12n
0675
(K12n
0675
)
A knot diagram
1
Linearized knot diagam
4 5 7 2 10 3 11 12 6 7 8 10
Solving Sequence
7,10
11 8 12
1,4
3 6 5 2 9
c
10
c
7
c
11
c
12
c
3
c
6
c
5
c
2
c
9
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h53226740u
26
− 42852864u
25
+ ··· + 64758649b + 11501166,
64856230u
26
− 108182252u
25
+ ··· + 129517298a − 547630611, u
27
− 4u
26
+ ··· + 4u + 1i
I
u
2
= hb + u, a + u + 2, u
2
+ u − 1i
I
u
3
= hb + u, a − 1, u
2
+ u − 1i
I
u
4
= hb + 1, a, u − 1i
* 4 irreducible components of dim
C
= 0, with total 32 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
= h5.32 × 10
7
u
26
− 4.29 × 10
7
u
25
+ · · · + 6.48 × 10
7
b + 1.15 × 10
7
, 6.49 ×
10
7
u
26
−1.08×10
8
u
25
+· · · +1.30×10
8
a−5.48×10
8
, u
27
−4u
26
+· · · +4u +1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
8
=
u
−u
3
+ u
a
12
=
−u
2
+ 1
u
4
− 2u
2
a
1
=
u
4
− 3u
2
+ 1
u
4
− 2u
2
a
4
=
−0.500753u
26
+ 0.835273u
25
+ ··· − 23.9653u + 4.22824
−0.821925u
26
+ 0.661732u
25
+ ··· − 3.64955u − 0.177600
a
3
=
−0.500753u
26
+ 0.835273u
25
+ ··· − 23.9653u + 4.22824
0.858098u
26
− 3.24135u
25
+ ··· − 8.82126u − 1.34534
a
6
=
2.71316u
26
− 6.18461u
25
+ ··· + 6.88730u − 3.99351
−2.21391u
26
+ 5.01988u
25
+ ··· + 8.14742u + 1.22175
a
5
=
0.499247u
26
− 1.16473u
25
+ ··· + 15.0347u − 2.77176
−2.21391u
26
+ 5.01988u
25
+ ··· + 8.14742u + 1.22175
a
2
=
−0.499247u
26
+ 1.16473u
25
+ ··· − 15.0347u + 2.77176
−1.71408u
26
+ 2.71888u
25
+ ··· − 0.116110u + 0.211149
a
9
=
−u
3
+ 2u
u
5
− 3u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes =
1678565237
129517298
u
26
−
2789533850
64758649
u
25
+ ··· −
5239074250
64758649
u −
3637568297
129517298
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
27
− 4u
26
+ ··· − 8u + 1
c
3
, c
6
u
27
+ 3u
26
+ ··· − 5u
2
+ 2
c
5
, c
9
u
27
+ 2u
26
+ ··· + 64u + 16
c
7
, c
8
, c
10
c
11
u
27
− 4u
26
+ ··· + 4u + 1
c
12
u
27
+ 18u
26
+ ··· + 3596u − 79
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
27
− 18y
26
+ ··· + 32y −1
c
3
, c
6
y
27
+ 3y
26
+ ··· + 20y −4
c
5
, c
9
y
27
+ 24y
26
+ ··· + 5760y −256
c
7
, c
8
, c
10
c
11
y
27
− 38y
26
+ ··· + 124y −1
c
12
y
27
− 98y
26
+ ··· + 19523924y −6241
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.121780 + 0.081011I
a = 0.927106 − 0.741069I
b = −0.180646 − 0.935031I
3.25561 − 1.59407I 1.48399 + 1.30043I
u = −1.121780 − 0.081011I
a = 0.927106 + 0.741069I
b = −0.180646 + 0.935031I
3.25561 + 1.59407I 1.48399 − 1.30043I
u = 0.676956 + 0.533355I
a = −0.448265 − 0.714880I
b = −0.426855 − 0.510105I
1.49150 + 0.51721I 4.06426 − 0.81218I
u = 0.676956 − 0.533355I
a = −0.448265 + 0.714880I
b = −0.426855 + 0.510105I
1.49150 − 0.51721I 4.06426 + 0.81218I
u = 0.791066 + 0.310151I
a = −0.264806 − 0.568080I
b = −0.454975 − 0.578171I
1.41628 + 0.49520I 5.40104 − 1.30639I
u = 0.791066 − 0.310151I
a = −0.264806 + 0.568080I
b = −0.454975 + 0.578171I
1.41628 − 0.49520I 5.40104 + 1.30639I
u = 0.228516 + 0.809567I
a = 0.877576 + 0.806986I
b = 0.241186 + 0.308813I
0.00983 + 4.15530I −1.52548 − 6.50197I
u = 0.228516 − 0.809567I
a = 0.877576 − 0.806986I
b = 0.241186 − 0.308813I
0.00983 − 4.15530I −1.52548 + 6.50197I
u = −1.100020 + 0.502260I
a = 0.243788 + 1.127340I
b = 0.07217 + 1.44072I
4.12391 − 8.58608I 0.18359 + 6.75545I
u = −1.100020 − 0.502260I
a = 0.243788 − 1.127340I
b = 0.07217 − 1.44072I
4.12391 + 8.58608I 0.18359 − 6.75545I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.240020 + 0.262480I
a = −0.444851 − 0.925878I
b = 0.124876 − 1.372680I
7.42745 − 3.21213I 3.92477 + 2.89567I
u = −1.240020 − 0.262480I
a = −0.444851 + 0.925878I
b = 0.124876 + 1.372680I
7.42745 + 3.21213I 3.92477 − 2.89567I
u = 0.687500
a = 0.446145
b = −3.22294
−0.443313 −52.6840
u = 1.42416
a = 0.788121
b = 1.37734
−1.56955 −5.83560
u = −0.505186
a = −3.09428
b = −0.715357
−8.08146 −27.4200
u = −1.62890
a = −0.330211
b = 2.82377
7.71976 −34.5090
u = 0.272815 + 0.206929I
a = −1.71928 − 0.05752I
b = 1.028070 + 0.618606I
−1.240020 + 0.678999I −6.54613 + 1.58470I
u = 0.272815 − 0.206929I
a = −1.71928 + 0.05752I
b = 1.028070 − 0.618606I
−1.240020 − 0.678999I −6.54613 − 1.58470I
u = 1.75657 + 0.14232I
a = −0.583385 + 0.775944I
b = −0.52277 + 2.72994I
14.1973 + 11.3192I 0
u = 1.75657 − 0.14232I
a = −0.583385 − 0.775944I
b = −0.52277 − 2.72994I
14.1973 − 11.3192I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.76109 + 0.08264I
a = −0.058331 − 0.610400I
b = 0.11130 − 2.01234I
11.07900 − 2.32684I 0
u = −1.76109 − 0.08264I
a = −0.058331 + 0.610400I
b = 0.11130 + 2.01234I
11.07900 + 2.32684I 0
u = 1.76627 + 0.02002I
a = −0.752423 − 0.715336I
b = −1.05997 − 2.28626I
13.78760 + 2.02303I 0
u = 1.76627 − 0.02002I
a = −0.752423 + 0.715336I
b = −1.05997 + 2.28626I
13.78760 − 2.02303I 0
u = 1.79047 + 0.06797I
a = 0.655019 − 0.749285I
b = 0.73536 − 2.51744I
18.4512 + 4.7028I 0
u = 1.79047 − 0.06797I
a = 0.655019 + 0.749285I
b = 0.73536 + 2.51744I
18.4512 − 4.7028I 0
u = −0.0970792
a = 6.32592
b = 0.401694
−0.870483 −12.0480
7
II. I
u
2
= hb + u, a + u + 2, u
2
+ u − 1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
u − 1
a
8
=
u
−u + 1
a
12
=
u
−u
a
1
=
0
−u
a
4
=
−u − 2
−u
a
3
=
−u − 2
−u + 1
a
6
=
2u + 3
0
a
5
=
2u + 3
0
a
2
=
2u + 3
−u + 1
a
9
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 16
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
10
, c
11
, c
12
u
2
+ u − 1
c
4
, c
6
, c
7
c
8
u
2
− u − 1
c
5
, c
9
u
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
10
, c
11
c
12
y
2
− 3y + 1
c
5
, c
9
y
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −2.61803
b = −0.618034
−7.89568 16.0000
u = −1.61803
a = −0.381966
b = 1.61803
7.89568 16.0000
11
III. I
u
3
= hb + u, a − 1, u
2
+ u − 1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
u − 1
a
8
=
u
−u + 1
a
12
=
u
−u
a
1
=
0
−u
a
4
=
1
−u
a
3
=
1
−1
a
6
=
u
0
a
5
=
u
0
a
2
=
u
−1
a
9
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 1
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
10
, c
11
, c
12
u
2
+ u − 1
c
4
, c
6
, c
7
c
8
u
2
− u − 1
c
5
, c
9
u
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
10
, c
11
c
12
y
2
− 3y + 1
c
5
, c
9
y
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 1.00000
b = −0.618034
0 1.00000
u = −1.61803
a = 1.00000
b = 1.61803
0 1.00000
15
IV. I
u
4
= hb + 1, a, u − 1i
(i) Arc colorings
a
7
=
0
1
a
10
=
1
0
a
11
=
1
−1
a
8
=
1
0
a
12
=
0
−1
a
1
=
−1
−1
a
4
=
0
−1
a
3
=
0
−1
a
6
=
0
1
a
5
=
1
1
a
2
=
−1
−2
a
9
=
1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
9
c
10
, c
11
, c
12
u − 1
c
3
, c
6
u
c
4
, c
5
, c
7
c
8
u + 1
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
7
, c
8
c
9
, c
10
, c
11
c
12
y −1
c
3
, c
6
y
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
0 0
19
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)(u
2
+ u − 1)
2
(u
27
− 4u
26
+ ··· − 8u + 1)
c
3
u(u
2
+ u − 1)
2
(u
27
+ 3u
26
+ ··· − 5u
2
+ 2)
c
4
(u + 1)(u
2
− u − 1)
2
(u
27
− 4u
26
+ ··· − 8u + 1)
c
5
u
4
(u + 1)(u
27
+ 2u
26
+ ··· + 64u + 16)
c
6
u(u
2
− u − 1)
2
(u
27
+ 3u
26
+ ··· − 5u
2
+ 2)
c
7
, c
8
(u + 1)(u
2
− u − 1)
2
(u
27
− 4u
26
+ ··· + 4u + 1)
c
9
u
4
(u − 1)(u
27
+ 2u
26
+ ··· + 64u + 16)
c
10
, c
11
(u − 1)(u
2
+ u − 1)
2
(u
27
− 4u
26
+ ··· + 4u + 1)
c
12
(u − 1)(u
2
+ u − 1)
2
(u
27
+ 18u
26
+ ··· + 3596u − 79)
20
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)(y
2
− 3y + 1)
2
(y
27
− 18y
26
+ ··· + 32y −1)
c
3
, c
6
y(y
2
− 3y + 1)
2
(y
27
+ 3y
26
+ ··· + 20y −4)
c
5
, c
9
y
4
(y −1)(y
27
+ 24y
26
+ ··· + 5760y −256)
c
7
, c
8
, c
10
c
11
(y −1)(y
2
− 3y + 1)
2
(y
27
− 38y
26
+ ··· + 124y −1)
c
12
(y −1)(y
2
− 3y + 1)
2
(y
27
− 98y
26
+ ··· + 1.95239 × 10
7
y −6241)
21
