12n
0305
(K12n
0305
)
A knot diagram
1
Linearized knot diagam
3 6 7 10 2 5 11 12 4 7 8 10
Solving Sequence
7,10
11 8 12
1,5
4 3 6 2 9
c
10
c
7
c
11
c
12
c
4
c
3
c
6
c
2
c
9
c
1
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−3u
19
− 6u
18
+ ··· + 2b − 3, −u
19
− 12u
18
+ ··· + 4a + 11, u
20
+ 4u
19
+ ··· + 2u + 1i
I
u
2
= hb, a
3
− a
2
u + a
2
− 2au + 4a − 2u + 3, u
2
− u − 1i
* 2 irreducible components of dim
C
= 0, with total 26 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
=
h−3u
19
−6u
18
+· · ·+2b−3, −u
19
−12u
18
+· · ·+4a+11, u
20
+4u
19
+· · ·+2u+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
5
=
1
4
u
19
+ 3u
18
+ ··· − 15u −
11
4
3
2
u
19
+ 3u
18
+ ··· + 2u +
3
2
a
4
=
7
4
u
19
+ 6u
18
+ ··· − 13u −
5
4
3
2
u
19
+ 3u
18
+ ··· + 2u +
3
2
a
3
=
7
4
u
19
+ 6u
18
+ ··· − 13u −
5
4
−
5
4
u
19
−
11
4
u
18
+ ··· +
7
4
u +
1
2
a
6
=
−
1
4
u
18
−
3
4
u
17
+ ··· −
21
4
u −
3
4
u
6
− 4u
4
+ 3u
2
+ 2u
a
2
=
1
4
u
18
+
3
4
u
17
+ ··· +
17
4
u +
7
4
1
4
u
19
+
1
2
u
18
+ ··· −
1
2
u +
1
4
a
9
=
u
3
− 2u
u
5
− 3u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −2u
19
− 8u
18
+ 16u
17
+
191
2
u
16
− 22u
15
−
943
2
u
14
− 192u
13
+
2425
2
u
12
+
1981
2
u
11
−
1627u
10
−2079u
9
+
1641
2
u
8
+ 2156u
7
+ 431u
6
−908u
5
−571u
4
−
53
2
u
3
+ 72u
2
+ 29u −
21
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
20
+ 9u
19
+ ··· + 21u + 1
c
2
, c
5
u
20
+ 3u
19
+ ··· − 7u − 1
c
3
u
20
− 3u
19
+ ··· − 17u − 1
c
4
, c
9
u
20
− u
19
+ ··· − 224u − 64
c
7
, c
8
, c
10
c
11
u
20
+ 4u
19
+ ··· + 2u + 1
c
12
u
20
− 20u
19
+ ··· − 3324u − 239
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
20
+ 7y
19
+ ··· − 221y + 1
c
2
, c
5
y
20
− 9y
19
+ ··· − 21y + 1
c
3
y
20
− 53y
19
+ ··· − 69y + 1
c
4
, c
9
y
20
− 35y
19
+ ··· − 5120y + 4096
c
7
, c
8
, c
10
c
11
y
20
− 32y
19
+ ··· − 26y + 1
c
12
y
20
− 116y
19
+ ··· − 2215058y + 57121
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.995757 + 0.162502I
a = 0.409533 + 0.992912I
b = −0.250501 − 1.034700I
−1.72654 − 2.01165I −16.2599 + 2.9665I
u = −0.995757 − 0.162502I
a = 0.409533 − 0.992912I
b = −0.250501 + 1.034700I
−1.72654 + 2.01165I −16.2599 − 2.9665I
u = −0.618715 + 0.543682I
a = 1.42405 − 0.18504I
b = −1.183060 − 0.429676I
−1.91039 + 3.31491I −16.4023 − 5.7778I
u = −0.618715 − 0.543682I
a = 1.42405 + 0.18504I
b = −1.183060 + 0.429676I
−1.91039 − 3.31491I −16.4023 + 5.7778I
u = 1.332060 + 0.199912I
a = 0.389569 + 0.439368I
b = −1.42120 + 0.01301I
−6.30937 − 1.46809I −15.3912 + 0.5997I
u = 1.332060 − 0.199912I
a = 0.389569 − 0.439368I
b = −1.42120 − 0.01301I
−6.30937 + 1.46809I −15.3912 − 0.5997I
u = 1.37734 + 0.35940I
a = −0.841843 − 0.739118I
b = 1.77867 + 0.01269I
−8.39761 − 6.66344I −17.7359 + 5.0430I
u = 1.37734 − 0.35940I
a = −0.841843 + 0.739118I
b = 1.77867 − 0.01269I
−8.39761 + 6.66344I −17.7359 − 5.0430I
u = −0.221606 + 0.329450I
a = −0.272738 + 1.295930I
b = 0.671635 − 0.209494I
−0.674628 − 0.165298I −12.57313 − 0.79676I
u = −0.221606 − 0.329450I
a = −0.272738 − 1.295930I
b = 0.671635 + 0.209494I
−0.674628 + 0.165298I −12.57313 + 0.79676I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.370071
a = 0.180421
b = 0.474305
−0.654279 −14.9070
u = 1.63979 + 0.13284I
a = −0.602671 + 0.356588I
b = 0.982271 − 0.847239I
−10.89840 + 0.58168I −20.0520 − 2.9617I
u = 1.63979 − 0.13284I
a = −0.602671 − 0.356588I
b = 0.982271 + 0.847239I
−10.89840 − 0.58168I −20.0520 + 2.9617I
u = 0.328694 + 0.052343I
a = −0.15243 + 3.84880I
b = 0.004947 − 0.482870I
2.50647 + 2.74594I −1.72205 − 1.87916I
u = 0.328694 − 0.052343I
a = −0.15243 − 3.84880I
b = 0.004947 + 0.482870I
2.50647 − 2.74594I −1.72205 + 1.87916I
u = −1.84665 + 0.06383I
a = −0.420429 + 0.524003I
b = 2.10366 − 0.42974I
−18.2504 + 2.8693I −15.5779 − 0.3444I
u = −1.84665 − 0.06383I
a = −0.420429 − 0.524003I
b = 2.10366 + 0.42974I
−18.2504 − 2.8693I −15.5779 + 0.3444I
u = −1.85353 + 0.10925I
a = 0.515285 − 0.864581I
b = −2.14536 + 0.73132I
19.1451 + 9.0847I −17.3799 − 4.2904I
u = −1.85353 − 0.10925I
a = 0.515285 + 0.864581I
b = −2.14536 − 0.73132I
19.1451 − 9.0847I −17.3799 + 4.2904I
u = −1.91314
a = 0.922913
b = −2.55644
14.2074 −19.9050
6
II. I
u
2
= hb, a
3
− a
2
u + a
2
− 2au + 4a − 2u + 3, 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
5
=
a
0
a
4
=
a
0
a
3
=
a
−au − a
a
6
=
a
2
u
u
a
2
=
a
2
u
−2a
2
u − a
2
− u
a
9
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2a
2
u + a
2
+ 2au − a + 3u − 19
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
4
, c
9
u
6
c
5
(u
3
− u
2
+ 1)
2
c
6
(u
3
+ u
2
+ 2u + 1)
2
c
7
, c
8
(u
2
+ u − 1)
3
c
10
, c
11
, c
12
(u
2
− u − 1)
3
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
5
(y
3
− y
2
+ 2y −1)
2
c
4
, c
9
y
6
c
7
, c
8
, c
10
c
11
, c
12
(y
2
− 3y + 1)
3
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = −0.922021
b = 0
−2.10041 −18.9930
u = −0.618034
a = −0.34801 + 2.11500I
b = 0
2.03717 + 2.82812I −19.0485 − 4.3818I
u = −0.618034
a = −0.34801 − 2.11500I
b = 0
2.03717 − 2.82812I −19.0485 + 4.3818I
u = 1.61803
a = 0.132927 + 0.807858I
b = 0
−5.85852 − 2.82812I −16.5384 + 2.7162I
u = 1.61803
a = 0.132927 − 0.807858I
b = 0
−5.85852 + 2.82812I −16.5384 − 2.7162I
u = 1.61803
a = 0.352181
b = 0
−9.99610 −12.8330
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
3
− u
2
+ 2u − 1)
2
)(u
20
+ 9u
19
+ ··· + 21u + 1)
c
2
((u
3
+ u
2
− 1)
2
)(u
20
+ 3u
19
+ ··· − 7u − 1)
c
3
((u
3
− u
2
+ 2u − 1)
2
)(u
20
− 3u
19
+ ··· − 17u − 1)
c
4
, c
9
u
6
(u
20
− u
19
+ ··· − 224u − 64)
c
5
((u
3
− u
2
+ 1)
2
)(u
20
+ 3u
19
+ ··· − 7u − 1)
c
6
((u
3
+ u
2
+ 2u + 1)
2
)(u
20
+ 9u
19
+ ··· + 21u + 1)
c
7
, c
8
((u
2
+ u − 1)
3
)(u
20
+ 4u
19
+ ··· + 2u + 1)
c
10
, c
11
((u
2
− u − 1)
3
)(u
20
+ 4u
19
+ ··· + 2u + 1)
c
12
((u
2
− u − 1)
3
)(u
20
− 20u
19
+ ··· − 3324u − 239)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
((y
3
+ 3y
2
+ 2y −1)
2
)(y
20
+ 7y
19
+ ··· − 221y + 1)
c
2
, c
5
((y
3
− y
2
+ 2y −1)
2
)(y
20
− 9y
19
+ ··· − 21y + 1)
c
3
((y
3
+ 3y
2
+ 2y −1)
2
)(y
20
− 53y
19
+ ··· − 69y + 1)
c
4
, c
9
y
6
(y
20
− 35y
19
+ ··· − 5120y + 4096)
c
7
, c
8
, c
10
c
11
((y
2
− 3y + 1)
3
)(y
20
− 32y
19
+ ··· − 26y + 1)
c
12
((y
2
− 3y + 1)
3
)(y
20
− 116y
19
+ ··· − 2215058y + 57121)
12
