12n
0045
(K12n
0045
)
A knot diagram
1
Linearized knot diagam
3 5 6 9 2 12 11 5 6 7 10 8
Solving Sequence
2,6
5
3,8
9 10 1 4 12 7 11
c
5
c
2
c
8
c
9
c
1
c
4
c
12
c
6
c
11
c
3
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−148u
29
− 963u
28
+ ··· + 32b − 233, −153u
29
− 960u
28
+ ··· + 32a − 98, u
30
+ 7u
29
+ ··· + 7u + 1i
I
u
2
= h−au + b + a, a
6
+ a
5
u − a
4
u + a
4
+ 2a
3
− au + a + 1, u
2
− u + 1i
* 2 irreducible components of dim
C
= 0, with total 42 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−148u
29
− 963u
28
+ · · · + 32b − 233, −153u
29
− 960u
28
+ · · · +
32a − 98, u
30
+ 7u
29
+ · · · + 7u + 1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
5
=
1
u
2
a
3
=
u
u
3
+ u
a
8
=
4.78125u
29
+ 30u
28
+ ··· + 28.9688u + 3.06250
4.62500u
29
+ 30.0938u
28
+ ··· + 43.3750u + 7.28125
a
9
=
1.90625u
29
+ 11.6875u
28
+ ··· + 5.09375u − 0.750000
3.56250u
29
+ 23.0313u
28
+ ··· + 33.5625u + 5.46875
a
10
=
5.46875u
29
+ 34.7188u
28
+ ··· + 38.6563u + 4.71875
3.56250u
29
+ 23.0313u
28
+ ··· + 33.5625u + 5.46875
a
1
=
u
3
u
5
+ u
3
+ u
a
4
=
−u
3
u
3
+ u
a
12
=
−0.0312500u
28
− 0.187500u
27
+ ··· − 2.18750u + 0.968750
0.0312500u
29
+ 0.218750u
28
+ ··· + 0.218750u + 0.0312500
a
7
=
−
9
32
u
29
− 2u
28
+ ··· −
81
32
u +
5
8
0.281250u
29
+ 1.71875u
28
+ ··· + 0.531250u + 0.0312500
a
11
=
9
32
u
28
+
27
16
u
27
+ ··· + 2u +
11
32
−0.281250u
29
− 1.65625u
28
+ ··· + 0.843750u + 0.0312500
(ii) Obstruction class = −1
(iii) Cusp Shapes =
45
4
u
29
+
1169
16
u
28
+ ··· + 103u +
99
8
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
30
+ 3u
29
+ ··· + 3u + 1
c
2
, c
5
u
30
+ 7u
29
+ ··· + 7u + 1
c
3
u
30
− 7u
29
+ ··· + 123187u + 9881
c
4
, c
8
u
30
+ u
29
+ ··· + 8192u + 4096
c
6
u
30
+ 9u
29
+ ··· + 57u + 17
c
7
, c
10
u
30
+ 3u
29
+ ··· + u + 1
c
9
, c
12
u
30
− 3u
29
+ ··· − 3u + 1
c
11
u
30
+ 13u
29
+ ··· − 7u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
30
+ 55y
29
+ ··· + 63y + 1
c
2
, c
5
y
30
+ 3y
29
+ ··· + 3y + 1
c
3
y
30
+ 107y
29
+ ··· + 1844116003y + 97634161
c
4
, c
8
y
30
+ 65y
29
+ ··· + 161480704y
2
+ 16777216
c
6
y
30
− 9y
29
+ ··· + 2531y + 289
c
7
, c
10
y
30
− 13y
29
+ ··· + 7y + 1
c
9
, c
12
y
30
− 53y
29
+ ··· + 7y + 1
c
11
y
30
+ 11y
29
+ ··· − 89y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.156149 + 0.923590I
a = 1.55521 − 0.59601I
b = −0.641878 + 0.220216I
−2.03388 + 4.90769I −5.49958 − 7.49846I
u = 0.156149 − 0.923590I
a = 1.55521 + 0.59601I
b = −0.641878 − 0.220216I
−2.03388 − 4.90769I −5.49958 + 7.49846I
u = 0.399490 + 0.989549I
a = −0.655955 + 0.923964I
b = −0.207702 − 0.507122I
−0.76957 + 1.31164I 0.008085 − 1.158902I
u = 0.399490 − 0.989549I
a = −0.655955 − 0.923964I
b = −0.207702 + 0.507122I
−0.76957 − 1.31164I 0.008085 + 1.158902I
u = 0.918172 + 0.122403I
a = −0.292882 + 0.357499I
b = −0.60497 + 1.73623I
2.37348 + 2.46946I 2.98715 − 3.45316I
u = 0.918172 − 0.122403I
a = −0.292882 − 0.357499I
b = −0.60497 − 1.73623I
2.37348 − 2.46946I 2.98715 + 3.45316I
u = 0.780975 + 0.876399I
a = −1.049250 − 0.803341I
b = −0.003320 − 1.369010I
1.59076 + 0.61155I 0.819771 + 0.666887I
u = 0.780975 − 0.876399I
a = −1.049250 + 0.803341I
b = −0.003320 + 1.369010I
1.59076 − 0.61155I 0.819771 − 0.666887I
u = 0.362356 + 0.695450I
a = −0.993697 − 0.057570I
b = −0.0321747 + 0.0227484I
−0.194740 + 1.399730I −2.02435 − 4.86797I
u = 0.362356 − 0.695450I
a = −0.993697 + 0.057570I
b = −0.0321747 − 0.0227484I
−0.194740 − 1.399730I −2.02435 + 4.86797I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.724731 + 1.015800I
a = 0.79644 + 1.28595I
b = −1.11851 + 1.13827I
1.10886 + 5.31866I −0.13308 − 5.18400I
u = 0.724731 − 1.015800I
a = 0.79644 − 1.28595I
b = −1.11851 − 1.13827I
1.10886 − 5.31866I −0.13308 + 5.18400I
u = −0.624839 + 0.318859I
a = −1.57129 − 0.30420I
b = −1.44565 + 0.98598I
0.78177 − 6.39042I −0.72567 + 4.22016I
u = −0.624839 − 0.318859I
a = −1.57129 + 0.30420I
b = −1.44565 − 0.98598I
0.78177 + 6.39042I −0.72567 − 4.22016I
u = 0.021362 + 0.665268I
a = 1.74798 + 0.18763I
b = −0.284692 − 0.540613I
−2.74789 − 1.44408I −8.12661 + 0.68826I
u = 0.021362 − 0.665268I
a = 1.74798 − 0.18763I
b = −0.284692 + 0.540613I
−2.74789 + 1.44408I −8.12661 − 0.68826I
u = −0.626826 + 0.181498I
a = 1.68869 + 0.16006I
b = 1.59366 − 0.59325I
2.71723 − 1.30741I 2.25892 + 0.04222I
u = −0.626826 − 0.181498I
a = 1.68869 − 0.16006I
b = 1.59366 + 0.59325I
2.71723 + 1.30741I 2.25892 − 0.04222I
u = −1.03232 + 1.03051I
a = −1.33615 − 1.25080I
b = −2.95729 + 0.40290I
11.18100 − 3.78919I −2.00000 + 1.99020I
u = −1.03232 − 1.03051I
a = −1.33615 + 1.25080I
b = −2.95729 − 0.40290I
11.18100 + 3.78919I −2.00000 − 1.99020I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.13894 + 0.98249I
a = −0.85062 − 1.72090I
b = −4.00129 − 1.44921I
16.0192 + 4.0819I 0
u = −1.13894 − 0.98249I
a = −0.85062 + 1.72090I
b = −4.00129 + 1.44921I
16.0192 − 4.0819I 0
u = −1.00742 + 1.12472I
a = −1.90712 − 1.30827I
b = −3.23563 + 2.16136I
15.4793 − 11.9246I 0. + 6.30873I
u = −1.00742 − 1.12472I
a = −1.90712 + 1.30827I
b = −3.23563 − 2.16136I
15.4793 + 11.9246I 0. − 6.30873I
u = −1.12156 + 1.02634I
a = 1.12316 + 1.72616I
b = 4.29671 + 0.57523I
17.7688 − 1.7617I 0
u = −1.12156 − 1.02634I
a = 1.12316 − 1.72616I
b = 4.29671 − 0.57523I
17.7688 + 1.7617I 0
u = −1.04184 + 1.10803I
a = 1.74324 + 1.46487I
b = 3.74797 − 1.61674I
17.4515 − 6.1631I 0
u = −1.04184 − 1.10803I
a = 1.74324 − 1.46487I
b = 3.74797 + 1.61674I
17.4515 + 6.1631I 0
u = −0.269490 + 0.299415I
a = −1.99775 − 0.56157I
b = −0.605235 + 0.631449I
−1.76892 + 0.41411I −5.79951 − 1.41452I
u = −0.269490 − 0.299415I
a = −1.99775 + 0.56157I
b = −0.605235 − 0.631449I
−1.76892 − 0.41411I −5.79951 + 1.41452I
7
II. I
u
2
= h−au + b + a, a
6
+ a
5
u − a
4
u + a
4
+ 2a
3
− au + a + 1, u
2
− u + 1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
5
=
1
u − 1
a
3
=
u
u − 1
a
8
=
a
au − a
a
9
=
a
au − a
a
10
=
au
au − a
a
1
=
−1
0
a
4
=
1
u − 1
a
12
=
−a
2
u + a
2
− 1
a
2
u
a
7
=
−a
4
+ a
2
u + 1
−a
4
u + a
4
a
11
=
−a
4
u − a
2
u − 1
−a
4
u + a
4
(ii) Obstruction class = 1
(iii) Cusp Shapes = −a
5
u + a
5
−4a
4
u + 4a
4
−2a
3
u −5a
2
u + 2a
2
−4au + 2a −5u − 2
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
6
c
2
(u
2
+ u + 1)
6
c
4
, c
8
u
12
c
6
, c
11
(u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)
2
c
7
, c
9
, c
12
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)
2
c
10
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
(y
2
+ y + 1)
6
c
4
, c
8
y
12
c
6
, c
11
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
c
7
, c
9
, c
10
c
12
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 0.245150 + 1.015700I
b = −1.002190 − 0.295542I
1.89061 + 1.10558I 1.81693 − 2.49433I
u = 0.500000 + 0.866025I
a = 0.757043 + 0.720154I
b = −1.002190 + 0.295542I
1.89061 + 2.95419I −0.06995 − 4.17815I
u = 0.500000 + 0.866025I
a = −0.789622 − 0.038604I
b = 0.428243 − 0.664531I
−1.89061 + 1.10558I −7.01188 − 1.87706I
u = 0.500000 + 0.866025I
a = 0.361379 − 0.703135I
b = 0.428243 + 0.664531I
−1.89061 + 2.95419I −3.50232 − 4.35344I
u = 0.500000 + 0.866025I
a = −1.020870 − 0.650692I
b = 1.073950 − 0.558752I
− 3.66314I −1.09315 + 2.75648I
u = 0.500000 + 0.866025I
a = −0.053081 − 1.209440I
b = 1.073950 + 0.558752I
7.72290I −4.13964 − 8.90605I
u = 0.500000 − 0.866025I
a = 0.245150 − 1.015700I
b = −1.002190 + 0.295542I
1.89061 − 1.10558I 1.81693 + 2.49433I
u = 0.500000 − 0.866025I
a = 0.757043 − 0.720154I
b = −1.002190 − 0.295542I
1.89061 − 2.95419I −0.06995 + 4.17815I
u = 0.500000 − 0.866025I
a = −0.789622 + 0.038604I
b = 0.428243 + 0.664531I
−1.89061 − 1.10558I −7.01188 + 1.87706I
u = 0.500000 − 0.866025I
a = 0.361379 + 0.703135I
b = 0.428243 − 0.664531I
−1.89061 − 2.95419I −3.50232 + 4.35344I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 − 0.866025I
a = −1.020870 + 0.650692I
b = 1.073950 + 0.558752I
3.66314I −1.09315 − 2.75648I
u = 0.500000 − 0.866025I
a = −0.053081 + 1.209440I
b = 1.073950 − 0.558752I
− 7.72290I −4.13964 + 8.90605I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
6
)(u
30
+ 3u
29
+ ··· + 3u + 1)
c
2
((u
2
+ u + 1)
6
)(u
30
+ 7u
29
+ ··· + 7u + 1)
c
3
((u
2
− u + 1)
6
)(u
30
− 7u
29
+ ··· + 123187u + 9881)
c
4
, c
8
u
12
(u
30
+ u
29
+ ··· + 8192u + 4096)
c
5
((u
2
− u + 1)
6
)(u
30
+ 7u
29
+ ··· + 7u + 1)
c
6
((u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)
2
)(u
30
+ 9u
29
+ ··· + 57u + 17)
c
7
((u
6
+ u
5
− u
4
− 2u
3
+ u + 1)
2
)(u
30
+ 3u
29
+ ··· + u + 1)
c
9
, c
12
((u
6
+ u
5
− u
4
− 2u
3
+ u + 1)
2
)(u
30
− 3u
29
+ ··· − 3u + 1)
c
10
((u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
)(u
30
+ 3u
29
+ ··· + u + 1)
c
11
((u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)
2
)(u
30
+ 13u
29
+ ··· − 7u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
6
)(y
30
+ 55y
29
+ ··· + 63y + 1)
c
2
, c
5
((y
2
+ y + 1)
6
)(y
30
+ 3y
29
+ ··· + 3y + 1)
c
3
((y
2
+ y + 1)
6
)(y
30
+ 107y
29
+ ··· + 1.84412 × 10
9
y + 9.76342 × 10
7
)
c
4
, c
8
y
12
(y
30
+ 65y
29
+ ··· + 1.61481 × 10
8
y
2
+ 1.67772 × 10
7
)
c
6
((y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
)(y
30
− 9y
29
+ ··· + 2531y + 289)
c
7
, c
10
((y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
)(y
30
− 13y
29
+ ··· + 7y + 1)
c
9
, c
12
((y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
)(y
30
− 53y
29
+ ··· + 7y + 1)
c
11
((y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
)(y
30
+ 11y
29
+ ··· − 89y + 1)
14
