12n
0248
(K12n
0248
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 12 11 10 3 5 7 6 9
Solving Sequence
6,11
7 12
3,5
2 1 4 10 8 9
c
6
c
11
c
5
c
2
c
1
c
4
c
10
c
7
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
12
+ 2u
11
− 9u
10
+ 15u
9
− 29u
8
+ 38u
7
− 40u
6
+ 37u
5
− 22u
4
+ 12u
3
− 3u
2
+ b − u + 1,
u
16
− 2u
15
+ ··· + a − 2, u
17
− 2u
16
+ ··· − 3u + 1i
I
u
2
= hb + u + 1, −u
4
− u
3
− 4u
2
+ a − 2u − 2, u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1i
* 2 irreducible components of dim
C
= 0, with total 22 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
12
+2u
11
+· · ·+b+1, u
16
−2u
15
+· · ·+a−2, u
17
−2u
16
+· · ·−3u+1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
−u
2
a
12
=
u
u
a
3
=
−u
16
+ 2u
15
+ ··· − 8u + 2
u
12
− 2u
11
+ ··· + u − 1
a
5
=
u
2
+ 1
u
2
a
2
=
−u
16
+ 2u
15
+ ··· − 8u + 1
−u
13
+ 2u
12
+ ··· + u − 1
a
1
=
−u
13
− 8u
11
− 23u
9
− 30u
7
− 20u
5
− 6u
3
− u
−u
13
− 7u
11
− 15u
9
− 8u
7
+ 4u
5
+ 3u
3
− u
a
4
=
−u
16
+ 2u
15
+ ··· − 6u + 1
u
14
− 2u
13
+ ··· + 2u − 1
a
10
=
−u
u
3
+ u
a
8
=
u
2
+ 1
−u
4
− 2u
2
a
9
=
−u
7
− 4u
5
− 4u
3
− 2u
−u
7
− 3u
5
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
16
+ 2u
15
− 14u
14
+ 24u
13
− 78u
12
+ 110u
11
− 216u
10
+
235u
9
− 298u
8
+ 219u
7
− 164u
6
+ 39u
5
+ 14u
4
− 52u
3
+ 34u
2
− 21u + 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
17
+ 28u
16
+ ··· + 47u + 1
c
2
, c
4
u
17
− 6u
16
+ ··· + 11u − 1
c
3
, c
8
u
17
+ u
16
+ ··· + 32u − 32
c
5
, c
6
, c
7
c
10
, c
11
u
17
+ 2u
16
+ ··· − 3u − 1
c
9
u
17
+ 2u
16
+ ··· − 20u − 100
c
12
u
17
+ 18u
15
+ ··· − u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
17
− 72y
16
+ ··· + 2319y −1
c
2
, c
4
y
17
− 28y
16
+ ··· + 47y −1
c
3
, c
8
y
17
+ 33y
16
+ ··· + 8704y −1024
c
5
, c
6
, c
7
c
10
, c
11
y
17
+ 24y
16
+ ··· − 3y −1
c
9
y
17
+ 24y
16
+ ··· − 163800y −10000
c
12
y
17
+ 36y
16
+ ··· − 3y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.172919 + 0.910697I
a = 0.318120 − 0.364787I
b = −0.026588 − 0.519308I
−2.10523 − 1.77554I 4.36935 + 3.95696I
u = −0.172919 − 0.910697I
a = 0.318120 + 0.364787I
b = −0.026588 + 0.519308I
−2.10523 + 1.77554I 4.36935 − 3.95696I
u = 0.076795 + 1.100920I
a = −0.899122 + 0.716334I
b = −0.46350 + 1.81350I
−6.05879 + 1.56653I −2.65237 − 1.48388I
u = 0.076795 − 1.100920I
a = −0.899122 − 0.716334I
b = −0.46350 − 1.81350I
−6.05879 − 1.56653I −2.65237 + 1.48388I
u = 0.363317 + 1.143140I
a = 0.676702 − 0.745881I
b = 1.01514 − 2.32860I
−17.4586 + 5.5119I −1.67992 − 3.43806I
u = 0.363317 − 1.143140I
a = 0.676702 + 0.745881I
b = 1.01514 + 2.32860I
−17.4586 − 5.5119I −1.67992 + 3.43806I
u = 0.640058 + 0.377809I
a = −1.49477 + 0.62802I
b = 0.659184 − 0.737438I
−12.69310 + 2.07755I 1.82746 − 2.83280I
u = 0.640058 − 0.377809I
a = −1.49477 − 0.62802I
b = 0.659184 + 0.737438I
−12.69310 − 2.07755I 1.82746 + 2.83280I
u = −0.352123
a = 0.596606
b = 0.192432
0.659166 15.3270
u = 0.197187 + 0.287158I
a = 0.27267 − 2.06821I
b = −0.661431 + 0.441892I
−1.63254 + 0.69110I −1.76799 − 2.88115I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.197187 − 0.287158I
a = 0.27267 + 2.06821I
b = −0.661431 − 0.441892I
−1.63254 − 0.69110I −1.76799 + 2.88115I
u = −0.04510 + 1.70602I
a = −0.122781 − 0.847756I
b = −0.383269 − 1.042710I
−11.46630 − 2.62660I 3.17531 + 1.46591I
u = −0.04510 − 1.70602I
a = −0.122781 + 0.847756I
b = −0.383269 + 1.042710I
−11.46630 + 2.62660I 3.17531 − 1.46591I
u = 0.01853 + 1.75791I
a = −0.40235 + 2.31946I
b = 0.01866 + 3.08154I
−16.4472 + 1.9633I −2.49608 − 1.09020I
u = 0.01853 − 1.75791I
a = −0.40235 − 2.31946I
b = 0.01866 − 3.08154I
−16.4472 − 1.9633I −2.49608 + 1.09020I
u = 0.09819 + 1.76530I
a = 1.35323 − 2.62476I
b = 1.24559 − 3.74596I
11.60450 + 7.50472I −2.43940 − 2.60727I
u = 0.09819 − 1.76530I
a = 1.35323 + 2.62476I
b = 1.24559 + 3.74596I
11.60450 − 7.50472I −2.43940 + 2.60727I
6
II.
I
u
2
= hb + u + 1, −u
4
− u
3
− 4u
2
+ a − 2u − 2, u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
−u
2
a
12
=
u
u
a
3
=
u
4
+ u
3
+ 4u
2
+ 2u + 2
−u − 1
a
5
=
u
2
+ 1
u
2
a
2
=
u
4
+ u
3
+ 3u
2
+ 2u + 1
−u
2
− u − 1
a
1
=
−u
2
− 1
−u
2
a
4
=
u
4
+ u
3
+ 4u
2
+ 2u + 2
−u − 1
a
10
=
−u
u
3
+ u
a
8
=
u
2
+ 1
−u
4
− 2u
2
a
9
=
u
2
+ 1
−u
4
− 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
4
+ 3u
3
+ 12u
2
+ 10u + 7
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
5
c
3
, c
8
u
5
c
4
(u + 1)
5
c
5
, c
6
, c
7
u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1
c
9
, c
12
u
5
− u
4
+ u
2
+ u − 1
c
10
, c
11
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y − 1)
5
c
3
, c
8
y
5
c
5
, c
6
, c
7
c
10
, c
11
y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y − 1
c
9
, c
12
y
5
− y
4
+ 4y
3
− 3y
2
+ 3y − 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.233677 + 0.885557I
a = −0.487744 + 0.170166I
b = −0.766323 − 0.885557I
−3.46474 − 2.21397I −1.39794 + 4.05273I
u = −0.233677 − 0.885557I
a = −0.487744 − 0.170166I
b = −0.766323 + 0.885557I
−3.46474 + 2.21397I −1.39794 − 4.05273I
u = −0.416284
a = 1.81849
b = −0.583716
−0.762751 4.79030
u = −0.05818 + 1.69128I
a = −0.92150 − 1.10071I
b = −0.94182 − 1.69128I
−12.60320 − 3.33174I −1.99723 + 3.46299I
u = −0.05818 − 1.69128I
a = −0.92150 + 1.10071I
b = −0.94182 + 1.69128I
−12.60320 + 3.33174I −1.99723 − 3.46299I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
17
+ 28u
16
+ ··· + 47u + 1)
c
2
((u − 1)
5
)(u
17
− 6u
16
+ ··· + 11u − 1)
c
3
, c
8
u
5
(u
17
+ u
16
+ ··· + 32u − 32)
c
4
((u + 1)
5
)(u
17
− 6u
16
+ ··· + 11u − 1)
c
5
, c
6
, c
7
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)(u
17
+ 2u
16
+ ··· − 3u − 1)
c
9
(u
5
− u
4
+ u
2
+ u − 1)(u
17
+ 2u
16
+ ··· − 20u − 100)
c
10
, c
11
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)(u
17
+ 2u
16
+ ··· − 3u − 1)
c
12
(u
5
− u
4
+ u
2
+ u − 1)(u
17
+ 18u
15
+ ··· − u − 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y − 1)
5
)(y
17
− 72y
16
+ ··· + 2319y − 1)
c
2
, c
4
((y − 1)
5
)(y
17
− 28y
16
+ ··· + 47y − 1)
c
3
, c
8
y
5
(y
17
+ 33y
16
+ ··· + 8704y − 1024)
c
5
, c
6
, c
7
c
10
, c
11
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y − 1)(y
17
+ 24y
16
+ ··· − 3y − 1)
c
9
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y − 1)(y
17
+ 24y
16
+ ··· − 163800y − 10000)
c
12
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y − 1)(y
17
+ 36y
16
+ ··· − 3y − 1)
12
