12n
0355
(K12n
0355
)
A knot diagram
1
Linearized knot diagam
3 5 12 11 2 10 3 4 7 5 9 8
Solving Sequence
4,12 3,9
8 1 7 11 5 2 6 10
c
3
c
8
c
12
c
7
c
11
c
4
c
2
c
5
c
10
c
1
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h1448u
11
− 622u
10
+ ··· + 1059b − 1564, a + 1,
u
12
− u
11
+ 2u
9
+ 4u
8
− 4u
7
− 3u
6
+ 9u
5
+ u
4
− 4u
3
+ 5u
2
− 2u + 1i
I
u
2
= hu
4
+ u
3
+ 2u
2
+ b + u + 2, a + 1, u
5
+ u
4
+ 2u
3
+ 2u − 1i
I
u
3
= h−28u
9
− 75u
8
− 50u
7
+ 61u
6
− 14u
5
− 212u
4
− 214u
3
− 47u
2
+ 23b − 10u + 13,
20u
9
+ 70u
8
+ 108u
7
+ 55u
6
+ 10u
5
+ 89u
4
+ 304u
3
+ 372u
2
+ 23a + 224u + 17,
u
10
+ 4u
9
+ 6u
8
+ 2u
7
− u
6
+ 7u
5
+ 18u
4
+ 17u
3
+ 9u
2
+ 3u + 1i
I
u
4
= hb − 1, a + 1, u
2
+ u + 1i
I
u
5
= h38u
9
− 23u
8
+ 194u
7
− 237u
6
+ 606u
5
+ 194u
4
+ 1010u
3
+ 389u
2
+ 563b + 442u + 545,
412u
9
− 842u
8
+ 1570u
7
− 1651u
6
+ 4022u
5
− 2371u
4
+ 4076u
3
− 2894u
2
+ 1689a + 1592u − 3425,
u
10
− 2u
9
+ 4u
8
− 4u
7
+ 11u
6
− 7u
5
+ 14u
4
− 5u
3
+ 11u
2
− 5u + 3i
* 5 irreducible components of dim
C
= 0, with total 39 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
= h1448u
11
− 622u
10
+ · · · + 1059b − 1564, a + 1, u
12
− u
11
+ · · · − 2u + 1i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
u
a
3
=
1
−u
2
a
9
=
−1
−1.36733u
11
+ 0.587347u
10
+ ··· − 1.97734u + 1.47686
a
8
=
−1.36733u
11
+ 0.587347u
10
+ ··· − 1.97734u + 0.476865
−1.36733u
11
+ 0.587347u
10
+ ··· − 1.97734u + 1.47686
a
1
=
−1.41171u
11
+ 0.928234u
10
+ ··· − 3.16997u + 2.25685
−2.19169u
11
+ 1.66383u
10
+ ··· − 4.42776u + 3.62417
a
7
=
0.0443815u
11
− 0.340888u
10
+ ··· + 0.192635u − 1.77998
−1.20113u
11
+ 0.864023u
10
+ ··· − 1.53258u + 1.96034
a
11
=
u
0.779981u
11
− 0.735600u
10
+ ··· + 2.25779u − 1.36733
a
5
=
−0.0443815u
11
+ 0.340888u
10
+ ··· − 0.192635u + 1.77998
0.483475u
11
− 0.649669u
10
+ ··· + 0.566572u − 1.41171
a
2
=
0.613787u
11
− 1.01228u
10
+ ··· + 0.813031u − 1.85080
−1.47498u
11
+ 1.66950u
10
+ ··· − 2.57224u + 3.70916
a
6
=
−0.972616u
11
+ 2.61945u
10
+ ··· − 1.79603u + 6.62512
3.39754u
11
− 4.62795u
10
+ ··· + 6.51275u − 9.75260
a
10
=
−0.613787u
11
+ 1.01228u
10
+ ··· − 0.813031u + 1.85080
1.37205u
11
− 1.68744u
10
+ ··· + 3.52975u − 3.64495
(ii) Obstruction class = −1
(iii) Cusp Shapes =
2459
1059
u
11
+
3149
1059
u
10
−
3035
1059
u
9
+
1248
353
u
8
+
20198
1059
u
7
+
17837
1059
u
6
−
17579
1059
u
5
+
1070
1059
u
4
+
14089
353
u
3
+
14917
1059
u
2
+
238
353
u +
13700
1059
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
12
− 12u
11
+ ··· + u + 1
c
2
, c
5
, c
6
c
9
u
12
− 6u
10
+ ··· − u + 1
c
3
, c
11
u
12
− u
11
+ 2u
9
+ 4u
8
− 4u
7
− 3u
6
+ 9u
5
+ u
4
− 4u
3
+ 5u
2
− 2u + 1
c
4
, c
10
, c
12
u
12
+ u
10
+ 12u
9
+ 21u
8
+ 30u
7
+ 38u
6
+ 39u
5
+ 33u
4
+ 4u
3
+ 4u + 4
c
7
u
12
− u
11
+ ··· − 184u + 141
c
8
u
12
− 15u
11
+ ··· − 576u + 96
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
12
+ 36y
11
+ ··· + 133y + 1
c
2
, c
5
, c
6
c
9
y
12
− 12y
11
+ ··· + y + 1
c
3
, c
11
y
12
− y
11
+ ··· + 6y + 1
c
4
, c
10
, c
12
y
12
+ 2y
11
+ ··· − 16y + 16
c
7
y
12
− 23y
11
+ ··· + 130832y + 19881
c
8
y
12
− 13y
11
+ ··· + 10752y + 9216
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.059120 + 0.307461I
a = −1.00000
b = 0.644594 − 0.475500I
−3.44334 + 1.44611I −3.96135 − 4.34110I
u = −1.059120 − 0.307461I
a = −1.00000
b = 0.644594 + 0.475500I
−3.44334 − 1.44611I −3.96135 + 4.34110I
u = 0.962226 + 0.662974I
a = −1.00000
b = 1.29438 + 1.01679I
−8.42695 − 4.69761I −5.39873 + 4.74294I
u = 0.962226 − 0.662974I
a = −1.00000
b = 1.29438 − 1.01679I
−8.42695 + 4.69761I −5.39873 − 4.74294I
u = 0.441542 + 0.466191I
a = −1.00000
b = 2.80322 − 1.33490I
7.39753 − 0.41539I 6.7393 + 12.7489I
u = 0.441542 − 0.466191I
a = −1.00000
b = 2.80322 + 1.33490I
7.39753 + 0.41539I 6.7393 − 12.7489I
u = −0.93584 + 1.09970I
a = −1.00000
b = 1.28519 − 0.74878I
−2.29460 + 6.83767I 0.069399 − 0.397217I
u = −0.93584 − 1.09970I
a = −1.00000
b = 1.28519 + 0.74878I
−2.29460 − 6.83767I 0.069399 + 0.397217I
u = 0.037712 + 0.516478I
a = −1.00000
b = 0.213856 + 0.762227I
0.825162 + 0.816210I 7.49070 − 5.15552I
u = 0.037712 − 0.516478I
a = −1.00000
b = 0.213856 − 0.762227I
0.825162 − 0.816210I 7.49070 + 5.15552I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.05347 + 1.22560I
a = −1.00000
b = 1.25877 + 1.59603I
10.0545 − 12.2992I 3.06065 + 5.11148I
u = 1.05347 − 1.22560I
a = −1.00000
b = 1.25877 − 1.59603I
10.0545 + 12.2992I 3.06065 − 5.11148I
6
II. I
u
2
= hu
4
+ u
3
+ 2u
2
+ b + u + 2, a + 1, u
5
+ u
4
+ 2u
3
+ 2u − 1i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
u
a
3
=
1
−u
2
a
9
=
−1
−u
4
− u
3
− 2u
2
− u − 2
a
8
=
−u
4
− u
3
− 2u
2
− u − 3
−u
4
− u
3
− 2u
2
− u − 2
a
1
=
u
4
+ 2u
3
+ 4u
2
+ 3u + 4
u
4
+ 2u
3
+ 3u
2
+ 3u + 3
a
7
=
u
3
+ u
2
+ u − 1
−u
4
− u
3
− 2u
2
− 3
a
11
=
u
u
2
+ u + 1
a
5
=
−u
3
− u
2
− u + 1
−u
4
− 2u
3
− 3u
2
− 2u − 1
a
2
=
−u
4
− u
3
− u
2
+ u
u
4
+ 2u
3
+ 3u
2
+ 3u + 4
a
6
=
0
−5u
4
− 7u
3
− 13u
2
− 5u − 12
a
10
=
u
4
+ u
3
+ u
2
− u
2u
4
+ 3u
3
+ 5u
2
+ u + 4
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
4
+ 10u
2
+ u + 10
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
5
+ 4u
4
− 10u
3
+ 8u
2
− 3u + 1
c
2
, c
6
u
5
+ 2u
4
+ 2u
2
− u + 1
c
3
, c
11
u
5
+ u
4
+ 2u
3
+ 2u − 1
c
4
, c
12
u
5
− u
4
+ 2u
3
− 3u
2
− 4
c
5
, c
9
u
5
− 2u
4
− 2u
2
− u − 1
c
7
u
5
− 2u
4
− 4u
3
− 3u
2
− 3u − 5
c
8
u
5
− u
4
− 5u
3
− 2u
2
+ 4u − 1
c
10
u
5
+ u
4
+ 2u
3
+ 3u
2
+ 4
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
5
− 36y
4
+ 30y
3
− 12y
2
− 7y −1
c
2
, c
5
, c
6
c
9
y
5
− 4y
4
− 10y
3
− 8y
2
− 3y −1
c
3
, c
11
y
5
+ 3y
4
+ 8y
3
+ 10y
2
+ 4y −1
c
4
, c
10
, c
12
y
5
+ 3y
4
− 2y
3
− 17y
2
− 24y −16
c
7
y
5
− 12y
4
− 2y
3
− 5y
2
− 21y −25
c
8
y
5
− 11y
4
+ 29y
3
− 46y
2
+ 12y −1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.205345 + 1.022070I
a = −1.00000
b = −0.394292 − 0.081621I
−4.76566 − 1.63339I 1.00951 + 4.37803I
u = 0.205345 − 1.022070I
a = −1.00000
b = −0.394292 + 0.081621I
−4.76566 + 1.63339I 1.00951 − 4.37803I
u = −0.91068 + 1.18795I
a = −1.00000
b = 1.31714 − 0.65774I
−2.30891 + 7.29116I −1.0723 − 17.9309I
u = −0.91068 − 1.18795I
a = −1.00000
b = 1.31714 + 0.65774I
−2.30891 − 7.29116I −1.0723 + 17.9309I
u = 0.410675
a = −1.00000
b = −2.84569
7.56942 12.1260
10
III. I
u
3
= h−28u
9
− 75u
8
+ · · · + 23b + 13, 20u
9
+ 70u
8
+ · · · + 23a +
17, u
10
+ 4u
9
+ · · · + 3u + 1i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
u
a
3
=
1
−u
2
a
9
=
−0.869565u
9
− 3.04348u
8
+ ··· − 9.73913u − 0.739130
1.21739u
9
+ 3.26087u
8
+ ··· + 0.434783u − 0.565217
a
8
=
0.347826u
9
+ 0.217391u
8
+ ··· − 9.30435u − 1.30435
1.21739u
9
+ 3.26087u
8
+ ··· + 0.434783u − 0.565217
a
1
=
−2.91304u
9
− 11.6957u
8
+ ··· − 20.8261u − 2.82609
−0.130435u
9
+ 0.0434783u
8
+ ··· − 1.26087u − 0.260870
a
7
=
−0.956522u
9
− 4.34783u
8
+ ··· − 12.9130u − 1.91304
1.04348u
9
+ 3.65217u
8
+ ··· + 1.08696u + 0.0869565
a
11
=
−1.13043u
9
− 5.95652u
8
+ ··· − 11.2609u − 0.260870
−1.65217u
9
− 5.78261u
8
+ ··· − 6.30435u − 2.30435
a
5
=
−2.82609u
9
− 9.39130u
8
+ ··· + 3.34783u + 2.34783
−0.565217u
9
− 1.47826u
8
+ ··· − 1.13043u + 0.869565
a
2
=
−3.30435u
9
− 13.5652u
8
+ ··· − 22.6087u − 2.60870
−0.782609u
9
− 1.73913u
8
+ ··· − 2.56522u − 0.565217
a
6
=
−2.65217u
9
− 9.78261u
8
+ ··· − 16.3043u − 6.30435
−0.304348u
9
+ 0.434783u
8
+ ··· + 1.39130u + 1.39130
a
10
=
−1.34783u
9
− 3.21739u
8
+ ··· + 3.30435u + 3.30435
1.95652u
9
+ 6.34783u
8
+ ··· + 3.91304u + 1.91304
(ii) Obstruction class = 1
(iii) Cusp Shapes
= −
122
23
u
9
−
289
23
u
8
−
139
23
u
7
+
297
23
u
6
−
107
23
u
5
−
835
23
u
4
−
755
23
u
3
−
75
23
u
2
+
101
23
u +
124
23
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
10
− 9u
9
+ 31u
8
− 56u
7
+ 73u
6
− 86u
5
+ 65u
4
− 12u
3
− 9u
2
+ 2u + 1
c
2
, c
6
u
10
+ u
9
+ 5u
8
+ 4u
7
+ 7u
6
+ 2u
5
+ 3u
4
− 4u
3
+ u
2
− 2u + 1
c
3
, c
11
u
10
+ 4u
9
+ 6u
8
+ 2u
7
− u
6
+ 7u
5
+ 18u
4
+ 17u
3
+ 9u
2
+ 3u + 1
c
4
, c
12
u
10
+ 3u
9
+ 6u
8
+ 8u
7
+ 7u
6
+ 2u
5
+ 6u
4
+ 5u
3
+ 15u
2
+ 7u + 7
c
5
, c
9
u
10
− u
9
+ 5u
8
− 4u
7
+ 7u
6
− 2u
5
+ 3u
4
+ 4u
3
+ u
2
+ 2u + 1
c
7
u
10
− 3u
9
+ ··· − 66u + 17
c
8
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
c
10
u
10
− 3u
9
+ 6u
8
− 8u
7
+ 7u
6
− 2u
5
+ 6u
4
− 5u
3
+ 15u
2
− 7u + 7
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
10
− 19y
9
+ ··· − 22y + 1
c
2
, c
5
, c
6
c
9
y
10
+ 9y
9
+ 31y
8
+ 56y
7
+ 73y
6
+ 86y
5
+ 65y
4
+ 12y
3
− 9y
2
− 2y + 1
c
3
, c
11
y
10
− 4y
9
+ 18y
8
− 36y
7
+ 71y
6
− 67y
5
+ 68y
4
− 9y
3
+ 15y
2
+ 9y + 1
c
4
, c
10
, c
12
y
10
+ 3y
9
+ ··· + 161y + 49
c
7
y
10
+ 21y
9
+ ··· + 302y + 289
c
8
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.637527 + 0.563270I
a = 1.53949 + 0.13288I
b = −1.23271 + 1.09381I
−7.51750 + 4.40083I 4.55516 − 1.78781I
u = −0.637527 − 0.563270I
a = 1.53949 − 0.13288I
b = −1.23271 − 1.09381I
−7.51750 − 4.40083I 4.55516 + 1.78781I
u = −1.108870 + 0.598693I
a = −0.548579 − 0.836099I
b = 0.588022
−4.04602 −7.96494 + 0.I
u = −1.108870 − 0.598693I
a = −0.548579 + 0.836099I
b = 0.588022
−4.04602 −7.96494 + 0.I
u = 1.056310 + 0.782435I
a = 0.644763 + 0.055651I
b = −1.23271 − 1.09381I
−7.51750 − 4.40083I 4.55516 + 1.78781I
u = 1.056310 − 0.782435I
a = 0.644763 − 0.055651I
b = −1.23271 + 1.09381I
−7.51750 + 4.40083I 4.55516 − 1.78781I
u = −0.008215 + 0.434693I
a = 2.20767 − 3.03625I
b = −0.561306 − 0.557752I
−1.97403 − 1.53058I 4.42731 + 4.45807I
u = −0.008215 − 0.434693I
a = 2.20767 + 3.03625I
b = −0.561306 + 0.557752I
−1.97403 + 1.53058I 4.42731 − 4.45807I
u = −1.30170 + 0.98460I
a = 0.156654 − 0.215449I
b = −0.561306 + 0.557752I
−1.97403 + 1.53058I 4.42731 − 4.45807I
u = −1.30170 − 0.98460I
a = 0.156654 + 0.215449I
b = −0.561306 − 0.557752I
−1.97403 − 1.53058I 4.42731 + 4.45807I
14
IV. I
u
4
= hb − 1, a + 1, u
2
+ u + 1i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
u
a
3
=
1
u + 1
a
9
=
−1
1
a
8
=
0
1
a
1
=
0
u
a
7
=
−1
−u
a
11
=
u
0
a
5
=
1
0
a
2
=
−u
u + 1
a
6
=
0
−u
a
10
=
u
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8u + 2
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
9
u
2
− u + 1
c
2
, c
3
, c
6
c
11
u
2
+ u + 1
c
4
, c
10
, c
12
u
2
c
7
, c
8
(u + 1)
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
9
c
11
y
2
+ y + 1
c
4
, c
10
, c
12
y
2
c
7
, c
8
(y −1)
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = −1.00000
b = 1.00000
4.05977I 6.00000 − 6.92820I
u = −0.500000 − 0.866025I
a = −1.00000
b = 1.00000
− 4.05977I 6.00000 + 6.92820I
18
V. I
u
5
= h38u
9
− 23u
8
+ · · · + 563b + 545, 412u
9
− 842u
8
+ · · · + 1689a −
3425, u
10
− 2u
9
+ · · · − 5u + 3i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
u
a
3
=
1
−u
2
a
9
=
−0.243931u
9
+ 0.498520u
8
+ ··· − 0.942570u + 2.02783
−0.0674956u
9
+ 0.0408526u
8
+ ··· − 0.785080u − 0.968028
a
8
=
−0.311427u
9
+ 0.539372u
8
+ ··· − 1.72765u + 1.05980
−0.0674956u
9
+ 0.0408526u
8
+ ··· − 0.785080u − 0.968028
a
1
=
−0.461220u
9
+ 1.11249u
8
+ ··· − 3.36471u + 2.88514
−0.346359u
9
+ 0.499112u
8
+ ··· − 0.765542u + 0.216696
a
7
=
−0.0485494u
9
+ 0.0118413u
8
+ ··· − 1.45944u + 1.77738
0.277087u
9
− 0.799290u
8
+ ··· + 0.0124334u − 0.973357
a
11
=
0.169331u
9
+ 0.178212u
8
+ ··· + 1.28538u + 1.84962
−0.284192u
9
+ 0.435169u
8
+ ··· − 1.88455u + 0.818828
a
5
=
−0.366489u
9
+ 0.967436u
8
+ ··· − 1.73653u + 0.612197
−0.165187u
9
+ 0.284192u
8
+ ··· − 1.02664u + 0.657194
a
2
=
−0.985198u
9
+ 1.50858u
8
+ ··· − 4.93310u + 3.23860
−0.884547u
9
+ 0.166963u
8
+ ··· + 0.921847u − 1.73890
a
6
=
−4.03848u
9
+ 6.27768u
8
+ ··· − 10.9739u + 2.57963
−1.03197u
9
− 0.138544u
8
+ ··· + 0.575488u − 4.19538
a
10
=
−1.07697u
9
+ 2.55536u
8
+ ··· − 1.94790u + 3.15927
−1.22025u
9
+ 1.71226u
8
+ ··· − 4.03552u + 0.209591
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
630
563
u
9
−
1211
563
u
8
+
2357
563
u
7
−
1855
563
u
6
+
5691
563
u
5
−
2147
563
u
4
+
6107
563
u
3
+
997
563
u
2
+
3861
563
u +
2724
563
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
10
− 19u
9
+ ··· + 2834u + 441
c
2
, c
5
, c
6
c
9
u
10
− 3u
9
+ ··· + 8u + 21
c
3
, c
11
u
10
− 2u
9
+ 4u
8
− 4u
7
+ 11u
6
− 7u
5
+ 14u
4
− 5u
3
+ 11u
2
− 5u + 3
c
4
, c
10
, c
12
u
10
+ u
9
+ ··· + 55u + 43
c
7
u
10
+ 5u
9
+ ··· + 248u + 59
c
8
(u
5
− u
4
+ u
2
+ u − 1)
2
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
10
+ 173y
9
+ ··· + 10034450y + 194481
c
2
, c
5
, c
6
c
9
y
10
− 19y
9
+ ··· + 2834y + 441
c
3
, c
11
y
10
+ 4y
9
+ ··· + 41y + 9
c
4
, c
10
, c
12
y
10
+ 7y
9
+ ··· − 875y + 1849
c
7
y
10
− 15y
9
+ ··· + 9178y + 3481
c
8
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)
2
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −0.597158 + 0.899620I
a = 0.542169 + 0.003339I
b = −0.758138 + 0.584034I
0.17487 + 2.21397I 5.11913 − 4.04855I
u = −0.597158 − 0.899620I
a = 0.542169 − 0.003339I
b = −0.758138 − 0.584034I
0.17487 − 2.21397I 5.11913 + 4.04855I
u = 0.453573 + 1.045560I
a = 0.50995 + 1.56353I
b = 0.935538 + 0.903908I
9.31336 − 3.33174I 4.71334 + 2.53508I
u = 0.453573 − 1.045560I
a = 0.50995 − 1.56353I
b = 0.935538 − 0.903908I
9.31336 + 3.33174I 4.71334 − 2.53508I
u = −0.586646 + 1.140630I
a = 0.581627 − 0.813455I
b = 0.645200
−2.52712 4.33506 + 0.I
u = −0.586646 − 1.140630I
a = 0.581627 + 0.813455I
b = 0.645200
−2.52712 4.33506 + 0.I
u = 0.326764 + 0.485752I
a = 1.84438 + 0.01136I
b = −0.758138 − 0.584034I
0.17487 − 2.21397I 5.11913 + 4.04855I
u = 0.326764 − 0.485752I
a = 1.84438 − 0.01136I
b = −0.758138 + 0.584034I
0.17487 + 2.21397I 5.11913 − 4.04855I
u = 1.40347 + 1.24236I
a = 0.188544 + 0.578083I
b = 0.935538 − 0.903908I
9.31336 + 3.33174I 4.71334 − 2.53508I
u = 1.40347 − 1.24236I
a = 0.188544 − 0.578083I
b = 0.935538 + 0.903908I
9.31336 − 3.33174I 4.71334 + 2.53508I
22
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)(u
5
+ 4u
4
− 10u
3
+ 8u
2
− 3u + 1)
· (u
10
− 19u
9
+ ··· + 2834u + 441)
· (u
10
− 9u
9
+ 31u
8
− 56u
7
+ 73u
6
− 86u
5
+ 65u
4
− 12u
3
− 9u
2
+ 2u + 1)
· (u
12
− 12u
11
+ ··· + u + 1)
c
2
, c
6
(u
2
+ u + 1)(u
5
+ 2u
4
+ 2u
2
− u + 1)(u
10
− 3u
9
+ ··· + 8u + 21)
· (u
10
+ u
9
+ 5u
8
+ 4u
7
+ 7u
6
+ 2u
5
+ 3u
4
− 4u
3
+ u
2
− 2u + 1)
· (u
12
− 6u
10
+ ··· − u + 1)
c
3
, c
11
(u
2
+ u + 1)(u
5
+ u
4
+ 2u
3
+ 2u − 1)
· (u
10
− 2u
9
+ 4u
8
− 4u
7
+ 11u
6
− 7u
5
+ 14u
4
− 5u
3
+ 11u
2
− 5u + 3)
· (u
10
+ 4u
9
+ 6u
8
+ 2u
7
− u
6
+ 7u
5
+ 18u
4
+ 17u
3
+ 9u
2
+ 3u + 1)
· (u
12
− u
11
+ 2u
9
+ 4u
8
− 4u
7
− 3u
6
+ 9u
5
+ u
4
− 4u
3
+ 5u
2
− 2u + 1)
c
4
, c
12
u
2
(u
5
− u
4
+ 2u
3
− 3u
2
− 4)(u
10
+ u
9
+ ··· + 55u + 43)
· (u
10
+ 3u
9
+ 6u
8
+ 8u
7
+ 7u
6
+ 2u
5
+ 6u
4
+ 5u
3
+ 15u
2
+ 7u + 7)
· (u
12
+ u
10
+ 12u
9
+ 21u
8
+ 30u
7
+ 38u
6
+ 39u
5
+ 33u
4
+ 4u
3
+ 4u + 4)
c
5
, c
9
(u
2
− u + 1)(u
5
− 2u
4
− 2u
2
− u − 1)(u
10
− 3u
9
+ ··· + 8u + 21)
· (u
10
− u
9
+ 5u
8
− 4u
7
+ 7u
6
− 2u
5
+ 3u
4
+ 4u
3
+ u
2
+ 2u + 1)
· (u
12
− 6u
10
+ ··· − u + 1)
c
7
((u + 1)
2
)(u
5
− 2u
4
+ ··· − 3u − 5)(u
10
− 3u
9
+ ··· − 66u + 17)
· (u
10
+ 5u
9
+ ··· + 248u + 59)(u
12
− u
11
+ ··· − 184u + 141)
c
8
(u + 1)
2
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
(u
5
− u
4
+ u
2
+ u − 1)
2
· (u
5
− u
4
− 5u
3
− 2u
2
+ 4u − 1)(u
12
− 15u
11
+ ··· − 576u + 96)
c
10
u
2
(u
5
+ u
4
+ 2u
3
+ 3u
2
+ 4)
· (u
10
− 3u
9
+ 6u
8
− 8u
7
+ 7u
6
− 2u
5
+ 6u
4
− 5u
3
+ 15u
2
− 7u + 7)
· (u
10
+ u
9
+ ··· + 55u + 43)
· (u
12
+ u
10
+ 12u
9
+ 21u
8
+ 30u
7
+ 38u
6
+ 39u
5
+ 33u
4
+ 4u
3
+ 4u + 4)
23
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
2
+ y + 1)(y
5
− 36y
4
+ 30y
3
− 12y
2
− 7y −1)
· (y
10
− 19y
9
+ ··· − 22y + 1)
· (y
10
+ 173y
9
+ ··· + 10034450y + 194481)
· (y
12
+ 36y
11
+ ··· + 133y + 1)
c
2
, c
5
, c
6
c
9
(y
2
+ y + 1)(y
5
− 4y
4
− 10y
3
− 8y
2
− 3y −1)
· (y
10
− 19y
9
+ ··· + 2834y + 441)
· (y
10
+ 9y
9
+ 31y
8
+ 56y
7
+ 73y
6
+ 86y
5
+ 65y
4
+ 12y
3
− 9y
2
− 2y + 1)
· (y
12
− 12y
11
+ ··· + y + 1)
c
3
, c
11
(y
2
+ y + 1)(y
5
+ 3y
4
+ 8y
3
+ 10y
2
+ 4y −1)
· (y
10
− 4y
9
+ 18y
8
− 36y
7
+ 71y
6
− 67y
5
+ 68y
4
− 9y
3
+ 15y
2
+ 9y + 1)
· (y
10
+ 4y
9
+ ··· + 41y + 9)(y
12
− y
11
+ ··· + 6y + 1)
c
4
, c
10
, c
12
y
2
(y
5
+ 3y
4
+ ··· − 24y −16)(y
10
+ 3y
9
+ ··· + 161y + 49)
· (y
10
+ 7y
9
+ ··· − 875y + 1849)(y
12
+ 2y
11
+ ··· − 16y + 16)
c
7
(y −1)
2
(y
5
− 12y
4
− 2y
3
− 5y
2
− 21y −25)
· (y
10
− 15y
9
+ ··· + 9178y + 3481)(y
10
+ 21y
9
+ ··· + 302y + 289)
· (y
12
− 23y
11
+ ··· + 130832y + 19881)
c
8
(y −1)
2
(y
5
− 11y
4
+ 29y
3
− 46y
2
+ 12y −1)
· (y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)
2
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
· (y
12
− 13y
11
+ ··· + 10752y + 9216)
24
