12n
0199
(K12n
0199
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 10 11 3 12 5 6 9 8
Solving Sequence
5,10
6 11
3,7
2 1 4 9 12 8
c
5
c
10
c
6
c
2
c
1
c
4
c
9
c
11
c
8
c
3
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
10
+ u
9
+ 5u
8
− 3u
7
− 9u
6
+ 7u
4
+ 4u
3
− 3u
2
+ b + u + 1,
u
10
− u
9
− 5u
8
+ 3u
7
+ 9u
6
− 7u
4
− 5u
3
+ 3u
2
+ a + u − 1,
u
11
− 2u
10
− 4u
9
+ 8u
8
+ 6u
7
− 8u
6
− 8u
5
+ 9u
3
− 2u
2
+ 1i
I
u
2
= hb + 1, −u
3
+ a + 2u − 1, u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1i
* 2 irreducible components of dim
C
= 0, with total 17 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
10
+ u
9
+ · · · + b + 1, u
10
− u
9
+ · · · + a − 1, u
11
− 2u
10
+ · · · − 2u
2
+ 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
−u
2
a
11
=
u
−u
3
+ u
a
3
=
−u
10
+ u
9
+ 5u
8
− 3u
7
− 9u
6
+ 7u
4
+ 5u
3
− 3u
2
− u + 1
u
10
− u
9
− 5u
8
+ 3u
7
+ 9u
6
− 7u
4
− 4u
3
+ 3u
2
− u − 1
a
7
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
u
3
− 2u
u
10
− u
9
− 5u
8
+ 3u
7
+ 9u
6
− 7u
4
− 4u
3
+ 3u
2
− u − 1
a
1
=
4u
10
− 3u
9
− 20u
8
+ 8u
7
+ 32u
6
+ 5u
5
− 16u
4
− 18u
3
+ 3u
2
− 3u − 2
−2u
10
+ u
9
+ 12u
8
− 2u
7
− 24u
6
− 5u
5
+ 16u
4
+ 10u
3
− u
2
+ u + 1
a
4
=
u
10
− u
9
− 4u
8
+ 3u
7
+ 4u
6
− 4u
3
+ u
2
− u
u
10
− u
9
− 4u
8
+ 3u
7
+ 5u
6
+ u
5
− 3u
4
− 7u
3
+ 3u
2
− 1
a
9
=
−u
u
a
12
=
−u
5
+ 2u
3
+ u
u
5
− 3u
3
+ u
a
8
=
−u
9
+ 4u
7
− 3u
5
− 2u
3
− u
u
9
− 5u
7
+ 7u
5
− 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
10
− 5u
9
− 15u
8
+ 17u
7
+ 14u
6
− 9u
5
− 4u
4
− 3u
3
+ 13u
2
− 18u + 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
11
+ 27u
10
+ ··· + 133u + 1
c
2
, c
4
u
11
− 7u
10
+ ··· − 13u + 1
c
3
, c
7
u
11
− u
10
+ ··· − 64u + 64
c
5
, c
6
, c
9
c
10
u
11
+ 2u
10
− 4u
9
− 8u
8
+ 6u
7
+ 8u
6
− 8u
5
+ 9u
3
+ 2u
2
− 1
c
8
, c
11
, c
12
u
11
+ 12u
9
+ 38u
7
+ 10u
5
− 11u
3
− 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
11
− 159y
10
+ ··· + 14833y −1
c
2
, c
4
y
11
− 27y
10
+ ··· + 133y −1
c
3
, c
7
y
11
+ 51y
10
+ ··· + 49152y −4096
c
5
, c
6
, c
9
c
10
y
11
− 12y
10
+ ··· + 4y −1
c
8
, c
11
, c
12
y
11
+ 24y
10
+ ··· + 4y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.556675 + 0.808029I
a = −0.55576 − 1.52077I
b = 2.58699 + 0.12834I
16.4992 − 2.6778I 2.70707 + 2.34778I
u = −0.556675 − 0.808029I
a = −0.55576 + 1.52077I
b = 2.58699 − 0.12834I
16.4992 + 2.6778I 2.70707 − 2.34778I
u = 1.26079
a = 1.16164
b = −1.67909
1.10399 6.07670
u = −1.44218 + 0.13979I
a = 0.14841 + 1.46791I
b = −0.179069 − 0.877965I
4.02973 − 3.04693I 7.61574 + 3.00651I
u = −1.44218 − 0.13979I
a = 0.14841 − 1.46791I
b = −0.179069 + 0.877965I
4.02973 + 3.04693I 7.61574 − 3.00651I
u = 0.263767 + 0.414640I
a = 0.051496 − 1.271950I
b = −0.696724 + 0.457926I
−1.53989 + 1.03784I 0.63702 − 4.26648I
u = 0.263767 − 0.414640I
a = 0.051496 + 1.271950I
b = −0.696724 − 0.457926I
−1.53989 − 1.03784I 0.63702 + 4.26648I
u = 1.52082
a = 0.186924
b = 0.288918
7.24960 14.5180
u = −0.426077
a = 0.683970
b = 0.0908333
0.618683 16.2830
u = 1.55733 + 0.28677I
a = −2.16041 + 1.82801I
b = 2.43848 − 0.33867I
−16.0730 + 6.7220I 5.60131 − 2.60237I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.55733 − 0.28677I
a = −2.16041 − 1.82801I
b = 2.43848 + 0.33867I
−16.0730 − 6.7220I 5.60131 + 2.60237I
6
II. I
u
2
= hb + 1, −u
3
+ a + 2u − 1, u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
−u
2
a
11
=
u
−u
3
+ u
a
3
=
u
3
− 2u + 1
−1
a
7
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
u
3
− 2u
−1
a
1
=
−1
0
a
4
=
u
3
− 2u + 1
−1
a
9
=
−u
u
a
12
=
−u
5
+ 2u
3
+ u
u
5
− 3u
3
+ u
a
8
=
−u
2
+ 1
u
4
− 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
5
+ u
4
− 6u
3
− u
2
− 2u + 2
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
6
c
3
, c
7
u
6
c
4
(u + 1)
6
c
5
, c
6
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1
c
8
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1
c
9
, c
10
u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
11
, c
12
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
7
y
6
c
5
, c
6
, c
9
c
10
y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1
c
8
, c
11
, c
12
y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.493180 + 0.575288I
a = −0.356069 − 0.921195I
b = −1.00000
−4.60518 + 1.97241I 2.71215 − 3.88360I
u = 0.493180 − 0.575288I
a = −0.356069 + 0.921195I
b = −1.00000
−4.60518 − 1.97241I 2.71215 + 3.88360I
u = −0.483672
a = 1.85419
b = −1.00000
−0.906083 3.38760
u = −1.52087 + 0.16310I
a = 0.645284 + 0.801205I
b = −1.00000
2.05064 − 4.59213I 6.49628 + 3.92496I
u = −1.52087 − 0.16310I
a = 0.645284 − 0.801205I
b = −1.00000
2.05064 + 4.59213I 6.49628 − 3.92496I
u = 1.53904
a = 1.56737
b = −1.00000
6.01515 6.19550
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
11
+ 27u
10
+ ··· + 133u + 1)
c
2
((u − 1)
6
)(u
11
− 7u
10
+ ··· − 13u + 1)
c
3
, c
7
u
6
(u
11
− u
10
+ ··· − 64u + 64)
c
4
((u + 1)
6
)(u
11
− 7u
10
+ ··· − 13u + 1)
c
5
, c
6
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)
· (u
11
+ 2u
10
− 4u
9
− 8u
8
+ 6u
7
+ 8u
6
− 8u
5
+ 9u
3
+ 2u
2
− 1)
c
8
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)
· (u
11
+ 12u
9
+ 38u
7
+ 10u
5
− 11u
3
− 2u + 1)
c
9
, c
10
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)
· (u
11
+ 2u
10
− 4u
9
− 8u
8
+ 6u
7
+ 8u
6
− 8u
5
+ 9u
3
+ 2u
2
− 1)
c
11
, c
12
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1)
· (u
11
+ 12u
9
+ 38u
7
+ 10u
5
− 11u
3
− 2u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
11
− 159y
10
+ ··· + 14833y −1)
c
2
, c
4
((y −1)
6
)(y
11
− 27y
10
+ ··· + 133y −1)
c
3
, c
7
y
6
(y
11
+ 51y
10
+ ··· + 49152y −4096)
c
5
, c
6
, c
9
c
10
(y
6
− 7y
5
+ ··· − 5y + 1)(y
11
− 12y
10
+ ··· + 4y −1)
c
8
, c
11
, c
12
(y
6
+ 5y
5
+ ··· − 5y + 1)(y
11
+ 24y
10
+ ··· + 4y −1)
12
