12n
0440
(K12n
0440
)
A knot diagram
1
Linearized knot diagam
3 6 12 7 2 10 11 12 3 5 8 10
Solving Sequence
2,6 3,10
7 1 5 11 4 9 12 8
c
2
c
6
c
1
c
5
c
10
c
4
c
9
c
12
c
8
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−238u
25
− 929u
24
+ ··· + 288b − 10325, −1220u
25
− 4477u
24
+ ··· + 2976a − 43144,
u
26
+ 4u
25
+ ··· + 112u + 31i
I
u
2
= h−3u
7
+ u
6
+ 2u
5
− 2u
4
− 7u
3
+ 3u
2
+ 2b + 3u − 3, −3u
7
+ 3u
6
+ 2u
5
− 2u
4
− 7u
3
+ 7u
2
+ 2a + 3u − 3,
u
8
− u
6
+ 3u
4
− 2u
2
+ 1i
I
u
3
= hu
11
+ u
10
− 3u
9
− u
8
+ 3u
7
− u
6
+ 4u
5
+ 4u
4
− 5u
3
+ u
2
+ 4b − 4u,
u
10
+ u
9
− 3u
8
− 3u
7
+ 5u
6
+ 3u
5
− 2u
4
+ 4u
3
− u
2
+ 4a − 9u + 4,
u
12
− 4u
10
+ u
9
+ 6u
8
− 3u
7
+ u
6
+ 3u
5
− 9u
4
− 2u
3
+ 5u
2
+ u + 2i
I
u
4
= h−a
2
+ b − 2a, a
3
+ 2a
2
+ a − 1, u − 1i
I
u
5
= hb
2
a − 2a
2
b + a
3
− b + 2a − 1, u − 1i
I
v
1
= ha, b
3
+ b + 1, v − 1i
* 5 irreducible components of dim
C
= 0, with total 52 representations.
* 1 irreducible components of dim
C
= 1
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−238u
25
− 929u
24
+ · · · + 288b − 10325, −1220u
25
− 4477u
24
+ · · · +
2976a − 43144, u
26
+ 4u
25
+ · · · + 112u + 31i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
10
=
0.409946u
25
+ 1.50437u
24
+ ··· + 39.5783u + 14.4973
0.826389u
25
+ 3.22569u
24
+ ··· + 93.5139u + 35.8507
a
7
=
0.353831u
25
+ 1.25907u
24
+ ··· + 37.8982u + 14.2540
0.291667u
25
+ 1.11458u
24
+ ··· + 35.6354u + 14.5208
a
1
=
−u
2
+ 1
−u
4
a
5
=
u
u
a
11
=
1.13564u
25
+ 2.78909u
24
+ ··· + 58.7102u + 16.2195
1.55208u
25
+ 4.51042u
24
+ ··· + 112.646u + 37.5729
a
4
=
0.663306u
25
+ 1.65323u
24
+ ··· + 36.2944u + 11.7278
0.937500u
25
+ 2.37500u
24
+ ··· + 54.1875u + 16.6875
a
9
=
−0.833109u
25
− 2.14841u
24
+ ··· − 51.4773u − 17.1555
−0.291667u
25
− 0.739583u
24
+ ··· − 15.7292u − 5.05208
a
12
=
0.0413306u
25
+ 0.290323u
24
+ ··· + 12.3044u + 6.87903
−0.270833u
25
− 0.572917u
24
+ ··· − 9.11458u − 0.979167
a
8
=
−0.573029u
25
− 1.26434u
24
+ ··· − 24.9541u − 7.52643
−
1
2
u
25
−
11
8
u
24
+ ··· −
65
2
u −
87
8
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
323
108
u
25
−
262
27
u
24
+ ··· −
7238
27
u −
2566
27
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
26
+ 14u
25
+ ··· + 1446u + 961
c
2
, c
5
u
26
− 4u
25
+ ··· − 112u + 31
c
3
2(2u
26
− 10u
25
+ ··· + 864u + 183)
c
4
2(2u
26
+ 6u
25
+ ··· + 6u + 93)
c
6
2(2u
26
+ 6u
25
+ ··· + 828u + 216)
c
7
, c
8
, c
11
u
26
+ 4u
25
+ ··· + 104u + 31
c
9
3(3u
26
+ 6u
25
+ ··· + 28u + 8)
c
10
3(3u
26
+ 6u
25
+ ··· − 28u + 8)
c
12
2(2u
26
+ 2u
25
+ ··· + 552u + 264)
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
26
− 2y
25
+ ··· − 2335010y + 923521
c
2
, c
5
y
26
− 14y
25
+ ··· − 1446y + 961
c
3
4(4y
26
− 152y
25
+ ··· + 146544y + 33489)
c
4
4(4y
26
+ 56y
25
+ ··· + 74364y + 8649)
c
6
4(4y
26
− 88y
25
+ ··· − 322704y + 46656)
c
7
, c
8
, c
11
y
26
− 22y
25
+ ··· + 282y + 961
c
9
9(9y
26
+ 276y
25
+ ··· + 4752y + 64)
c
10
9(9y
26
+ 60y
25
+ ··· + 400y + 64)
c
12
4(4y
26
+ 184y
25
+ ··· + 708000y + 69696)
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.207245 + 0.967760I
a = −0.149721 + 1.284220I
b = −0.516236 + 0.281803I
−3.60172 − 2.61761I −0.26262 + 1.62312I
u = 0.207245 − 0.967760I
a = −0.149721 − 1.284220I
b = −0.516236 − 0.281803I
−3.60172 + 2.61761I −0.26262 − 1.62312I
u = 0.820953 + 0.396871I
a = −0.464795 + 0.712990I
b = −0.578506 + 0.981502I
−1.82983 − 1.27994I −6.43006 + 3.42409I
u = 0.820953 − 0.396871I
a = −0.464795 − 0.712990I
b = −0.578506 − 0.981502I
−1.82983 + 1.27994I −6.43006 − 3.42409I
u = 0.093519 + 1.101170I
a = 0.479969 + 1.064420I
b = −0.492737 + 0.320156I
−1.83826 + 8.69089I 1.85327 − 5.19152I
u = 0.093519 − 1.101170I
a = 0.479969 − 1.064420I
b = −0.492737 − 0.320156I
−1.83826 − 8.69089I 1.85327 + 5.19152I
u = −0.875136 + 0.787433I
a = 0.012821 + 0.379761I
b = 0.682772 − 0.320729I
8.20037 + 2.96512I 10.04842 − 2.30654I
u = −0.875136 − 0.787433I
a = 0.012821 − 0.379761I
b = 0.682772 + 0.320729I
8.20037 − 2.96512I 10.04842 + 2.30654I
u = −1.161750 + 0.364732I
a = 1.216800 − 0.327659I
b = 2.00120 − 0.76731I
−2.84479 + 4.92637I −3.47765 − 6.11098I
u = −1.161750 − 0.364732I
a = 1.216800 + 0.327659I
b = 2.00120 + 0.76731I
−2.84479 − 4.92637I −3.47765 + 6.11098I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.641989 + 0.388321I
a = −0.17877 − 1.49759I
b = −0.812488 − 0.481122I
1.35413 + 1.46610I 7.33690 − 4.93932I
u = −0.641989 − 0.388321I
a = −0.17877 + 1.49759I
b = −0.812488 + 0.481122I
1.35413 − 1.46610I 7.33690 + 4.93932I
u = −0.128992 + 0.733180I
a = 1.24585 − 1.03021I
b = −0.0120395 − 0.0178498I
4.13697 − 4.56838I 6.59727 + 4.70609I
u = −0.128992 − 0.733180I
a = 1.24585 + 1.03021I
b = −0.0120395 + 0.0178498I
4.13697 + 4.56838I 6.59727 − 4.70609I
u = −1.183540 + 0.424868I
a = −1.024870 + 0.400952I
b = −2.00747 + 0.99437I
0.97162 + 8.87300I 1.52140 − 7.94866I
u = −1.183540 − 0.424868I
a = −1.024870 − 0.400952I
b = −2.00747 − 0.99437I
0.97162 − 8.87300I 1.52140 + 7.94866I
u = 0.940665 + 0.925328I
a = 0.514581 − 0.495678I
b = 0.869814 − 0.281626I
5.50559 − 3.38335I −3.13078 + 3.33250I
u = 0.940665 − 0.925328I
a = 0.514581 + 0.495678I
b = 0.869814 + 0.281626I
5.50559 + 3.38335I −3.13078 − 3.33250I
u = −1.342860 + 0.390773I
a = 0.978142 − 0.178240I
b = 1.78351 + 0.39115I
−8.47827 + 7.26693I −2.82019 − 4.69299I
u = −1.342860 − 0.390773I
a = 0.978142 + 0.178240I
b = 1.78351 − 0.39115I
−8.47827 − 7.26693I −2.82019 + 4.69299I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.31172 + 0.58643I
a = 1.376260 − 0.051091I
b = 2.25303 + 0.42993I
−10.39120 − 9.04784I −3.60025 + 5.37589I
u = 1.31172 − 0.58643I
a = 1.376260 + 0.051091I
b = 2.25303 − 0.42993I
−10.39120 + 9.04784I −3.60025 − 5.37589I
u = −1.37849 + 0.40779I
a = −0.815957 + 0.087138I
b = −1.41656 − 0.61555I
−11.75740 + 1.91795I −5.35471 − 0.48729I
u = −1.37849 − 0.40779I
a = −0.815957 − 0.087138I
b = −1.41656 + 0.61555I
−11.75740 − 1.91795I −5.35471 + 0.48729I
u = 1.33866 + 0.56921I
a = −1.399980 − 0.153122I
b = −2.25429 − 0.76615I
−5.7462 − 14.6212I −0.28099 + 7.59450I
u = 1.33866 − 0.56921I
a = −1.399980 + 0.153122I
b = −2.25429 + 0.76615I
−5.7462 + 14.6212I −0.28099 − 7.59450I
7
II.
I
u
2
= h−3u
7
+u
6
+· · ·+2b−3, −3u
7
+3u
6
+· · ·+2a−3, u
8
−u
6
+3u
4
−2u
2
+1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
10
=
3
2
u
7
−
3
2
u
6
+ ··· −
3
2
u +
3
2
3
2
u
7
−
1
2
u
6
+ ··· −
3
2
u +
3
2
a
7
=
2u
7
−
5
2
u
6
+ u
4
+ 5u
3
−
13
2
u
2
+ u +
3
2
5
2
u
7
− 2u
6
+ ··· −
1
2
u + 2
a
1
=
−u
2
+ 1
−u
4
a
5
=
u
u
a
11
=
3
2
u
7
−
5
2
u
6
+ ··· −
3
2
u +
5
2
3
2
u
7
−
3
2
u
6
+ ··· −
3
2
u +
5
2
a
4
=
−
21
4
u
7
+
15
4
u
6
+ ··· +
27
4
u −
25
4
−
17
4
u
7
+
9
4
u
6
+ ··· +
27
4
u −
23
4
a
9
=
1
2
u
7
−
3
2
u
6
+ ··· −
3
2
u +
3
2
1
2
u
7
−
1
2
u
6
+ ··· −
1
2
u +
3
2
a
12
=
3
2
u
7
− 2u
6
+ u
4
+
7
2
u
3
− 6u
2
+
1
2
u + 2
2u
7
−
3
2
u
6
− u
5
+ 5u
3
−
7
2
u
2
− u +
3
2
a
8
=
3
2
u
7
− 2u
6
+ u
4
+
5
2
u
3
− 5u
2
+
1
2
u + 1
2u
7
−
3
2
u
6
− u
5
+ u
4
+ 4u
3
−
7
2
u
2
+
3
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
6
− 4u
4
+ 12u
2
− 4
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
2
, c
5
u
8
− u
6
+ 3u
4
− 2u
2
+ 1
c
3
2(2u
8
+ 10u
7
+ 29u
6
+ 48u
5
+ 58u
4
+ 50u
3
+ 27u
2
+ 8u + 1)
c
4
2(2u
8
+ 6u
7
+ 17u
6
+ 24u
5
+ 30u
4
+ 16u
3
+ 11u
2
+ 2u + 1)
c
6
2(2u
8
− 2u
7
− 3u
6
+ 12u
5
+ 14u
4
+ 8u
3
+ 7u
2
+ 2u + 1)
c
7
, c
8
, c
11
u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1
c
9
, c
10
(u
2
+ 1)
4
c
12
2(2u
8
− 2u
7
− 7u
6
+ 16u
5
+ 2u
4
− 26u
3
+ 23u
2
− 8u + 1)
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
c
2
, c
5
(y
4
− y
3
+ 3y
2
− 2y + 1)
2
c
3
4(4y
8
+ 16y
7
+ 113y
6
+ 168y
5
− 26y
4
− 78y
3
+ 45y
2
− 10y + 1)
c
4
4(4y
8
+ 32y
7
+ ··· + 18y + 1)
c
6
4(4y
8
− 16y
7
+ 113y
6
− 168y
5
− 26y
4
+ 78y
3
+ 45y
2
+ 10y + 1)
c
7
, c
8
, c
11
(y
4
− 5y
3
+ 7y
2
− 2y + 1)
2
c
9
, c
10
(y + 1)
8
c
12
4(4y
8
− 32y
7
+ ··· − 18y + 1)
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.720342 + 0.351808I
a = −0.433324 − 0.562927I
b = 0.114099 + 0.557947I
−0.21101 − 1.41510I 0.17326 + 4.90874I
u = 0.720342 − 0.351808I
a = −0.433324 + 0.562927I
b = 0.114099 − 0.557947I
−0.21101 + 1.41510I 0.17326 − 4.90874I
u = −0.720342 + 0.351808I
a = 1.19439 + 2.50547I
b = 1.74182 + 1.38460I
−0.21101 + 1.41510I 0.17326 − 4.90874I
u = −0.720342 − 0.351808I
a = 1.19439 − 2.50547I
b = 1.74182 − 1.38460I
−0.21101 − 1.41510I 0.17326 + 4.90874I
u = 0.911292 + 0.851808I
a = 0.352886 + 0.149146I
b = −0.194538 − 0.436506I
6.79074 − 3.16396I 3.82674 + 2.56480I
u = 0.911292 − 0.851808I
a = 0.352886 − 0.149146I
b = −0.194538 + 0.436506I
6.79074 + 3.16396I 3.82674 − 2.56480I
u = −0.911292 + 0.851808I
a = −0.613954 − 0.706599I
b = −1.161380 − 0.120947I
6.79074 + 3.16396I 3.82674 − 2.56480I
u = −0.911292 − 0.851808I
a = −0.613954 + 0.706599I
b = −1.161380 + 0.120947I
6.79074 − 3.16396I 3.82674 + 2.56480I
11
III.
I
u
3
= hu
11
+ u
10
+ · · · +4b − 4u, u
10
+ u
9
+ · · · +4a + 4, u
12
− 4u
10
+ · · · +u + 2i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
10
=
−
1
4
u
10
−
1
4
u
9
+ ··· +
9
4
u − 1
−
1
4
u
11
−
1
4
u
10
+ ··· −
1
4
u
2
+ u
a
7
=
−
1
4
u
10
+
5
4
u
8
+ ··· −
1
2
u −
1
2
−
1
4
u
9
+
1
2
u
8
+ ··· +
5
4
u +
1
2
a
1
=
−u
2
+ 1
−u
4
a
5
=
u
u
a
11
=
−
1
4
u
9
+
3
4
u
7
+ ··· +
7
4
u − 1
−
1
4
u
11
+
3
4
u
9
+ ··· −
3
2
u
2
+
1
2
u
a
4
=
1
2
u
11
−
1
4
u
10
+ ··· −
1
2
u +
3
2
1
2
u
11
− u
9
+ ··· −
1
4
u +
1
2
a
9
=
−
1
4
u
10
+
3
4
u
8
+ ··· +
3
2
u −
1
2
−
1
4
u
10
+
3
4
u
8
+ ··· +
1
4
u
2
+ u
a
12
=
−
1
4
u
9
+
1
4
u
7
+ ··· −
5
4
u + 1
−
1
2
u
9
+ u
7
+ ··· +
1
2
u
2
−
1
2
u
a
8
=
1
2
u
8
− u
6
+ ··· +
1
2
u −
1
2
1
4
u
10
−
1
4
u
8
+ ··· +
3
4
u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
9
+ 6u
7
− 2u
6
− 6u
5
+ 4u
4
− 6u
3
− 2u
2
+ 8u + 2
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
12
+ 8u
11
+ ··· − 19u + 4
c
2
, c
5
, c
7
c
8
, c
11
u
12
− 4u
10
− u
9
+ 6u
8
+ 3u
7
+ u
6
− 3u
5
− 9u
4
+ 2u
3
+ 5u
2
− u + 2
c
3
2(2u
12
+ 2u
11
+ ··· − 56u + 8)
c
4
2(2u
12
+ 10u
11
+ ··· + 12u + 8)
c
6
2(2u
12
− 2u
11
+ ··· + 66u + 47)
c
9
(u
4
− u
3
+ 3u
2
− 2u + 1)
3
c
10
(u
4
− u
3
+ u
2
+ 1)
3
c
12
2(2u
12
− 6u
11
+ ··· − 150u + 103)
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
12
− 8y
11
+ ··· − 417y + 16
c
2
, c
5
, c
7
c
8
, c
11
y
12
− 8y
11
+ ··· + 19y + 4
c
3
4(4y
12
− 88y
11
+ ··· + 2080y + 64)
c
4
4(4y
12
+ 16y
11
+ ··· + 688y + 64)
c
6
4(4y
12
− 56y
11
+ ··· − 4168y + 2209)
c
9
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
3
c
10
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
3
c
12
4(4y
12
+ 80y
11
+ ··· + 116344y + 10609)
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.152073 + 1.050790I
a = −0.216032 − 1.219810I
b = 0.518508 − 0.302871I
−6.79074 + 3.16396I −1.82674 − 2.56480I
u = 0.152073 − 1.050790I
a = −0.216032 + 1.219810I
b = 0.518508 + 0.302871I
−6.79074 − 3.16396I −1.82674 + 2.56480I
u = −1.043920 + 0.280279I
a = −1.82414 − 0.12229I
b = −2.31130 + 0.43695I
0.21101 + 1.41510I 1.82674 − 4.90874I
u = −1.043920 − 0.280279I
a = −1.82414 + 0.12229I
b = −2.31130 − 0.43695I
0.21101 − 1.41510I 1.82674 + 4.90874I
u = 1.131290 + 0.219998I
a = 0.169450 − 1.077300I
b = 0.34978 − 1.74911I
0.21101 + 1.41510I 1.82674 − 4.90874I
u = 1.131290 − 0.219998I
a = 0.169450 + 1.077300I
b = 0.34978 + 1.74911I
0.21101 − 1.41510I 1.82674 + 4.90874I
u = 1.272550 + 0.614267I
a = −1.238380 + 0.267755I
b = −2.06161 − 0.05098I
−6.79074 − 3.16396I −1.82674 + 2.56480I
u = 1.272550 − 0.614267I
a = −1.238380 − 0.267755I
b = −2.06161 + 0.05098I
−6.79074 + 3.16396I −1.82674 − 2.56480I
u = −1.42462 + 0.43653I
a = 0.618232 − 0.037304I
b = 0.985647 + 0.714952I
−6.79074 − 3.16396I −1.82674 + 2.56480I
u = −1.42462 − 0.43653I
a = 0.618232 + 0.037304I
b = 0.985647 − 0.714952I
−6.79074 + 3.16396I −1.82674 − 2.56480I
15
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.087368 + 0.500278I
a = −1.25913 + 1.24198I
b = 0.018967 + 0.315552I
0.21101 − 1.41510I 1.82674 + 4.90874I
u = −0.087368 − 0.500278I
a = −1.25913 − 1.24198I
b = 0.018967 − 0.315552I
0.21101 + 1.41510I 1.82674 − 4.90874I
16
IV. I
u
4
= h−a
2
+ b − 2a, a
3
+ 2a
2
+ a − 1, u − 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
1
a
3
=
1
1
a
10
=
a
a
2
+ 2a
a
7
=
−a
2
a
a
1
=
0
−1
a
5
=
1
1
a
11
=
−a
2
a
a
4
=
−a
2
− 2a + 2
−a
2
− a + 2
a
9
=
−a
2
a
a
12
=
a
2
−a
a
8
=
−a
2
a
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
(u + 1)
3
c
3
, c
4
, c
9
c
10
u
3
+ u + 1
c
6
u
3
+ 2u
2
+ u − 1
c
7
, c
8
, c
11
u
3
c
12
u
3
− 2u
2
+ u + 1
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y − 1)
3
c
3
, c
4
, c
9
c
10
y
3
+ 2y
2
+ y − 1
c
6
, c
12
y
3
− 2y
2
+ 5y − 1
c
7
, c
8
, c
11
y
3
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.23279 + 0.79255I
b = −1.57395 − 0.36899I
−1.64493 −6.00000
u = 1.00000
a = −1.23279 − 0.79255I
b = −1.57395 + 0.36899I
−1.64493 −6.00000
u = 1.00000
a = 0.465571
b = 1.14790
−1.64493 −6.00000
20
V. I
u
5
= hb
2
a − 2a
2
b + a
3
− b + 2a − 1, u − 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
1
a
3
=
1
1
a
10
=
a
b
a
7
=
−a
2
−ba + 1
a
1
=
0
−1
a
5
=
1
1
a
11
=
−b + 2a
a
a
4
=
a
3
b − a
4
− a
2
+ 1
a
3
b − a
4
− ba − a
2
+ a + 2
a
9
=
−b + 2a
a
a
12
=
a
2
ba − 1
a
8
=
−a
2
− b + 2a
−ba + a + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
(iv) u-Polynomials at the component : It cannot be defined for a positive
dimension component.
(v) Riley Polynomials at the component : It cannot be defined for a positive
dimension component.
21
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
0 0
22
VI. I
v
1
= ha, b
3
+ b + 1, v − 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
1
0
a
3
=
1
0
a
10
=
0
b
a
7
=
1
b
2
a
1
=
1
0
a
5
=
1
0
a
11
=
−b
b
a
4
=
b
2
+ 1
−b
2
− b
a
9
=
−b
b
a
12
=
1
b
2
a
8
=
−b + 1
b
2
+ b
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
23
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
u
3
c
3
u
3
+ 2u
2
+ u − 1
c
4
u
3
− 2u
2
+ u + 1
c
6
, c
9
, c
10
c
12
u
3
+ u − 1
c
7
, c
8
, c
11
(u − 1)
3
24
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
y
3
c
3
, c
4
y
3
− 2y
2
+ 5y − 1
c
6
, c
9
, c
10
c
12
y
3
+ 2y
2
+ y − 1
c
7
, c
8
, c
11
(y − 1)
3
25
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = 0.341164 + 1.161540I
1.64493 6.00000
v = 1.00000
a = 0
b = 0.341164 − 1.161540I
1.64493 6.00000
v = 1.00000
a = 0
b = −0.682328
1.64493 6.00000
26
VII. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
3
(u + 1)
3
(u
4
− u
3
+ ··· − 2u + 1)
2
(u
12
+ 8u
11
+ ··· − 19u + 4)
· (u
26
+ 14u
25
+ ··· + 1446u + 961)
c
2
, c
5
u
3
(u + 1)
3
(u
8
− u
6
+ 3u
4
− 2u
2
+ 1)
· (u
12
− 4u
10
− u
9
+ 6u
8
+ 3u
7
+ u
6
− 3u
5
− 9u
4
+ 2u
3
+ 5u
2
− u + 2)
· (u
26
− 4u
25
+ ··· − 112u + 31)
c
3
8(u
3
+ u + 1)(u
3
+ 2u
2
+ u − 1)
· (2u
8
+ 10u
7
+ 29u
6
+ 48u
5
+ 58u
4
+ 50u
3
+ 27u
2
+ 8u + 1)
· (2u
12
+ 2u
11
+ ··· − 56u + 8)(2u
26
− 10u
25
+ ··· + 864u + 183)
c
4
8(u
3
+ u + 1)(u
3
− 2u
2
+ u + 1)
· (2u
8
+ 6u
7
+ 17u
6
+ 24u
5
+ 30u
4
+ 16u
3
+ 11u
2
+ 2u + 1)
· (2u
12
+ 10u
11
+ ··· + 12u + 8)(2u
26
+ 6u
25
+ ··· + 6u + 93)
c
6
8(u
3
+ u − 1)(u
3
+ 2u
2
+ u − 1)
· (2u
8
− 2u
7
− 3u
6
+ 12u
5
+ 14u
4
+ 8u
3
+ 7u
2
+ 2u + 1)
· (2u
12
− 2u
11
+ ··· + 66u + 47)(2u
26
+ 6u
25
+ ··· + 828u + 216)
c
7
, c
8
, c
11
u
3
(u − 1)
3
(u
8
− 5u
6
+ 7u
4
− 2u
2
+ 1)
· (u
12
− 4u
10
− u
9
+ 6u
8
+ 3u
7
+ u
6
− 3u
5
− 9u
4
+ 2u
3
+ 5u
2
− u + 2)
· (u
26
+ 4u
25
+ ··· + 104u + 31)
c
9
3(u
2
+ 1)
4
(u
3
+ u − 1)(u
3
+ u + 1)(u
4
− u
3
+ 3u
2
− 2u + 1)
3
· (3u
26
+ 6u
25
+ ··· + 28u + 8)
c
10
3(u
2
+ 1)
4
(u
3
+ u − 1)(u
3
+ u + 1)(u
4
− u
3
+ u
2
+ 1)
3
· (3u
26
+ 6u
25
+ ··· − 28u + 8)
c
12
8(u
3
+ u − 1)(u
3
− 2u
2
+ u + 1)
· (2u
8
− 2u
7
− 7u
6
+ 16u
5
+ 2u
4
− 26u
3
+ 23u
2
− 8u + 1)
· (2u
12
− 6u
11
+ ··· − 150u + 103)(2u
26
+ 2u
25
+ ··· + 552u + 264)
27
VIII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
3
(y − 1)
3
(y
4
+ 5y
3
+ ··· + 2y + 1)
2
(y
12
− 8y
11
+ ··· − 417y + 16)
· (y
26
− 2y
25
+ ··· − 2335010y + 923521)
c
2
, c
5
y
3
(y − 1)
3
(y
4
− y
3
+ ··· − 2y + 1)
2
(y
12
− 8y
11
+ ··· + 19y + 4)
· (y
26
− 14y
25
+ ··· − 1446y + 961)
c
3
64(y
3
− 2y
2
+ 5y − 1)(y
3
+ 2y
2
+ y − 1)
· (4y
8
+ 16y
7
+ 113y
6
+ 168y
5
− 26y
4
− 78y
3
+ 45y
2
− 10y + 1)
· (4y
12
− 88y
11
+ ··· + 2080y + 64)
· (4y
26
− 152y
25
+ ··· + 146544y + 33489)
c
4
64(y
3
− 2y
2
+ 5y − 1)(y
3
+ 2y
2
+ y − 1)
· (4y
8
+ 32y
7
+ 121y
6
+ 296y
5
+ 486y
4
+ 342y
3
+ 117y
2
+ 18y + 1)
· (4y
12
+ 16y
11
+ ··· + 688y + 64)
· (4y
26
+ 56y
25
+ ··· + 74364y + 8649)
c
6
64(y
3
− 2y
2
+ 5y − 1)(y
3
+ 2y
2
+ y − 1)
· (4y
8
− 16y
7
+ 113y
6
− 168y
5
− 26y
4
+ 78y
3
+ 45y
2
+ 10y + 1)
· (4y
12
− 56y
11
+ ··· − 4168y + 2209)
· (4y
26
− 88y
25
+ ··· − 322704y + 46656)
c
7
, c
8
, c
11
y
3
(y − 1)
3
(y
4
− 5y
3
+ ··· − 2y + 1)
2
(y
12
− 8y
11
+ ··· + 19y + 4)
· (y
26
− 22y
25
+ ··· + 282y + 961)
c
9
9(y + 1)
8
(y
3
+ 2y
2
+ y − 1)
2
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
3
· (9y
26
+ 276y
25
+ ··· + 4752y + 64)
c
10
9(y + 1)
8
(y
3
+ 2y
2
+ y − 1)
2
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
3
· (9y
26
+ 60y
25
+ ··· + 400y + 64)
c
12
64(y
3
− 2y
2
+ 5y − 1)(y
3
+ 2y
2
+ y − 1)
· (4y
8
− 32y
7
+ 121y
6
− 296y
5
+ 486y
4
− 342y
3
+ 117y
2
− 18y + 1)
· (4y
12
+ 80y
11
+ ··· + 116344y + 10609)
· (4y
26
+ 184y
25
+ ··· + 708000y + 69696)
28
