12n
0168
(K12n
0168
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 10 11 4 5 12 6 7 9
Solving Sequence
6,10
11 7 12
2,5
3 1 4 9 8
c
10
c
6
c
11
c
5
c
2
c
1
c
4
c
9
c
8
c
3
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
20
+ 10u
18
+ ··· + b + 2u, u
23
− u
22
+ ··· + a − 1, u
24
− 2u
23
+ ··· − 8u
2
+ 1i
I
u
2
= hu
4
− 2u
2
+ b + 2u, −u
5
+ 3u
3
+ a + 1, u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1i
* 2 irreducible components of dim
C
= 0, with total 30 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
20
+10 u
18
+ · · ·+b+2u, u
23
−u
22
+ · · ·+a−1, u
24
−2u
23
+ · · ·−8u
2
+1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
7
=
−u
−u
3
+ u
a
12
=
−u
2
+ 1
−u
4
+ 2u
2
a
2
=
−u
23
+ u
22
+ ··· + 5u + 1
u
20
− 10u
18
+ ··· − 6u
2
− 2u
a
5
=
u
u
a
3
=
−2u
23
+ u
22
+ ··· + 6u + 2
−u
23
+ 12u
21
+ ··· − u + 1
a
1
=
u
10
− 5u
8
+ 8u
6
− 5u
4
+ 3u
2
− 1
u
12
− 6u
10
+ 12u
8
− 8u
6
+ u
4
− 2u
2
a
4
=
−u
19
+ 10u
17
+ ··· + 5u + 1
u
23
− u
22
+ ··· − 2u − 1
a
9
=
−u
6
+ 3u
4
− 2u
2
+ 1
−u
8
+ 4u
6
− 4u
4
a
8
=
−u
10
+ 5u
8
− 8u
6
+ 5u
4
− 3u
2
+ 1
−u
10
+ 4u
8
− 3u
6
− 2u
4
− u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
23
+ 7u
22
+ 45u
21
− 76u
20
− 213u
19
+ 331u
18
+ 563u
17
−
727u
16
− 961u
15
+ 840u
14
+ 1203u
13
− 550u
12
− 1159u
11
+ 373u
10
+ 788u
9
− 284u
8
−
414u
7
+ 79u
6
+ 239u
5
− 78u
4
− 63u
3
+ 41u
2
+ 22u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
24
+ u
23
+ ··· + 33u + 1
c
2
, c
4
u
24
− 7u
23
+ ··· − 9u + 1
c
3
, c
7
u
24
− u
23
+ ··· + 64u + 64
c
5
, c
6
, c
10
c
11
u
24
− 2u
23
+ ··· − 8u
2
+ 1
c
8
u
24
+ 2u
23
+ ··· + 7956u + 4721
c
9
, c
12
u
24
− 2u
23
+ ··· + 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
24
+ 51y
23
+ ··· − 789y + 1
c
2
, c
4
y
24
− y
23
+ ··· − 33y + 1
c
3
, c
7
y
24
− 39y
23
+ ··· − 65536y + 4096
c
5
, c
6
, c
10
c
11
y
24
− 26y
23
+ ··· − 16y + 1
c
8
y
24
+ 94y
23
+ ··· − 805269180y + 22287841
c
9
, c
12
y
24
+ 34y
23
+ ··· − 16y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.546136 + 0.704998I
a = 0.218919 + 0.545111I
b = −1.58637 + 0.30753I
13.8157 + 6.6372I −5.51662 − 4.82221I
u = −0.546136 − 0.704998I
a = 0.218919 − 0.545111I
b = −1.58637 − 0.30753I
13.8157 − 6.6372I −5.51662 + 4.82221I
u = −0.490937 + 0.721217I
a = −0.42824 + 1.49833I
b = −0.020820 − 0.317218I
13.98230 − 1.87133I −5.11127 − 0.61495I
u = −0.490937 − 0.721217I
a = −0.42824 − 1.49833I
b = −0.020820 + 0.317218I
13.98230 + 1.87133I −5.11127 + 0.61495I
u = 0.581256 + 0.532393I
a = 0.124238 + 0.261688I
b = 1.180950 + 0.505513I
2.81700 − 3.48253I −5.09473 + 5.87430I
u = 0.581256 − 0.532393I
a = 0.124238 − 0.261688I
b = 1.180950 − 0.505513I
2.81700 + 3.48253I −5.09473 − 5.87430I
u = 0.371044 + 0.593072I
a = 0.614735 + 1.078340I
b = −0.138383 + 0.111177I
3.46440 − 0.36969I −3.22680 + 1.74990I
u = 0.371044 − 0.593072I
a = 0.614735 − 1.078340I
b = −0.138383 − 0.111177I
3.46440 + 0.36969I −3.22680 − 1.74990I
u = −0.560055
a = −0.358547
b = −0.698358
−0.920303 −10.4000
u = −1.44707 + 0.16024I
a = −0.1025080 + 0.0199673I
b = −0.075688 − 0.613179I
−2.38092 + 3.00213I −7.53165 − 2.42924I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.44707 − 0.16024I
a = −0.1025080 − 0.0199673I
b = −0.075688 + 0.613179I
−2.38092 − 3.00213I −7.53165 + 2.42924I
u = 1.46661 + 0.06456I
a = −1.18298 + 1.35435I
b = −1.81615 + 1.04191I
−6.76285 − 2.21001I −11.76840 + 2.54579I
u = 1.46661 − 0.06456I
a = −1.18298 − 1.35435I
b = −1.81615 − 1.04191I
−6.76285 + 2.21001I −11.76840 − 2.54579I
u = −1.47571
a = 3.35878
b = 4.21798
−8.10337 −9.82550
u = 1.50201 + 0.24602I
a = 0.543477 − 0.750165I
b = 1.09231 − 1.65332I
7.50585 − 1.64558I −8.27015 + 0.73963I
u = 1.50201 − 0.24602I
a = 0.543477 + 0.750165I
b = 1.09231 + 1.65332I
7.50585 + 1.64558I −8.27015 − 0.73963I
u = −0.344587 + 0.302443I
a = −1.119750 + 0.097114I
b = 0.366639 + 0.663744I
−0.814955 + 1.024630I −8.91038 − 6.27818I
u = −0.344587 − 0.302443I
a = −1.119750 − 0.097114I
b = 0.366639 − 0.663744I
−0.814955 − 1.024630I −8.91038 + 6.27818I
u = 1.53592 + 0.23426I
a = 2.64852 − 0.49160I
b = 3.50928 + 0.08824I
6.98523 − 10.07590I −8.82304 + 4.75008I
u = 1.53592 − 0.23426I
a = 2.64852 + 0.49160I
b = 3.50928 − 0.08824I
6.98523 + 10.07590I −8.82304 − 4.75008I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.55778
a = 2.20275
b = 2.47794
−8.18010 −9.41090
u = −1.55100 + 0.14681I
a = −2.45672 − 0.03517I
b = −2.88742 + 0.30740I
−4.30018 + 5.91154I −8.53274 − 5.50143I
u = −1.55100 − 0.14681I
a = −2.45672 + 0.03517I
b = −2.88742 − 0.30740I
−4.30018 − 5.91154I −8.53274 + 5.50143I
u = 0.323756
a = 2.07762
b = −1.24627
−2.07138 3.20840
7
II.
I
u
2
= hu
4
− 2u
2
+ b + 2u, −u
5
+ 3u
3
+ a + 1, u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
7
=
−u
−u
3
+ u
a
12
=
−u
2
+ 1
−u
4
+ 2u
2
a
2
=
u
5
− 3u
3
− 1
−u
4
+ 2u
2
− 2u
a
5
=
u
u
a
3
=
u
5
− 3u
3
+ u − 1
−u
4
+ 2u
2
− u
a
1
=
−u
−u
a
4
=
u
5
− 3u
3
+ u − 1
−u
4
+ 2u
2
− u
a
9
=
u
5
− 2u
3
− u
u
5
− 3u
3
+ u
a
8
=
−u
−u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
5
+ u
4
− 14u
3
− u
2
+ 14u − 18
8
(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
, c
12
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1
c
9
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
10
, c
11
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1
9
(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
10
c
11
y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1
c
8
, c
9
, c
12
y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.493180 + 0.575288I
a = 0.011399 − 0.918055I
b = −0.847526 + 0.083869I
1.31531 − 1.97241I −6.43930 + 3.48596I
u = 0.493180 − 0.575288I
a = 0.011399 + 0.918055I
b = −0.847526 − 0.083869I
1.31531 + 1.97241I −6.43930 − 3.48596I
u = −0.483672
a = −0.687021
b = 1.38049
−2.38379 −23.4460
u = −1.52087 + 0.16310I
a = 1.98288 + 0.88048I
b = 2.63293 + 0.95019I
−5.34051 + 4.59213I −10.66600 − 2.48468I
u = −1.52087 − 0.16310I
a = 1.98288 − 0.88048I
b = 2.63293 − 0.95019I
−5.34051 − 4.59213I −10.66600 + 2.48468I
u = 1.53904
a = −3.30155
b = −3.95130
−9.30502 −18.3430
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
24
+ u
23
+ ··· + 33u + 1)
c
2
((u − 1)
6
)(u
24
− 7u
23
+ ··· − 9u + 1)
c
3
, c
7
u
6
(u
24
− u
23
+ ··· + 64u + 64)
c
4
((u + 1)
6
)(u
24
− 7u
23
+ ··· − 9u + 1)
c
5
, c
6
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
24
− 2u
23
+ ··· − 8u
2
+ 1)
c
8
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
24
+ 2u
23
+ ··· + 7956u + 4721)
c
9
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
24
− 2u
23
+ ··· + 2u + 1)
c
10
, c
11
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
24
− 2u
23
+ ··· − 8u
2
+ 1)
c
12
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
24
− 2u
23
+ ··· + 2u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
24
+ 51y
23
+ ··· − 789y + 1)
c
2
, c
4
((y −1)
6
)(y
24
− y
23
+ ··· − 33y + 1)
c
3
, c
7
y
6
(y
24
− 39y
23
+ ··· − 65536y + 4096)
c
5
, c
6
, c
10
c
11
(y
6
− 7y
5
+ ··· − 5y + 1)(y
24
− 26y
23
+ ··· − 16y + 1)
c
8
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1)
· (y
24
+ 94y
23
+ ··· − 805269180y + 22287841)
c
9
, c
12
(y
6
+ 5y
5
+ ··· − 5y + 1)(y
24
+ 34y
23
+ ··· − 16y + 1)
13
