12n
0805
(K12n
0805
)
A knot diagram
1
Linearized knot diagam
4 7 10 1 11 3 1 12 6 4 9 8
Solving Sequence
9,12
8
1,4
2 5 7 11 6 10 3
c
8
c
12
c
1
c
4
c
7
c
11
c
5
c
10
c
3
c
2
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−3u
21
− 14u
20
+ ··· + 2b − 6, −5u
21
− 28u
20
+ ··· + 4a − 36, u
22
+ 6u
21
+ ··· + 22u + 4i
I
u
2
= h−7988u
7
a
3
+ 5153u
7
a
2
+ ··· − 10905a + 11031, u
7
a
2
+ 4u
7
a + ··· − 2a + 7,
u
8
− u
7
+ 5u
6
− 4u
5
+ 7u
4
− 4u
3
+ 2u
2
+ 1i
I
u
3
= h−u
11
+ u
10
− 8u
9
+ 7u
8
− 23u
7
+ 17u
6
− 29u
5
+ 16u
4
− 16u
3
+ 3u
2
+ b − 3u − 1,
− u
10
+ 2u
9
− 9u
8
+ 14u
7
− 29u
6
+ 33u
5
− 40u
4
+ 30u
3
− 22u
2
+ a + 9u − 3,
u
12
− u
11
+ 9u
10
− 8u
9
+ 30u
8
− 23u
7
+ 45u
6
− 28u
5
+ 30u
4
− 12u
3
+ 9u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 66 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−3u
21
− 14u
20
+ · · · + 2b − 6, −5u
21
− 28u
20
+ · · · + 4a − 36, u
22
+
6u
21
+ · · · + 22u + 4i
(i) Arc colorings
a
9
=
1
0
a
12
=
0
u
a
8
=
1
−u
2
a
1
=
−u
u
3
+ u
a
4
=
5
4
u
21
+ 7u
20
+ ··· +
125
4
u + 9
3
2
u
21
+ 7u
20
+ ··· +
29
2
u + 3
a
2
=
−
1
2
u
21
−
5
2
u
20
+ ··· −
3
2
u +
5
2
−
1
2
u
21
− 3u
20
+ ··· − 20u
2
−
11
2
u
a
5
=
5
4
u
21
+ 7u
20
+ ··· +
109
4
u + 7
1
2
u
21
+ 4u
20
+ ··· +
37
2
u + 5
a
7
=
u
2
+ 1
−u
4
− 2u
2
a
11
=
u
u
a
6
=
−
3
4
u
21
− 4u
20
+ ··· −
11
4
u + 1
−
3
2
u
21
− 7u
20
+ ··· −
23
2
u − 1
a
10
=
−
1
2
u
21
−
5
2
u
20
+ ··· −
3
2
u +
1
2
1
2
u
21
+ 3u
20
+ ··· +
37
2
u + 4
a
3
=
1
2
u
20
+ 3u
19
+ ··· + 18u +
13
2
−
1
2
u
21
− 2u
20
+ ··· − 6u
2
−
3
2
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 5u
21
+ 27u
20
+ 137u
19
+ 457u
18
+ 1341u
17
+ 3187u
16
+ 6649u
15
+
11918u
14
+ 18804u
13
+ 25932u
12
+ 31399u
11
+ 33139u
10
+ 30322u
9
+ 23720u
8
+
15633u
7
+ 8484u
6
+ 3756u
5
+ 1394u
4
+ 536u
3
+ 233u
2
+ 84u + 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
22
− u
21
+ ··· − 5u + 1
c
2
, c
6
u
22
+ 17u
21
+ ··· + 2816u + 256
c
3
, c
9
, c
10
u
22
− u
21
+ ··· − u + 1
c
5
u
22
+ 4u
20
+ ··· − 4u + 1
c
7
, c
8
, c
11
c
12
u
22
− 6u
21
+ ··· − 22u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
22
− 23y
21
+ ··· + 13y + 1
c
2
, c
6
y
22
+ 9y
21
+ ··· + 393216y + 65536
c
3
, c
9
, c
10
y
22
− 13y
21
+ ··· − y + 1
c
5
y
22
+ 8y
21
+ ··· + 12y + 1
c
7
, c
8
, c
11
c
12
y
22
+ 26y
21
+ ··· + 36y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.672231 + 0.669430I
a = −0.646053 + 0.197647I
b = −1.12221 − 1.29155I
1.57285 + 10.92870I −0.09523 − 8.13846I
u = −0.672231 − 0.669430I
a = −0.646053 − 0.197647I
b = −1.12221 + 1.29155I
1.57285 − 10.92870I −0.09523 + 8.13846I
u = −0.535508 + 0.711181I
a = 0.314853 + 0.241466I
b = 0.606532 + 1.248090I
−1.54704 + 3.14824I 0.77612 − 6.02980I
u = −0.535508 − 0.711181I
a = 0.314853 − 0.241466I
b = 0.606532 − 1.248090I
−1.54704 − 3.14824I 0.77612 + 6.02980I
u = −0.759597 + 0.320799I
a = −1.058800 − 0.555950I
b = 0.598202 − 0.762166I
0.54148 − 6.16925I −2.10987 + 3.70957I
u = −0.759597 − 0.320799I
a = −1.058800 + 0.555950I
b = 0.598202 + 0.762166I
0.54148 + 6.16925I −2.10987 − 3.70957I
u = −0.288943 + 1.232890I
a = 0.412257 + 0.459691I
b = −0.222633 − 0.193104I
5.51465 − 2.42737I 0.51879 + 3.32610I
u = −0.288943 − 1.232890I
a = 0.412257 − 0.459691I
b = −0.222633 + 0.193104I
5.51465 + 2.42737I 0.51879 − 3.32610I
u = 0.068378 + 1.330140I
a = 0.746060 − 0.076338I
b = 0.327439 − 0.582001I
5.16986 − 2.20242I 0.58597 + 3.53079I
u = 0.068378 − 1.330140I
a = 0.746060 + 0.076338I
b = 0.327439 + 0.582001I
5.16986 + 2.20242I 0.58597 − 3.53079I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.540881 + 0.284720I
a = 1.42566 + 0.38957I
b = 0.256071 + 0.621077I
−2.84705 + 0.58927I −4.02070 + 2.06270I
u = −0.540881 − 0.284720I
a = 1.42566 − 0.38957I
b = 0.256071 − 0.621077I
−2.84705 − 0.58927I −4.02070 − 2.06270I
u = −0.11745 + 1.45301I
a = −1.65572 − 0.36277I
b = −1.100210 − 0.121955I
2.80552 + 2.73021I −1.72854 − 0.56419I
u = −0.11745 − 1.45301I
a = −1.65572 + 0.36277I
b = −1.100210 + 0.121955I
2.80552 − 2.73021I −1.72854 + 0.56419I
u = 0.276614 + 0.310933I
a = −0.535008 − 0.626026I
b = −0.121326 + 0.413923I
−0.116282 − 0.855526I −2.84227 + 8.01769I
u = 0.276614 − 0.310933I
a = −0.535008 + 0.626026I
b = −0.121326 − 0.413923I
−0.116282 + 0.855526I −2.84227 − 8.01769I
u = −0.21959 + 1.59057I
a = 1.99989 + 0.71251I
b = 1.71567 + 1.64082I
9.0925 + 14.2705I 2.92033 − 7.26786I
u = −0.21959 − 1.59057I
a = 1.99989 − 0.71251I
b = 1.71567 − 1.64082I
9.0925 − 14.2705I 2.92033 + 7.26786I
u = −0.17706 + 1.62106I
a = −1.41137 − 1.11105I
b = −1.29958 − 1.77102I
6.38360 + 5.88492I 3.26398 − 4.41155I
u = −0.17706 − 1.62106I
a = −1.41137 + 1.11105I
b = −1.29958 + 1.77102I
6.38360 − 5.88492I 3.26398 + 4.41155I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.03373 + 1.75344I
a = 0.158232 + 0.724477I
b = 0.362036 + 1.269890I
16.1982 − 1.4224I −2.26858 + 6.72406I
u = −0.03373 − 1.75344I
a = 0.158232 − 0.724477I
b = 0.362036 − 1.269890I
16.1982 + 1.4224I −2.26858 − 6.72406I
7
II. I
u
2
= h−7988u
7
a
3
+ 5153u
7
a
2
+ · · · − 10905a + 11031, u
7
a
2
+ 4u
7
a +
· · · − 2a + 7, u
8
− u
7
+ 5u
6
− 4u
5
+ 7u
4
− 4u
3
+ 2u
2
+ 1i
(i) Arc colorings
a
9
=
1
0
a
12
=
0
u
a
8
=
1
−u
2
a
1
=
−u
u
3
+ u
a
4
=
a
0.148683a
3
u
7
− 0.0959144a
2
u
7
+ ··· + 0.202978a − 0.205323
a
2
=
0.368581a
3
u
7
+ 0.700102a
2
u
7
+ ··· − 0.361657a − 0.327110
0.140679a
3
u
7
+ 0.558902a
2
u
7
+ ··· − 0.629502a + 0.738092
a
5
=
−0.136194a
3
u
7
+ 0.0847278a
2
u
7
+ ··· + 1.22531a + 0.554751
0.227231a
3
u
7
− 0.0604560a
2
u
7
+ ··· + 1.39283a − 0.244691
a
7
=
u
2
+ 1
−u
4
− 2u
2
a
11
=
u
u
a
6
=
−0.725305a
3
u
7
− 0.681210a
2
u
7
+ ··· + 0.801396a + 0.350005
−0.361880a
3
u
7
− 0.826394a
2
u
7
+ ··· + 0.968916a − 0.449437
a
10
=
−0.368581a
3
u
7
− 0.700102a
2
u
7
+ ··· + 0.361657a + 0.327110
0.0635458a
3
u
7
− 0.209567a
2
u
7
+ ··· + 0.980270a − 0.811875
a
3
=
−0.463993a
3
u
7
+ 0.141796a
2
u
7
+ ··· − 0.563611a + 0.158883
−0.312611a
3
u
7
− 0.467287a
2
u
7
+ ··· + 0.616101a + 1.29158
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
6416
7675
u
7
a
3
−
10404
7675
u
7
a
2
+ ··· +
4668
1535
a +
12542
7675
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
32
− 7u
31
+ ··· − 110u + 37
c
2
, c
6
(u
2
− u + 1)
16
c
3
, c
9
, c
10
u
32
+ u
31
+ ··· + 300u + 343
c
5
u
32
+ u
31
+ ··· − 2776u + 343
c
7
, c
8
, c
11
c
12
(u
8
+ u
7
+ 5u
6
+ 4u
5
+ 7u
4
+ 4u
3
+ 2u
2
+ 1)
4
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
32
− 5y
31
+ ··· − 28528y + 1369
c
2
, c
6
(y
2
+ y + 1)
16
c
3
, c
9
, c
10
y
32
− 21y
31
+ ··· − 1386540y + 117649
c
5
y
32
− 13y
31
+ ··· − 8932744y + 117649
c
7
, c
8
, c
11
c
12
(y
8
+ 9y
7
+ 31y
6
+ 50y
5
+ 39y
4
+ 22y
3
+ 18y
2
+ 4y + 1)
4
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.647085 + 0.502738I
a = 0.592287 − 0.833358I
b = −0.453639 − 0.557953I
−1.67479 − 0.15547I −1.58319 − 0.32355I
u = 0.647085 + 0.502738I
a = −1.270790 + 0.209688I
b = −0.057820 + 1.153500I
−1.67479 − 4.21524I −1.58319 + 6.60465I
u = 0.647085 + 0.502738I
a = −0.630581 − 0.097668I
b = −0.355056 + 1.312410I
−1.67479 − 0.15547I −1.58319 − 0.32355I
u = 0.647085 + 0.502738I
a = 0.483643 + 0.288989I
b = 1.115550 − 0.830375I
−1.67479 − 4.21524I −1.58319 + 6.60465I
u = 0.647085 − 0.502738I
a = 0.592287 + 0.833358I
b = −0.453639 + 0.557953I
−1.67479 + 0.15547I −1.58319 + 0.32355I
u = 0.647085 − 0.502738I
a = −1.270790 − 0.209688I
b = −0.057820 − 1.153500I
−1.67479 + 4.21524I −1.58319 − 6.60465I
u = 0.647085 − 0.502738I
a = −0.630581 + 0.097668I
b = −0.355056 − 1.312410I
−1.67479 + 0.15547I −1.58319 + 0.32355I
u = 0.647085 − 0.502738I
a = 0.483643 − 0.288989I
b = 1.115550 + 0.830375I
−1.67479 + 4.21524I −1.58319 − 6.60465I
u = −0.283060 + 0.443755I
a = 0.498434 + 1.041700I
b = −0.57514 − 1.33218I
4.93480 + 3.07589I 2.00000 − 10.14955I
u = −0.283060 + 0.443755I
a = −0.537114 + 1.186060I
b = −1.91169 − 0.11910I
4.93480 − 0.98388I 2.00000 − 3.22135I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.283060 + 0.443755I
a = −1.55933 − 2.06959I
b = 0.389762 + 0.054060I
4.93480 − 0.98388I 2.00000 − 3.22135I
u = −0.283060 + 0.443755I
a = 1.31495 − 2.41550I
b = 1.392430 + 0.046674I
4.93480 + 3.07589I 2.00000 − 10.14955I
u = −0.283060 − 0.443755I
a = 0.498434 − 1.041700I
b = −0.57514 + 1.33218I
4.93480 − 3.07589I 2.00000 + 10.14955I
u = −0.283060 − 0.443755I
a = −0.537114 − 1.186060I
b = −1.91169 + 0.11910I
4.93480 + 0.98388I 2.00000 + 3.22135I
u = −0.283060 − 0.443755I
a = −1.55933 + 2.06959I
b = 0.389762 − 0.054060I
4.93480 + 0.98388I 2.00000 + 3.22135I
u = −0.283060 − 0.443755I
a = 1.31495 + 2.41550I
b = 1.392430 − 0.046674I
4.93480 − 3.07589I 2.00000 + 10.14955I
u = −0.06382 + 1.51723I
a = −0.112835 + 0.153959I
b = −0.343676 − 0.971058I
11.54440 + 0.15547I 5.58319 + 0.32355I
u = −0.06382 + 1.51723I
a = 0.82329 + 1.83665I
b = 0.76874 + 2.76112I
11.54440 + 4.21524I 5.58319 − 6.60465I
u = −0.06382 + 1.51723I
a = −2.24271 + 0.60948I
b = −1.65098 − 0.26388I
11.54440 + 4.21524I 5.58319 − 6.60465I
u = −0.06382 + 1.51723I
a = 2.94096 − 0.14777I
b = 2.94748 + 0.48648I
11.54440 + 0.15547I 5.58319 + 0.32355I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.06382 − 1.51723I
a = −0.112835 − 0.153959I
b = −0.343676 + 0.971058I
11.54440 − 0.15547I 5.58319 − 0.32355I
u = −0.06382 − 1.51723I
a = 0.82329 − 1.83665I
b = 0.76874 − 2.76112I
11.54440 − 4.21524I 5.58319 + 6.60465I
u = −0.06382 − 1.51723I
a = −2.24271 − 0.60948I
b = −1.65098 + 0.26388I
11.54440 − 4.21524I 5.58319 + 6.60465I
u = −0.06382 − 1.51723I
a = 2.94096 + 0.14777I
b = 2.94748 − 0.48648I
11.54440 − 0.15547I 5.58319 − 0.32355I
u = 0.19980 + 1.51366I
a = −0.413833 + 0.373606I
b = −0.166931 + 0.023712I
4.93480 − 3.20880I 2.00000 − 0.42152I
u = 0.19980 + 1.51366I
a = 1.37518 − 0.69793I
b = 0.87012 − 1.31414I
4.93480 − 3.20880I 2.00000 − 0.42152I
u = 0.19980 + 1.51366I
a = 1.34760 − 0.78701I
b = 0.439335 − 0.833174I
4.93480 − 7.26857I 2.00000 + 6.50668I
u = 0.19980 + 1.51366I
a = −2.10915 + 0.11662I
b = −1.90847 + 0.86941I
4.93480 − 7.26857I 2.00000 + 6.50668I
u = 0.19980 − 1.51366I
a = −0.413833 − 0.373606I
b = −0.166931 − 0.023712I
4.93480 + 3.20880I 2.00000 + 0.42152I
u = 0.19980 − 1.51366I
a = 1.37518 + 0.69793I
b = 0.87012 + 1.31414I
4.93480 + 3.20880I 2.00000 + 0.42152I
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.19980 − 1.51366I
a = 1.34760 + 0.78701I
b = 0.439335 + 0.833174I
4.93480 + 7.26857I 2.00000 − 6.50668I
u = 0.19980 − 1.51366I
a = −2.10915 − 0.11662I
b = −1.90847 − 0.86941I
4.93480 + 7.26857I 2.00000 − 6.50668I
14
III.
I
u
3
= h−u
11
+u
10
+· · ·+b−1, −u
10
+2u
9
+· · ·+a−3, u
12
−u
11
+· · ·+9u
2
+1i
(i) Arc colorings
a
9
=
1
0
a
12
=
0
u
a
8
=
1
−u
2
a
1
=
−u
u
3
+ u
a
4
=
u
10
− 2u
9
+ 9u
8
− 14u
7
+ 29u
6
− 33u
5
+ 40u
4
− 30u
3
+ 22u
2
− 9u + 3
u
11
− u
10
+ ··· + 3u + 1
a
2
=
−u
11
+ u
10
− 8u
9
+ 6u
8
− 22u
7
+ 11u
6
− 23u
5
+ 5u
4
− 6u
3
− 4u
2
− 4
−u
11
+ 2u
10
+ ··· − 4u + 1
a
5
=
−u
9
+ 2u
8
− 8u
7
+ 13u
6
− 22u
5
+ 27u
4
− 24u
3
+ 19u
2
− 9u + 3
u
11
− 2u
10
+ ··· − 9u
2
+ 3u
a
7
=
u
2
+ 1
−u
4
− 2u
2
a
11
=
u
u
a
6
=
u
10
− 2u
9
+ ··· − 10u + 4
u
11
− u
10
+ ··· + 2u + 1
a
10
=
u
11
− u
10
+ ··· + 12u − 4
−u
9
+ u
8
− 7u
7
+ 6u
6
− 16u
5
+ 11u
4
− 13u
3
+ 6u
2
− 3u − 1
a
3
=
−u
11
− 7u
9
− u
8
− 16u
7
− 6u
6
− 11u
5
− 12u
4
+ 3u
3
− 11u
2
+ 2u − 5
−u
11
+ u
10
+ ··· + 6u
2
− 4u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
8
+ 17u
6
− u
5
+ 29u
4
− 3u
3
+ 14u
2
− 2u + 7
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
12
− u
11
− 4u
9
+ 6u
8
− 2u
7
+ 6u
6
− 9u
5
+ 3u
4
− 4u
3
+ 4u
2
+ 1
c
2
u
12
+ 4u
10
− 4u
9
+ 3u
8
− 9u
7
+ 6u
6
− 2u
5
+ 6u
4
− 4u
3
− u + 1
c
3
, c
9
u
12
+ u
11
− 5u
10
− 5u
9
+ 10u
8
+ 10u
7
− 8u
6
− 9u
5
+ u
4
+ 3u
3
+ u
2
+ 1
c
4
u
12
+ u
11
+ 4u
9
+ 6u
8
+ 2u
7
+ 6u
6
+ 9u
5
+ 3u
4
+ 4u
3
+ 4u
2
+ 1
c
5
u
12
− u
10
+ ··· + 15u + 37
c
6
u
12
+ 4u
10
+ 4u
9
+ 3u
8
+ 9u
7
+ 6u
6
+ 2u
5
+ 6u
4
+ 4u
3
+ u + 1
c
7
, c
8
u
12
− u
11
+ ··· + 9u
2
+ 1
c
10
u
12
− u
11
− 5u
10
+ 5u
9
+ 10u
8
− 10u
7
− 8u
6
+ 9u
5
+ u
4
− 3u
3
+ u
2
+ 1
c
11
, c
12
u
12
+ u
11
+ ··· + 9u
2
+ 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
12
− y
11
+ ··· + 8y + 1
c
2
, c
6
y
12
+ 8y
11
+ ··· − y + 1
c
3
, c
9
, c
10
y
12
− 11y
11
+ ··· + 2y + 1
c
5
y
12
− 2y
11
+ ··· − 373y + 1369
c
7
, c
8
, c
11
c
12
y
12
+ 17y
11
+ ··· + 18y + 1
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.089013 + 0.906809I
a = −0.411424 + 0.831833I
b = −0.881210 − 0.394305I
7.14530 − 1.50712I 6.80370 + 0.78584I
u = −0.089013 − 0.906809I
a = −0.411424 − 0.831833I
b = −0.881210 + 0.394305I
7.14530 + 1.50712I 6.80370 − 0.78584I
u = 0.560380 + 0.536459I
a = −0.646516 + 0.336204I
b = −0.397396 + 0.941015I
−2.38168 − 1.90999I −3.16998 + 3.64025I
u = 0.560380 − 0.536459I
a = −0.646516 − 0.336204I
b = −0.397396 − 0.941015I
−2.38168 + 1.90999I −3.16998 − 3.64025I
u = −0.03756 + 1.51189I
a = −1.91798 − 0.42547I
b = −1.68049 − 1.38079I
11.54940 + 2.73635I 5.59363 − 1.06316I
u = −0.03756 − 1.51189I
a = −1.91798 + 0.42547I
b = −1.68049 + 1.38079I
11.54940 − 2.73635I 5.59363 + 1.06316I
u = 0.19341 + 1.55032I
a = 1.29683 − 0.61636I
b = 0.971343 − 0.961955I
4.61882 − 4.69602I 0.54451 + 4.43635I
u = 0.19341 − 1.55032I
a = 1.29683 + 0.61636I
b = 0.971343 + 0.961955I
4.61882 + 4.69602I 0.54451 − 4.43635I
u = −0.106406 + 0.331105I
a = 1.14337 − 3.27977I
b = 1.29750 + 0.61995I
5.16484 + 2.20517I 5.88472 − 1.20023I
u = −0.106406 − 0.331105I
a = 1.14337 + 3.27977I
b = 1.29750 − 0.61995I
5.16484 − 2.20517I 5.88472 + 1.20023I
18
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.02080 + 1.72148I
a = 0.535722 + 0.647392I
b = 0.69025 + 1.26677I
16.6716 − 1.0667I 9.34342 − 1.65312I
u = −0.02080 − 1.72148I
a = 0.535722 − 0.647392I
b = 0.69025 − 1.26677I
16.6716 + 1.0667I 9.34342 + 1.65312I
19
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
12
− u
11
− 4u
9
+ 6u
8
− 2u
7
+ 6u
6
− 9u
5
+ 3u
4
− 4u
3
+ 4u
2
+ 1)
· (u
22
− u
21
+ ··· − 5u + 1)(u
32
− 7u
31
+ ··· − 110u + 37)
c
2
(u
2
− u + 1)
16
· (u
12
+ 4u
10
− 4u
9
+ 3u
8
− 9u
7
+ 6u
6
− 2u
5
+ 6u
4
− 4u
3
− u + 1)
· (u
22
+ 17u
21
+ ··· + 2816u + 256)
c
3
, c
9
(u
12
+ u
11
− 5u
10
− 5u
9
+ 10u
8
+ 10u
7
− 8u
6
− 9u
5
+ u
4
+ 3u
3
+ u
2
+ 1)
· (u
22
− u
21
+ ··· − u + 1)(u
32
+ u
31
+ ··· + 300u + 343)
c
4
(u
12
+ u
11
+ 4u
9
+ 6u
8
+ 2u
7
+ 6u
6
+ 9u
5
+ 3u
4
+ 4u
3
+ 4u
2
+ 1)
· (u
22
− u
21
+ ··· − 5u + 1)(u
32
− 7u
31
+ ··· − 110u + 37)
c
5
(u
12
− u
10
+ ··· + 15u + 37)(u
22
+ 4u
20
+ ··· − 4u + 1)
· (u
32
+ u
31
+ ··· − 2776u + 343)
c
6
(u
2
− u + 1)
16
· (u
12
+ 4u
10
+ 4u
9
+ 3u
8
+ 9u
7
+ 6u
6
+ 2u
5
+ 6u
4
+ 4u
3
+ u + 1)
· (u
22
+ 17u
21
+ ··· + 2816u + 256)
c
7
, c
8
((u
8
+ u
7
+ ··· + 2u
2
+ 1)
4
)(u
12
− u
11
+ ··· + 9u
2
+ 1)
· (u
22
− 6u
21
+ ··· − 22u + 4)
c
10
(u
12
− u
11
− 5u
10
+ 5u
9
+ 10u
8
− 10u
7
− 8u
6
+ 9u
5
+ u
4
− 3u
3
+ u
2
+ 1)
· (u
22
− u
21
+ ··· − u + 1)(u
32
+ u
31
+ ··· + 300u + 343)
c
11
, c
12
((u
8
+ u
7
+ ··· + 2u
2
+ 1)
4
)(u
12
+ u
11
+ ··· + 9u
2
+ 1)
· (u
22
− 6u
21
+ ··· − 22u + 4)
20
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
12
− y
11
+ ··· + 8y + 1)(y
22
− 23y
21
+ ··· + 13y + 1)
· (y
32
− 5y
31
+ ··· − 28528y + 1369)
c
2
, c
6
((y
2
+ y + 1)
16
)(y
12
+ 8y
11
+ ··· − y + 1)
· (y
22
+ 9y
21
+ ··· + 393216y + 65536)
c
3
, c
9
, c
10
(y
12
− 11y
11
+ ··· + 2y + 1)(y
22
− 13y
21
+ ··· − y + 1)
· (y
32
− 21y
31
+ ··· − 1386540y + 117649)
c
5
(y
12
− 2y
11
+ ··· − 373y + 1369)(y
22
+ 8y
21
+ ··· + 12y + 1)
· (y
32
− 13y
31
+ ··· − 8932744y + 117649)
c
7
, c
8
, c
11
c
12
(y
8
+ 9y
7
+ 31y
6
+ 50y
5
+ 39y
4
+ 22y
3
+ 18y
2
+ 4y + 1)
4
· (y
12
+ 17y
11
+ ··· + 18y + 1)(y
22
+ 26y
21
+ ··· + 36y + 16)
21
