12n
0488
(K12n
0488
)
A knot diagram
1
Linearized knot diagam
3 6 9 11 2 11 12 4 6 5 7 9
Solving Sequence
2,6 3,11
7 1 5 4 10 9 12 8
c
2
c
6
c
1
c
5
c
4
c
10
c
9
c
12
c
7
c
3
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
8
− 2u
7
+ 3u
6
− 3u
5
+ 4u
4
− 3u
3
+ 2u
2
+ b − u, u
6
− 2u
5
+ 2u
4
− u
3
+ 2u
2
+ a − 2u,
u
9
− 3u
8
+ 5u
7
− 5u
6
+ 6u
5
− 7u
4
+ 6u
3
− 2u
2
+ u − 1i
I
u
2
= hu
6
+ 2u
5
+ u
4
− 2u
3
− 2u
2
+ b, u
6
+ 2u
5
+ 2u
4
− u
3
− 2u
2
+ a − 2u, u
7
+ 2u
6
+ 2u
5
− u
4
− u
3
− u
2
− 1i
* 2 irreducible components of dim
C
= 0, with total 16 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
= hu
8
−2u
7
+· · ·+b−u, u
6
−2u
5
+2u
4
−u
3
+2u
2
+a−2u, u
9
−3u
8
+· · ·+u−1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
11
=
−u
6
+ 2u
5
− 2u
4
+ u
3
− 2u
2
+ 2u
−u
8
+ 2u
7
− 3u
6
+ 3u
5
− 4u
4
+ 3u
3
− 2u
2
+ u
a
7
=
−u
7
+ u
6
− u
5
+ u
4
− 3u
3
+ 2u
2
− u + 1
−u
8
+ u
7
− u
6
+ u
5
− 2u
4
+ u
2
+ u
a
1
=
−u
2
+ 1
−u
4
a
5
=
u
u
a
4
=
u
8
− 2u
7
+ 2u
6
− 2u
5
+ 3u
4
− 3u
3
+ u
2
+ 1
u
8
− 3u
7
+ 3u
6
− 3u
5
+ 4u
4
− 5u
3
+ 2u
2
+ 1
a
10
=
u
7
− 2u
6
+ 3u
5
− 3u
4
+ 4u
3
− 3u
2
+ 2u − 1
−u
8
+ 3u
7
− 4u
6
+ 4u
5
− 5u
4
+ 6u
3
− 3u
2
+ u − 1
a
9
=
u
7
− 2u
6
+ 3u
5
− 3u
4
+ 4u
3
− 3u
2
+ 2u − 1
u
7
− 2u
6
+ 2u
5
− u
4
+ 2u
3
− 2u
2
a
12
=
u
7
− u
6
+ u
5
− u
4
+ 2u
3
− 2u
2
+ u
u
8
− u
7
+ u
6
− u
5
+ u
4
− u
2
a
8
=
−2u
7
+ 4u
6
− 5u
5
+ 4u
4
− 8u
3
+ 5u
2
− 2u + 2
−u
8
+ u
7
− u
5
− 3u
4
+ 2u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −5u
8
+ 12u
7
− 17u
6
+ 12u
5
− 20u
4
+ 20u
3
− 13u
2
− 6u − 3
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
9
− u
8
+ 7u
7
− 5u
6
+ 16u
5
− 7u
4
+ 10u
3
+ 6u
2
− 3u + 1
c
2
, c
5
u
9
+ 3u
8
+ 5u
7
+ 5u
6
+ 6u
5
+ 7u
4
+ 6u
3
+ 2u
2
+ u + 1
c
3
, c
4
, c
8
c
10
u
9
− 2u
7
+ 4u
6
+ 17u
5
+ 9u
4
+ 9u
3
+ 2u
2
+ 2u + 1
c
6
, c
7
, c
11
u
9
+ 8u
8
+ 27u
7
+ 46u
6
+ 33u
5
− 10u
4
− 27u
3
− u
2
+ 14u + 4
c
9
, c
12
u
9
− 4u
8
+ 8u
7
− 3u
6
− 2u
4
+ 24u
3
− 23u
2
+ 9u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
9
+ 13y
8
+ 71y
7
+ 205y
6
+ 332y
5
+ 291y
4
+ 98y
3
− 82y
2
− 3y −1
c
2
, c
5
y
9
+ y
8
+ 7y
7
+ 5y
6
+ 16y
5
+ 7y
4
+ 10y
3
− 6y
2
− 3y −1
c
3
, c
4
, c
8
c
10
y
9
− 4y
8
+ 38y
7
− 66y
6
+ 185y
5
+ 201y
4
+ 105y
3
+ 14y
2
− 1
c
6
, c
7
, c
11
y
9
− 10y
8
+ ··· + 204y −16
c
9
, c
12
y
9
+ 40y
7
+ 23y
6
+ 206y
5
+ 10y
4
+ 490y
3
− 93y
2
+ 127y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.543663 + 0.958634I
a = −0.62928 − 1.29649I
b = −1.044710 − 0.790946I
8.80377 + 2.34106I 3.55048 − 3.71378I
u = −0.543663 − 0.958634I
a = −0.62928 + 1.29649I
b = −1.044710 + 0.790946I
8.80377 − 2.34106I 3.55048 + 3.71378I
u = 0.780042
a = 0.429639
b = −0.0889832
−1.02700 −12.4430
u = 1.022250 + 0.813773I
a = 1.249250 − 0.023619I
b = 0.74446 + 1.23201I
−2.68542 + 1.09922I 0.831621 − 0.481760I
u = 1.022250 − 0.813773I
a = 1.249250 + 0.023619I
b = 0.74446 − 1.23201I
−2.68542 − 1.09922I 0.831621 + 0.481760I
u = 0.877200 + 1.062120I
a = 0.148693 − 1.372070I
b = 1.77486 − 1.66508I
−1.87595 − 8.01095I 1.68345 + 4.08979I
u = 0.877200 − 1.062120I
a = 0.148693 + 1.372070I
b = 1.77486 + 1.66508I
−1.87595 + 8.01095I 1.68345 − 4.08979I
u = −0.245807 + 0.515171I
a = 0.016526 + 1.227710I
b = 0.569881 + 0.456696I
1.20590 + 0.78253I 4.15601 − 2.65874I
u = −0.245807 − 0.515171I
a = 0.016526 − 1.227710I
b = 0.569881 − 0.456696I
1.20590 − 0.78253I 4.15601 + 2.65874I
5
II. I
u
2
= hu
6
+ 2u
5
+ u
4
− 2u
3
− 2u
2
+ b, u
6
+ 2u
5
+ 2u
4
− u
3
− 2u
2
+ a −
2u, u
7
+ 2u
6
+ 2u
5
− u
4
− u
3
− u
2
− 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
11
=
−u
6
− 2u
5
− 2u
4
+ u
3
+ 2u
2
+ 2u
−u
6
− 2u
5
− u
4
+ 2u
3
+ 2u
2
a
7
=
u
6
+ 3u
5
+ 4u
4
− 3u
2
− 3u
u
6
+ 2u
5
+ 2u
4
− 2u
3
− 2u
2
+ 1
a
1
=
−u
2
+ 1
−u
4
a
5
=
u
u
a
4
=
u
5
+ 2u
4
+ 2u
3
− u
2
− u − 1
u
6
+ 2u
5
+ 2u
4
− u
3
− u
2
− u
a
10
=
−u
5
− 2u
4
− u
3
+ 2u
2
+ 2u
−u
5
− u
4
+ 2u
2
a
9
=
−u
5
− 2u
4
− u
3
+ 2u
2
+ 2u
u
3
+ u
2
− 1
a
12
=
u
6
+ 3u
5
+ 4u
4
+ u
3
− 3u
2
− 3u − 1
u
6
+ 2u
5
+ u
4
− 2u
3
− 2u
2
− u + 1
a
8
=
−u
5
− 2u
4
− 2u
3
+ u
2
+ 2u + 2
−u
4
+ u
2
+ 2u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
6
+ 5u
5
+ 3u
4
− 5u
3
− u
2
− u + 3
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
7
+ 6u
5
− u
4
+ 3u
3
− 3u
2
− 2u − 1
c
2
u
7
+ 2u
6
+ 2u
5
− u
4
− u
3
− u
2
− 1
c
3
, c
10
u
7
+ u
6
+ 4u
5
+ 3u
4
+ 6u
3
+ 2u
2
+ 3u − 1
c
4
, c
8
u
7
− u
6
+ 4u
5
− 3u
4
+ 6u
3
− 2u
2
+ 3u + 1
c
5
u
7
− 2u
6
+ 2u
5
+ u
4
− u
3
+ u
2
+ 1
c
6
, c
7
u
7
− 5u
5
+ 7u
3
− u + 1
c
9
, c
12
u
7
+ 3u
6
− 8u
4
− 4u
3
+ 8u
2
+ 4u − 3
c
11
u
7
− 5u
5
+ 7u
3
− u − 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
7
+ 12y
6
+ 42y
5
+ 31y
4
− 21y
3
− 23y
2
− 2y −1
c
2
, c
5
y
7
+ 6y
5
− y
4
+ 3y
3
− 3y
2
− 2y −1
c
3
, c
4
, c
8
c
10
y
7
+ 7y
6
+ 22y
5
+ 41y
4
+ 50y
3
+ 38y
2
+ 13y −1
c
6
, c
7
, c
11
y
7
− 10y
6
+ 39y
5
− 72y
4
+ 59y
3
− 14y
2
+ y −1
c
9
, c
12
y
7
− 9y
6
+ 40y
5
− 104y
4
+ 162y
3
− 144y
2
+ 64y −9
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.679131 + 0.739231I
a = −0.090421 + 0.468757I
b = 1.067080 − 0.219644I
4.71019 + 2.40933I 0.29860 − 3.62563I
u = −0.679131 − 0.739231I
a = −0.090421 − 0.468757I
b = 1.067080 + 0.219644I
4.71019 − 2.40933I 0.29860 + 3.62563I
u = 0.939920
a = 0.759448
b = 0.490461
−0.502424 5.10270
u = 0.273857 + 0.616814I
a = −0.63288 + 2.30893I
b = −1.49346 + 0.77301I
9.14166 − 1.05666I 5.26558 − 1.27318I
u = 0.273857 − 0.616814I
a = −0.63288 − 2.30893I
b = −1.49346 − 0.77301I
9.14166 + 1.05666I 5.26558 + 1.27318I
u = −1.06469 + 1.08838I
a = −0.656418 − 0.759679I
b = −1.318850 − 0.288008I
11.07340 + 3.97449I 4.88445 − 3.30547I
u = −1.06469 − 1.08838I
a = −0.656418 + 0.759679I
b = −1.318850 + 0.288008I
11.07340 − 3.97449I 4.88445 + 3.30547I
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
7
+ 6u
5
− u
4
+ 3u
3
− 3u
2
− 2u − 1)
· (u
9
− u
8
+ 7u
7
− 5u
6
+ 16u
5
− 7u
4
+ 10u
3
+ 6u
2
− 3u + 1)
c
2
(u
7
+ 2u
6
+ 2u
5
− u
4
− u
3
− u
2
− 1)
· (u
9
+ 3u
8
+ 5u
7
+ 5u
6
+ 6u
5
+ 7u
4
+ 6u
3
+ 2u
2
+ u + 1)
c
3
, c
10
(u
7
+ u
6
+ 4u
5
+ 3u
4
+ 6u
3
+ 2u
2
+ 3u − 1)
· (u
9
− 2u
7
+ 4u
6
+ 17u
5
+ 9u
4
+ 9u
3
+ 2u
2
+ 2u + 1)
c
4
, c
8
(u
7
− u
6
+ 4u
5
− 3u
4
+ 6u
3
− 2u
2
+ 3u + 1)
· (u
9
− 2u
7
+ 4u
6
+ 17u
5
+ 9u
4
+ 9u
3
+ 2u
2
+ 2u + 1)
c
5
(u
7
− 2u
6
+ 2u
5
+ u
4
− u
3
+ u
2
+ 1)
· (u
9
+ 3u
8
+ 5u
7
+ 5u
6
+ 6u
5
+ 7u
4
+ 6u
3
+ 2u
2
+ u + 1)
c
6
, c
7
(u
7
− 5u
5
+ 7u
3
− u + 1)
· (u
9
+ 8u
8
+ 27u
7
+ 46u
6
+ 33u
5
− 10u
4
− 27u
3
− u
2
+ 14u + 4)
c
9
, c
12
(u
7
+ 3u
6
− 8u
4
− 4u
3
+ 8u
2
+ 4u − 3)
· (u
9
− 4u
8
+ 8u
7
− 3u
6
− 2u
4
+ 24u
3
− 23u
2
+ 9u + 1)
c
11
(u
7
− 5u
5
+ 7u
3
− u − 1)
· (u
9
+ 8u
8
+ 27u
7
+ 46u
6
+ 33u
5
− 10u
4
− 27u
3
− u
2
+ 14u + 4)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
7
+ 12y
6
+ 42y
5
+ 31y
4
− 21y
3
− 23y
2
− 2y −1)
· (y
9
+ 13y
8
+ 71y
7
+ 205y
6
+ 332y
5
+ 291y
4
+ 98y
3
− 82y
2
− 3y −1)
c
2
, c
5
(y
7
+ 6y
5
− y
4
+ 3y
3
− 3y
2
− 2y −1)
· (y
9
+ y
8
+ 7y
7
+ 5y
6
+ 16y
5
+ 7y
4
+ 10y
3
− 6y
2
− 3y −1)
c
3
, c
4
, c
8
c
10
(y
7
+ 7y
6
+ 22y
5
+ 41y
4
+ 50y
3
+ 38y
2
+ 13y −1)
· (y
9
− 4y
8
+ 38y
7
− 66y
6
+ 185y
5
+ 201y
4
+ 105y
3
+ 14y
2
− 1)
c
6
, c
7
, c
11
(y
7
− 10y
6
+ 39y
5
− 72y
4
+ 59y
3
− 14y
2
+ y −1)
· (y
9
− 10y
8
+ ··· + 204y −16)
c
9
, c
12
(y
7
− 9y
6
+ 40y
5
− 104y
4
+ 162y
3
− 144y
2
+ 64y −9)
· (y
9
+ 40y
7
+ 23y
6
+ 206y
5
+ 10y
4
+ 490y
3
− 93y
2
+ 127y −1)
11
