12n
0149
(K12n
0149
)
A knot diagram
1
Linearized knot diagam
3 4 9 2 10 3 12 11 6 4 8 7
Solving Sequence
3,9 4,6
7 10 11 2 1 5 8 12
c
3
c
6
c
9
c
10
c
2
c
1
c
5
c
8
c
12
c
4
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
7
− 2u
6
+ 6u
5
+ 12u
4
− 8u
3
− 12u
2
+ 2b + 13u + 8,
7u
7
+ 9u
6
− 38u
5
− 59u
4
+ 48u
3
+ 59u
2
+ 20a − 59u − 48,
u
8
+ 2u
7
− 4u
6
− 12u
5
− u
4
+ 12u
3
− 2u
2
− 14u − 5i
I
u
2
= h−u
3
b − 2u
2
b + b
2
+ bu + 2u
2
− u − 2, u
2
+ a, u
4
− u
2
+ 1i
I
u
3
= hb + u, a + u − 1, u
2
− u + 1i
* 3 irreducible components of dim
C
= 0, with total 18 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
7
−2u
6
+· · ·+2b+8, 7u
7
+9u
6
+· · ·+20a−48, u
8
+2u
7
+· · ·−14u−5i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−u
2
a
6
=
−0.350000u
7
− 0.450000u
6
+ ··· + 2.95000u + 2.40000
1
2
u
7
+ u
6
− 3u
5
− 6u
4
+ 4u
3
+ 6u
2
−
13
2
u − 4
a
7
=
3
20
u
7
+
11
20
u
6
+ ··· −
71
20
u −
8
5
1
2
u
7
+ u
6
− 3u
5
− 6u
4
+ 4u
3
+ 6u
2
−
13
2
u − 4
a
10
=
−
9
20
u
7
−
3
20
u
6
+ ··· +
53
20
u +
4
5
2u
7
+
5
4
u
6
+ ··· −
57
4
u −
27
4
a
11
=
11
20
u
7
+
7
20
u
6
+ ··· −
67
20
u −
11
5
−
1
4
u
6
+
1
4
u
5
+ ··· +
7
4
u +
3
4
a
2
=
−u
2
+ 1
u
4
a
1
=
u
4
− u
2
+ 1
u
4
a
5
=
u
4
− u
2
+ 1
−u
6
− u
2
a
8
=
−0.450000u
7
− 0.650000u
6
+ ··· + 4.65000u + 2.80000
−
1
2
u
7
−
1
2
u
6
+ ··· + 5u + 2
a
12
=
13
20
u
7
+
1
20
u
6
+ ··· −
81
20
u −
3
5
1
2
u
7
−
1
4
u
6
+ ··· −
9
4
u −
1
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
3
2
u
7
− 2u
6
+
15
2
u
5
+
27
2
u
4
−
15
2
u
3
−
33
2
u
2
+ 11u +
49
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
8
− 20u
7
+ ··· − 13476u + 625
c
2
, c
4
u
8
− 12u
7
+ 62u
6
− 188u
5
+ 351u
4
− 436u
3
+ 350u
2
− 176u + 25
c
3
u
8
− 2u
7
− 4u
6
+ 12u
5
− u
4
− 12u
3
− 2u
2
+ 14u − 5
c
5
, c
9
u
8
+ 2u
7
− 3u
6
+ 8u
5
+ 19u
4
+ 26u
3
+ 11u
2
+ 4u − 4
c
6
u
8
− 12u
7
+ 21u
6
+ 102u
5
− 554u
4
− 2292u
3
+ 749u
2
− 502u + 179
c
7
, c
8
, c
11
c
12
u
8
+ 2u
7
+ 6u
6
+ 8u
5
+ 5u
4
+ 8u
3
− 12u
2
+ 6u − 1
c
10
u
8
− 2u
7
− 3u
6
− 92u
5
− 70u
4
− 350u
3
− 705u
2
− 144u + 193
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
8
− 332y
7
+ ··· − 198380076y + 390625
c
2
, c
4
y
8
− 20y
7
+ ··· − 13476y + 625
c
3
y
8
− 12y
7
+ 62y
6
− 188y
5
+ 351y
4
− 436y
3
+ 350y
2
− 176y + 25
c
5
, c
9
y
8
− 10y
7
+ 15y
6
− 260y
5
− 145y
4
− 298y
3
− 239y
2
− 104y + 16
c
6
y
8
− 102y
7
+ ··· + 16138y + 32041
c
7
, c
8
, c
11
c
12
y
8
+ 8y
7
+ 14y
6
− 60y
5
− 273y
4
− 292y
3
+ 38y
2
− 12y + 1
c
10
y
8
− 10y
7
+ ··· − 292866y + 37249
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.872971 + 0.618128I
a = −0.215446 − 0.731489I
b = −0.005364 + 0.460904I
−1.46286 + 2.16790I 9.57172 − 4.32976I
u = 0.872971 − 0.618128I
a = −0.215446 + 0.731489I
b = −0.005364 − 0.460904I
−1.46286 − 2.16790I 9.57172 + 4.32976I
u = −1.162380 + 0.411109I
a = −0.420013 − 0.734093I
b = 2.69985 + 1.42341I
−8.79021 − 1.33537I 7.32369 + 0.78408I
u = −1.162380 − 0.411109I
a = −0.420013 + 0.734093I
b = 2.69985 − 1.42341I
−8.79021 + 1.33537I 7.32369 − 0.78408I
u = −0.458955
a = 0.746207
b = −0.337573
0.592549 17.1350
u = −1.56303 + 0.67202I
a = 1.198220 − 0.500803I
b = −3.72972 + 3.74449I
4.57005 − 8.46981I 6.36910 + 3.46503I
u = −1.56303 − 0.67202I
a = 1.198220 + 0.500803I
b = −3.72972 − 3.74449I
4.57005 + 8.46981I 6.36910 − 3.46503I
u = 2.16384
a = 1.52826
b = −9.59194
10.7735 8.33600
5
II. I
u
2
= h−u
3
b − 2u
2
b + b
2
+ bu + 2u
2
− u − 2, u
2
+ a, u
4
− u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−u
2
a
6
=
−u
2
b
a
7
=
−u
2
+ b
b
a
10
=
u
3
− u
−u
3
b + u
a
11
=
−u
3
b + u
3
− u
−bu + u
a
2
=
−u
2
+ 1
u
2
− 1
a
1
=
0
u
2
− 1
a
5
=
0
−u
2
+ 1
a
8
=
−u
3
+ u
2
− b + u
u
3
b + u
2
b − 2u
3
− b + 2u + 1
a
12
=
u
3
b − u
3
− u
2
+ u
u
3
b − u
3
− u
2
+ b + u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
+ 4
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
2
− u + 1)
4
c
2
(u
2
+ u + 1)
4
c
3
(u
4
− u
2
+ 1)
2
c
5
, c
9
(u
2
+ 1)
4
c
6
u
8
+ 4u
7
+ 7u
6
+ 16u
5
+ 36u
4
+ 50u
3
+ 55u
2
+ 50u + 25
c
7
, c
8
, c
11
c
12
(u
4
+ 3u
2
+ 1)
2
c
10
u
8
+ 2u
7
+ 3u
6
+ 2u
5
− 4u
4
− 20u
3
− 5u
2
+ 25
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y
2
+ y + 1)
4
c
3
(y
2
− y + 1)
4
c
5
, c
9
(y + 1)
8
c
6
y
8
− 2y
7
− 7y
6
− 42y
5
+ 116y
4
+ 210y
3
− 175y
2
+ 250y + 625
c
7
, c
8
, c
11
c
12
(y
2
+ 3y + 1)
4
c
10
y
8
+ 2y
7
− 7y
6
+ 42y
5
+ 116y
4
− 210y
3
− 175y
2
− 250y + 625
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.866025 + 0.500000I
a = −0.500000 − 0.866025I
b = 1.035230 + 0.557008I
−2.63189 + 2.02988I 2.00000 − 3.46410I
u = 0.866025 + 0.500000I
a = −0.500000 − 0.866025I
b = −0.90126 + 1.67504I
−10.52760 + 2.02988I 2.00000 − 3.46410I
u = 0.866025 − 0.500000I
a = −0.500000 + 0.866025I
b = 1.035230 − 0.557008I
−2.63189 − 2.02988I 2.00000 + 3.46410I
u = 0.866025 − 0.500000I
a = −0.500000 + 0.866025I
b = −0.90126 − 1.67504I
−10.52760 − 2.02988I 2.00000 + 3.46410I
u = −0.866025 + 0.500000I
a = −0.500000 + 0.866025I
b = −0.035233 − 1.175040I
−2.63189 − 2.02988I 2.00000 + 3.46410I
u = −0.866025 + 0.500000I
a = −0.500000 + 0.866025I
b = 1.90126 − 0.05701I
−10.52760 − 2.02988I 2.00000 + 3.46410I
u = −0.866025 − 0.500000I
a = −0.500000 − 0.866025I
b = −0.035233 + 1.175040I
−2.63189 + 2.02988I 2.00000 − 3.46410I
u = −0.866025 − 0.500000I
a = −0.500000 − 0.866025I
b = 1.90126 + 0.05701I
−10.52760 + 2.02988I 2.00000 − 3.46410I
9
III. I
u
3
= hb + u, a + u − 1, u
2
− u + 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−u + 1
a
6
=
−u + 1
−u
a
7
=
−2u + 1
−u
a
10
=
−u + 1
0
a
11
=
1
1
a
2
=
−u + 2
−u
a
1
=
−2u + 2
−u
a
5
=
−2u + 2
−u
a
8
=
−u
0
a
12
=
−u
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u + 8
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
10
u
2
+ u + 1
c
5
, c
9
(u + 1)
2
c
6
, c
7
, c
8
c
11
, c
12
u
2
− u + 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
10
, c
11
c
12
y
2
+ y + 1
c
5
, c
9
(y −1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 0.500000 − 0.866025I
b = −0.500000 − 0.866025I
−1.64493 + 2.02988I 6.00000 − 3.46410I
u = 0.500000 − 0.866025I
a = 0.500000 + 0.866025I
b = −0.500000 + 0.866025I
−1.64493 − 2.02988I 6.00000 + 3.46410I
13
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
4
)(u
2
+ u + 1)(u
8
− 20u
7
+ ··· − 13476u + 625)
c
2
(u
2
+ u + 1)
5
· (u
8
− 12u
7
+ 62u
6
− 188u
5
+ 351u
4
− 436u
3
+ 350u
2
− 176u + 25)
c
3
(u
2
+ u + 1)(u
4
− u
2
+ 1)
2
· (u
8
− 2u
7
− 4u
6
+ 12u
5
− u
4
− 12u
3
− 2u
2
+ 14u − 5)
c
4
(u
2
− u + 1)
4
(u
2
+ u + 1)
· (u
8
− 12u
7
+ 62u
6
− 188u
5
+ 351u
4
− 436u
3
+ 350u
2
− 176u + 25)
c
5
, c
9
(u + 1)
2
(u
2
+ 1)
4
· (u
8
+ 2u
7
− 3u
6
+ 8u
5
+ 19u
4
+ 26u
3
+ 11u
2
+ 4u − 4)
c
6
(u
2
− u + 1)
· (u
8
− 12u
7
+ 21u
6
+ 102u
5
− 554u
4
− 2292u
3
+ 749u
2
− 502u + 179)
· (u
8
+ 4u
7
+ 7u
6
+ 16u
5
+ 36u
4
+ 50u
3
+ 55u
2
+ 50u + 25)
c
7
, c
8
, c
11
c
12
(u
2
− u + 1)(u
4
+ 3u
2
+ 1)
2
· (u
8
+ 2u
7
+ 6u
6
+ 8u
5
+ 5u
4
+ 8u
3
− 12u
2
+ 6u − 1)
c
10
(u
2
+ u + 1)
· (u
8
− 2u
7
− 3u
6
− 92u
5
− 70u
4
− 350u
3
− 705u
2
− 144u + 193)
· (u
8
+ 2u
7
+ 3u
6
+ 2u
5
− 4u
4
− 20u
3
− 5u
2
+ 25)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
5
)(y
8
− 332y
7
+ ··· − 1.98380 × 10
8
y + 390625)
c
2
, c
4
((y
2
+ y + 1)
5
)(y
8
− 20y
7
+ ··· − 13476y + 625)
c
3
(y
2
− y + 1)
4
(y
2
+ y + 1)
· (y
8
− 12y
7
+ 62y
6
− 188y
5
+ 351y
4
− 436y
3
+ 350y
2
− 176y + 25)
c
5
, c
9
(y −1)
2
(y + 1)
8
· (y
8
− 10y
7
+ 15y
6
− 260y
5
− 145y
4
− 298y
3
− 239y
2
− 104y + 16)
c
6
(y
2
+ y + 1)(y
8
− 102y
7
+ ··· + 16138y + 32041)
· (y
8
− 2y
7
− 7y
6
− 42y
5
+ 116y
4
+ 210y
3
− 175y
2
+ 250y + 625)
c
7
, c
8
, c
11
c
12
(y
2
+ y + 1)(y
2
+ 3y + 1)
4
· (y
8
+ 8y
7
+ 14y
6
− 60y
5
− 273y
4
− 292y
3
+ 38y
2
− 12y + 1)
c
10
(y
2
+ y + 1)(y
8
− 10y
7
+ ··· − 292866y + 37249)
· (y
8
+ 2y
7
− 7y
6
+ 42y
5
+ 116y
4
− 210y
3
− 175y
2
− 250y + 625)
15
