12n
0604
(K12n
0604
)
A knot diagram
1
Linearized knot diagam
3 7 10 8 9 2 12 5 3 4 7 11
Solving Sequence
7,11
12
4,8
5 1 10 3 2 6 9
c
11
c
7
c
4
c
12
c
10
c
3
c
2
c
6
c
9
c
1
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= ⟨6141u
12
+ 35469u
11
+ ··· + 4348b − 14051, −17214u
12
− 43917u
11
+ ··· + 47828a − 232915,
3u
13
+ 18u
12
+ 47u
11
+ 60u
10
+ 14u
9
− 84u
8
− 153u
7
− 114u
6
+ 14u
5
+ 105u
4
+ 80u
3
+ 2u
2
− 25u − 11⟩
I
u
2
= ⟨u
9
− 2u
8
+ u
7
+ 2u
6
− 2u
5
− u
4
+ u
2
a + 3u
3
− au + b − 2u + 1, −2u
10
a − 13u
10
+ ··· − 2a − 35,
u
11
− 2u
10
+ 4u
8
− 2u
7
− 4u
6
+ 5u
5
+ 2u
4
− 5u
3
+ u
2
+ 3u − 1⟩
I
u
3
= ⟨b + 2a + 1, 2a
2
+ 2a − 1, u − 1⟩
I
u
4
= ⟨−2au + 2b − 2a + u + 3, 4a
2
− 4a + 1, u
2
+ 2u + 1⟩
I
u
5
= ⟨b, a − 1, u
3
− u + 1⟩
I
u
6
= ⟨b + 1, a + u, u
3
− u + 1⟩
I
u
7
= ⟨b, a − 1, u − 1⟩
I
u
8
= ⟨b + 1, u
2
a − au − 1⟩
I
u
9
= ⟨b + 1, u + 1⟩
* 7 irreducible components of dim
C
= 0, with total 48 representations.
* 2 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
= ⟨6141u
12
+ 35469u
11
+ · · · + 4348b − 14051, −1.72 × 10
4
u
12
− 4.39 ×
10
4
u
11
+ · · · + 4.78 × 10
4
a − 2.33 × 10
5
, 3u
13
+ 18u
12
+ · · · − 25u − 11⟩
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
4
=
0.359915u
12
+ 0.918228u
11
+ ··· + 7.51984u + 4.86985
−1.41237u
12
− 8.15754u
11
+ ··· + 1.15501u + 3.23160
a
8
=
u
−u
3
+ u
a
5
=
0.676612u
12
+ 3.37246u
11
+ ··· − 1.01834u − 0.308857
1.52001u
12
+ 5.01127u
11
+ ··· − 13.1615u − 3.97861
a
1
=
−u
2
+ 1
−u
2
a
10
=
−1.17283u
12
− 6.60021u
11
+ ··· − 6.76142u − 0.483336
0.411914u
12
+ 3.59407u
11
+ ··· + 11.9177u + 2.69894
a
3
=
−0.863344u
12
− 5.14142u
11
+ ··· − 4.86443u + 1.08965
−0.00965961u
12
+ 1.66697u
11
+ ··· + 20.0262u + 7.04140
a
2
=
−0.863344u
12
− 5.14142u
11
+ ··· − 4.86443u + 1.08965
0.890064u
12
+ 5.82889u
11
+ ··· + 22.8698u + 6.89972
a
6
=
0.234089u
12
+ 2.30426u
11
+ ··· + 5.28669u + 0.892866
−0.449862u
12
− 3.58096u
11
+ ··· − 21.4218u − 6.42916
a
9
=
0.0502425u
12
+ 0.933470u
11
+ ··· + 1.62986u − 0.0270135
0.638224u
12
+ 1.82498u
11
+ ··· − 21.8395u − 8.12810
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
9825
2174
u
12
−
54711
2174
u
11
+ ··· −
20596
1087
u +
3441
2174
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
9(9u
13
+ 114u
12
+ ··· + 1485u + 121)
c
2
, c
6
3(3u
13
− 18u
12
+ ··· + 11u − 11)
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
13
+ 2u
12
+ ··· − 2u + 2
c
7
, c
11
3(3u
13
+ 18u
12
+ ··· − 25u − 11)
c
12
9(9u
13
− 42u
12
+ ··· + 669u − 121)
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
81(81y
13
− 1962y
12
+ ··· + 662717y −14641)
c
2
, c
6
9(9y
13
− 114y
12
+ ··· + 1485y −121)
c
3
, c
4
, c
5
c
8
, c
9
, c
10
y
13
− 12y
12
+ ··· + 16y −4
c
7
, c
11
9(9y
13
− 42y
12
+ ··· + 669y −121)
c
12
81(81y
13
+ 630y
12
+ ··· + 37613y −14641)
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.15461
a = −0.667204
b = 1.65982
14.2281 19.7580
u = −0.875852 + 0.854559I
a = −0.480641 − 0.877360I
b = 0.160333 − 1.044863I
−8.47998 − 3.13363I 2.30766 + 3.10183I
u = −0.875852 − 0.854559I
a = −0.480641 + 0.877360I
b = 0.160333 + 1.044863I
−8.47998 + 3.13363I 2.30766 − 3.10183I
u = −0.195045 + 1.209339I
a = 0.484703 + 0.539351I
b = 1.182797 + 0.515971I
−2.29018 + 7.80194I 7.00595 − 5.86285I
u = −0.195045 − 1.209339I
a = 0.484703 − 0.539351I
b = 1.182797 − 0.515971I
−2.29018 − 7.80194I 7.00595 + 5.86285I
u = −0.602765 + 0.436556I
a = 0.289714 + 1.181214I
b = 0.018598 + 0.533941I
−0.98838 − 1.46599I 1.41277 + 4.35204I
u = −0.602765 − 0.436556I
a = 0.289714 − 1.181214I
b = 0.018598 − 0.533941I
−0.98838 + 1.46599I 1.41277 − 4.35204I
u = 0.734365
a = 0.253961
b = −0.323459
0.880574 13.4560
u = 0.692816
a = 1.53700
b = −1.80261
12.2154 2.46050
u = −1.32583 + 0.68048I
a = 0.433863 + 1.256459I
b = −1.39870 + 0.52665I
1.1831 − 14.4275I 9.59825 + 7.66969I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.32583 − 0.68048I
a = 0.433863 − 1.256459I
b = −1.39870 − 0.52665I
1.1831 + 14.4275I 9.59825 − 7.66969I
u = −1.29140 + 0.76843I
a = −0.107698 − 1.063230I
b = 1.270099 − 0.358705I
6.78301 − 8.58406I 12.8385 + 7.0528I
u = −1.29140 − 0.76843I
a = −0.107698 + 1.063230I
b = 1.270099 + 0.358705I
6.78301 + 8.58406I 12.8385 − 7.0528I
6
II. I
u
2
=
⟨u
9
−2u
8
+· · · + b +1, −2u
10
a−13u
10
+· · · − 2a − 35, u
11
−2u
10
+· · · +3u − 1⟩
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
4
=
a
−u
9
+ 2u
8
− u
7
− 2u
6
+ 2u
5
+ u
4
− u
2
a − 3u
3
+ au + 2u − 1
a
8
=
u
−u
3
+ u
a
5
=
−u
9
+ 2u
8
− 3u
6
+ 2u
5
+ u
3
a + 2u
4
− u
2
a − 3u
3
+ a + 3u − 1
−2u
9
+ 3u
8
+ ··· + 2u − 1
a
1
=
−u
2
+ 1
−u
2
a
10
=
−u
10
a + u
10
+ ··· − u + 5
−u
8
a + u
7
a − u
5
a − u
2
a + u
3
− u
2
+ 1
a
3
=
u
10
− 2u
9
− u
8
+ 5u
7
− u
6
− 6u
5
+ 4u
4
+ 4u
3
− 5u
2
− u + 3
u
10
− 2u
9
− u
8
+ 4u
7
− u
6
− 4u
5
+ 2u
4
+ 2u
3
− 3u
2
− u + 1
a
2
=
u
10
− 2u
9
− u
8
+ 5u
7
− u
6
− 6u
5
+ 4u
4
+ 4u
3
− 5u
2
− u + 3
2u
10
− 3u
9
− 2u
8
+ 6u
7
− 6u
5
+ 2u
4
+ 4u
3
− 3u
2
− 2u + 1
a
6
=
u
10
− u
9
− u
8
+ 3u
7
− 3u
5
+ 3u
4
+ 2u
3
− 3u
2
+ 2
−u
10
+ u
9
+ 3u
8
− 3u
7
− 4u
6
+ 7u
5
+ 2u
4
− 6u
3
+ 2u
2
+ 4u − 1
a
9
=
u
10
− u
9
+ ··· + a + 4
u
10
a − u
9
a + ··· + a + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
10
+ 6u
9
− 10u
7
+ 4u
6
+ 6u
5
− 12u
4
− 4u
3
+ 8u
2
− 6u + 4
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
11
+ 12u
10
+ ··· − 5u + 1)
2
c
2
, c
6
(u
11
+ 2u
10
− 4u
9
− 8u
8
+ 6u
7
+ 8u
6
− 7u
5
+ 2u
4
+ 7u
3
− 3u
2
− u − 1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
22
+ 2u
21
+ ··· − 54u − 23
c
7
, c
11
(u
11
− 2u
10
+ 4u
8
− 2u
7
− 4u
6
+ 5u
5
+ 2u
4
− 5u
3
+ u
2
+ 3u − 1)
2
c
12
(u
11
− 4u
10
+ ··· + 11u − 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
11
− 24y
10
+ ··· − 13y −1)
2
c
2
, c
6
(y
11
− 12y
10
+ ··· − 5y −1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
y
22
− 16y
21
+ ··· − 1306y + 529
c
7
, c
11
(y
11
− 4y
10
+ ··· + 11y −1)
2
c
12
(y
11
+ 8y
10
+ ··· + 67y −1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.952018 + 0.226513I
a = −0.347172 − 0.939025I
b = −0.930670 − 0.421418I
5.02081 − 0.74196I 15.5393 + 1.1191I
u = −0.952018 + 0.226513I
a = −1.36874 − 1.45440I
b = 1.254363 − 0.162092I
5.02081 − 0.74196I 15.5393 + 1.1191I
u = −0.952018 − 0.226513I
a = −0.347172 + 0.939025I
b = −0.930670 + 0.421418I
5.02081 + 0.74196I 15.5393 − 1.1191I
u = −0.952018 − 0.226513I
a = −1.36874 + 1.45440I
b = 1.254363 + 0.162092I
5.02081 + 0.74196I 15.5393 − 1.1191I
u = −0.850023 + 0.614930I
a = 0.76372 + 1.33350I
b = −1.49337 + 0.42695I
0.08426 − 2.41892I 7.07184 + 2.88947I
u = −0.850023 + 0.614930I
a = 0.077370 + 0.159281I
b = 1.27603 + 0.68990I
0.08426 − 2.41892I 7.07184 + 2.88947I
u = −0.850023 − 0.614930I
a = 0.76372 − 1.33350I
b = −1.49337 − 0.42695I
0.08426 + 2.41892I 7.07184 − 2.88947I
u = −0.850023 − 0.614930I
a = 0.077370 − 0.159281I
b = 1.27603 − 0.68990I
0.08426 + 2.41892I 7.07184 − 2.88947I
u = 0.523691 + 0.948055I
a = −0.534548 + 1.013410I
b = 0.196709 + 0.952827I
−5.32590 − 2.58451I 3.80806 + 1.01660I
u = 0.523691 + 0.948055I
a = 0.382826 − 0.290327I
b = 1.191516 − 0.585396I
−5.32590 − 2.58451I 3.80806 + 1.01660I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.523691 − 0.948055I
a = −0.534548 − 1.013410I
b = 0.196709 − 0.952827I
−5.32590 + 2.58451I 3.80806 − 1.01660I
u = 0.523691 − 0.948055I
a = 0.382826 + 0.290327I
b = 1.191516 + 0.585396I
−5.32590 + 2.58451I 3.80806 − 1.01660I
u = 0.978643 + 0.595733I
a = −0.010571 − 0.992888I
b = −0.109176 − 0.710565I
2.61864 + 4.69742I 9.08124 − 5.88322I
u = 0.978643 + 0.595733I
a = −0.17727 + 1.45820I
b = 1.225999 + 0.305614I
2.61864 + 4.69742I 9.08124 − 5.88322I
u = 0.978643 − 0.595733I
a = −0.010571 + 0.992888I
b = −0.109176 + 0.710565I
2.61864 − 4.69742I 9.08124 + 5.88322I
u = 0.978643 − 0.595733I
a = −0.17727 − 1.45820I
b = 1.225999 − 0.305614I
2.61864 − 4.69742I 9.08124 + 5.88322I
u = 1.126055 + 0.711355I
a = −0.407610 + 0.813195I
b = 0.097811 + 1.107042I
−3.47965 + 8.65115I 6.21430 − 5.57892I
u = 1.126055 + 0.711355I
a = 0.531471 − 1.289216I
b = −1.43289 − 0.49484I
−3.47965 + 8.65115I 6.21430 − 5.57892I
u = 1.126055 − 0.711355I
a = −0.407610 − 0.813195I
b = 0.097811 − 1.107042I
−3.47965 − 8.65115I 6.21430 + 5.57892I
u = 1.126055 − 0.711355I
a = 0.531471 + 1.289216I
b = −1.43289 + 0.49484I
−3.47965 − 8.65115I 6.21430 + 5.57892I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.347303
a = −3.28286
b = −1.15435
2.16369 2.57060
u = 0.347303
a = 4.46391
b = 0.601712
2.16369 2.57060
12
III. I
u
3
= ⟨b + 2a + 1, 2a
2
+ 2a − 1, u − 1⟩
(i) Arc colorings
a
7
=
0
1
a
11
=
1
0
a
12
=
1
−1
a
4
=
a
−2a − 1
a
8
=
1
0
a
5
=
−a − 1
−2a − 1
a
1
=
0
−1
a
10
=
a
3
a
3
=
1
4a + 2
a
2
=
1
4a + 1
a
6
=
1
4a + 2
a
9
=
−a − 1
−3
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
11
c
12
(u − 1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
2
− 3
c
6
, c
7
(u + 1)
2
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
, c
12
(y −1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
(y −3)
2
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.36603
b = 1.73205
13.1595 12.0000
u = 1.00000
a = 0.366025
b = −1.73205
13.1595 12.0000
16
IV. I
u
4
= ⟨−2au + 2b − 2a + u + 3, 4a
2
− 4a + 1, u
2
+ 2u + 1⟩
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
2u + 1
a
4
=
a
au + a −
1
2
u −
3
2
a
8
=
u
−2u − 2
a
5
=
−au +
3
2
u +
1
2
au + a −
5
2
u −
7
2
a
1
=
2u + 2
2u + 1
a
10
=
1
2
au −
1
2
a −
1
4
u +
3
4
−2au − 2a + u + 2
a
3
=
1
−2au − 2a + u + 1
a
2
=
1
−2au − 2a + 3u + 2
a
6
=
u
2au + 2a − 3u − 3
a
9
=
3
2
au +
1
2
a −
3
4
u −
3
4
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
6
c
7
, c
8
, c
12
(u − 1)
4
c
2
, c
4
, c
5
c
9
, c
10
, c
11
(u + 1)
4
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
(y −1)
4
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0.500000
b = −1.00000
3.28987 12.0000
u = −1.00000
a = 0.500000
b = −1.00000
3.28987 12.0000
u = −1.00000
a = 0.500000
b = −1.00000
3.28987 12.0000
u = −1.00000
a = 0.500000
b = −1.00000
3.28987 12.0000
20
V. I
u
5
= ⟨b, a − 1, u
3
− u + 1⟩
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
4
=
1
0
a
8
=
u
1
a
5
=
−u + 1
−1
a
1
=
−u
2
+ 1
−u
2
a
10
=
1
0
a
3
=
1
0
a
2
=
1
−u
2
a
6
=
u
1
a
9
=
1
0
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
21
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
3
+ 2u
2
+ u + 1
c
2
, c
6
, c
7
c
11
u
3
− u + 1
c
3
, c
9
, c
10
u
3
c
4
, c
5
, c
8
(u − 1)
3
c
12
u
3
− 2u
2
+ u − 1
22
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
12
y
3
− 2y
2
− 3y −1
c
2
, c
6
, c
7
c
11
y
3
− 2y
2
+ y −1
c
3
, c
9
, c
10
y
3
c
4
, c
5
, c
8
(y −1)
3
23
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 0.662359 + 0.562280I
a = 1.00000
b = 0
1.64493 6.00000
u = 0.662359 − 0.562280I
a = 1.00000
b = 0
1.64493 6.00000
u = −1.32472
a = 1.00000
b = 0
1.64493 6.00000
24
VI. I
u
6
= ⟨b + 1, a + u, u
3
− u + 1⟩
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
4
=
−u
−1
a
8
=
u
1
a
5
=
−u
−1
a
1
=
−u
2
+ 1
−u
2
a
10
=
u + 1
1
a
3
=
1
0
a
2
=
1
−u
2
a
6
=
u
1
a
9
=
u
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
25
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
3
+ 2u
2
+ u + 1
c
2
, c
6
, c
7
c
11
u
3
− u + 1
c
3
, c
9
, c
10
(u − 1)
3
c
4
, c
5
, c
8
u
3
c
12
u
3
− 2u
2
+ u − 1
26
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
12
y
3
− 2y
2
− 3y −1
c
2
, c
6
, c
7
c
11
y
3
− 2y
2
+ y −1
c
3
, c
9
, c
10
(y −1)
3
c
4
, c
5
, c
8
y
3
27
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = 0.662359 + 0.562280I
a = −0.662359 − 0.562280I
b = −1.00000
1.64493 6.00000
u = 0.662359 − 0.562280I
a = −0.662359 + 0.562280I
b = −1.00000
1.64493 6.00000
u = −1.32472
a = 1.32472
b = −1.00000
1.64493 6.00000
28
VII. I
u
7
= ⟨b, a − 1, u − 1⟩
(i) Arc colorings
a
7
=
0
1
a
11
=
1
0
a
12
=
1
−1
a
4
=
1
0
a
8
=
1
0
a
5
=
1
0
a
1
=
0
−1
a
10
=
1
0
a
3
=
1
0
a
2
=
1
−1
a
6
=
1
0
a
9
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
29
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
11
c
12
u − 1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
c
6
, c
7
u + 1
30
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
, c
12
y −1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
y
31
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
7
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 0
0 0
32
VIII. I
u
8
= ⟨b + 1, u
2
a − au − 1⟩
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
4
=
a
−1
a
8
=
u
−u
3
+ u
a
5
=
a + u
−u
3
+ u − 1
a
1
=
−u
2
+ 1
−u
2
a
10
=
−a + 1
1
a
3
=
1
0
a
2
=
1
−u
2
a
6
=
u
−u
3
+ u
a
9
=
−a
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
(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.
33
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
8
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
3.28987 12.0000
34
IX. I
u
9
= ⟨b + 1, u + 1⟩
(i) Arc colorings
a
7
=
0
−1
a
11
=
1
0
a
12
=
1
−1
a
4
=
a
−1
a
8
=
−1
0
a
5
=
a − 1
−1
a
1
=
0
−1
a
10
=
−a + 1
1
a
3
=
1
0
a
2
=
1
−1
a
6
=
−1
0
a
9
=
−a
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
(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.
35
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
9
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
3.28987 12.0000
36
X. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
9(u − 1)
7
(u
3
+ 2u
2
+ u + 1)
2
(u
11
+ 12u
10
+ ··· − 5u + 1)
2
· (9u
13
+ 114u
12
+ ··· + 1485u + 121)
c
2
3(u − 1)
3
(u + 1)
4
(u
3
− u + 1)
2
· (u
11
+ 2u
10
− 4u
9
− 8u
8
+ 6u
7
+ 8u
6
− 7u
5
+ 2u
4
+ 7u
3
− 3u
2
− u − 1)
2
· (3u
13
− 18u
12
+ ··· + 11u − 11)
c
3
, c
8
u
4
(u − 1)
7
(u
2
− 3)(u
13
+ 2u
12
+ ··· − 2u + 2)
· (u
22
+ 2u
21
+ ··· − 54u − 23)
c
4
, c
5
, c
9
c
10
u
4
(u − 1)
3
(u + 1)
4
(u
2
− 3)(u
13
+ 2u
12
+ ··· − 2u + 2)
· (u
22
+ 2u
21
+ ··· − 54u − 23)
c
6
3(u − 1)
4
(u + 1)
3
(u
3
− u + 1)
2
· (u
11
+ 2u
10
− 4u
9
− 8u
8
+ 6u
7
+ 8u
6
− 7u
5
+ 2u
4
+ 7u
3
− 3u
2
− u − 1)
2
· (3u
13
− 18u
12
+ ··· + 11u − 11)
c
7
3(u − 1)
4
(u + 1)
3
(u
3
− u + 1)
2
· (u
11
− 2u
10
+ 4u
8
− 2u
7
− 4u
6
+ 5u
5
+ 2u
4
− 5u
3
+ u
2
+ 3u − 1)
2
· (3u
13
+ 18u
12
+ ··· − 25u − 11)
c
11
3(u − 1)
3
(u + 1)
4
(u
3
− u + 1)
2
· (u
11
− 2u
10
+ 4u
8
− 2u
7
− 4u
6
+ 5u
5
+ 2u
4
− 5u
3
+ u
2
+ 3u − 1)
2
· (3u
13
+ 18u
12
+ ··· − 25u − 11)
c
12
9(u − 1)
7
(u
3
− 2u
2
+ u − 1)
2
(u
11
− 4u
10
+ ··· + 11u − 1)
2
· (9u
13
− 42u
12
+ ··· + 669u − 121)
37
XI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
81(y −1)
7
(y
3
− 2y
2
− 3y − 1)
2
(y
11
− 24y
10
+ ··· − 13y −1)
2
· (81y
13
− 1962y
12
+ ··· + 662717y −14641)
c
2
, c
6
9(y −1)
7
(y
3
− 2y
2
+ y −1)
2
(y
11
− 12y
10
+ ··· − 5y −1)
2
· (9y
13
− 114y
12
+ ··· + 1485y −121)
c
3
, c
4
, c
5
c
8
, c
9
, c
10
y
4
(y −3)
2
(y −1)
7
(y
13
− 12y
12
+ ··· + 16y −4)
· (y
22
− 16y
21
+ ··· − 1306y + 529)
c
7
, c
11
9(y −1)
7
(y
3
− 2y
2
+ y −1)
2
(y
11
− 4y
10
+ ··· + 11y −1)
2
· (9y
13
− 42y
12
+ ··· + 669y −121)
c
12
81(y −1)
7
(y
3
− 2y
2
− 3y − 1)
2
(y
11
+ 8y
10
+ ··· + 67y −1)
2
· (81y
13
+ 630y
12
+ ··· + 37613y −14641)
38
