
=
0
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
24
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u + 1
c
2
, c
3
, c
4
c
5
, c
6
, c
9
c
10
, c
11
u − 1
c
7
, c
8
, c
12
u
25
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
9
, c
10
, c
11
y − 1
c
7
, c
8
, c
12
y
26
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 1.00000
1.64493 6.00000
27
VII. I
u
7
= hb, a − 1, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
6
=
1
0
a
7
=
1
1
a
3
=
1
0
a
12
=
1
0
a
8
=
2
1
a
1
=
−1
−1
a
9
=
1
0
a
11
=
1
0
a
5
=
1
0
a
10
=
1
0
a
4
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
28
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
12
u − 1
c
3
, c
4
, c
5
c
9
, c
10
, c
11
u
c
6
, c
7
, c
8
u + 1
29
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
8
, c
12
y − 1
c
3
, c
4
, c
5
c
9
, c
10
, c
11
y
30
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
7
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 1.00000
b = 0
0 0
31
VIII. I
v
1
= ha, b + 1, v + 1i
(i) Arc colorings
a
2
=
−1
0
a
6
=
1
0
a
7
=
1
0
a
3
=
−1
0
a
12
=
0
−1
a
8
=
1
−1
a
1
=
−1
0
a
9
=
0
−1
a
11
=
1
−1
a
5
=
0
1
a
10
=
1
0
a
4
=
−1
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 18
32
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
u
c
3
, c
4
, c
5
c
7
, c
8
, c
9
c
10
, c
11
, c
12
u + 1
33
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
y
c
3
, c
4
, c
5
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y − 1
34
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = −1.00000
4.93480 18.0000
35
IX. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u(u − 1)
5
(u + 1)
· (u
9
+ 6u
8
+ 15u
7
+ 25u
6
+ 35u
5
+ 36u
4
+ 27u
3
+ 17u
2
+ 6u + 1)
· (u
25
+ 7u
24
+ ··· + 739u + 49)
c
2
u(u − 1)
4
(u + 1)
2
(u
9
− 3u
7
+ u
6
+ 3u
5
− 2u
4
− 3u
3
+ u
2
+ 2u − 1)
· (u
25
− u
24
+ ··· − 9u − 7)
c
3
, c
4
, c
5
c
9
, c
10
, c
11
u(u − 1)
2
(u + 1)
2
(u
2
− 3)(u
3
− u
2
− 2u + 1)
3
· (u
25
+ 2u
24
+ ··· + 2u + 2)
c
6
u(u − 1)
3
(u + 1)
3
(u
9
− 3u
7
+ u
6
+ 3u
5
− 2u
4
− 3u
3
+ u
2
+ 2u − 1)
· (u
25
− u
24
+ ··· − 9u − 7)
c
7
, c
8
u(u − 1)
2
(u + 1)
4
(u
9
− 3u
7
+ u
6
+ 3u
5
− 2u
4
− 3u
3
+ u
2
+ 2u − 1)
· (u
25
+ u
24
+ ··· + 39u − 9)
c
12
u(u − 1)
3
(u + 1)
3
(u
9
− 3u
7
+ u
6
+ 3u
5
− 2u
4
− 3u
3
+ u
2
+ 2u − 1)
· (u
25
+ u
24
+ ··· + 39u − 9)
36
X. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y(y − 1)
6
· (y
9
− 6y
8
− 5y
7
+ 47y
6
+ 43y
5
− 88y
4
− 125y
3
− 37y
2
+ 2y − 1)
· (y
25
+ 29y
24
+ ··· + 138539y − 2401)
c
2
, c
6
y(y − 1)
6
· (y
9
− 6y
8
+ 15y
7
− 25y
6
+ 35y
5
− 36y
4
+ 27y
3
− 17y
2
+ 6y − 1)
· (y
25
− 7y
24
+ ··· + 739y − 49)
c
3
, c
4
, c
5
c
9
, c
10
, c
11
y(y − 3)
2
(y − 1)
4
(y
3
− 5y
2
+ 6y − 1)
3
· (y
25
− 36y
24
+ ··· + 147y
2
− 4)
c
7
, c
8
, c
12
y(y − 1)
6
· (y
9
− 6y
8
+ 15y
7
− 25y
6
+ 35y
5
− 36y
4
+ 27y
3
− 17y
2
+ 6y − 1)
· (y
25
− 31y
24
+ ··· + 1755y − 81)
37
