12n
0321
(K12n
0321
)
A knot diagram
1
Linearized knot diagam
3 6 12 7 2 10 3 1 12 5 7 9
Solving Sequence
3,6
2 1
5,9
8 7 4 12 10 11
c
2
c
1
c
5
c
8
c
7
c
4
c
12
c
9
c
10
c
3
, c
6
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−28u
8
+ 32u
7
− 37u
6
− 26u
5
− 112u
4
− 40u
3
− 37u
2
+ 59b + 86u − 72,
− 7u
8
+ 8u
7
− 24u
6
+ 23u
5
− 28u
4
− 69u
3
− 24u
2
+ 59a + 110u − 18,
u
9
− 2u
8
+ 2u
7
+ u
6
+ 2u
5
− 2u
4
+ u
3
− 3u
2
+ 4u − 1i
I
u
2
= h−u
2
+ b − u + 2, a + 1, u
4
+ u
3
− u
2
− u + 1i
I
u
3
= h−2u
7
− 3u
6
− 2u
5
+ 4u
4
+ 7u
3
+ 5u
2
+ b − 5u − 5, −2u
7
− 2u
6
− u
5
+ 5u
4
+ 5u
3
+ 3u
2
+ a − 6u − 2,
u
8
+ 2u
7
+ 2u
6
− u
5
− 4u
4
− 4u
3
+ u
2
+ 3u + 1i
I
u
4
= h−3u
5
+ u
3
− 11u
2
+ 19b + 7u − 18, −9u
5
+ 19u
4
− 35u
3
+ 43u
2
+ 19a − 74u + 22,
u
6
− 3u
5
+ 6u
4
− 8u
3
+ 12u
2
− 6u + 1i
I
u
5
= hb, a + u + 1, u
2
+ u + 1i
* 5 irreducible components of dim
C
= 0, with total 29 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−28u
8
+32u
7
+· · ·+59b −72, −7u
8
+8u
7
+· · ·+59a −18, u
9
−2u
8
+· · ·+4u −1i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
2
=
1
−u
2
a
1
=
−u
2
+ 1
−u
2
a
5
=
u
−u
3
+ u
a
9
=
0.118644u
8
− 0.135593u
7
+ ··· − 1.86441u + 0.305085
0.474576u
8
− 0.542373u
7
+ ··· − 1.45763u + 1.22034
a
8
=
0.355932u
8
− 0.406780u
7
+ ··· − 1.59322u + 0.915254
0.355932u
8
− 0.406780u
7
+ ··· − 0.593220u + 0.915254
a
7
=
−u
0.355932u
8
− 0.406780u
7
+ ··· − 0.593220u + 0.915254
a
4
=
−0.118644u
8
+ 0.135593u
7
+ ··· + 1.86441u − 0.305085
0.440678u
8
− 0.932203u
7
+ ··· − 1.06780u + 0.847458
a
12
=
0.389831u
8
− 1.01695u
7
+ ··· − 0.983051u + 1.28814
−0.355932u
8
+ 0.406780u
7
+ ··· + 0.593220u − 0.915254
a
10
=
−1
0.915254u
8
− 1.47458u
7
+ ··· − 2.52542u + 2.06780
a
11
=
−0.305085u
8
+ 0.491525u
7
+ ··· + 0.508475u − 1.35593
0.711864u
8
− 0.813559u
7
+ ··· − 2.18644u + 1.83051
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
15
59
u
8
+
93
59
u
7
−
43
59
u
6
−
94
59
u
5
+
235
59
u
4
+
282
59
u
3
+
193
59
u
2
−
93
59
u −
342
59
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
9
+ 12u
7
− u
6
+ 8u
5
− 18u
4
+ 7u
3
+ 5u
2
+ 10u + 1
c
2
, c
5
, c
6
u
9
+ 2u
8
+ 2u
7
− u
6
+ 2u
5
+ 2u
4
+ u
3
+ 3u
2
+ 4u + 1
c
3
, c
8
, c
9
c
10
, c
12
u
9
+ 7u
7
+ 5u
6
+ 15u
5
+ 13u
4
+ 11u
3
+ 7u
2
+ u + 1
c
4
u
9
+ 2u
8
− 7u
7
− 15u
6
+ 13u
5
+ 53u
4
+ 71u
3
+ 61u
2
+ 29u + 5
c
7
u
9
− 7u
8
+ 5u
7
+ 42u
6
+ 92u
5
+ 125u
4
+ 125u
3
+ 88u
2
+ 39u + 9
c
11
u
9
+ u
8
+ 14u
7
+ 10u
6
+ 39u
5
− 44u
4
− 45u
3
+ 4u
2
+ 20u + 9
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
9
+ 24y
8
+ ··· + 90y −1
c
2
, c
5
, c
6
y
9
+ 12y
7
+ y
6
+ 8y
5
+ 18y
4
+ 7y
3
− 5y
2
+ 10y −1
c
3
, c
8
, c
9
c
10
, c
12
y
9
+ 14y
8
+ 79y
7
+ 207y
6
+ 251y
5
+ 105y
4
− 41y
3
− 53y
2
− 13y −1
c
4
y
9
− 18y
8
+ ··· + 231y −25
c
7
y
9
− 39y
8
+ ··· − 63y −81
c
11
y
9
+ 27y
8
+ ··· + 328y −81
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.236649 + 0.987655I
a = 1.59024 − 3.04254I
b = 1.32330 − 0.67194I
9.89350 + 2.31667I 0.86831 − 3.48815I
u = −0.236649 − 0.987655I
a = 1.59024 + 3.04254I
b = 1.32330 + 0.67194I
9.89350 − 2.31667I 0.86831 + 3.48815I
u = −0.948444 + 0.610151I
a = 0.291757 + 0.057514I
b = −0.400025 − 0.194724I
1.50288 + 4.69117I −4.69492 − 7.49384I
u = −0.948444 − 0.610151I
a = 0.291757 − 0.057514I
b = −0.400025 + 0.194724I
1.50288 − 4.69117I −4.69492 + 7.49384I
u = 0.731097 + 0.406841I
a = −0.967602 + 0.291558I
b = −0.297854 + 1.104540I
−1.11949 − 1.41007I −6.06702 + 5.32264I
u = 0.731097 − 0.406841I
a = −0.967602 − 0.291558I
b = −0.297854 − 1.104540I
−1.11949 + 1.41007I −6.06702 − 5.32264I
u = 0.323158
a = −0.211236
b = 0.860492
−1.09696 −5.76550
u = 1.29242 + 1.30359I
a = 1.69122 + 1.25994I
b = 2.44434 + 0.36799I
−13.8407 − 9.8067I −4.22361 + 3.66185I
u = 1.29242 − 1.30359I
a = 1.69122 − 1.25994I
b = 2.44434 − 0.36799I
−13.8407 + 9.8067I −4.22361 − 3.66185I
5
II. I
u
2
= h−u
2
+ b − u + 2, a + 1, u
4
+ u
3
− u
2
− u + 1i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
2
=
1
−u
2
a
1
=
−u
2
+ 1
−u
2
a
5
=
u
−u
3
+ u
a
9
=
−1
u
2
+ u − 2
a
8
=
u
3
+ u
2
− u − 1
u
3
+ u
2
− 1
a
7
=
−u
u
3
+ u
2
− 1
a
4
=
1
−u
3
− u
2
+ u
a
12
=
0
u
2
− 1
a
10
=
−1
u
3
+ 2u
2
− 2
a
11
=
−u
3
+ u − 1
u
3
+ 2u
2
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
3
+ 5u
2
− 3u − 13
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
4
− 3u
3
+ 5u
2
− 3u + 1
c
2
, c
6
u
4
+ u
3
− u
2
− u + 1
c
3
, c
12
u
4
+ u
3
+ 2u
2
+ 2u + 1
c
4
u
4
− 3u
3
+ 2u
2
+ 1
c
5
u
4
− u
3
− u
2
+ u + 1
c
7
(u
2
− u + 1)
2
c
8
, c
9
, c
10
u
4
− u
3
+ 2u
2
− 2u + 1
c
11
u
4
+ 2u
2
+ 3u + 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
4
+ y
3
+ 9y
2
+ y + 1
c
2
, c
5
, c
6
y
4
− 3y
3
+ 5y
2
− 3y + 1
c
3
, c
8
, c
9
c
10
, c
12
y
4
+ 3y
3
+ 2y
2
+ 1
c
4
y
4
− 5y
3
+ 6y
2
+ 4y + 1
c
7
(y
2
+ y + 1)
2
c
11
y
4
+ 4y
3
+ 6y
2
− 5y + 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.692440 + 0.318148I
a = −1.00000
b = −0.929304 + 0.758745I
−1.74699 − 0.56550I −12.94255 + 2.09940I
u = 0.692440 − 0.318148I
a = −1.00000
b = −0.929304 − 0.758745I
−1.74699 + 0.56550I −12.94255 − 2.09940I
u = −1.192440 + 0.547877I
a = −1.00000
b = −2.07070 − 0.75874I
5.03685 + 4.62527I −5.05745 − 3.83145I
u = −1.192440 − 0.547877I
a = −1.00000
b = −2.07070 + 0.75874I
5.03685 − 4.62527I −5.05745 + 3.83145I
9
III. I
u
3
= h−2u
7
− 3u
6
+ · · · + b − 5, −2u
7
− 2u
6
+ · · · + a − 2, u
8
+ 2u
7
+
2u
6
− u
5
− 4u
4
− 4u
3
+ u
2
+ 3u + 1i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
2
=
1
−u
2
a
1
=
−u
2
+ 1
−u
2
a
5
=
u
−u
3
+ u
a
9
=
2u
7
+ 2u
6
+ u
5
− 5u
4
− 5u
3
− 3u
2
+ 6u + 2
2u
7
+ 3u
6
+ 2u
5
− 4u
4
− 7u
3
− 5u
2
+ 5u + 5
a
8
=
4u
7
+ 5u
6
+ 4u
5
− 8u
4
− 12u
3
− 9u
2
+ 11u + 6
u
7
+ 2u
6
+ 2u
5
− u
4
− 4u
3
− 4u
2
+ 2u + 3
a
7
=
3u
7
+ 3u
6
+ 2u
5
− 7u
4
− 8u
3
− 5u
2
+ 9u + 3
u
7
+ 2u
6
+ 2u
5
− u
4
− 4u
3
− 4u
2
+ 2u + 3
a
4
=
−10u
7
− 13u
6
− 11u
5
+ 17u
4
+ 27u
3
+ 20u
2
− 23u − 12
−u
7
− 2u
6
− 2u
5
+ u
4
+ 4u
3
+ 4u
2
− u − 4
a
12
=
6u
7
+ 9u
6
+ 7u
5
− 11u
4
− 20u
3
− 14u
2
+ 15u + 13
−5u
7
− 7u
6
− 6u
5
+ 8u
4
+ 15u
3
+ 11u
2
− 11u − 8
a
10
=
−7u
7
− 10u
6
− 8u
5
+ 12u
4
+ 22u
3
+ 16u
2
− 16u − 12
3u
7
+ 4u
6
+ 3u
5
− 6u
4
− 9u
3
− 6u
2
+ 8u + 5
a
11
=
−10u
7
− 14u
6
− 11u
5
+ 17u
4
+ 31u
3
+ 22u
2
− 23u − 17
u
7
+ 2u
6
+ u
5
− 3u
4
− 4u
3
− 2u
2
+ 4u + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
7
+ 3u
6
+ 3u
5
− 2u
4
− 5u
3
− 4u
2
+ 3u − 2
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
8
+ u
5
+ 2u
4
− 14u
3
+ 17u
2
− 7u + 1
c
2
, c
6
u
8
+ 2u
7
+ 2u
6
− u
5
− 4u
4
− 4u
3
+ u
2
+ 3u + 1
c
3
, c
12
u
8
− u
7
+ 6u
6
− 5u
5
+ 12u
4
− 8u
3
+ 9u
2
− 4u + 1
c
4
(u
4
+ 3u
3
+ u
2
− 2u + 1)
2
c
5
u
8
− 2u
7
+ 2u
6
+ u
5
− 4u
4
+ 4u
3
+ u
2
− 3u + 1
c
7
u
8
+ u
7
+ 5u
6
+ 8u
5
+ 7u
4
+ 11u
3
+ 10u
2
+ 1
c
8
, c
9
, c
10
u
8
+ u
7
+ 6u
6
+ 5u
5
+ 12u
4
+ 8u
3
+ 9u
2
+ 4u + 1
c
11
u
8
− 2u
7
+ 5u
6
+ 3u
5
+ 4u
4
+ 27u
3
− 3u
2
− 16u + 52
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
8
+ 4y
6
+ 33y
5
+ 34y
4
− 114y
3
+ 97y
2
− 15y + 1
c
2
, c
5
, c
6
y
8
+ y
5
+ 2y
4
− 14y
3
+ 17y
2
− 7y + 1
c
3
, c
8
, c
9
c
10
, c
12
y
8
+ 11y
7
+ 50y
6
+ 121y
5
+ 166y
4
+ 124y
3
+ 41y
2
+ 2y + 1
c
4
(y
4
− 7y
3
+ 15y
2
− 2y + 1)
2
c
7
y
8
+ 9y
7
+ 23y
6
+ 4y
5
− 25y
4
+ 29y
3
+ 114y
2
+ 20y + 1
c
11
y
8
+ 6y
7
+ 45y
6
+ 133y
5
− 136y
4
− 137y
3
+ 1289y
2
− 568y + 2704
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.963269 + 0.149069I
a = −0.307661 − 1.178810I
b = 0.148192 − 0.911292I
1.43393 − 3.16396I −4.10488 + 1.55249I
u = 0.963269 − 0.149069I
a = −0.307661 + 1.178810I
b = 0.148192 + 0.911292I
1.43393 + 3.16396I −4.10488 − 1.55249I
u = −1.006590 + 0.790269I
a = 0.550701 + 0.903791I
b = 0.148192 + 0.911292I
1.43393 + 3.16396I −4.10488 − 1.55249I
u = −1.006590 − 0.790269I
a = 0.550701 − 0.903791I
b = 0.148192 − 0.911292I
1.43393 − 3.16396I −4.10488 + 1.55249I
u = −0.384833 + 1.326500I
a = 1.24719 − 2.12175I
b = 1.35181 − 0.72034I
8.43568 + 1.41510I −4.39512 − 0.50684I
u = −0.384833 − 1.326500I
a = 1.24719 + 2.12175I
b = 1.35181 + 0.72034I
8.43568 − 1.41510I −4.39512 + 0.50684I
u = −0.571852 + 0.099314I
a = −1.99023 + 0.84034I
b = 1.35181 + 0.72034I
8.43568 − 1.41510I −4.39512 + 0.50684I
u = −0.571852 − 0.099314I
a = −1.99023 − 0.84034I
b = 1.35181 − 0.72034I
8.43568 + 1.41510I −4.39512 − 0.50684I
13
IV. I
u
4
= h−3u
5
+ u
3
− 11u
2
+ 19b + 7u − 18, −9u
5
+ 19u
4
+ · · · + 19a +
22, u
6
− 3u
5
+ 6u
4
− 8u
3
+ 12u
2
− 6u + 1i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
2
=
1
−u
2
a
1
=
−u
2
+ 1
−u
2
a
5
=
u
−u
3
+ u
a
9
=
9
19
u
5
− u
4
+ ··· +
74
19
u −
22
19
0.157895u
5
− 0.0526316u
3
+ ··· − 0.368421u + 0.947368
a
8
=
9
19
u
5
− 2u
4
+ ··· +
55
19
u −
3
19
3
19
u
5
− u
4
+ ··· −
7
19
u +
18
19
a
7
=
6
19
u
5
− u
4
+ ··· +
62
19
u −
21
19
3
19
u
5
− u
4
+ ··· −
7
19
u +
18
19
a
4
=
8
19
u
5
− u
4
+ ··· +
51
19
u −
9
19
11
19
u
5
− u
4
+ ··· +
6
19
u +
9
19
a
12
=
−
6
19
u
5
+ u
4
+ ··· −
81
19
u +
40
19
−0.105263u
5
+ 0.368421u
3
+ ··· − 0.421053u − 0.631579
a
10
=
0.684211u
5
− 2u
4
+ ··· + 7.73684u − 2.89474
−0.105263u
5
+ 0.368421u
3
+ ··· + 0.578947u + 1.36842
a
11
=
7
19
u
5
− u
4
+ ··· +
123
19
u −
53
19
−1.57895u
5
+ 3u
4
+ ··· − 1.31579u + 1.52632
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
2
19
u
5
+
7
19
u
3
−
20
19
u
2
+
11
19
u −
107
19
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
6
− 3u
5
+ 12u
4
− 46u
3
+ 60u
2
+ 12u + 1
c
2
, c
5
, c
6
u
6
+ 3u
5
+ 6u
4
+ 8u
3
+ 12u
2
+ 6u + 1
c
3
, c
8
, c
9
c
10
, c
12
u
6
+ 9u
4
+ 8u
3
+ 27u
2
+ 45u + 19
c
4
(u
3
− 3u − 1)
2
c
7
u
6
+ 9u
5
+ 48u
4
+ 349u
3
+ 1647u
2
+ 2058u + 757
c
11
u
6
+ 9u
4
− 9u
3
+ 54u
2
+ 81u + 27
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
6
+ 15y
5
− 12y
4
− 602y
3
+ 4728y
2
− 24y + 1
c
2
, c
5
, c
6
y
6
+ 3y
5
+ 12y
4
+ 46y
3
+ 60y
2
− 12y + 1
c
3
, c
8
, c
9
c
10
, c
12
y
6
+ 18y
5
+ 135y
4
+ 460y
3
+ 351y
2
− 999y + 361
c
4
(y
3
− 6y
2
+ 9y −1)
2
c
7
y
6
+ 15y
5
− 684y
4
+ 781y
3
+ 1348797y
2
− 1741806y + 573049
c
11
y
6
+ 18y
5
+ 189y
4
+ 945y
3
+ 4860y
2
− 3645y + 729
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.26604 + 1.50881I
a = −1.17365 + 0.98481I
b = −1.34730
12.0628 −2.12061 + 0.I
u = −0.26604 − 1.50881I
a = −1.17365 − 0.98481I
b = −1.34730
12.0628 −2.12061 + 0.I
u = 0.326352 + 0.118782I
a = −0.060307 + 0.342020I
b = 0.879385
−1.09662 −5.53209 + 0.I
u = 0.326352 − 0.118782I
a = −0.060307 − 0.342020I
b = 0.879385
−1.09662 −5.53209 + 0.I
u = 1.43969 + 1.20805I
a = −1.76604 − 0.64279I
b = −2.53209
−14.2561 −4.34730 + 0.I
u = 1.43969 − 1.20805I
a = −1.76604 + 0.64279I
b = −2.53209
−14.2561 −4.34730 + 0.I
17
V. I
u
5
= hb, a + u + 1, u
2
+ u + 1i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
2
=
1
u + 1
a
1
=
u + 2
u + 1
a
5
=
u
u − 1
a
9
=
−u − 1
0
a
8
=
−2
−1
a
7
=
−1
−1
a
4
=
u + 1
u
a
12
=
u + 1
u + 1
a
10
=
−u
1
a
11
=
−u − 1
−u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −3
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
8
c
9
, c
10
, c
12
u
2
− u + 1
c
4
, c
7
(u − 1)
2
c
11
u
2
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
8
c
9
, c
10
, c
12
y
2
+ y + 1
c
4
, c
7
(y − 1)
2
c
11
y
2
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = −0.500000 − 0.866025I
b = 0
3.28987 −3.00000
u = −0.500000 − 0.866025I
a = −0.500000 + 0.866025I
b = 0
3.28987 −3.00000
21
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)(u
4
− 3u
3
+ 5u
2
− 3u + 1)
· (u
6
− 3u
5
+ 12u
4
− 46u
3
+ 60u
2
+ 12u + 1)
· (u
8
+ u
5
+ 2u
4
− 14u
3
+ 17u
2
− 7u + 1)
· (u
9
+ 12u
7
− u
6
+ 8u
5
− 18u
4
+ 7u
3
+ 5u
2
+ 10u + 1)
c
2
, c
6
(u
2
− u + 1)(u
4
+ u
3
− u
2
− u + 1)(u
6
+ 3u
5
+ ··· + 6u + 1)
· (u
8
+ 2u
7
+ 2u
6
− u
5
− 4u
4
− 4u
3
+ u
2
+ 3u + 1)
· (u
9
+ 2u
8
+ 2u
7
− u
6
+ 2u
5
+ 2u
4
+ u
3
+ 3u
2
+ 4u + 1)
c
3
, c
12
(u
2
− u + 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)(u
6
+ 9u
4
+ ··· + 45u + 19)
· (u
8
− u
7
+ 6u
6
− 5u
5
+ 12u
4
− 8u
3
+ 9u
2
− 4u + 1)
· (u
9
+ 7u
7
+ 5u
6
+ 15u
5
+ 13u
4
+ 11u
3
+ 7u
2
+ u + 1)
c
4
((u − 1)
2
)(u
3
− 3u − 1)
2
(u
4
− 3u
3
+ 2u
2
+ 1)(u
4
+ 3u
3
+ ··· − 2u + 1)
2
· (u
9
+ 2u
8
− 7u
7
− 15u
6
+ 13u
5
+ 53u
4
+ 71u
3
+ 61u
2
+ 29u + 5)
c
5
(u
2
− u + 1)(u
4
− u
3
− u
2
+ u + 1)(u
6
+ 3u
5
+ ··· + 6u + 1)
· (u
8
− 2u
7
+ 2u
6
+ u
5
− 4u
4
+ 4u
3
+ u
2
− 3u + 1)
· (u
9
+ 2u
8
+ 2u
7
− u
6
+ 2u
5
+ 2u
4
+ u
3
+ 3u
2
+ 4u + 1)
c
7
(u − 1)
2
(u
2
− u + 1)
2
· (u
6
+ 9u
5
+ 48u
4
+ 349u
3
+ 1647u
2
+ 2058u + 757)
· (u
8
+ u
7
+ 5u
6
+ 8u
5
+ 7u
4
+ 11u
3
+ 10u
2
+ 1)
· (u
9
− 7u
8
+ 5u
7
+ 42u
6
+ 92u
5
+ 125u
4
+ 125u
3
+ 88u
2
+ 39u + 9)
c
8
, c
9
, c
10
(u
2
− u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
6
+ 9u
4
+ ··· + 45u + 19)
· (u
8
+ u
7
+ 6u
6
+ 5u
5
+ 12u
4
+ 8u
3
+ 9u
2
+ 4u + 1)
· (u
9
+ 7u
7
+ 5u
6
+ 15u
5
+ 13u
4
+ 11u
3
+ 7u
2
+ u + 1)
c
11
u
2
(u
4
+ 2u
2
+ 3u + 1)(u
6
+ 9u
4
− 9u
3
+ 54u
2
+ 81u + 27)
· (u
8
− 2u
7
+ 5u
6
+ 3u
5
+ 4u
4
+ 27u
3
− 3u
2
− 16u + 52)
· (u
9
+ u
8
+ 14u
7
+ 10u
6
+ 39u
5
− 44u
4
− 45u
3
+ 4u
2
+ 20u + 9)
22
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
2
+ y + 1)(y
4
+ y
3
+ 9y
2
+ y + 1)
· (y
6
+ 15y
5
− 12y
4
− 602y
3
+ 4728y
2
− 24y + 1)
· (y
8
+ 4y
6
+ 33y
5
+ 34y
4
− 114y
3
+ 97y
2
− 15y + 1)
· (y
9
+ 24y
8
+ ··· + 90y − 1)
c
2
, c
5
, c
6
(y
2
+ y + 1)(y
4
− 3y
3
+ 5y
2
− 3y + 1)
· (y
6
+ 3y
5
+ 12y
4
+ 46y
3
+ 60y
2
− 12y + 1)
· (y
8
+ y
5
+ 2y
4
− 14y
3
+ 17y
2
− 7y + 1)
· (y
9
+ 12y
7
+ y
6
+ 8y
5
+ 18y
4
+ 7y
3
− 5y
2
+ 10y − 1)
c
3
, c
8
, c
9
c
10
, c
12
(y
2
+ y + 1)(y
4
+ 3y
3
+ 2y
2
+ 1)
· (y
6
+ 18y
5
+ 135y
4
+ 460y
3
+ 351y
2
− 999y + 361)
· (y
8
+ 11y
7
+ 50y
6
+ 121y
5
+ 166y
4
+ 124y
3
+ 41y
2
+ 2y + 1)
· (y
9
+ 14y
8
+ 79y
7
+ 207y
6
+ 251y
5
+ 105y
4
− 41y
3
− 53y
2
− 13y − 1)
c
4
(y − 1)
2
(y
3
− 6y
2
+ 9y − 1)
2
(y
4
− 7y
3
+ 15y
2
− 2y + 1)
2
· (y
4
− 5y
3
+ 6y
2
+ 4y + 1)(y
9
− 18y
8
+ ··· + 231y − 25)
c
7
(y − 1)
2
(y
2
+ y + 1)
2
· (y
6
+ 15y
5
− 684y
4
+ 781y
3
+ 1348797y
2
− 1741806y + 573049)
· (y
8
+ 9y
7
+ 23y
6
+ 4y
5
− 25y
4
+ 29y
3
+ 114y
2
+ 20y + 1)
· (y
9
− 39y
8
+ ··· − 63y − 81)
c
11
y
2
(y
4
+ 4y
3
+ 6y
2
− 5y + 1)
· (y
6
+ 18y
5
+ 189y
4
+ 945y
3
+ 4860y
2
− 3645y + 729)
· (y
8
+ 6y
7
+ 45y
6
+ 133y
5
− 136y
4
− 137y
3
+ 1289y
2
− 568y + 2704)
· (y
9
+ 27y
8
+ ··· + 328y − 81)
23
