12n
0446
(K12n
0446
)
A knot diagram
1
Linearized knot diagam
3 5 9 11 2 4 12 3 7 5 10 8
Solving Sequence
5,10
11 12
4,7
6 9 3 2 1 8
c
10
c
11
c
4
c
6
c
9
c
3
c
2
c
1
c
8
c
5
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−9u
16
+ 75u
15
+ ··· + 4b + 48, −8u
16
+ 61u
15
+ ··· + 4a + 16, u
17
− 9u
16
+ ··· − 16u + 8i
I
u
2
= hu
12
+ u
11
+ 3u
10
+ 2u
9
+ 7u
8
+ 4u
7
+ 9u
6
+ 3u
5
+ 9u
4
+ 2u
3
+ 5u
2
+ b + u + 2,
− 3u
13
− 3u
12
− 8u
11
− 5u
10
− 18u
9
− 10u
8
− 23u
7
− 8u
6
− 23u
5
− 6u
4
− 14u
3
− 4u
2
+ a − 5u + 1,
u
14
+ u
13
+ 3u
12
+ 2u
11
+ 7u
10
+ 4u
9
+ 10u
8
+ 4u
7
+ 11u
6
+ 3u
5
+ 8u
4
+ 2u
3
+ 4u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 31 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−9u
16
+ 75u
15
+ · · · + 4b + 48, −8u
16
+ 61u
15
+ · · · + 4a + 16, u
17
−
9u
16
+ · · · − 16u + 8i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
12
=
u
2
+ 1
u
2
a
4
=
u
u
3
+ u
a
7
=
2u
16
−
61
4
u
15
+ ··· +
53
4
u − 4
9
4
u
16
−
75
4
u
15
+ ··· + 23u − 12
a
6
=
−
3
2
u
16
+
45
4
u
15
+ ··· −
35
4
u + 2
−
17
4
u
16
+
151
4
u
15
+ ··· − 51u + 34
a
9
=
15
8
u
16
−
117
8
u
15
+ ··· + 14u −
5
2
9
4
u
16
−
77
4
u
15
+ ··· +
55
2
u − 15
a
3
=
1
4
u
16
−
9
4
u
15
+ ··· +
7
2
u −
5
2
−
1
2
u
15
+
7
2
u
14
+ ··· +
5
2
u − 2
a
2
=
1
4
u
16
−
9
4
u
15
+ ··· +
7
2
u −
5
2
−
1
2
u
16
+ 3u
15
+ ··· +
1
2
u − 2
a
1
=
−
13
8
u
16
+
119
8
u
15
+ ··· − 22u +
31
2
5
4
u
16
−
29
4
u
15
+ ··· −
3
2
u + 11
a
8
=
9
4
u
16
− 18u
15
+ ··· +
73
4
u − 6
9
4
u
16
−
79
4
u
15
+ ··· + 29u − 16
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6u
16
+ 48u
15
− 204u
14
+ 579u
13
− 1201u
12
+ 1920u
11
−
2456u
10
+ 2621u
9
−2430u
8
+ 2003u
7
−1452u
6
+ 891u
5
−445u
4
+ 148u
3
+ 6u
2
−44u + 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
17
− 22u
16
+ ··· + 54u − 1
c
2
, c
5
u
17
+ 2u
16
+ ··· − 2u + 1
c
3
, c
8
u
17
− u
16
+ ··· − u + 1
c
4
, c
10
u
17
− 9u
16
+ ··· − 16u + 8
c
6
u
17
+ 4u
16
+ ··· + 24694u + 2511
c
7
, c
12
u
17
+ 7u
15
+ ··· − 226u + 111
c
9
u
17
− 3u
16
+ ··· − 4u + 1
c
11
u
17
− 5u
16
+ ··· + 160u + 64
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
17
+ 62y
16
+ ··· + 426y −1
c
2
, c
5
y
17
− 22y
16
+ ··· + 54y −1
c
3
, c
8
y
17
+ 21y
16
+ ··· − 7y −1
c
4
, c
10
y
17
+ 5y
16
+ ··· + 160y −64
c
6
y
17
− 22y
16
+ ··· + 769543456y −6305121
c
7
, c
12
y
17
+ 14y
16
+ ··· − 22850y −12321
c
9
y
17
− 3y
16
+ ··· + 10y −1
c
11
y
17
+ 13y
16
+ ··· + 156160y −4096
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.743107 + 0.737300I
a = −0.957729 + 0.743386I
b = −1.065040 + 0.659072I
−3.59959 + 0.58481I −8.87613 + 1.72757I
u = 0.743107 − 0.737300I
a = −0.957729 − 0.743386I
b = −1.065040 − 0.659072I
−3.59959 − 0.58481I −8.87613 − 1.72757I
u = −0.072889 + 0.887618I
a = 0.312452 − 0.912707I
b = −0.304982 + 0.710279I
1.67215 + 1.49188I 1.61333 − 4.98502I
u = −0.072889 − 0.887618I
a = 0.312452 + 0.912707I
b = −0.304982 − 0.710279I
1.67215 − 1.49188I 1.61333 + 4.98502I
u = −0.474839 + 0.714353I
a = 1.293010 + 0.446550I
b = 0.470362 − 0.202610I
−0.29498 + 1.80975I −2.70110 − 3.65617I
u = −0.474839 − 0.714353I
a = 1.293010 − 0.446550I
b = 0.470362 + 0.202610I
−0.29498 − 1.80975I −2.70110 + 3.65617I
u = 0.699430 + 0.964772I
a = −1.65451 + 0.74282I
b = −1.001640 − 0.878315I
−2.90799 − 6.08549I −5.32420 + 3.68722I
u = 0.699430 − 0.964772I
a = −1.65451 − 0.74282I
b = −1.001640 + 0.878315I
−2.90799 + 6.08549I −5.32420 − 3.68722I
u = 0.864249 + 0.915420I
a = 1.71758 − 0.46850I
b = 0.779025 + 0.016840I
−7.95991 − 3.20234I −15.1724 + 0.4148I
u = 0.864249 − 0.915420I
a = 1.71758 + 0.46850I
b = 0.779025 − 0.016840I
−7.95991 + 3.20234I −15.1724 − 0.4148I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.408880 + 0.063065I
a = 1.183850 − 0.166931I
b = 0.955388 − 0.931462I
6.30330 + 3.42896I −6.06062 − 2.16109I
u = 1.408880 − 0.063065I
a = 1.183850 + 0.166931I
b = 0.955388 + 0.931462I
6.30330 − 3.42896I −6.06062 + 2.16109I
u = −0.395985
a = −0.300112
b = −0.572653
−0.874933 −11.5210
u = 0.72984 + 1.44188I
a = 0.168023 − 0.244503I
b = 0.909775 − 1.029340I
10.83540 − 3.92897I −4.57004 + 0.99423I
u = 0.72984 − 1.44188I
a = 0.168023 + 0.244503I
b = 0.909775 + 1.029340I
10.83540 + 3.92897I −4.57004 − 0.99423I
u = 0.80021 + 1.43594I
a = 1.33738 − 0.75405I
b = 1.043440 + 0.934400I
10.3710 − 11.0996I −5.14830 + 4.90846I
u = 0.80021 − 1.43594I
a = 1.33738 + 0.75405I
b = 1.043440 − 0.934400I
10.3710 + 11.0996I −5.14830 − 4.90846I
6
II.
I
u
2
= hu
12
+u
11
+· · ·+b+2, −3u
13
−3u
12
+· · ·+a+1, u
14
+u
13
+· · ·+4u
2
+1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
12
=
u
2
+ 1
u
2
a
4
=
u
u
3
+ u
a
7
=
3u
13
+ 3u
12
+ ··· + 5u − 1
−u
12
− u
11
+ ··· − u − 2
a
6
=
2u
13
+ 2u
12
+ ··· + 3u − 2
−u
13
− 2u
12
+ ··· − 2u − 3
a
9
=
2u
13
+ 2u
12
+ ··· + 2u − 3
−u
12
− 2u
11
+ ··· − 3u − 2
a
3
=
u
12
+ u
11
+ ··· + 2u + 3
u
13
+ u
12
+ ··· + u
2
+ 4u
a
2
=
u
12
+ u
11
+ ··· + 2u + 3
u
13
+ u
12
+ ··· + 4u − 1
a
1
=
−2u
13
− 2u
12
+ ··· − 5u + 1
−u
13
− u
12
+ ··· − 3u
2
+ u
a
8
=
2u
13
+ 3u
12
+ ··· + 4u − 1
−u
12
− u
11
+ ··· − 8u
2
− 3
(ii) Obstruction class = 1
(iii) Cusp Shapes
= −u
13
+3u
12
+2u
11
+12u
10
+9u
9
+32u
8
+21u
7
+42u
6
+25u
5
+45u
4
+22u
3
+24u
2
+10u
7
(iv) u-Polynomials at the component
8
Crossings u-Polynomials at each crossing
c
1
u
14
− 11u
13
+ ··· − 4u + 1
c
2
u
14
+ u
13
+ 6u
12
+ 4u
11
+ 9u
10
+ 3u
8
− 5u
7
− 4u
5
+ 2u
4
− u
3
+ 2u
2
+ 1
c
3
u
14
− 5u
12
+ 9u
10
− 2u
9
− 6u
8
+ 6u
7
− 7u
5
+ 4u
4
+ 4u
3
− 3u
2
− u + 1
c
4
u
14
− u
13
+ ··· + 4u
2
+ 1
c
5
u
14
− u
13
+ 6u
12
− 4u
11
+ 9u
10
+ 3u
8
+ 5u
7
+ 4u
5
+ 2u
4
+ u
3
+ 2u
2
+ 1
c
6
u
14
− u
13
+ ··· + 8u + 67
c
7
u
14
− u
13
+ ··· + 4u + 1
c
8
u
14
− 5u
12
+ 9u
10
+ 2u
9
− 6u
8
− 6u
7
+ 7u
5
+ 4u
4
− 4u
3
− 3u
2
+ u + 1
c
9
u
14
+ 6u
13
+ ··· + 4u + 1
c
10
u
14
+ u
13
+ ··· + 4u
2
+ 1
c
11
u
14
− 5u
13
+ ··· − 8u + 1
c
12
u
14
+ u
13
+ ··· − 4u + 1
9
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
14
− 29y
13
+ ··· + 12y
2
+ 1
c
2
, c
5
y
14
+ 11y
13
+ ··· + 4y + 1
c
3
, c
8
y
14
− 10y
13
+ ··· − 7y + 1
c
4
, c
10
y
14
+ 5y
13
+ ··· + 8y + 1
c
6
y
14
+ 11y
13
+ ··· − 466y + 4489
c
7
, c
12
y
14
− 13y
13
+ ··· − 4y + 1
c
9
y
14
− 2y
13
+ ··· − 8y + 1
c
11
y
14
+ 13y
13
+ ··· + 32y
2
+ 1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.105110 + 0.959669I
a = 0.448698 − 0.035015I
b = −0.594085 + 0.956154I
0.09953 + 1.96463I −3.48083 − 3.91633I
u = 0.105110 − 0.959669I
a = 0.448698 + 0.035015I
b = −0.594085 − 0.956154I
0.09953 − 1.96463I −3.48083 + 3.91633I
u = 0.694518 + 0.776039I
a = −0.573318 + 1.021770I
b = −1.16230 + 0.95610I
−3.84339 + 1.43715I −12.12204 − 5.82210I
u = 0.694518 − 0.776039I
a = −0.573318 − 1.021770I
b = −1.16230 − 0.95610I
−3.84339 − 1.43715I −12.12204 + 5.82210I
u = −0.584040 + 0.656749I
a = 0.75333 − 1.27705I
b = 0.796480 + 0.277469I
−6.62904 + 0.87099I −9.32156 + 1.93932I
u = −0.584040 − 0.656749I
a = 0.75333 + 1.27705I
b = 0.796480 − 0.277469I
−6.62904 − 0.87099I −9.32156 − 1.93932I
u = 0.672935 + 0.942423I
a = −1.78530 + 0.66319I
b = −1.09781 − 1.12323I
−3.32221 − 6.71387I −11.7069 + 12.3294I
u = 0.672935 − 0.942423I
a = −1.78530 − 0.66319I
b = −1.09781 + 1.12323I
−3.32221 + 6.71387I −11.7069 − 12.3294I
u = −0.645970 + 1.039580I
a = 0.263318 + 0.909593I
b = 0.491954 − 0.568665I
−5.34937 + 4.06327I −5.63140 − 4.63388I
u = −0.645970 − 1.039580I
a = 0.263318 − 0.909593I
b = 0.491954 + 0.568665I
−5.34937 − 4.06327I −5.63140 + 4.63388I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.925465 + 0.908624I
a = −1.45765 − 0.16567I
b = −0.528256 + 0.113442I
−7.43274 + 3.38328I −1.12875 − 4.65800I
u = −0.925465 − 0.908624I
a = −1.45765 + 0.16567I
b = −0.528256 − 0.113442I
−7.43274 − 3.38328I −1.12875 + 4.65800I
u = 0.182913 + 0.587851I
a = −2.14907 + 1.62071I
b = −0.905989 − 0.618123I
−1.48665 − 3.24685I −5.60851 + 4.03314I
u = 0.182913 − 0.587851I
a = −2.14907 − 1.62071I
b = −0.905989 + 0.618123I
−1.48665 + 3.24685I −5.60851 − 4.03314I
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
14
− 11u
13
+ ··· − 4u + 1)(u
17
− 22u
16
+ ··· + 54u − 1)
c
2
(u
14
+ u
13
+ 6u
12
+ 4u
11
+ 9u
10
+ 3u
8
− 5u
7
− 4u
5
+ 2u
4
− u
3
+ 2u
2
+ 1)
· (u
17
+ 2u
16
+ ··· − 2u + 1)
c
3
(u
14
− 5u
12
+ 9u
10
− 2u
9
− 6u
8
+ 6u
7
− 7u
5
+ 4u
4
+ 4u
3
− 3u
2
− u + 1)
· (u
17
− u
16
+ ··· − u + 1)
c
4
(u
14
− u
13
+ ··· + 4u
2
+ 1)(u
17
− 9u
16
+ ··· − 16u + 8)
c
5
(u
14
− u
13
+ 6u
12
− 4u
11
+ 9u
10
+ 3u
8
+ 5u
7
+ 4u
5
+ 2u
4
+ u
3
+ 2u
2
+ 1)
· (u
17
+ 2u
16
+ ··· − 2u + 1)
c
6
(u
14
− u
13
+ ··· + 8u + 67)(u
17
+ 4u
16
+ ··· + 24694u + 2511)
c
7
(u
14
− u
13
+ ··· + 4u + 1)(u
17
+ 7u
15
+ ··· − 226u + 111)
c
8
(u
14
− 5u
12
+ 9u
10
+ 2u
9
− 6u
8
− 6u
7
+ 7u
5
+ 4u
4
− 4u
3
− 3u
2
+ u + 1)
· (u
17
− u
16
+ ··· − u + 1)
c
9
(u
14
+ 6u
13
+ ··· + 4u + 1)(u
17
− 3u
16
+ ··· − 4u + 1)
c
10
(u
14
+ u
13
+ ··· + 4u
2
+ 1)(u
17
− 9u
16
+ ··· − 16u + 8)
c
11
(u
14
− 5u
13
+ ··· − 8u + 1)(u
17
− 5u
16
+ ··· + 160u + 64)
c
12
(u
14
+ u
13
+ ··· − 4u + 1)(u
17
+ 7u
15
+ ··· − 226u + 111)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
14
− 29y
13
+ ··· + 12y
2
+ 1)(y
17
+ 62y
16
+ ··· + 426y −1)
c
2
, c
5
(y
14
+ 11y
13
+ ··· + 4y + 1)(y
17
− 22y
16
+ ··· + 54y −1)
c
3
, c
8
(y
14
− 10y
13
+ ··· − 7y + 1)(y
17
+ 21y
16
+ ··· − 7y −1)
c
4
, c
10
(y
14
+ 5y
13
+ ··· + 8y + 1)(y
17
+ 5y
16
+ ··· + 160y −64)
c
6
(y
14
+ 11y
13
+ ··· − 466y + 4489)
· (y
17
− 22y
16
+ ··· + 769543456y −6305121)
c
7
, c
12
(y
14
− 13y
13
+ ··· − 4y + 1)(y
17
+ 14y
16
+ ··· − 22850y −12321)
c
9
(y
14
− 2y
13
+ ··· − 8y + 1)(y
17
− 3y
16
+ ··· + 10y −1)
c
11
(y
14
+ 13y
13
+ ··· + 32y
2
+ 1)(y
17
+ 13y
16
+ ··· + 156160y −4096)
15
