11n
86
(K11n
86
)
A knot diagram
1
Linearized knot diagam
6 1 8 9 2 3 10 11 1 3 4
Solving Sequence
3,8 4,11
1 2 10 7 6 5 9
c
3
c
11
c
2
c
10
c
7
c
6
c
5
c
9
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−43u
9
+ 25u
8
+ 66u
7
− 10u
6
− 255u
5
+ 76u
4
+ 202u
3
+ 72u
2
+ 77b − 63u − 107,
− 200u
9
+ 93u
8
+ 264u
7
− 68u
6
− 1041u
5
+ 255u
4
+ 773u
3
+ 197u
2
+ 77a − 105u − 435,
u
10
− u
9
− u
8
+ u
7
+ 5u
6
− 4u
5
− 3u
4
+ u
3
+ u
2
+ 2u − 1i
I
u
2
= h−u
4
+ u
2
+ b − u − 1, −3u
4
− u
3
+ 2u
2
+ a − 2u − 3, u
5
− u
3
+ u
2
+ u − 1i
I
u
3
= h624u
13
+ 464u
12
+ ··· + 481b − 103, 879u
13
+ 406u
12
+ ··· + 481a − 2102,
u
14
+ 6u
10
− u
9
− u
8
− 4u
7
+ 12u
6
− 4u
5
+ 5u
4
− 10u
3
+ 11u
2
− 5u + 1i
I
u
4
= hu
3
− u
2
+ b − u + 1, a, u
4
− u
3
− u
2
+ u + 1i
I
u
5
= hb + 1, a, u + 1i
* 5 irreducible components of dim
C
= 0, with total 34 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−43u
9
+ 25u
8
+ · · · + 77b − 107, −200u
9
+ 93u
8
+ · · · + 77a −
435, u
10
− u
9
+ · · · + 2u − 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
4
=
1
u
2
a
11
=
2.59740u
9
− 1.20779u
8
+ ··· + 1.36364u + 5.64935
0.558442u
9
− 0.324675u
8
+ ··· + 0.818182u + 1.38961
a
1
=
2.59740u
9
− 1.20779u
8
+ ··· + 0.363636u + 5.64935
0.558442u
9
− 0.324675u
8
+ ··· + 0.818182u + 1.38961
a
2
=
−2.63636u
9
+ 1.09091u
8
+ ··· − 1.90909u − 4.90909
−1.41558u
9
+ 0.753247u
8
+ ··· + 0.181818u − 2.10390
a
10
=
2.03896u
9
− 0.883117u
8
+ ··· + 0.545455u + 4.25974
0.558442u
9
− 0.324675u
8
+ ··· + 0.818182u + 1.38961
a
7
=
2.18182u
9
− 1.45455u
8
+ ··· + 2.54545u + 5.54545
0.831169u
9
− 0.506494u
8
+ ··· + 1.63636u + 2.20779
a
6
=
3.01299u
9
− 1.96104u
8
+ ··· + 4.18182u + 7.75325
0.831169u
9
− 0.506494u
8
+ ··· + 1.63636u + 2.20779
a
5
=
6.70130u
9
− 3.89610u
8
+ ··· + 4.81818u + 14.6753
2.11688u
9
− 1.64935u
8
+ ··· + 1.63636u + 4.77922
a
9
=
31
7
u
9
−
19
7
u
8
+ ··· + 4u +
76
7
1.41558u
9
− 0.753247u
8
+ ··· + 1.81818u + 3.10390
a
9
=
31
7
u
9
−
19
7
u
8
+ ··· + 4u +
76
7
1.41558u
9
− 0.753247u
8
+ ··· + 1.81818u + 3.10390
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
73
11
u
9
−
34
11
u
8
− 7u
7
+
18
11
u
6
+
360
11
u
5
−
95
11
u
4
−
192
11
u
3
−
68
11
u
2
−
12
11
u +
109
11
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
10
− 5u
9
+ 13u
8
− 21u
7
+ 27u
6
− 32u
5
+ 35u
4
− 27u
3
+ 11u
2
− 3
c
2
u
10
+ u
9
+ ··· − 66u + 9
c
3
, c
11
u
10
− u
9
− u
8
+ u
7
+ 5u
6
− 4u
5
− 3u
4
+ u
3
+ u
2
+ 2u − 1
c
4
, c
10
u
10
− 8u
8
− u
7
+ 19u
6
+ 4u
5
− 4u
4
− 10u
3
− 8u
2
− 3u − 1
c
6
u
10
+ 2u
9
+ ··· − 1236u − 471
c
7
, c
9
u
10
+ 10u
8
+ 11u
7
+ 13u
6
+ 54u
5
− 66u
4
− 220u
3
− 96u
2
+ 13u − 1
c
8
u
10
+ 9u
9
+ ··· − 33u − 3
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
10
+ y
9
+ ··· − 66y + 9
c
2
y
10
+ 25y
9
+ ··· − 5958y + 81
c
3
, c
11
y
10
− 3y
9
+ 13y
8
− 25y
7
+ 43y
6
− 48y
5
+ 25y
4
− y
3
+ 3y
2
− 6y + 1
c
4
, c
10
y
10
− 16y
9
+ ··· + 7y + 1
c
6
y
10
+ 46y
9
+ ··· − 1142418y + 221841
c
7
, c
9
y
10
+ 20y
9
+ ··· + 23y + 1
c
8
y
10
− 19y
9
+ ··· − 183y + 9
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.959690 + 0.284587I
a = −1.388110 − 0.012888I
b = −0.199305 − 0.484467I
−3.79026 − 3.69224I −7.80243 + 4.12303I
u = 0.959690 − 0.284587I
a = −1.388110 + 0.012888I
b = −0.199305 + 0.484467I
−3.79026 + 3.69224I −7.80243 − 4.12303I
u = −0.891654
a = 0.375214
b = −0.593341
−1.69527 −5.20410
u = −0.291247 + 0.679656I
a = −0.051370 + 0.427907I
b = −0.102468 + 0.538626I
−0.05612 + 1.78093I −0.00118 − 2.91964I
u = −0.291247 − 0.679656I
a = −0.051370 − 0.427907I
b = −0.102468 − 0.538626I
−0.05612 − 1.78093I −0.00118 + 2.91964I
u = −1.07634 + 0.95572I
a = 0.65860 + 1.36757I
b = 1.89867 − 0.06406I
12.08100 + 3.47973I 0.63239 − 2.31358I
u = −1.07634 − 0.95572I
a = 0.65860 − 1.36757I
b = 1.89867 + 0.06406I
12.08100 − 3.47973I 0.63239 + 2.31358I
u = 1.13781 + 0.99669I
a = −0.425016 + 1.320730I
b = −1.98575 + 0.43054I
11.8608 − 11.7195I 0.24253 + 5.99452I
u = 1.13781 − 0.99669I
a = −0.425016 − 1.320730I
b = −1.98575 − 0.43054I
11.8608 + 11.7195I 0.24253 − 5.99452I
u = 0.431833
a = 5.03658
b = 1.37105
2.62770 7.06150
5
II.
I
u
2
= h−u
4
+ u
2
+ b − u − 1, −3u
4
− u
3
+ 2u
2
+ a − 2u − 3, u
5
− u
3
+ u
2
+ u − 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
4
=
1
u
2
a
11
=
3u
4
+ u
3
− 2u
2
+ 2u + 3
u
4
− u
2
+ u + 1
a
1
=
3u
4
+ u
3
− 2u
2
+ 3u + 3
u
4
+ u
3
− u
2
+ u + 1
a
2
=
−4u
4
− 3u
3
+ u
2
− u − 6
−2u
4
− u − 2
a
10
=
2u
4
+ u
3
− u
2
+ u + 2
u
4
− u
2
+ u + 1
a
7
=
3u
4
+ 2u
3
− u
2
+ 2u + 4
2u
4
+ u
3
− u
2
+ 2u + 2
a
6
=
5u
4
+ 3u
3
− 2u
2
+ 4u + 6
2u
4
+ u
3
− u
2
+ 2u + 2
a
5
=
9u
4
+ 8u
3
− 5u
2
+ 5u + 15
3u
4
+ 3u
3
− u
2
+ u + 6
a
9
=
8u
4
+ 4u
3
− 4u
2
+ 5u + 9
3u
4
+ u
3
− 2u
2
+ 3u + 3
a
9
=
8u
4
+ 4u
3
− 4u
2
+ 5u + 9
3u
4
+ u
3
− 2u
2
+ 3u + 3
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
4
− 6u
3
+ 6u
2
+ 2u − 12
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
5
− 2u
4
+ 3u
3
− 3u
2
+ u − 1
c
2
u
5
+ 2u
4
− u
3
− 7u
2
− 5u − 1
c
3
, c
11
u
5
− u
3
+ u
2
+ u − 1
c
4
, c
10
u
5
+ u
4
− u
3
− u
2
+ 1
c
5
u
5
+ 2u
4
+ 3u
3
+ 3u
2
+ u + 1
c
6
u
5
+ u
4
− 8u
3
+ 7u
2
− u + 1
c
7
, c
9
u
5
+ 3u
4
+ 3u
3
+ 3u
2
+ 2u + 1
c
8
u
5
+ 8u
4
+ 25u
3
+ 40u
2
+ 34u + 13
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
5
+ 2y
4
− y
3
− 7y
2
− 5y −1
c
2
y
5
− 6y
4
+ 19y
3
− 35y
2
+ 11y −1
c
3
, c
11
y
5
− 2y
4
+ 3y
3
− 3y
2
+ 3y −1
c
4
, c
10
y
5
− 3y
4
+ 3y
3
− 3y
2
+ 2y −1
c
6
y
5
− 17y
4
+ 48y
3
− 35y
2
− 13y −1
c
7
, c
9
y
5
− 3y
4
− 5y
3
− 3y
2
− 2y −1
c
8
y
5
− 14y
4
+ 53y
3
− 108y
2
+ 116y −169
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.699311 + 0.811268I
a = −0.078457 − 1.141870I
b = 0.609585 − 0.707177I
0.29233 − 3.70382I −1.60688 + 5.64419I
u = 0.699311 − 0.811268I
a = −0.078457 + 1.141870I
b = 0.609585 + 0.707177I
0.29233 + 3.70382I −1.60688 − 5.64419I
u = −1.045750 + 0.405588I
a = −1.14636 − 0.95711I
b = −0.831219 − 0.322384I
−3.01018 + 5.17259I −5.18262 − 7.13326I
u = −1.045750 − 0.405588I
a = −1.14636 + 0.95711I
b = −0.831219 + 0.322384I
−3.01018 − 5.17259I −5.18262 + 7.13326I
u = 0.692872
a = 4.44963
b = 1.44327
2.14584 −10.4210
9
III. I
u
3
= h624u
13
+ 464u
12
+ · · · + 481b − 103, 879u
13
+ 406u
12
+ · · · +
481a − 2102, u
14
+ 6u
10
+ · · · − 5u + 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
4
=
1
u
2
a
11
=
−1.82744u
13
− 0.844075u
12
+ ··· − 14.1351u + 4.37006
−1.29730u
13
− 0.964657u
12
+ ··· − 5.52807u + 0.214137
a
1
=
−u
13
− 6u
9
+ u
8
+ u
7
+ 4u
6
− 12u
5
+ 4u
4
− 5u
3
+ 10u
2
− 11u + 5
−0.827443u
13
− 0.844075u
12
+ ··· − 2.13514u − 0.629938
a
2
=
−0.629938u
13
+ 0.827443u
12
+ ··· − 10.7069u + 6.28482
−0.871102u
13
− 0.207900u
12
+ ··· − 7.56341u + 1.53222
a
10
=
−0.530146u
13
+ 0.120582u
12
+ ··· − 8.60707u + 4.15593
−1.29730u
13
− 0.964657u
12
+ ··· − 5.52807u + 0.214137
a
7
=
0.538462u
12
+ 0.153846u
11
+ ··· − 6.69231u + 3.61538
−0.787942u
13
− 1.07900u
12
+ ··· − 0.403326u − 0.754678
a
6
=
−0.787942u
13
− 0.540541u
12
+ ··· − 7.09563u + 2.86071
−0.787942u
13
− 1.07900u
12
+ ··· − 0.403326u − 0.754678
a
5
=
3.86694u
13
+ 0.916840u
12
+ ··· + 28.3285u − 6.57173
−1.28690u
13
− 1.23701u
12
+ ··· − 3.44075u + 0.812890
a
9
=
−1.50728u
13
− 1.50936u
12
+ ··· − 7.66112u + 1.18087
−0.719335u
13
− 0.968815u
12
+ ··· + 1.43451u − 1.67983
a
9
=
−1.50728u
13
− 1.50936u
12
+ ··· − 7.66112u + 1.18087
−0.719335u
13
− 0.968815u
12
+ ··· + 1.43451u − 1.67983
(ii) Obstruction class = −1
(iii) Cusp Shapes =
3807
481
u
13
−
317
481
u
12
+ ··· +
38069
481
u −
12692
481
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u
7
+ 2u
6
+ 2u
5
− u
2
− 2u − 1)
2
c
2
(u
7
+ 4u
5
− 4u
3
− u
2
+ 2u − 1)
2
c
3
, c
11
u
14
+ 6u
10
− u
9
− u
8
− 4u
7
+ 12u
6
− 4u
5
+ 5u
4
− 10u
3
+ 11u
2
− 5u + 1
c
4
, c
10
u
14
− 10u
12
+ ··· + 143u + 43
c
6
(u
7
− 2u
6
+ 10u
5
+ 8u
4
− 18u
3
− 39u
2
− 22u − 5)
2
c
7
, c
9
u
14
− 5u
13
+ ··· − 198u + 121
c
8
(u
7
− 3u
6
+ 2u
5
+ 5u
4
− 9u
3
+ u
2
+ 6u − 4)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
(y
7
+ 4y
5
− 4y
3
− y
2
+ 2y −1)
2
c
2
(y
7
+ 8y
6
+ 8y
5
− 28y
4
+ 32y
3
− 17y
2
+ 2y −1)
2
c
3
, c
11
y
14
+ 12y
12
+ ··· − 3y + 1
c
4
, c
10
y
14
− 20y
13
+ ··· − 12623y + 1849
c
6
(y
7
+ 16y
6
+ 96y
5
− 624y
4
+ 488y
3
− 649y
2
+ 94y −25)
2
c
7
, c
9
y
14
+ 23y
13
+ ··· − 22264y + 14641
c
8
(y
7
− 5y
6
+ 16y
5
− 43y
4
+ 71y
3
− 69y
2
+ 44y −16)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.872006 + 0.599247I
a = 0.26301 − 1.49380I
b = 1.15078 − 1.28311I
1.69011 − 4.26740I 3.53857 + 7.16930I
u = 0.872006 − 0.599247I
a = 0.26301 + 1.49380I
b = 1.15078 + 1.28311I
1.69011 + 4.26740I 3.53857 − 7.16930I
u = −0.515925 + 0.958517I
a = 0.43660 − 1.40817I
b = −0.805651 − 0.112130I
1.69011 + 4.26740I 3.53857 − 7.16930I
u = −0.515925 − 0.958517I
a = 0.43660 + 1.40817I
b = −0.805651 + 0.112130I
1.69011 − 4.26740I 3.53857 + 7.16930I
u = 0.455596 + 0.508546I
a = 1.43288 − 1.59941I
b = 1.123580 + 0.237077I
2.45915 4.25058 + 0.I
u = 0.455596 − 0.508546I
a = 1.43288 + 1.59941I
b = 1.123580 − 0.237077I
2.45915 4.25058 + 0.I
u = −1.185390 + 0.692372I
a = −0.269101 − 0.619577I
b = −0.906225 − 0.384044I
−1.50295 + 3.09849I −4.37162 − 6.44758I
u = −1.185390 − 0.692372I
a = −0.269101 + 0.619577I
b = −0.906225 + 0.384044I
−1.50295 − 3.09849I −4.37162 + 6.44758I
u = −0.933345 + 1.046640I
a = 0.472546 + 0.634897I
b = 2.08158 + 0.10145I
12.56520 + 3.87242I 1.20776 − 2.37795I
u = −0.933345 − 1.046640I
a = 0.472546 − 0.634897I
b = 2.08158 − 0.10145I
12.56520 − 3.87242I 1.20776 + 2.37795I
13
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.94715 + 1.16711I
a = −0.514111 + 0.530046I
b = −1.86610 − 0.33648I
12.56520 + 3.87242I 1.20776 − 2.37795I
u = 0.94715 − 1.16711I
a = −0.514111 − 0.530046I
b = −1.86610 + 0.33648I
12.56520 − 3.87242I 1.20776 + 2.37795I
u = 0.359911 + 0.252178I
a = 0.67818 − 1.99812I
b = −0.777963 − 0.701026I
−1.50295 − 3.09849I −4.37162 + 6.44758I
u = 0.359911 − 0.252178I
a = 0.67818 + 1.99812I
b = −0.777963 + 0.701026I
−1.50295 + 3.09849I −4.37162 − 6.44758I
14
IV. I
u
4
= hu
3
− u
2
+ b − u + 1, a, u
4
− u
3
− u
2
+ u + 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
4
=
1
u
2
a
11
=
0
−u
3
+ u
2
+ u − 1
a
1
=
u
3
− u
2
− u + 1
−u
3
+ u
2
− 1
a
2
=
u
3
− 2u
2
+ 2
−u
3
+ u
2
a
10
=
u
3
− u
2
− u + 1
−u
3
+ u
2
+ u − 1
a
7
=
−u
3
+ u
2
+ u − 1
u
3
− u
2
+ 1
a
6
=
u
u
3
− u
2
+ 1
a
5
=
1
0
a
9
=
0
u
a
9
=
0
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
3
− u
2
− 3
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
(u
2
+ u + 1)
2
c
3
, c
4
, c
10
c
11
u
4
− u
3
− u
2
+ u + 1
c
5
(u
2
− u + 1)
2
c
7
, c
9
(u − 1)
4
c
8
u
4
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
(y
2
+ y + 1)
2
c
3
, c
4
, c
10
c
11
y
4
− 3y
3
+ 5y
2
− 3y + 1
c
7
, c
9
(y −1)
4
c
8
y
4
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.692440 + 0.318148I
a = 0
b = −1.192440 − 0.547877I
−1.64493 − 2.02988I −3.50000 + 0.86603I
u = −0.692440 − 0.318148I
a = 0
b = −1.192440 + 0.547877I
−1.64493 + 2.02988I −3.50000 − 0.86603I
u = 1.192440 + 0.547877I
a = 0
b = 0.692440 − 0.318148I
−1.64493 − 2.02988I −3.50000 + 0.86603I
u = 1.192440 − 0.547877I
a = 0
b = 0.692440 + 0.318148I
−1.64493 + 2.02988I −3.50000 − 0.86603I
18
V. I
u
5
= hb + 1, a, u + 1i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
−1
a
4
=
1
1
a
11
=
0
−1
a
1
=
1
0
a
2
=
1
0
a
10
=
1
−1
a
7
=
−1
0
a
6
=
−1
0
a
5
=
−1
0
a
9
=
0
−1
a
9
=
0
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
8
u
c
3
, c
4
, c
7
c
9
, c
10
, c
11
u + 1
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
8
y
c
3
, c
4
, c
7
c
9
, c
10
, c
11
y −1
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0
b = −1.00000
−1.64493 −6.00000
22
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u(u
2
+ u + 1)
2
(u
5
− 2u
4
+ 3u
3
− 3u
2
+ u − 1)
· (u
7
+ 2u
6
+ 2u
5
− u
2
− 2u − 1)
2
· (u
10
− 5u
9
+ 13u
8
− 21u
7
+ 27u
6
− 32u
5
+ 35u
4
− 27u
3
+ 11u
2
− 3)
c
2
u(u
2
+ u + 1)
2
(u
5
+ 2u
4
− u
3
− 7u
2
− 5u − 1)
· ((u
7
+ 4u
5
− 4u
3
− u
2
+ 2u − 1)
2
)(u
10
+ u
9
+ ··· − 66u + 9)
c
3
, c
11
(u + 1)(u
4
− u
3
− u
2
+ u + 1)(u
5
− u
3
+ u
2
+ u − 1)
· (u
10
− u
9
− u
8
+ u
7
+ 5u
6
− 4u
5
− 3u
4
+ u
3
+ u
2
+ 2u − 1)
· (u
14
+ 6u
10
− u
9
− u
8
− 4u
7
+ 12u
6
− 4u
5
+ 5u
4
− 10u
3
+ 11u
2
− 5u + 1)
c
4
, c
10
(u + 1)(u
4
− u
3
− u
2
+ u + 1)(u
5
+ u
4
− u
3
− u
2
+ 1)
· (u
10
− 8u
8
− u
7
+ 19u
6
+ 4u
5
− 4u
4
− 10u
3
− 8u
2
− 3u − 1)
· (u
14
− 10u
12
+ ··· + 143u + 43)
c
5
u(u
2
− u + 1)
2
(u
5
+ 2u
4
+ 3u
3
+ 3u
2
+ u + 1)
· (u
7
+ 2u
6
+ 2u
5
− u
2
− 2u − 1)
2
· (u
10
− 5u
9
+ 13u
8
− 21u
7
+ 27u
6
− 32u
5
+ 35u
4
− 27u
3
+ 11u
2
− 3)
c
6
u(u
2
+ u + 1)
2
(u
5
+ u
4
− 8u
3
+ 7u
2
− u + 1)
· (u
7
− 2u
6
+ 10u
5
+ 8u
4
− 18u
3
− 39u
2
− 22u − 5)
2
· (u
10
+ 2u
9
+ ··· − 1236u − 471)
c
7
, c
9
(u − 1)
4
(u + 1)(u
5
+ 3u
4
+ 3u
3
+ 3u
2
+ 2u + 1)
· (u
10
+ 10u
8
+ 11u
7
+ 13u
6
+ 54u
5
− 66u
4
− 220u
3
− 96u
2
+ 13u − 1)
· (u
14
− 5u
13
+ ··· − 198u + 121)
c
8
u
5
(u
5
+ 8u
4
+ 25u
3
+ 40u
2
+ 34u + 13)
· (u
7
− 3u
6
+ 2u
5
+ 5u
4
− 9u
3
+ u
2
+ 6u − 4)
2
· (u
10
+ 9u
9
+ ··· − 33u − 3)
23
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
y(y
2
+ y + 1)
2
(y
5
+ 2y
4
− y
3
− 7y
2
− 5y −1)
· ((y
7
+ 4y
5
− 4y
3
− y
2
+ 2y −1)
2
)(y
10
+ y
9
+ ··· − 66y + 9)
c
2
y(y
2
+ y + 1)
2
(y
5
− 6y
4
+ 19y
3
− 35y
2
+ 11y −1)
· (y
7
+ 8y
6
+ 8y
5
− 28y
4
+ 32y
3
− 17y
2
+ 2y −1)
2
· (y
10
+ 25y
9
+ ··· − 5958y + 81)
c
3
, c
11
(y −1)(y
4
− 3y
3
+ 5y
2
− 3y + 1)(y
5
− 2y
4
+ 3y
3
− 3y
2
+ 3y −1)
· (y
10
− 3y
9
+ 13y
8
− 25y
7
+ 43y
6
− 48y
5
+ 25y
4
− y
3
+ 3y
2
− 6y + 1)
· (y
14
+ 12y
12
+ ··· − 3y + 1)
c
4
, c
10
(y −1)(y
4
− 3y
3
+ 5y
2
− 3y + 1)(y
5
− 3y
4
+ 3y
3
− 3y
2
+ 2y −1)
· (y
10
− 16y
9
+ ··· + 7y + 1)(y
14
− 20y
13
+ ··· − 12623y + 1849)
c
6
y(y
2
+ y + 1)
2
(y
5
− 17y
4
+ 48y
3
− 35y
2
− 13y −1)
· (y
7
+ 16y
6
+ 96y
5
− 624y
4
+ 488y
3
− 649y
2
+ 94y −25)
2
· (y
10
+ 46y
9
+ ··· − 1142418y + 221841)
c
7
, c
9
((y −1)
5
)(y
5
− 3y
4
+ ··· − 2y −1)(y
10
+ 20y
9
+ ··· + 23y + 1)
· (y
14
+ 23y
13
+ ··· − 22264y + 14641)
c
8
y
5
(y
5
− 14y
4
+ 53y
3
− 108y
2
+ 116y −169)
· (y
7
− 5y
6
+ 16y
5
− 43y
4
+ 71y
3
− 69y
2
+ 44y −16)
2
· (y
10
− 19y
9
+ ··· − 183y + 9)
24
