12n
0244
(K12n
0244
)
A knot diagram
1
Linearized knot diagam
3 5 9 2 12 11 10 3 5 6 7 8
Solving Sequence
7,12
11 6
3,5
2 1 4 10 8 9
c
11
c
6
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
= hu
19
+ u
18
+ ··· + u
2
+ b, −u
19
− u
18
+ ··· + a − 2, u
20
+ 2u
19
+ ··· + 2u + 1i
I
u
2
= hb − 1, u
6
− 3u
4
− u
3
+ 2u
2
+ a + 2u + 1, u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 28 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
19
+u
18
+· · ·+u
2
+b, −u
19
−u
18
+· · ·+a−2, u
20
+2u
19
+· · ·+2u+1i
(i) Arc colorings
a
7
=
0
u
a
12
=
1
0
a
11
=
1
−u
2
a
6
=
u
−u
3
+ u
a
3
=
u
19
+ u
18
+ ··· + 3u + 2
−u
19
− u
18
+ ··· − 8u
3
− u
2
a
5
=
−u
3
+ 2u
−u
3
+ u
a
2
=
2u
19
+ 2u
18
+ ··· + 6u + 3
−u
19
− u
18
+ ··· − 12u
3
+ u
a
1
=
−u
12
+ 5u
10
− 9u
8
+ 6u
6
− u
2
+ 1
u
14
− 6u
12
+ 13u
10
− 10u
8
− 2u
6
+ 4u
4
+ u
2
a
4
=
−3u
19
− 3u
18
+ ··· − 6u − 3
u
19
+ u
18
+ ··· − 2u
2
− u
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
8
=
−u
5
+ 2u
3
− u
u
7
− 3u
5
+ 2u
3
+ u
a
9
=
u
10
− 5u
8
+ 8u
6
− 3u
4
− 3u
2
+ 1
u
10
− 4u
8
+ 5u
6
− 3u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 5u
19
+ 4u
18
−38u
17
−21u
16
+ 118u
15
+ 27u
14
−176u
13
+ 40u
12
+
88u
11
− 121u
10
+ 80u
9
+ 58u
8
− 89u
7
+ 52u
6
− 36u
5
− 27u
4
+ 46u
3
− 16u
2
+ 12u − 11
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
20
+ 39u
19
+ ··· + 18u + 1
c
2
, c
4
u
20
− 9u
19
+ ··· + 9u
2
− 1
c
3
, c
8
u
20
+ u
19
+ ··· − 640u − 256
c
5
, c
7
u
20
− 6u
19
+ ··· + 2u − 5
c
6
, c
10
, c
11
u
20
+ 2u
19
+ ··· + 2u + 1
c
9
, c
12
u
20
− 2u
19
+ ··· + 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
20
− 155y
19
+ ··· − 690y + 1
c
2
, c
4
y
20
− 39y
19
+ ··· − 18y + 1
c
3
, c
8
y
20
− 51y
19
+ ··· + 16384y + 65536
c
5
, c
7
y
20
+ 6y
19
+ ··· − 194y + 25
c
6
, c
10
, c
11
y
20
− 18y
19
+ ··· − 6y + 1
c
9
, c
12
y
20
− 42y
19
+ ··· − 6y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.911170 + 0.459727I
a = −1.34870 − 0.62192I
b = −2.12337 + 0.10108I
−17.2402 + 0.6353I −16.0469 + 1.1994I
u = 0.911170 − 0.459727I
a = −1.34870 + 0.62192I
b = −2.12337 − 0.10108I
−17.2402 − 0.6353I −16.0469 − 1.1994I
u = 0.247662 + 0.821626I
a = −2.43556 − 1.84521I
b = −2.12150 − 0.21202I
−15.1547 − 5.2095I −13.60177 + 3.17253I
u = 0.247662 − 0.821626I
a = −2.43556 + 1.84521I
b = −2.12150 + 0.21202I
−15.1547 + 5.2095I −13.60177 − 3.17253I
u = −1.208090 + 0.243596I
a = 0.570598 + 0.429334I
b = −0.218634 + 0.234930I
−1.33162 + 1.52088I −9.67152 − 0.73849I
u = −1.208090 − 0.243596I
a = 0.570598 − 0.429334I
b = −0.218634 − 0.234930I
−1.33162 − 1.52088I −9.67152 + 0.73849I
u = −0.102862 + 0.701439I
a = 0.082707 − 0.900739I
b = −0.141682 − 0.411106I
1.97616 + 1.89773I −7.07242 − 3.81165I
u = −0.102862 − 0.701439I
a = 0.082707 + 0.900739I
b = −0.141682 + 0.411106I
1.97616 − 1.89773I −7.07242 + 3.81165I
u = 1.312190 + 0.118081I
a = 0.496056 − 0.398382I
b = 0.691084 − 0.636911I
−4.98968 − 0.65533I −18.6254 + 0.2318I
u = 1.312190 − 0.118081I
a = 0.496056 + 0.398382I
b = 0.691084 + 0.636911I
−4.98968 + 0.65533I −18.6254 − 0.2318I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.335820 + 0.290040I
a = −0.158410 + 0.393649I
b = −0.140101 + 0.577407I
−2.56262 − 5.49819I −13.3788 + 5.1703I
u = 1.335820 − 0.290040I
a = −0.158410 − 0.393649I
b = −0.140101 − 0.577407I
−2.56262 + 5.49819I −13.3788 − 5.1703I
u = −1.371860 + 0.209407I
a = −0.86878 − 1.49091I
b = 1.54182 − 0.34669I
−6.47245 + 3.72845I −18.3956 − 2.9383I
u = −1.371860 − 0.209407I
a = −0.86878 + 1.49091I
b = 1.54182 + 0.34669I
−6.47245 − 3.72845I −18.3956 + 2.9383I
u = −1.41274 + 0.33785I
a = −0.17556 + 2.27560I
b = −2.15361 + 0.28774I
19.0447 + 9.4000I −17.5612 − 4.3303I
u = −1.41274 − 0.33785I
a = −0.17556 − 2.27560I
b = −2.15361 − 0.28774I
19.0447 − 9.4000I −17.5612 + 4.3303I
u = 0.188195 + 0.497650I
a = 0.89217 + 2.08322I
b = 1.230350 + 0.275114I
−1.49234 − 1.05642I −12.43000 + 2.14230I
u = 0.188195 − 0.497650I
a = 0.89217 − 2.08322I
b = 1.230350 − 0.275114I
−1.49234 + 1.05642I −12.43000 − 2.14230I
u = −1.49219
a = 0.836574
b = −2.31363
14.2449 −19.8440
u = −0.306795
a = 1.05438
b = 0.184912
−0.567629 −17.5890
6
II.
I
u
2
= hb−1, u
6
−3u
4
−u
3
+2u
2
+a+2u +1, u
8
−u
7
−3u
6
+2u
5
+3u
4
−2u−1i
(i) Arc colorings
a
7
=
0
u
a
12
=
1
0
a
11
=
1
−u
2
a
6
=
u
−u
3
+ u
a
3
=
−u
6
+ 3u
4
+ u
3
− 2u
2
− 2u − 1
1
a
5
=
−u
3
+ 2u
−u
3
+ u
a
2
=
−u
6
+ 3u
4
+ 2u
3
− 2u
2
− 4u − 1
u
3
− u + 1
a
1
=
u
3
− 2u
u
3
− u
a
4
=
−u
6
+ 3u
4
+ u
3
− 2u
2
− 2u − 1
1
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
8
=
−u
5
+ 2u
3
− u
u
7
− 3u
5
+ 2u
3
+ u
a
9
=
−u
5
+ 2u
3
− u
u
7
− 3u
5
+ 2u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
7
− 2u
6
+ 2u
5
+ 8u
4
+ 3u
3
− 7u
2
− 8u − 19
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
8
c
3
, c
8
u
8
c
4
(u + 1)
8
c
5
, c
7
u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1
c
6
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
c
9
, c
12
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1
c
10
, c
11
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
8
c
3
, c
8
y
8
c
5
, c
7
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
c
6
, c
10
, c
11
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
c
9
, c
12
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.180120 + 0.268597I
a = 0.646194 − 0.127698I
b = 1.00000
−2.68559 + 1.13123I −15.9046 − 0.8051I
u = −1.180120 − 0.268597I
a = 0.646194 + 0.127698I
b = 1.00000
−2.68559 − 1.13123I −15.9046 + 0.8051I
u = −0.108090 + 0.747508I
a = 1.43073 − 0.89199I
b = 1.00000
0.51448 + 2.57849I −11.78039 − 3.88175I
u = −0.108090 − 0.747508I
a = 1.43073 + 0.89199I
b = 1.00000
0.51448 − 2.57849I −11.78039 + 3.88175I
u = 1.37100
a = −0.966009
b = 1.00000
−8.14766 −19.8290
u = 1.334530 + 0.318930I
a = 0.142888 + 1.323540I
b = 1.00000
−4.02461 − 6.44354I −16.5091 + 6.0410I
u = 1.334530 − 0.318930I
a = 0.142888 − 1.323540I
b = 1.00000
−4.02461 + 6.44354I −16.5091 − 6.0410I
u = −0.463640
a = −0.473616
b = 1.00000
−2.48997 −16.7830
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
20
+ 39u
19
+ ··· + 18u + 1)
c
2
((u − 1)
8
)(u
20
− 9u
19
+ ··· + 9u
2
− 1)
c
3
, c
8
u
8
(u
20
+ u
19
+ ··· − 640u − 256)
c
4
((u + 1)
8
)(u
20
− 9u
19
+ ··· + 9u
2
− 1)
c
5
, c
7
(u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
· (u
20
− 6u
19
+ ··· + 2u − 5)
c
6
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
20
+ 2u
19
+ ··· + 2u + 1)
c
9
, c
12
(u
8
+ u
7
+ ··· − 2u − 1)(u
20
− 2u
19
+ ··· + 2u + 1)
c
10
, c
11
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)(u
20
+ 2u
19
+ ··· + 2u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
8
)(y
20
− 155y
19
+ ··· − 690y + 1)
c
2
, c
4
((y −1)
8
)(y
20
− 39y
19
+ ··· − 18y + 1)
c
3
, c
8
y
8
(y
20
− 51y
19
+ ··· + 16384y + 65536)
c
5
, c
7
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
20
+ 6y
19
+ ··· − 194y + 25)
c
6
, c
10
, c
11
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
20
− 18y
19
+ ··· − 6y + 1)
c
9
, c
12
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
20
− 42y
19
+ ··· − 6y + 1)
12
