12n
0046
(K12n
0046
)
A knot diagram
1
Linearized knot diagam
3 5 6 9 2 12 11 5 6 8 7 10
Solving Sequence
6,12 2,7
5 3 1 11 8 9 4 10
c
6
c
5
c
2
c
1
c
11
c
7
c
8
c
4
c
10
c
3
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
27
− 2u
26
+ ··· + 2b − 6u, u
26
+ 2u
25
+ ··· + 2a + 4, u
28
+ 3u
27
+ ··· + 6u + 1i
I
u
2
= h−au + b − u, u
3
a − u
2
a − u
3
+ a
2
+ 3au − 2u, u
4
− u
3
+ 3u
2
− 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 36 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−u
27
−2u
26
+· · ·+2b −6u, u
26
+2u
25
+· · ·+2a +4, u
28
+3u
27
+· · ·+6u +1i
(i) Arc colorings
a
6
=
1
0
a
12
=
0
u
a
2
=
−
1
2
u
26
− u
25
+ ··· −
33
2
u
2
− 2
1
2
u
27
+ u
26
+ ··· + u
2
+ 3u
a
7
=
1
u
2
a
5
=
−
1
2
u
27
−
3
2
u
26
+ ··· − 15u − 2
−
1
2
u
27
− u
26
+ ··· − 3u − 1
a
3
=
u
27
+
7
2
u
26
+ ··· + 20u + 3
−
1
2
u
27
− 2u
26
+ ··· − 14u
2
− 2u
a
1
=
u
7
+ 4u
5
+ 4u
3
u
9
+ 5u
7
+ 7u
5
+ 2u
3
+ u
a
11
=
u
u
3
+ u
a
8
=
u
2
+ 1
u
4
+ 2u
2
a
9
=
−u
5
− 2u
3
+ u
u
5
+ 3u
3
+ u
a
4
=
−
3
2
u
27
−
11
2
u
26
+ ··· − 22u − 3
1
2
u
27
+ 2u
26
+ ··· + 14u
2
+ 2u
a
10
=
u
3
+ 2u
u
5
+ 3u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes =
5
2
u
27
+ 5u
26
+ ··· + 21u +
1
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
28
+ 5u
27
+ ··· + 14u + 1
c
2
, c
5
u
28
+ 5u
27
+ ··· + 4u + 1
c
3
u
28
− 5u
27
+ ··· + 5562u + 1321
c
4
, c
8
u
28
+ u
27
+ ··· + 384u + 256
c
6
, c
7
, c
10
c
11
u
28
− 3u
27
+ ··· − 6u + 1
c
9
u
28
− 3u
27
+ ··· − 2u + 1
c
12
u
28
+ 11u
27
+ ··· + 184u + 209
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
28
+ 41y
27
+ ··· + 14y + 1
c
2
, c
5
y
28
+ 5y
27
+ ··· + 14y + 1
c
3
y
28
+ 77y
27
+ ··· + 102223598y + 1745041
c
4
, c
8
y
28
+ 45y
27
+ ··· + 344064y + 65536
c
6
, c
7
, c
10
c
11
y
28
+ 35y
27
+ ··· + 6y + 1
c
9
y
28
− 49y
27
+ ··· + 6y + 1
c
12
y
28
− 29y
27
+ ··· + 2808126y + 43681
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.546864 + 0.864620I
a = −1.78266 + 0.81365I
b = 0.938673 + 1.030660I
12.0428 + 7.8502I 0.54802 − 5.73315I
u = −0.546864 − 0.864620I
a = −1.78266 − 0.81365I
b = 0.938673 − 1.030660I
12.0428 − 7.8502I 0.54802 + 5.73315I
u = −0.512075 + 0.914815I
a = −0.310810 + 0.834490I
b = 1.006070 − 0.921537I
12.40950 + 0.72573I 1.20101 − 1.26627I
u = −0.512075 − 0.914815I
a = −0.310810 − 0.834490I
b = 1.006070 + 0.921537I
12.40950 − 0.72573I 1.20101 + 1.26627I
u = 0.041750 + 0.816332I
a = 0.984967 − 0.816170I
b = −0.730216 + 0.546904I
2.67787 − 1.51352I 3.15826 + 2.96332I
u = 0.041750 − 0.816332I
a = 0.984967 + 0.816170I
b = −0.730216 − 0.546904I
2.67787 + 1.51352I 3.15826 − 2.96332I
u = −0.755205 + 0.031373I
a = −0.154324 − 0.783184I
b = 0.955403 − 0.970517I
9.53090 − 3.51075I −2.36490 + 2.10810I
u = −0.755205 − 0.031373I
a = −0.154324 + 0.783184I
b = 0.955403 + 0.970517I
9.53090 + 3.51075I −2.36490 − 2.10810I
u = 0.429610 + 0.590805I
a = −1.30185 − 0.60397I
b = 0.291696 − 0.394438I
−0.15101 − 2.02920I −3.29658 + 3.20774I
u = 0.429610 − 0.590805I
a = −1.30185 + 0.60397I
b = 0.291696 + 0.394438I
−0.15101 + 2.02920I −3.29658 − 3.20774I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.179883 + 0.692364I
a = 2.11541 − 0.80616I
b = −0.514946 − 1.029420I
1.04465 + 3.25872I 1.94579 − 3.88394I
u = −0.179883 − 0.692364I
a = 2.11541 + 0.80616I
b = −0.514946 + 1.029420I
1.04465 − 3.25872I 1.94579 + 3.88394I
u = 0.398241 + 0.347741I
a = −0.595350 + 0.678601I
b = −0.007486 + 0.515758I
−0.844911 − 0.963937I −7.23227 + 5.09608I
u = 0.398241 − 0.347741I
a = −0.595350 − 0.678601I
b = −0.007486 − 0.515758I
−0.844911 + 0.963937I −7.23227 − 5.09608I
u = 0.05307 + 1.53702I
a = −0.565559 + 0.333640I
b = 0.007766 + 0.841204I
5.52882 − 2.13387I −4.00000 + 3.29212I
u = 0.05307 − 1.53702I
a = −0.565559 − 0.333640I
b = 0.007766 − 0.841204I
5.52882 + 2.13387I −4.00000 − 3.29212I
u = 0.12702 + 1.57248I
a = −1.339830 − 0.272553I
b = 0.478138 − 0.445759I
7.19200 − 4.05999I 0
u = 0.12702 − 1.57248I
a = −1.339830 + 0.272553I
b = 0.478138 + 0.445759I
7.19200 + 4.05999I 0
u = −0.04195 + 1.63425I
a = 1.62168 + 0.08443I
b = −0.578450 − 1.150140I
9.22292 + 4.03870I 0. − 2.65080I
u = −0.04195 − 1.63425I
a = 1.62168 − 0.08443I
b = −0.578450 + 1.150140I
9.22292 − 4.03870I 0. + 2.65080I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.01177 + 1.65893I
a = 1.41681 − 0.67494I
b = −0.955702 + 0.548085I
11.38100 − 1.72426I 3.45594 + 0.I
u = 0.01177 − 1.65893I
a = 1.41681 + 0.67494I
b = −0.955702 − 0.548085I
11.38100 + 1.72426I 3.45594 + 0.I
u = −0.16140 + 1.66866I
a = −1.98113 + 0.05979I
b = 0.93469 + 1.08649I
−18.7492 + 10.6138I 0. − 4.77955I
u = −0.16140 − 1.66866I
a = −1.98113 − 0.05979I
b = 0.93469 − 1.08649I
−18.7492 − 10.6138I 0. + 4.77955I
u = −0.14287 + 1.68669I
a = −1.04698 + 1.01839I
b = 1.072160 − 0.890015I
−18.0712 + 3.2992I 0
u = −0.14287 − 1.68669I
a = −1.04698 − 1.01839I
b = 1.072160 + 0.890015I
−18.0712 − 3.2992I 0
u = −0.221218 + 0.191391I
a = −2.06037 + 1.41739I
b = −0.397798 + 0.843645I
−0.31537 − 1.65529I −2.65586 + 5.38450I
u = −0.221218 − 0.191391I
a = −2.06037 − 1.41739I
b = −0.397798 − 0.843645I
−0.31537 + 1.65529I −2.65586 − 5.38450I
7
II.
I
u
2
= h−au + b − u, u
3
a − u
2
a − u
3
+ a
2
+ 3au − 2u, u
4
− u
3
+ 3u
2
− 2u + 1i
(i) Arc colorings
a
6
=
1
0
a
12
=
0
u
a
2
=
a
au + u
a
7
=
1
u
2
a
5
=
u
3
− au − u
2
+ a + 2u
au + u − 1
a
3
=
u
3
− u
2
+ a + 3u − 1
au + u − 1
a
1
=
−1
0
a
11
=
u
u
3
+ u
a
8
=
u
2
+ 1
u
3
− u
2
+ 2u − 1
a
9
=
u
2
+ 1
u
3
− u
2
+ 2u − 1
a
4
=
u
3
− au − u
2
+ a + 2u
au + u − 1
a
10
=
u
3
+ 2u
u
3
− u
2
+ 2u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
2
a + 4u
3
− 4au − 5u
2
+ a + 9u − 5
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
4
c
2
(u
2
+ u + 1)
4
c
4
, c
8
u
8
c
6
, c
7
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
9
, c
12
(u
4
+ u
3
+ u
2
+ 1)
2
c
10
, c
11
(u
4
+ u
3
+ 3u
2
+ 2u + 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)
4
c
4
, c
8
y
8
c
6
, c
7
, c
10
c
11
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
c
9
, c
12
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.395123 + 0.506844I
a = 0.541116 + 0.214920I
b = 0.500000 + 0.866025I
−0.211005 + 0.614778I −1.64912 + 1.57080I
u = 0.395123 + 0.506844I
a = −1.58443 − 1.44211I
b = 0.500000 − 0.866025I
−0.21101 − 3.44499I −4.65255 + 7.52635I
u = 0.395123 − 0.506844I
a = 0.541116 − 0.214920I
b = 0.500000 − 0.866025I
−0.211005 − 0.614778I −1.64912 − 1.57080I
u = 0.395123 − 0.506844I
a = −1.58443 + 1.44211I
b = 0.500000 + 0.866025I
−0.21101 + 3.44499I −4.65255 − 7.52635I
u = 0.10488 + 1.55249I
a = −0.423047 − 0.283088I
b = 0.500000 + 0.866025I
6.79074 − 1.13408I 1.80063 − 0.49697I
u = 0.10488 + 1.55249I
a = −1.53364 − 0.35811I
b = 0.500000 − 0.866025I
6.79074 − 5.19385I −1.99896 + 6.53786I
u = 0.10488 − 1.55249I
a = −0.423047 + 0.283088I
b = 0.500000 − 0.866025I
6.79074 + 1.13408I 1.80063 + 0.49697I
u = 0.10488 − 1.55249I
a = −1.53364 + 0.35811I
b = 0.500000 + 0.866025I
6.79074 + 5.19385I −1.99896 − 6.53786I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
4
)(u
28
+ 5u
27
+ ··· + 14u + 1)
c
2
((u
2
+ u + 1)
4
)(u
28
+ 5u
27
+ ··· + 4u + 1)
c
3
((u
2
− u + 1)
4
)(u
28
− 5u
27
+ ··· + 5562u + 1321)
c
4
, c
8
u
8
(u
28
+ u
27
+ ··· + 384u + 256)
c
5
((u
2
− u + 1)
4
)(u
28
+ 5u
27
+ ··· + 4u + 1)
c
6
, c
7
((u
4
− u
3
+ 3u
2
− 2u + 1)
2
)(u
28
− 3u
27
+ ··· − 6u + 1)
c
9
((u
4
+ u
3
+ u
2
+ 1)
2
)(u
28
− 3u
27
+ ··· − 2u + 1)
c
10
, c
11
((u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
)(u
28
− 3u
27
+ ··· − 6u + 1)
c
12
((u
4
+ u
3
+ u
2
+ 1)
2
)(u
28
+ 11u
27
+ ··· + 184u + 209)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
4
)(y
28
+ 41y
27
+ ··· + 14y + 1)
c
2
, c
5
((y
2
+ y + 1)
4
)(y
28
+ 5y
27
+ ··· + 14y + 1)
c
3
((y
2
+ y + 1)
4
)(y
28
+ 77y
27
+ ··· + 1.02224 × 10
8
y + 1745041)
c
4
, c
8
y
8
(y
28
+ 45y
27
+ ··· + 344064y + 65536)
c
6
, c
7
, c
10
c
11
((y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
)(y
28
+ 35y
27
+ ··· + 6y + 1)
c
9
((y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
)(y
28
− 49y
27
+ ··· + 6y + 1)
c
12
((y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
)(y
28
− 29y
27
+ ··· + 2808126y + 43681)
13
