12a
0923
(K12a
0923
)
A knot diagram
1
Linearized knot diagam
4 6 9 11 2 10 12 1 3 5 7 8
Solving Sequence
2,5
6
3,11
4 1 10 7 12 9 8
c
5
c
2
c
4
c
1
c
10
c
6
c
11
c
9
c
8
c
3
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−169316u
55
+ 765227u
54
+ ··· + 1990656b + 207504590,
64741301u
55
− 347996846u
54
+ ··· + 709337088a − 165503765534,
u
56
− 6u
55
+ ··· − 5065u + 2138i
I
u
2
= hu
5
+ b + u, −7u
5
− 2u
4
+ u
3
+ u
2
+ 5a − 8u − 3, u
6
+ u
4
+ 2u
2
+ 1i
I
u
3
= h−a
2
+ b − a, a
3
+ 2a
2
+ a − 1, u + 1i
I
u
4
= hb
4
a
2
− 2b
3
a + 2b
2
a
2
− b
2
a + b
2
− 2ba + a
2
+ b − a − 1, u + 1i
I
v
1
= ha, b
6
+ 2b
4
+ b
3
+ b
2
+ b − 1, v − 1i
* 4 irreducible components of dim
C
= 0, with total 71 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−1.69×10
5
u
55
+7.65×10
5
u
54
+· · · +1.99×10
6
b+2.08×10
8
, 6.47×10
7
u
55
−
3.48 × 10
8
u
54
+ · · · + 7.09 × 10
8
a − 1.66 × 10
11
, u
56
− 6u
55
+ · · · − 5065u + 2138i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
3
+ u
a
11
=
−0.0912701u
55
+ 0.490594u
54
+ ··· − 305.555u + 233.322
0.0850554u
55
− 0.384409u
54
+ ··· + 202.825u − 104.239
a
4
=
−0.0214370u
55
+ 0.107795u
54
+ ··· − 64.6193u + 46.3327
0.00435384u
55
− 0.0207994u
54
+ ··· + 12.3366u − 6.37981
a
1
=
0.0281056u
55
− 0.145026u
54
+ ··· + 85.8717u − 67.5545
−0.0278343u
55
+ 0.123449u
54
+ ··· − 62.7416u + 34.1313
a
10
=
−0.00621477u
55
+ 0.106185u
54
+ ··· − 102.730u + 129.082
0.0850554u
55
− 0.384409u
54
+ ··· + 202.825u − 104.239
a
7
=
0.0442234u
55
− 0.241015u
54
+ ··· + 145.317u − 135.156
−0.0182020u
55
+ 0.0910577u
54
+ ··· − 54.3352u + 39.5251
a
12
=
0.111606u
55
− 0.585341u
54
+ ··· + 348.486u − 276.321
0.0596452u
55
− 0.316842u
54
+ ··· + 200.960u − 123.452
a
9
=
0.0627283u
55
− 0.266879u
54
+ ··· + 135.607u − 49.3441
0.0142003u
55
− 0.0564472u
54
+ ··· + 22.6996u − 12.6041
a
8
=
0.180172u
55
− 0.822387u
54
+ ··· + 451.374u − 189.430
0.0792387u
55
− 0.456648u
54
+ ··· + 304.815u − 239.788
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
563051
2985984
u
55
+
2308397
2985984
u
54
+ ··· −
186951611
497664
u +
9690719
186624
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
16(16u
56
− 48u
55
+ ··· + 388278u + 882567)
c
2
, c
5
u
56
− 6u
55
+ ··· − 5065u + 2138
c
3
, c
9
9(9u
56
− 9u
55
+ ··· + 80u + 25)
c
4
, c
10
9(9u
56
− 9u
55
+ ··· − 50u + 25)
c
6
16(16u
56
+ 32u
55
+ ··· + 735282u + 119709)
c
7
, c
8
, c
11
c
12
u
56
+ 4u
55
+ ··· − 7u + 62
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
256
· (256y
56
− 11136y
55
+ ··· − 10663154787870y + 778924509489)
c
2
, c
5
y
56
− 32y
55
+ ··· − 6390845y + 4571044
c
3
, c
9
81(81y
56
+ 4239y
55
+ ··· + 12400y + 625)
c
4
, c
10
81(81y
56
+ 1971y
55
+ ··· + 16400y + 625)
c
6
256(256y
56
− 10624y
55
+ ··· − 1.77562 × 10
11
y + 1.43302 × 10
10
)
c
7
, c
8
, c
11
c
12
y
56
− 66y
55
+ ··· − 2901y + 3844
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.984706 + 0.213928I
a = 2.54376 − 1.38909I
b = 0.219398 + 0.692771I
−10.33120 − 0.88863I −11.9568 + 8.3028I
u = 0.984706 − 0.213928I
a = 2.54376 + 1.38909I
b = 0.219398 − 0.692771I
−10.33120 + 0.88863I −11.9568 − 8.3028I
u = 0.981741 + 0.367067I
a = −1.39029 + 1.32174I
b = −0.297204 − 0.885770I
−1.84811 − 1.66866I −10.40782 + 3.06587I
u = 0.981741 − 0.367067I
a = −1.39029 − 1.32174I
b = −0.297204 + 0.885770I
−1.84811 + 1.66866I −10.40782 − 3.06587I
u = −0.392152 + 0.823059I
a = 0.331863 + 0.870571I
b = −0.459949 − 0.211030I
−4.33948 + 2.58823I −15.9377 − 3.4312I
u = −0.392152 − 0.823059I
a = 0.331863 − 0.870571I
b = −0.459949 + 0.211030I
−4.33948 − 2.58823I −15.9377 + 3.4312I
u = −0.210486 + 0.885885I
a = −0.754309 − 0.859129I
b = 0.916464 + 0.273165I
−13.7069 + 4.2294I −15.0908 − 2.2409I
u = −0.210486 − 0.885885I
a = −0.754309 + 0.859129I
b = 0.916464 − 0.273165I
−13.7069 − 4.2294I −15.0908 + 2.2409I
u = −1.117840 + 0.121937I
a = 0.183222 − 0.754249I
b = −0.175598 − 0.720988I
−2.33703 − 0.81235I −9.18182 + 8.94342I
u = −1.117840 − 0.121937I
a = 0.183222 + 0.754249I
b = −0.175598 + 0.720988I
−2.33703 + 0.81235I −9.18182 − 8.94342I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.783032 + 0.807710I
a = −0.52560 + 1.38392I
b = −0.010851 − 1.124970I
−1.59922 − 2.81223I −5.39241 + 2.86664I
u = 0.783032 − 0.807710I
a = −0.52560 − 1.38392I
b = −0.010851 + 1.124970I
−1.59922 + 2.81223I −5.39241 − 2.86664I
u = −0.066075 + 1.135000I
a = −0.36094 − 1.62838I
b = 0.615579 + 1.197570I
−10.9441 − 9.8213I −12.63911 + 5.52696I
u = −0.066075 − 1.135000I
a = −0.36094 + 1.62838I
b = 0.615579 − 1.197570I
−10.9441 + 9.8213I −12.63911 − 5.52696I
u = 0.411231 + 0.747950I
a = 0.36943 − 1.47797I
b = −0.166040 + 1.104020I
3.36428 − 0.23401I −0.97991 + 2.43933I
u = 0.411231 − 0.747950I
a = 0.36943 + 1.47797I
b = −0.166040 − 1.104020I
3.36428 + 0.23401I −0.97991 − 2.43933I
u = 1.098780 + 0.455875I
a = 1.02438 − 1.04747I
b = 0.490597 + 1.118040I
1.18518 − 4.32062I −5.36534 + 4.17876I
u = 1.098780 − 0.455875I
a = 1.02438 + 1.04747I
b = 0.490597 − 1.118040I
1.18518 + 4.32062I −5.36534 − 4.17876I
u = −1.182510 + 0.227112I
a = −0.429835 + 0.746826I
b = 0.377689 + 0.830409I
−9.97809 − 1.88428I −12.95942 + 3.16374I
u = −1.182510 − 0.227112I
a = −0.429835 − 0.746826I
b = 0.377689 − 0.830409I
−9.97809 + 1.88428I −12.95942 − 3.16374I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.651226 + 1.017980I
a = 0.229212 + 1.095150I
b = −0.154199 − 0.511987I
−4.36511 + 2.70896I −17.8835 − 1.9240I
u = −0.651226 − 1.017980I
a = 0.229212 − 1.095150I
b = −0.154199 + 0.511987I
−4.36511 − 2.70896I −17.8835 + 1.9240I
u = −0.118014 + 1.203290I
a = 0.30642 + 1.47527I
b = −0.503344 − 1.061300I
−2.28016 − 6.66641I −11.44494 + 7.11180I
u = −0.118014 − 1.203290I
a = 0.30642 − 1.47527I
b = −0.503344 + 1.061300I
−2.28016 + 6.66641I −11.44494 − 7.11180I
u = 0.143056 + 0.772743I
a = 0.641494 − 1.196870I
b = −0.574548 + 1.166620I
−5.75586 + 5.22238I −8.67240 − 3.67476I
u = 0.143056 − 0.772743I
a = 0.641494 + 1.196870I
b = −0.574548 − 1.166620I
−5.75586 − 5.22238I −8.67240 + 3.67476I
u = 0.233014 + 0.741341I
a = −0.428118 + 1.316410I
b = 0.373389 − 1.116040I
2.07162 + 3.27052I −5.16362 − 5.02421I
u = 0.233014 − 0.741341I
a = −0.428118 − 1.316410I
b = 0.373389 + 1.116040I
2.07162 − 3.27052I −5.16362 + 5.02421I
u = 1.154880 + 0.445094I
a = −0.940220 + 0.970110I
b = −0.705436 − 1.185240I
−0.73558 − 7.71938I −8.00000 + 9.01590I
u = 1.154880 − 0.445094I
a = −0.940220 − 0.970110I
b = −0.705436 + 1.185240I
−0.73558 + 7.71938I −8.00000 − 9.01590I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.186190 + 0.441930I
a = 0.863048 − 0.971863I
b = 0.89596 + 1.24873I
−8.91014 − 9.70074I 0
u = 1.186190 − 0.441930I
a = 0.863048 + 0.971863I
b = 0.89596 − 1.24873I
−8.91014 + 9.70074I 0
u = −0.281565 + 1.254220I
a = −0.289928 − 1.294920I
b = 0.396705 + 0.868356I
−0.09114 − 1.67831I 0
u = −0.281565 − 1.254220I
a = −0.289928 + 1.294920I
b = 0.396705 − 0.868356I
−0.09114 + 1.67831I 0
u = 1.324820 + 0.384063I
a = 0.055551 + 0.278282I
b = −1.364170 + 0.268279I
−18.4368 − 8.6427I 0
u = 1.324820 − 0.384063I
a = 0.055551 − 0.278282I
b = −1.364170 − 0.268279I
−18.4368 + 8.6427I 0
u = 1.335470 + 0.358449I
a = 0.0527808 − 0.1215550I
b = 1.127220 − 0.238919I
−9.45136 − 6.69405I 0
u = 1.335470 − 0.358449I
a = 0.0527808 + 0.1215550I
b = 1.127220 + 0.238919I
−9.45136 + 6.69405I 0
u = −1.225400 + 0.652300I
a = −0.32305 − 1.43167I
b = −0.732798 + 0.721623I
−16.5150 + 1.3611I 0
u = −1.225400 − 0.652300I
a = −0.32305 + 1.43167I
b = −0.732798 − 0.721623I
−16.5150 − 1.3611I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.37863 + 0.31544I
a = −0.092380 − 0.165122I
b = −0.799954 + 0.318153I
−6.21382 − 3.21160I 0
u = 1.37863 − 0.31544I
a = −0.092380 + 0.165122I
b = −0.799954 − 0.318153I
−6.21382 + 3.21160I 0
u = −1.35351 + 0.56530I
a = −0.87548 − 1.35090I
b = −0.72269 + 1.34990I
−15.0022 + 15.8154I 0
u = −1.35351 − 0.56530I
a = −0.87548 + 1.35090I
b = −0.72269 − 1.34990I
−15.0022 − 15.8154I 0
u = −1.36457 + 0.58672I
a = 0.77901 + 1.28814I
b = 0.65208 − 1.26539I
−6.2885 + 12.9423I 0
u = −1.36457 − 0.58672I
a = 0.77901 − 1.28814I
b = 0.65208 + 1.26539I
−6.2885 − 12.9423I 0
u = −1.32133 + 0.68751I
a = 0.471952 + 1.284130I
b = 0.577036 − 0.948979I
−6.95153 + 3.85491I 0
u = −1.32133 − 0.68751I
a = 0.471952 − 1.284130I
b = 0.577036 + 0.948979I
−6.95153 − 3.85491I 0
u = 1.46368 + 0.35591I
a = −0.123992 + 0.336867I
b = 0.659559 − 0.634367I
−7.88342 + 0.96412I 0
u = 1.46368 − 0.35591I
a = −0.123992 − 0.336867I
b = 0.659559 + 0.634367I
−7.88342 − 0.96412I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.37201 + 0.62798I
a = −0.645034 − 1.241570I
b = −0.581916 + 1.144830I
−3.77550 + 8.37956I 0
u = −1.37201 − 0.62798I
a = −0.645034 + 1.241570I
b = −0.581916 − 1.144830I
−3.77550 − 8.37956I 0
u = 1.46550 + 0.44360I
a = 0.324608 − 0.372587I
b = −0.689131 + 0.909581I
−15.9597 + 4.0146I 0
u = 1.46550 − 0.44360I
a = 0.324608 + 0.372587I
b = −0.689131 − 0.909581I
−15.9597 − 4.0146I 0
u = −0.288024 + 0.269790I
a = 0.642999 − 0.968459I
b = 0.136152 − 0.381527I
−0.573993 + 0.867939I −10.17045 − 7.71087I
u = −0.288024 − 0.269790I
a = 0.642999 + 0.968459I
b = 0.136152 + 0.381527I
−0.573993 − 0.867939I −10.17045 + 7.71087I
10
II. I
u
2
= hu
5
+ b + u, −7u
5
− 2u
4
+ u
3
+ u
2
+ 5a − 8u − 3, u
6
+ u
4
+ 2u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
3
+ u
a
11
=
7
5
u
5
+
2
5
u
4
+ ··· +
8
5
u +
3
5
−u
5
− u
a
4
=
4
5
u
5
−
1
5
u
4
+ ··· +
6
5
u −
4
5
1
a
1
=
−0.240000u
5
+ 0.560000u
4
+ ··· − 0.160000u + 1.04000
−
1
5
u
5
−
1
5
u
4
+ ··· +
1
5
u −
4
5
a
10
=
2
5
u
5
+
2
5
u
4
+ ··· +
3
5
u +
3
5
−u
5
− u
a
7
=
−0.240000u
5
+ 0.560000u
4
+ ··· − 0.160000u + 2.04000
−
1
5
u
5
−
1
5
u
4
+ ··· −
4
5
u −
4
5
a
12
=
−0.0800000u
5
+ 0.520000u
4
+ ··· + 0.280000u + 0.680000
−
2
5
u
5
−
2
5
u
4
+ ··· +
2
5
u −
3
5
a
9
=
2
5
u
5
+
7
5
u
4
+ ··· +
3
5
u +
8
5
−u
5
− u
4
− 2u
2
− u − 2
a
8
=
−0.0800000u
5
+ 1.52000u
4
+ ··· + 0.280000u + 1.68000
−
2
5
u
5
−
7
5
u
4
+ ··· −
3
5
u −
8
5
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
4
+ 4u
2
− 4
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
5(5u
6
− 14u
5
+ 23u
4
− 24u
3
+ 16u
2
− 6u + 1)
c
2
, c
5
u
6
+ u
4
+ 2u
2
+ 1
c
3
, c
4
, c
9
c
10
(u
2
+ 1)
3
c
6
5(5u
6
+ 8u
5
+ 3u
4
+ 2u
3
+ 4u
2
+ 2u + 1)
c
7
, c
8
, c
11
c
12
u
6
− 3u
4
+ 2u
2
+ 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
25(25y
6
+ 34y
5
+ 17y
4
+ 2y
3
+ 14y
2
− 4y + 1)
c
2
, c
5
(y
3
+ y
2
+ 2y + 1)
2
c
3
, c
4
, c
9
c
10
(y + 1)
6
c
6
25(25y
6
− 34y
5
+ 17y
4
− 2y
3
+ 14y
2
+ 4y + 1)
c
7
, c
8
, c
11
c
12
(y
3
− 3y
2
+ 2y + 1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.744862 + 0.877439I
a = 0.38847 − 1.86784I
b = 1.000000I
−3.02413 + 2.82812I −11.50976 + 2.97945I
u = 0.744862 − 0.877439I
a = 0.38847 + 1.86784I
b = − 1.000000I
−3.02413 − 2.82812I −11.50976 − 2.97945I
u = −0.744862 + 0.877439I
a = −0.432328 − 0.895156I
b = 1.000000I
−3.02413 − 2.82812I −11.50976 − 2.97945I
u = −0.744862 − 0.877439I
a = −0.432328 + 0.895156I
b = − 1.000000I
−3.02413 + 2.82812I −11.50976 + 2.97945I
u = 0.754878I
a = 0.84386 + 1.63701I
b = − 1.000000I
1.11345 −4.98050
u = − 0.754878I
a = 0.84386 − 1.63701I
b = 1.000000I
1.11345 −4.98050
14
III. I
u
3
= h−a
2
+ b − a, a
3
+ 2a
2
+ a − 1, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
5
=
1
0
a
6
=
1
1
a
3
=
1
0
a
11
=
a
a
2
+ a
a
4
=
a
2
+ a
−a
a
1
=
−a
−a
2
− a
a
10
=
a
2
+ 2a
a
2
+ a
a
7
=
a
a
2
+ a
a
12
=
a
a
2
+ a
a
9
=
a
a
2
+ a
a
8
=
a
a
2
+ a
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
9
, c
10
u
3
+ u + 1
c
2
, c
5
(u + 1)
3
c
6
u
3
+ 2u
2
+ u − 1
c
7
, c
8
, c
11
c
12
u
3
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
9
, c
10
y
3
+ 2y
2
+ y − 1
c
2
, c
5
(y − 1)
3
c
6
y
3
− 2y
2
+ 5y − 1
c
7
, c
8
, c
11
c
12
y
3
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.23279 + 0.79255I
b = −0.341164 − 1.161540I
−1.64493 −6.00000
u = −1.00000
a = −1.23279 − 0.79255I
b = −0.341164 + 1.161540I
−1.64493 −6.00000
u = −1.00000
a = 0.465571
b = 0.682328
−1.64493 −6.00000
18
IV. I
u
4
= hb
4
a
2
− 2b
3
a + 2b
2
a
2
− b
2
a + b
2
− 2ba + a
2
+ b − a − 1, u + 1i
(i) Arc colorings
a
2
=
0
−1
a
5
=
1
0
a
6
=
1
1
a
3
=
1
0
a
11
=
a
b
a
4
=
−ba + 1
−b
2
a
1
=
−b
2
a
2
+ 2ba − 1
−b
3
a + b
2
− 1
a
10
=
b + a
b
a
7
=
−ba − a
2
+ 1
−ba + 1
a
12
=
−b
3
a
2
− a
3
b
2
+ 2b
2
a − a
3
− b + 2a
−b
3
a
2
+ 2b
2
a − a
2
b + a
a
9
=
a
b
a
8
=
b
3
a
2
+ a
3
b
2
− b
2
a
2
− 2b
2
a + a
3
+ ba − a
2
+ b − a
b
3
a
2
− b
3
a − 2b
2
a + a
2
b + b
2
− ba + b − a
(ii) Obstruction class = 1
(iii) Cusp Shapes = −16
(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.
19
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
−10.5276 −16.0000
20
V. I
v
1
= ha, b
6
+ 2b
4
+ b
3
+ b
2
+ b − 1, v − 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
1
0
a
6
=
1
0
a
3
=
1
0
a
11
=
0
b
a
4
=
1
−b
2
a
1
=
b
2
+ 1
−b
4
a
10
=
b
b
a
7
=
b
2
+ 1
b
2
a
12
=
−b
5
− 2b
3
− b
−b
5
− b
3
+ b
a
9
=
0
b
a
8
=
b
5
+ 2b
3
+ b
b
5
+ b
4
+ b
3
+ b
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −10
21
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
6
+ 4u
5
+ 6u
4
+ u
3
− 5u
2
− 3u + 1
c
2
, c
5
u
6
c
3
, c
4
, c
6
c
9
, c
10
u
6
+ 2u
4
− u
3
+ u
2
− u − 1
c
7
, c
8
, c
11
c
12
(u
2
− u − 1)
3
22
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
6
− 4y
5
+ 18y
4
− 35y
3
+ 43y
2
− 19y + 1
c
2
, c
5
y
6
c
3
, c
4
, c
6
c
9
, c
10
y
6
+ 4y
5
+ 6y
4
+ y
3
− 5y
2
− 3y + 1
c
7
, c
8
, c
11
c
12
(y
2
− 3y + 1)
3
23
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = −0.896795
−8.88264 −10.0000
v = 1.00000
a = 0
b = −0.248003 + 1.088360I
−0.986960 −10.0000
v = 1.00000
a = 0
b = −0.248003 − 1.088360I
−0.986960 −10.0000
v = 1.00000
a = 0
b = 0.448397 + 1.266170I
−8.88264 −10.0000
v = 1.00000
a = 0
b = 0.448397 − 1.266170I
−8.88264 −10.0000
v = 1.00000
a = 0
b = 0.496006
−0.986960 −10.0000
24
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
80(u
3
+ u + 1)(u
6
+ 4u
5
+ 6u
4
+ u
3
− 5u
2
− 3u + 1)
· (5u
6
− 14u
5
+ 23u
4
− 24u
3
+ 16u
2
− 6u + 1)
· (16u
56
− 48u
55
+ ··· + 388278u + 882567)
c
2
, c
5
u
6
(u + 1)
3
(u
6
+ u
4
+ 2u
2
+ 1)(u
56
− 6u
55
+ ··· − 5065u + 2138)
c
3
, c
9
9(u
2
+ 1)
3
(u
3
+ u + 1)(u
6
+ 2u
4
− u
3
+ u
2
− u − 1)
· (9u
56
− 9u
55
+ ··· + 80u + 25)
c
4
, c
10
9(u
2
+ 1)
3
(u
3
+ u + 1)(u
6
+ 2u
4
− u
3
+ u
2
− u − 1)
· (9u
56
− 9u
55
+ ··· − 50u + 25)
c
6
80(u
3
+ 2u
2
+ u − 1)(u
6
+ 2u
4
− u
3
+ u
2
− u − 1)
· (5u
6
+ 8u
5
+ 3u
4
+ 2u
3
+ 4u
2
+ 2u + 1)
· (16u
56
+ 32u
55
+ ··· + 735282u + 119709)
c
7
, c
8
, c
11
c
12
u
3
(u
2
− u − 1)
3
(u
6
− 3u
4
+ 2u
2
+ 1)(u
56
+ 4u
55
+ ··· − 7u + 62)
25
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
6400(y
3
+ 2y
2
+ y − 1)(y
6
− 4y
5
+ 18y
4
− 35y
3
+ 43y
2
− 19y + 1)
· (25y
6
+ 34y
5
+ 17y
4
+ 2y
3
+ 14y
2
− 4y + 1)
· (256y
56
− 11136y
55
+ ··· − 10663154787870y + 778924509489)
c
2
, c
5
y
6
(y − 1)
3
(y
3
+ y
2
+ 2y + 1)
2
· (y
56
− 32y
55
+ ··· − 6390845y + 4571044)
c
3
, c
9
81(y + 1)
6
(y
3
+ 2y
2
+ y − 1)(y
6
+ 4y
5
+ 6y
4
+ y
3
− 5y
2
− 3y + 1)
· (81y
56
+ 4239y
55
+ ··· + 12400y + 625)
c
4
, c
10
81(y + 1)
6
(y
3
+ 2y
2
+ y − 1)(y
6
+ 4y
5
+ 6y
4
+ y
3
− 5y
2
− 3y + 1)
· (81y
56
+ 1971y
55
+ ··· + 16400y + 625)
c
6
6400(y
3
− 2y
2
+ 5y − 1)(y
6
+ 4y
5
+ 6y
4
+ y
3
− 5y
2
− 3y + 1)
· (25y
6
− 34y
5
+ 17y
4
− 2y
3
+ 14y
2
+ 4y + 1)
· (256y
56
− 10624y
55
+ ··· − 177561504270y + 14330244681)
c
7
, c
8
, c
11
c
12
y
3
(y
2
− 3y + 1)
3
(y
3
− 3y
2
+ 2y + 1)
2
· (y
56
− 66y
55
+ ··· − 2901y + 3844)
26
