12n
0423
(K12n
0423
)
A knot diagram
1
Linearized knot diagam
3 5 7 12 2 4 6 11 5 4 9 10
Solving Sequence
3,5
2 6
1,10
9 12 4 7 8 11
c
2
c
5
c
1
c
9
c
12
c
4
c
6
c
7
c
11
c
3
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h1231339u
16
− 5163151u
15
+ ··· + 19124224b − 5476095,
− 9673313u
16
+ 16707789u
15
+ ··· + 9562112a − 54216803, u
17
− 2u
16
+ ··· + 5u + 1i
I
u
2
= h−3u
3
+ u
2
+ 4b − 2u − 1, −u
3
+ u
2
+ 2a − 2u + 1, u
4
+ u
2
+ u + 1i
I
u
3
= ha
4
+ a
3
u + 2a
2
+ au + b + u + 2, a
5
− a
4
+ 2a
3
− a
2
+ a − 1, u
2
+ 1i
I
u
4
= h−2048u
9
+ 33496u
8
+ ··· + 334809b + 864245,
535181u
9
− 69173u
8
+ ··· + 5691753a − 2597382,
u
10
− u
8
+ 15u
6
− u
5
+ 57u
4
+ 7u
3
+ 56u
2
+ 12u + 17i
I
u
5
= hu
5
+ u
3
− u
2
+ b, u
5
+ 2u
3
− u
2
+ a + u − 1, u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1i
* 5 irreducible components of dim
C
= 0, with total 47 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
= h1.23 × 10
6
u
16
− 5.16 × 10
6
u
15
+ · · · + 1.91 × 10
7
b − 5.48 × 10
6
, −9.67 ×
10
6
u
16
+1.67×10
7
u
15
+· · · +9.56×10
6
a−5.42×10
7
, u
17
−2u
16
+· · · +5u +1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
6
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
2
a
10
=
1.01163u
16
− 1.74729u
15
+ ··· + 11.6261u + 5.66996
−0.0643864u
16
+ 0.269980u
15
+ ··· − 1.56569u + 0.286343
a
9
=
1.01163u
16
− 1.74729u
15
+ ··· + 11.6261u + 5.66996
−0.299164u
16
+ 0.733070u
15
+ ··· − 3.95716u + 0.0103754
a
12
=
0.521430u
16
− 1.40337u
15
+ ··· + 2.34977u − 1.07163
0.297738u
16
− 0.653196u
15
+ ··· + 2.42019u + 0.284846
a
4
=
0.0312500u
15
− 0.0625000u
14
+ ··· + 0.156250u + 1.03125
−u
2
a
7
=
0.0312500u
16
− 0.0625000u
15
+ ··· + 0.156250u
2
+ 2.03125u
u
a
8
=
0.0312500u
16
− 0.0625000u
15
+ ··· + 0.156250u
2
+ 2.03125u
u
5
+ u
3
+ u
a
11
=
0.874375u
16
− 1.76166u
15
+ ··· + 11.3348u + 4.82131
−0.142742u
16
+ 0.397649u
15
+ ··· − 2.44474u + 0.0854782
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
45218375
76496896
u
16
+
24809901
10928128
u
15
+ ··· +
290991179
38248448
u +
522539755
76496896
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
17
+ 8u
15
+ ··· − 15u − 1
c
2
, c
3
, c
5
c
6
u
17
+ 2u
16
+ ··· + 5u − 1
c
4
u
17
+ 6u
16
+ ··· + 12u + 4
c
8
, c
11
u
17
+ 5u
16
+ ··· − 97u − 16
c
9
2(2u
17
− 5u
16
+ ··· + 16u + 568)
c
10
2(2u
17
− 7u
16
+ ··· − 6391u + 13778)
c
12
u
17
− 3u
16
+ ··· − 800u + 256
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
17
+ 16y
16
+ ··· − 243y −1
c
2
, c
3
, c
5
c
6
y
17
+ 8y
15
+ ··· − 15y −1
c
4
y
17
+ 2y
16
+ ··· − 104y −16
c
8
, c
11
y
17
− 25y
16
+ ··· − 2207y −256
c
9
4(4y
17
− 145y
16
+ ··· − 145152y −322624)
c
10
4(4y
17
− 109y
16
+ ··· − 6.94790 × 10
8
y −1.89833 × 10
8
)
c
12
y
17
+ 39y
16
+ ··· + 316416y −65536
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.717924 + 0.486358I
a = 0.853936 − 0.745729I
b = 0.307111 − 0.292541I
1.26458 − 1.13730I 3.35116 + 1.82167I
u = −0.717924 − 0.486358I
a = 0.853936 + 0.745729I
b = 0.307111 + 0.292541I
1.26458 + 1.13730I 3.35116 − 1.82167I
u = −0.026424 + 0.757016I
a = 0.71710 + 1.42840I
b = 0.234197 + 0.550904I
3.54267 − 4.48886I 11.22908 + 5.05916I
u = −0.026424 − 0.757016I
a = 0.71710 − 1.42840I
b = 0.234197 − 0.550904I
3.54267 + 4.48886I 11.22908 − 5.05916I
u = 0.711308 + 1.176950I
a = −0.256063 − 0.466323I
b = −0.293782 + 0.033385I
−1.05610 + 6.71672I 0.12255 − 2.58272I
u = 0.711308 − 1.176950I
a = −0.256063 + 0.466323I
b = −0.293782 − 0.033385I
−1.05610 − 6.71672I 0.12255 + 2.58272I
u = −1.48564
a = −2.63464
b = −2.25227
4.37099 −1.92960
u = 0.051104 + 0.476773I
a = −0.70597 − 1.39720I
b = −0.525015 − 0.475256I
−0.95916 − 1.44555I −0.42899 + 2.81466I
u = 0.051104 − 0.476773I
a = −0.70597 + 1.39720I
b = −0.525015 + 0.475256I
−0.95916 + 1.44555I −0.42899 − 2.81466I
u = −0.185924 + 0.218364I
a = 4.27912 + 0.21207I
b = 0.454654 − 0.748287I
1.91786 − 0.70287I 4.74657 − 1.86210I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.185924 − 0.218364I
a = 4.27912 − 0.21207I
b = 0.454654 + 0.748287I
1.91786 + 0.70287I 4.74657 + 1.86210I
u = 0.98623 + 1.52887I
a = −1.163130 + 0.777324I
b = −2.24771 − 0.01624I
17.7964 + 14.2831I 2.56357 − 5.88484I
u = 0.98623 − 1.52887I
a = −1.163130 − 0.777324I
b = −2.24771 + 0.01624I
17.7964 − 14.2831I 2.56357 + 5.88484I
u = −0.96033 + 1.57079I
a = 0.924952 + 0.534960I
b = 1.98481 + 0.05465I
17.6776 − 6.3252I 2.67569 + 2.04454I
u = −0.96033 − 1.57079I
a = 0.924952 − 0.534960I
b = 1.98481 − 0.05465I
17.6776 + 6.3252I 2.67569 − 2.04454I
u = 1.88478 + 0.57916I
a = 0.917377 − 0.559017I
b = 1.58688 + 0.23145I
7.80121 + 3.61191I 6.17391 − 2.86781I
u = 1.88478 − 0.57916I
a = 0.917377 + 0.559017I
b = 1.58688 − 0.23145I
7.80121 − 3.61191I 6.17391 + 2.86781I
6
II. I
u
2
= h−3u
3
+ u
2
+ 4b − 2u − 1, −u
3
+ u
2
+ 2a − 2u + 1 , u
4
+ u
2
+ u + 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
6
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
2
a
10
=
1
2
u
3
−
1
2
u
2
+ u −
1
2
3
4
u
3
−
1
4
u
2
+
1
2
u +
1
4
a
9
=
1
2
u
3
−
1
2
u
2
+ u −
1
2
5
4
u
3
−
3
4
u
2
+
1
2
u +
3
4
a
12
=
u
2
+ 1
u
2
a
4
=
u
3
− u
2
−u
2
a
7
=
−u
3
− u
2
− 1
u
a
8
=
−u
2
− 1
−u
2
a
11
=
1
2
u
3
+
1
2
u
2
+ u +
1
2
5
4
u
3
+
1
4
u
2
+
1
2
u +
3
4
(ii) Obstruction class = 1
(iii) Cusp Shapes =
79
16
u
3
−
85
16
u
2
−
21
8
u +
93
16
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
4
− 2u
3
+ 3u
2
− u + 1
c
2
, c
3
u
4
+ u
2
+ u + 1
c
4
u
4
− 3u
3
+ 4u
2
− 3u + 2
c
5
, c
6
u
4
+ u
2
− u + 1
c
7
u
4
+ 2u
3
+ 3u
2
+ u + 1
c
8
(u + 1)
4
c
9
, c
10
2(2u
4
+ 3u
3
+ 4u
2
+ 3u + 1)
c
11
(u − 1)
4
c
12
u
4
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
4
+ 2y
3
+ 7y
2
+ 5y + 1
c
2
, c
3
, c
5
c
6
y
4
+ 2y
3
+ 3y
2
+ y + 1
c
4
y
4
− y
3
+ 2y
2
+ 7y + 4
c
8
, c
11
(y − 1)
4
c
9
, c
10
4(4y
4
+ 7y
3
+ 2y
2
− y + 1)
c
12
y
4
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.547424 + 0.585652I
a = −0.826150 + 1.069070I
b = 0.286541 + 0.697356I
2.62503 − 1.39709I 9.45081 + 3.47689I
u = −0.547424 − 0.585652I
a = −0.826150 − 1.069070I
b = 0.286541 − 0.697356I
2.62503 + 1.39709I 9.45081 − 3.47689I
u = 0.547424 + 1.120870I
a = −0.423850 + 0.307015I
b = −0.661541 − 0.046758I
−0.98010 + 7.64338I 0.08044 − 11.43934I
u = 0.547424 − 1.120870I
a = −0.423850 − 0.307015I
b = −0.661541 + 0.046758I
−0.98010 − 7.64338I 0.08044 + 11.43934I
10
III. I
u
3
= ha
4
+ a
3
u + 2a
2
+ au + b + u + 2, a
5
− a
4
+ 2a
3
− a
2
+ a − 1, u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
−1
a
6
=
u
0
a
1
=
0
−1
a
10
=
a
−a
4
− a
3
u − 2a
2
− au − u − 2
a
9
=
a
−a
4
− a
3
u − 2a
2
− au − a − u − 2
a
12
=
−a
2
a
4
u + a
4
+ a
2
u + a
2
+ au + a
a
4
=
a
4
u
1
a
7
=
a
4
+ u
−u
a
8
=
a
4
−u
a
11
=
a
4
−a
4
− 2a
2
− u − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4a
3
+ 4a
2
− 4a
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
10
c
2
, c
3
, c
5
c
6
(u
2
+ 1)
5
c
4
u
10
+ u
8
+ 8u
6
+ 3u
4
+ 3u
2
+ 1
c
7
(u + 1)
10
c
8
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
c
9
u
10
− 3u
8
+ 4u
6
− u
4
− u
2
+ 1
c
10
u
10
+ 5u
8
+ 8u
6
+ 3u
4
− u
2
+ 1
c
11
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
c
12
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
(y − 1)
10
c
2
, c
3
, c
5
c
6
(y + 1)
10
c
4
(y
5
+ y
4
+ 8y
3
+ 3y
2
+ 3y + 1)
2
c
8
, c
11
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y − 1)
2
c
9
(y
5
− 3y
4
+ 4y
3
− y
2
− y + 1)
2
c
10
(y
5
+ 5y
4
+ 8y
3
+ 3y
2
− y + 1)
2
c
12
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y − 1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = −0.339110 + 0.822375I
b = −0.331455 − 0.820551I
−2.96077 + 1.53058I −3.48489 − 4.43065I
u = 1.000000I
a = −0.339110 − 0.822375I
b = −1.43128 − 1.79928I
−2.96077 − 1.53058I −3.48489 + 4.43065I
u = 1.000000I
a = 0.766826
b = −3.52181 − 2.21774I
−0.888787 −2.51890
u = 1.000000I
a = 0.455697 + 1.200150I
b = 0.361438 + 0.927855I
2.58269 − 4.40083I 0.74431 + 3.49859I
u = 1.000000I
a = 0.455697 − 1.200150I
b = −0.0768928 − 0.0902877I
2.58269 + 4.40083I 0.74431 − 3.49859I
u = − 1.000000I
a = −0.339110 + 0.822375I
b = −1.43128 + 1.79928I
−2.96077 + 1.53058I −3.48489 − 4.43065I
u = − 1.000000I
a = −0.339110 − 0.822375I
b = −0.331455 + 0.820551I
−2.96077 − 1.53058I −3.48489 + 4.43065I
u = − 1.000000I
a = 0.766826
b = −3.52181 + 2.21774I
−0.888787 −2.51890
u = − 1.000000I
a = 0.455697 + 1.200150I
b = −0.0768928 + 0.0902877I
2.58269 − 4.40083I 0.74431 + 3.49859I
u = − 1.000000I
a = 0.455697 − 1.200150I
b = 0.361438 − 0.927855I
2.58269 + 4.40083I 0.74431 − 3.49859I
14
IV. I
u
4
= h−2048u
9
+ 33496u
8
+ · · · + 334809b + 864245, 5.35 × 10
5
u
9
−
6.92 × 10
4
u
8
+ · · · + 5.69 × 10
6
a − 2.60 × 10
6
, u
10
− u
8
+ · · · + 12u + 17i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
6
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
2
a
10
=
−0.0940274u
9
+ 0.0121532u
8
+ ··· − 3.38339u + 0.456341
0.00611692u
9
− 0.100045u
8
+ ··· + 1.92072u − 2.58131
a
9
=
−0.0940274u
9
+ 0.0121532u
8
+ ··· − 3.38339u + 0.456341
0.0534394u
9
− 0.0632629u
8
+ ··· + 3.37335u − 2.78791
a
12
=
−0.106750u
9
+ 0.00431291u
8
+ ··· − 1.89053u − 0.993144
0.0711032u
9
− 0.0335863u
8
+ ··· + 3.72600u + 1.69219
a
4
=
0.0137314u
9
− 0.0104298u
8
+ ··· + 0.720925u − 1.04379
−0.00129029u
9
− 0.0413967u
8
+ ··· − 0.108277u + 1.17731
a
7
=
0.0588235u
9
− 0.0588235u
7
+ ··· + 3.29412u + 0.705882
−0.0104298u
9
− 0.00129029u
8
+ ··· − 2.20857u − 0.233435
a
8
=
0.100220u
9
− 0.00518803u
8
+ ··· + 2.10133u + 0.683947
−0.0521491u
9
− 0.00645144u
8
+ ··· − 4.04285u − 0.167173
a
11
=
−0.103990u
9
+ 0.0138855u
8
+ ··· − 3.10282u + 0.408093
0.0268153u
9
− 0.0470836u
8
+ ··· + 2.71321u − 1.70339
(ii) Obstruction class = −1
(iii) Cusp Shapes =
7390
111603
u
9
+
94492
111603
u
8
−
1453
111603
u
7
−
64107
37201
u
6
+
69416
111603
u
5
+
533230
37201
u
4
+
179147
37201
u
3
+
3799966
111603
u
2
+
279050
37201
u +
1780766
111603
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
10
− 2u
9
+ ··· + 1760u + 289
c
2
, c
3
, c
5
c
6
u
10
− u
8
+ 15u
6
+ u
5
+ 57u
4
− 7u
3
+ 56u
2
− 12u + 17
c
4
(u
5
− 2u
4
+ 2u
3
+ u − 1)
2
c
8
, c
11
(u
5
+ 4u
4
+ u
3
− 5u
2
+ 6u + 1)
2
c
9
(u
5
+ 8u
4
+ 21u
3
+ 19u
2
+ 2u − 4)
2
c
10
(u
5
− u
4
+ 28u
3
− 4u
2
− 6u − 1)
2
c
12
(u
5
− u
4
+ 17u
3
+ 4u
2
+ 20u + 8)
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
10
+ 58y
9
+ ··· − 261932y + 83521
c
2
, c
3
, c
5
c
6
y
10
− 2y
9
+ ··· + 1760y + 289
c
4
(y
5
+ 6y
3
+ y − 1)
2
c
8
, c
11
(y
5
− 14y
4
+ 53y
3
− 21y
2
+ 46y − 1)
2
c
9
(y
5
− 22y
4
+ 141y
3
− 213y
2
+ 156y − 16)
2
c
10
(y
5
+ 55y
4
+ 764y
3
− 354y
2
+ 28y − 1)
2
c
12
(y
5
+ 33y
4
+ 337y
3
+ 680y
2
+ 336y − 64)
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.223424 + 1.072270I
a = −0.064776 + 0.310879I
b = −0.554957 + 1.072270I
−4.19771 −7.55749 + 0.I
u = 0.223424 − 1.072270I
a = −0.064776 − 0.310879I
b = −0.554957 − 1.072270I
−4.19771 −7.55749 + 0.I
u = −0.005641 + 1.186120I
a = −0.334630 − 0.830732I
b = −1.203510 − 0.040685I
−1.58742 − 1.37362I 4.55634 + 3.01933I
u = −0.005641 − 1.186120I
a = −0.334630 + 0.830732I
b = −1.203510 + 0.040685I
−1.58742 + 1.37362I 4.55634 − 3.01933I
u = −0.232935 + 0.614344I
a = 0.02548 − 1.61663I
b = −1.43081 + 1.84114I
−1.58742 + 1.37362I 4.55634 − 3.01933I
u = −0.232935 − 0.614344I
a = 0.02548 + 1.61663I
b = −1.43081 − 1.84114I
−1.58742 − 1.37362I 4.55634 + 3.01933I
u = 1.84404 + 1.19233I
a = 1.26566 − 0.71520I
b = 1.93110 − 0.12690I
−19.3428 − 4.0569I 3.72240 + 1.88627I
u = 1.84404 − 1.19233I
a = 1.26566 + 0.71520I
b = 1.93110 + 0.12690I
−19.3428 + 4.0569I 3.72240 − 1.88627I
u = −1.82889 + 1.22222I
a = −1.15643 − 0.87684I
b = −1.74183 − 0.09702I
−19.3428 − 4.0569I 3.72240 + 1.88627I
u = −1.82889 − 1.22222I
a = −1.15643 + 0.87684I
b = −1.74183 + 0.09702I
−19.3428 + 4.0569I 3.72240 − 1.88627I
18
V.
I
u
5
= hu
5
+u
3
−u
2
+b, u
5
+2u
3
−u
2
+a+u−1, u
6
−u
5
+2u
4
−2u
3
+2u
2
−2u+1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
6
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
2
a
10
=
−u
5
− 2u
3
+ u
2
− u + 1
−u
5
− u
3
+ u
2
a
9
=
−u
5
− 2u
3
+ u
2
− u + 1
−2u
5
+ u
4
− 2u
3
+ 2u
2
− u + 1
a
12
=
u
2
+ 1
u
2
a
4
=
u
5
+ 2u
3
+ u
u
5
+ u
3
+ u
a
7
=
u
5
− u
4
+ 2u
3
− 2u
2
+ 2u − 2
u
5
+ 2u
3
− u
2
+ u − 1
a
8
=
−u
2
− 1
−u
2
a
11
=
−u
5
− 2u
3
+ 2u
2
− u + 2
−2u
5
+ u
4
− 2u
3
+ 3u
2
− u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
5
+ 5u
3
− 2u
2
+ 5u
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
6
− 3u
5
+ 4u
4
− 2u
3
+ 1
c
2
, c
3
u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1
c
4
(u
3
+ u
2
− 1)
2
c
5
, c
6
u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1
c
7
u
6
+ 3u
5
+ 4u
4
+ 2u
3
+ 1
c
8
(u + 1)
6
c
9
, c
10
(u
3
− u + 1)
2
c
11
(u − 1)
6
c
12
u
6
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1
c
2
, c
3
, c
5
c
6
y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1
c
4
(y
3
− y
2
+ 2y − 1)
2
c
8
, c
11
(y − 1)
6
c
9
, c
10
(y
3
− 2y
2
+ y − 1)
2
c
12
y
6
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −0.498832 + 1.001300I
a = −0.713912 − 0.305839I
b = −0.836473 + 0.439023I
1.37919 − 2.82812I 3.30760 + 3.35914I
u = −0.498832 − 1.001300I
a = −0.713912 + 0.305839I
b = −0.836473 − 0.439023I
1.37919 + 2.82812I 3.30760 − 3.35914I
u = 0.284920 + 1.115140I
a = −0.284920 + 1.115140I
b = −2.03980 + 1.11514I
−2.75839 2.38480 + 0.I
u = 0.284920 − 1.115140I
a = −0.284920 − 1.115140I
b = −2.03980 − 1.11514I
−2.75839 2.38480 + 0.I
u = 0.713912 + 0.305839I
a = 0.498832 − 1.001300I
b = 0.376271 − 0.256441I
1.37919 − 2.82812I 3.30760 + 3.35914I
u = 0.713912 − 0.305839I
a = 0.498832 + 1.001300I
b = 0.376271 + 0.256441I
1.37919 + 2.82812I 3.30760 − 3.35914I
22
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
10
(u
4
− 2u
3
+ 3u
2
− u + 1)(u
6
− 3u
5
+ 4u
4
− 2u
3
+ 1)
· (u
10
− 2u
9
+ ··· + 1760u + 289)(u
17
+ 8u
15
+ ··· − 15u − 1)
c
2
, c
3
(u
2
+ 1)
5
(u
4
+ u
2
+ u + 1)(u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1)
· (u
10
− u
8
+ 15u
6
+ u
5
+ 57u
4
− 7u
3
+ 56u
2
− 12u + 17)
· (u
17
+ 2u
16
+ ··· + 5u − 1)
c
4
(u
3
+ u
2
− 1)
2
(u
4
− 3u
3
+ 4u
2
− 3u + 2)(u
5
− 2u
4
+ 2u
3
+ u − 1)
2
· (u
10
+ u
8
+ 8u
6
+ 3u
4
+ 3u
2
+ 1)(u
17
+ 6u
16
+ ··· + 12u + 4)
c
5
, c
6
(u
2
+ 1)
5
(u
4
+ u
2
− u + 1)(u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1)
· (u
10
− u
8
+ 15u
6
+ u
5
+ 57u
4
− 7u
3
+ 56u
2
− 12u + 17)
· (u
17
+ 2u
16
+ ··· + 5u − 1)
c
7
(u + 1)
10
(u
4
+ 2u
3
+ 3u
2
+ u + 1)(u
6
+ 3u
5
+ 4u
4
+ 2u
3
+ 1)
· (u
10
− 2u
9
+ ··· + 1760u + 289)(u
17
+ 8u
15
+ ··· − 15u − 1)
c
8
((u + 1)
10
)(u
5
− u
4
+ ··· + u + 1)
2
(u
5
+ 4u
4
+ ··· + 6u + 1)
2
· (u
17
+ 5u
16
+ ··· − 97u − 16)
c
9
4(u
3
− u + 1)
2
(2u
4
+ 3u
3
+ 4u
2
+ 3u + 1)
· (u
5
+ 8u
4
+ 21u
3
+ 19u
2
+ 2u − 4)
2
(u
10
− 3u
8
+ 4u
6
− u
4
− u
2
+ 1)
· (2u
17
− 5u
16
+ ··· + 16u + 568)
c
10
4(u
3
− u + 1)
2
(2u
4
+ 3u
3
+ 4u
2
+ 3u + 1)
· (u
5
− u
4
+ 28u
3
− 4u
2
− 6u − 1)
2
(u
10
+ 5u
8
+ 8u
6
+ 3u
4
− u
2
+ 1)
· (2u
17
− 7u
16
+ ··· − 6391u + 13778)
c
11
((u − 1)
10
)(u
5
+ u
4
+ ··· + u − 1)
2
(u
5
+ 4u
4
+ ··· + 6u + 1)
2
· (u
17
+ 5u
16
+ ··· − 97u − 16)
c
12
u
10
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
(u
5
− u
4
+ 17u
3
+ 4u
2
+ 20u + 8)
2
· (u
17
− 3u
16
+ ··· − 800u + 256)
23
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
(y − 1)
10
(y
4
+ 2y
3
+ 7y
2
+ 5y + 1)(y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1)
· (y
10
+ 58y
9
+ ··· − 261932y + 83521)(y
17
+ 16y
16
+ ··· − 243y − 1)
c
2
, c
3
, c
5
c
6
(y + 1)
10
(y
4
+ 2y
3
+ 3y
2
+ y + 1)(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
· (y
10
− 2y
9
+ ··· + 1760y + 289)(y
17
+ 8y
15
+ ··· − 15y − 1)
c
4
(y
3
− y
2
+ 2y − 1)
2
(y
4
− y
3
+ 2y
2
+ 7y + 4)(y
5
+ 6y
3
+ y − 1)
2
· ((y
5
+ y
4
+ 8y
3
+ 3y
2
+ 3y + 1)
2
)(y
17
+ 2y
16
+ ··· − 104y − 16)
c
8
, c
11
(y − 1)
10
(y
5
− 14y
4
+ 53y
3
− 21y
2
+ 46y − 1)
2
· ((y
5
− 5y
4
+ 8y
3
− 3y
2
− y − 1)
2
)(y
17
− 25y
16
+ ··· − 2207y − 256)
c
9
16(y
3
− 2y
2
+ y − 1)
2
(4y
4
+ 7y
3
+ 2y
2
− y + 1)
· (y
5
− 22y
4
+ 141y
3
− 213y
2
+ 156y − 16)
2
· (y
5
− 3y
4
+ 4y
3
− y
2
− y + 1)
2
· (4y
17
− 145y
16
+ ··· − 145152y − 322624)
c
10
16(y
3
− 2y
2
+ y − 1)
2
(4y
4
+ 7y
3
+ 2y
2
− y + 1)
· (y
5
+ 5y
4
+ 8y
3
+ 3y
2
− y + 1)
2
· (y
5
+ 55y
4
+ 764y
3
− 354y
2
+ 28y − 1)
2
· (4y
17
− 109y
16
+ ··· − 694790095y − 189833284)
c
12
y
10
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y − 1)
2
· (y
5
+ 33y
4
+ 337y
3
+ 680y
2
+ 336y − 64)
2
· (y
17
+ 39y
16
+ ··· + 316416y − 65536)
24
