12n
0671
(K12n
0671
)
A knot diagram
1
Linearized knot diagam
4 5 7 2 8 3 11 5 12 1 8 10
Solving Sequence
1,4
2
5,10
11 12 9 8 6 7 3
c
1
c
4
c
10
c
12
c
9
c
8
c
5
c
7
c
3
c
2
, c
6
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hb + u, 2u
12
+ 7u
11
+ 2u
10
− 18u
9
− 18u
8
+ 6u
7
+ 9u
6
+ 6u
5
+ 19u
4
+ 11u
3
− 6u
2
+ 2a − 4u − 1,
u
13
+ 3u
12
− u
11
− 10u
10
− 4u
9
+ 9u
8
+ 3u
7
+ 8u
5
− 7u
3
− u
2
+ u − 1i
I
u
2
= h1.99592 × 10
42
u
43
+ 5.39326 × 10
42
u
42
+ ··· + 7.34448 × 10
41
b − 6.74459 × 10
41
,
2.00974 × 10
41
u
43
+ 3.00732 × 10
41
u
42
+ ··· + 1.83612 × 10
41
a − 1.49049 × 10
43
, u
44
+ 4u
43
+ ··· + 116u − 1i
I
u
3
= hb + 1, 2u
2
+ a + 4u + 4, u
3
+ u
2
− 1i
I
u
4
= h−a
2
+ 4b − 6a + 4, a
3
+ 4a
2
− 12a + 8, u − 1i
I
u
5
= hb + u, a − 2, u
2
+ u − 1i
I
u
6
= hb − u − 1, a + u + 1, u
2
+ u − 1i
* 6 irreducible components of dim
C
= 0, with total 67 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
= hb + u, 2u
12
+ 7u
11
+ · · · + 2a − 1, u
13
+ 3u
12
+ · · · + u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
10
=
−u
12
−
7
2
u
11
+ ··· + 2u +
1
2
−u
a
11
=
−u
12
−
7
2
u
11
+ ··· + 3u +
1
2
−u
a
12
=
−
1
2
u
12
− 2u
11
+ ··· + u
2
+
3
2
u
−u
2
a
9
=
−
1
2
u
12
− 2u
11
+ ··· +
3
2
u + 1
u
3
− u
a
8
=
−u
11
− 2u
10
+ 2u
9
+ 6u
8
+ u
7
− 3u
6
− 2u
5
− 3u
4
− 3u
3
+ 2u + 1
1
2
u
12
+
1
2
u
11
+ ··· −
1
2
u −
1
2
a
6
=
1
2
u
12
− 3u
10
+ ··· +
3
2
u − 1
3
2
u
12
+
7
2
u
11
+ ··· +
7
2
u −
3
2
a
7
=
1
2
u
11
+ u
10
+ ··· − u +
3
2
−u
11
− u
10
+ 4u
9
+ 3u
8
− 6u
7
− u
6
+ 3u
5
− 4u
4
+ u
3
+ 3u
2
− u
a
3
=
−u
2
+ 1
−u
4
+ 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −3u
12
−18u
11
−24u
10
+ 32u
9
+ 80u
8
+ 9u
7
−34u
6
+ 5u
5
−47u
4
−70u
3
+ 6u
2
+ 11u −18
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
9
, c
10
, c
12
u
13
− 3u
12
− u
11
+ 10u
10
− 4u
9
− 9u
8
+ 3u
7
+ 8u
5
− 7u
3
+ u
2
+ u + 1
c
3
, c
6
, c
7
c
11
u
13
+ u
12
+ ··· + 5u + 1
c
5
, c
8
u
13
+ 5u
12
+ ··· − 8u − 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
9
, c
10
, c
12
y
13
− 11y
12
+ ··· − y −1
c
3
, c
6
, c
7
c
11
y
13
− 3y
12
+ ··· + 7y −1
c
5
, c
8
y
13
− 5y
12
+ ··· + 96y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.920255
a = −5.92663
b = −0.920255
−2.84609 −65.8580
u = 0.217488 + 0.883339I
a = 0.154186 + 0.593271I
b = −0.217488 − 0.883339I
2.14237 − 5.68500I −7.77978 + 6.07128I
u = 0.217488 − 0.883339I
a = 0.154186 − 0.593271I
b = −0.217488 + 0.883339I
2.14237 + 5.68500I −7.77978 − 6.07128I
u = −0.795282 + 0.405757I
a = 1.200860 + 0.132364I
b = 0.795282 − 0.405757I
1.52283 + 3.56370I −3.66796 − 8.41026I
u = −0.795282 − 0.405757I
a = 1.200860 − 0.132364I
b = 0.795282 + 0.405757I
1.52283 − 3.56370I −3.66796 + 8.41026I
u = 1.266340 + 0.164860I
a = −3.55407 + 0.37923I
b = −1.266340 − 0.164860I
−3.86762 − 1.80054I −11.65148 + 0.61379I
u = 1.266340 − 0.164860I
a = −3.55407 − 0.37923I
b = −1.266340 + 0.164860I
−3.86762 + 1.80054I −11.65148 − 0.61379I
u = −1.38670 + 0.37744I
a = 1.43136 + 0.81847I
b = 1.38670 − 0.37744I
−10.71940 + 7.71547I −15.7360 − 5.7316I
u = −1.38670 − 0.37744I
a = 1.43136 − 0.81847I
b = 1.38670 + 0.37744I
−10.71940 − 7.71547I −15.7360 + 5.7316I
u = 0.240304 + 0.377267I
a = 1.39203 + 1.59640I
b = −0.240304 − 0.377267I
−0.98403 − 1.11558I −8.69395 + 6.01211I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.240304 − 0.377267I
a = 1.39203 − 1.59640I
b = −0.240304 + 0.377267I
−0.98403 + 1.11558I −8.69395 − 6.01211I
u = −1.50228 + 0.43298I
a = 1.83895 + 0.89236I
b = 1.50228 − 0.43298I
−8.8777 + 15.5620I −14.5417 − 7.8795I
u = −1.50228 − 0.43298I
a = 1.83895 − 0.89236I
b = 1.50228 + 0.43298I
−8.8777 − 15.5620I −14.5417 + 7.8795I
6
II. I
u
2
=
h2.00×10
42
u
43
+5.39×10
42
u
42
+· · ·+7.34×10
41
b−6.74×10
41
, 2.01×10
41
u
43
+
3.01 × 10
41
u
42
+ · · · + 1.84 × 10
41
a − 1.49 × 10
43
, u
44
+ 4u
43
+ · · · + 116u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
10
=
−1.09456u
43
− 1.63787u
42
+ ··· − 526.141u + 81.1760
−2.71758u
43
− 7.34328u
42
+ ··· − 193.776u + 0.918321
a
11
=
1.62303u
43
+ 5.70542u
42
+ ··· − 332.365u + 80.2576
−2.71758u
43
− 7.34328u
42
+ ··· − 193.776u + 0.918321
a
12
=
3.21805u
43
+ 8.47168u
42
+ ··· + 23.3865u + 57.2283
2.36406u
43
+ 6.27063u
42
+ ··· + 140.740u − 1.74238
a
9
=
3.61816u
43
+ 9.27065u
42
+ ··· + 182.489u + 32.8026
4.49904u
43
+ 11.8002u
42
+ ··· + 312.338u − 3.00702
a
8
=
1.96942u
43
+ 5.25944u
42
+ ··· + 65.5876u + 33.8026
2.07745u
43
+ 5.75738u
42
+ ··· + 127.873u − 1.42316
a
6
=
1.07148u
43
+ 2.89399u
42
+ ··· + 59.4605u + 9.96375
1.37364u
43
+ 3.38028u
42
+ ··· + 86.1371u − 0.830453
a
7
=
1.78926u
43
+ 4.62600u
42
+ ··· + 163.701u − 11.9689
−1.22781u
43
− 2.74973u
42
+ ··· − 70.8975u + 0.701920
a
3
=
−u
2
+ 1
−u
4
+ 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2.60530u
43
− 11.0854u
42
+ ··· + 553.011u − 13.5461
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
9
, c
10
, c
12
u
44
− 4u
43
+ ··· − 116u − 1
c
3
, c
6
, c
7
c
11
u
44
+ 3u
43
+ ··· − 44u + 8
c
5
, c
8
(u
22
− u
21
+ ··· − 9u − 2)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
9
, c
10
, c
12
y
44
− 40y
43
+ ··· − 12428y + 1
c
3
, c
6
, c
7
c
11
y
44
− 21y
43
+ ··· − 7760y + 64
c
5
, c
8
(y
22
− 15y
21
+ ··· − 113y + 4)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.090540 + 0.022158I
a = −1.92894 − 1.79340I
b = −0.643688 + 0.282110I
−2.83824 + 0.14755I −2.77483 − 4.21375I
u = 1.090540 − 0.022158I
a = −1.92894 + 1.79340I
b = −0.643688 − 0.282110I
−2.83824 − 0.14755I −2.77483 + 4.21375I
u = 0.109119 + 0.888646I
a = −0.127810 − 0.241019I
b = 1.351490 + 0.160264I
−5.93215 − 3.14286I −14.6418 + 3.7109I
u = 0.109119 − 0.888646I
a = −0.127810 + 0.241019I
b = 1.351490 − 0.160264I
−5.93215 + 3.14286I −14.6418 − 3.7109I
u = 0.344224 + 1.065750I
a = 0.631899 − 0.686719I
b = 1.40825 + 0.36939I
−3.01557 − 10.18830I −12.15400 + 6.99410I
u = 0.344224 − 1.065750I
a = 0.631899 + 0.686719I
b = 1.40825 − 0.36939I
−3.01557 + 10.18830I −12.15400 − 6.99410I
u = −1.134110 + 0.122816I
a = 0.989662 − 0.285685I
b = 0.748799 − 0.898808I
1.18895 + 3.23778I −15.5021 − 9.5411I
u = −1.134110 − 0.122816I
a = 0.989662 + 0.285685I
b = 0.748799 + 0.898808I
1.18895 − 3.23778I −15.5021 + 9.5411I
u = 1.036890 + 0.519128I
a = 0.888297 + 0.072748I
b = −0.117503 + 0.569726I
−0.357526 + 0.716312I −8.85937 − 2.91987I
u = 1.036890 − 0.519128I
a = 0.888297 − 0.072748I
b = −0.117503 − 0.569726I
−0.357526 − 0.716312I −8.85937 + 2.91987I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.748799 + 0.898808I
a = 0.887522 + 0.470312I
b = 1.134110 − 0.122816I
1.18895 + 3.23778I −15.5021 − 9.5411I
u = −0.748799 − 0.898808I
a = 0.887522 − 0.470312I
b = 1.134110 + 0.122816I
1.18895 − 3.23778I −15.5021 + 9.5411I
u = 0.238284 + 0.726491I
a = −0.740307 + 0.364992I
b = −1.358520 − 0.282419I
−1.09298 − 3.55787I −9.79859 + 4.38747I
u = 0.238284 − 0.726491I
a = −0.740307 − 0.364992I
b = −1.358520 + 0.282419I
−1.09298 + 3.55787I −9.79859 − 4.38747I
u = 0.736176
a = 0.933140
b = −0.00897213
−1.10346 −8.70720
u = −0.164222 + 0.700108I
a = −0.035754 − 0.152428I
b = 0.164222 + 0.700108I
3.71629 −3.80483 + 0.I
u = −0.164222 − 0.700108I
a = −0.035754 + 0.152428I
b = 0.164222 − 0.700108I
3.71629 −3.80483 + 0.I
u = 1.243520 + 0.352072I
a = 1.29291 − 1.34278I
b = 1.45462 + 0.06689I
−9.47192 − 1.36166I 0
u = 1.243520 − 0.352072I
a = 1.29291 + 1.34278I
b = 1.45462 − 0.06689I
−9.47192 + 1.36166I 0
u = 0.643688 + 0.282110I
a = −1.54817 + 3.78331I
b = −1.090540 + 0.022158I
−2.83824 − 0.14755I −2.77483 + 4.21375I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.643688 − 0.282110I
a = −1.54817 − 3.78331I
b = −1.090540 − 0.022158I
−2.83824 + 0.14755I −2.77483 − 4.21375I
u = 1.047760 + 0.864022I
a = 0.870174 − 0.473031I
b = 1.347140 − 0.234013I
−5.02280 + 3.68716I 0
u = 1.047760 − 0.864022I
a = 0.870174 + 0.473031I
b = 1.347140 + 0.234013I
−5.02280 − 3.68716I 0
u = −1.351490 + 0.160264I
a = −0.134000 − 0.119389I
b = −0.109119 + 0.888646I
−5.93215 + 3.14286I 0
u = −1.351490 − 0.160264I
a = −0.134000 + 0.119389I
b = −0.109119 − 0.888646I
−5.93215 − 3.14286I 0
u = −1.347140 + 0.234013I
a = −0.919401 − 0.349911I
b = −1.047760 − 0.864022I
−5.02280 + 3.68716I 0
u = −1.347140 − 0.234013I
a = −0.919401 + 0.349911I
b = −1.047760 + 0.864022I
−5.02280 − 3.68716I 0
u = 1.358520 + 0.282419I
a = −0.377703 − 0.253352I
b = −0.238284 − 0.726491I
−1.09298 − 3.55787I 0
u = 1.358520 − 0.282419I
a = −0.377703 + 0.253352I
b = −0.238284 + 0.726491I
−1.09298 + 3.55787I 0
u = 0.117503 + 0.569726I
a = −0.59666 − 1.67345I
b = −1.036890 + 0.519128I
−0.357526 − 0.716312I −8.85937 + 2.91987I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.117503 − 0.569726I
a = −0.59666 + 1.67345I
b = −1.036890 − 0.519128I
−0.357526 + 0.716312I −8.85937 − 2.91987I
u = −1.42436
a = 2.23200
b = 1.80378
−16.0009 0
u = −1.39535 + 0.29610I
a = −2.22418 − 0.65388I
b = −1.57069 + 0.28000I
−6.28468 + 7.27868I 0
u = −1.39535 − 0.29610I
a = −2.22418 + 0.65388I
b = −1.57069 − 0.28000I
−6.28468 − 7.27868I 0
u = −1.40825 + 0.36939I
a = −0.706930 + 0.124928I
b = −0.344224 + 1.065750I
−3.01557 + 10.18830I 0
u = −1.40825 − 0.36939I
a = −0.706930 − 0.124928I
b = −0.344224 − 1.065750I
−3.01557 − 10.18830I 0
u = −1.45462 + 0.06689I
a = −1.38893 − 0.89884I
b = −1.243520 + 0.352072I
−9.47192 + 1.36166I 0
u = −1.45462 − 0.06689I
a = −1.38893 + 0.89884I
b = −1.243520 − 0.352072I
−9.47192 − 1.36166I 0
u = 0.521744
a = −1.71909
b = 1.63226
−9.78452 30.1490
u = 1.57069 + 0.28000I
a = 2.00659 − 0.51930I
b = 1.39535 + 0.29610I
−6.28468 − 7.27868I 0
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.57069 − 0.28000I
a = 2.00659 + 0.51930I
b = 1.39535 − 0.29610I
−6.28468 + 7.27868I 0
u = −1.63226
a = 0.549499
b = −0.521744
−9.78452 0
u = −1.80378
a = 1.76252
b = 1.42436
−16.0009 0
u = 0.00897213
a = 76.5654
b = −0.736176
−1.10346 −8.70720
14
III. I
u
3
= hb + 1, 2u
2
+ a + 4u + 4, u
3
+ u
2
− 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
u
2
+ u − 1
a
10
=
−2u
2
− 4u − 4
−1
a
11
=
−2u
2
− 4u − 3
−1
a
12
=
−2u
2
− 4u − 3
−1
a
9
=
−1
0
a
8
=
−u
2u
2
+ u − 2
a
6
=
−2u + 1
5u
2
+ 2u − 4
a
7
=
−u
2u
2
+ u − 2
a
3
=
−u
2
+ 1
u
2
− u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 21u
2
+ 45u + 27
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
u
3
+ u
2
− 1
c
3
u
3
− u
2
+ 2u − 1
c
4
u
3
− u
2
+ 1
c
5
u
3
+ 3u
2
+ 2u − 1
c
6
u
3
+ u
2
+ 2u + 1
c
7
, c
11
u
3
c
8
u
3
− 3u
2
+ 2u + 1
c
9
, c
10
(u − 1)
3
c
12
(u + 1)
3
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
3
− y
2
+ 2y −1
c
3
, c
6
y
3
+ 3y
2
+ 2y −1
c
5
, c
8
y
3
− 5y
2
+ 10y −1
c
7
, c
11
y
3
c
9
, c
10
, c
12
(y −1)
3
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = −0.920404 − 0.365165I
b = −1.00000
1.37919 + 2.82812I −7.96807 + 6.06881I
u = −0.877439 − 0.744862I
a = −0.920404 + 0.365165I
b = −1.00000
1.37919 − 2.82812I −7.96807 − 6.06881I
u = 0.754878
a = −8.15919
b = −1.00000
−2.75839 72.9360
18
IV. I
u
4
= h−a
2
+ 4b − 6a + 4, a
3
+ 4a
2
− 12a + 8, u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
−1
0
a
10
=
a
1
4
a
2
+
3
2
a − 1
a
11
=
−
1
4
a
2
−
1
2
a + 1
1
4
a
2
+
3
2
a − 1
a
12
=
−
1
2
a
2
− 2a + 3
−
1
2
a
2
−
5
2
a + 3
a
9
=
−
1
4
a
2
−
1
2
a + 3
−
1
4
a
2
− a + 3
a
8
=
1
2
a
−
1
4
a
2
− a + 3
a
6
=
0
−
3
4
a
2
−
7
2
a + 7
a
7
=
0
−
3
4
a
2
−
7
2
a + 7
a
3
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
3
4
a
2
− 15a + 9
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
6
u
3
c
4
(u + 1)
3
c
5
u
3
+ 3u
2
+ 2u − 1
c
7
u
3
− u
2
+ 2u − 1
c
8
u
3
− 3u
2
+ 2u + 1
c
9
, c
10
u
3
+ u
2
− 1
c
11
u
3
+ u
2
+ 2u + 1
c
12
u
3
− u
2
+ 1
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
3
c
3
, c
6
y
3
c
5
, c
8
y
3
− 5y
2
+ 10y −1
c
7
, c
11
y
3
+ 3y
2
+ 2y −1
c
9
, c
10
, c
12
y
3
− y
2
+ 2y −1
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.079600 + 0.365165I
b = 0.877439 + 0.744862I
1.37919 − 2.82812I −7.96807 − 6.06881I
u = 1.00000
a = 1.079600 − 0.365165I
b = 0.877439 − 0.744862I
1.37919 + 2.82812I −7.96807 + 6.06881I
u = 1.00000
a = −6.15919
b = −0.754878
−2.75839 72.9360
22
V. I
u
5
= hb + u, a − 2, u
2
+ u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
−u + 1
a
5
=
−u
−u + 1
a
10
=
2
−u
a
11
=
u + 2
−u
a
12
=
2u + 1
u − 1
a
9
=
u
u − 1
a
8
=
u
u − 1
a
6
=
−u
−u + 1
a
7
=
−1
0
a
3
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −20
23
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
7
, c
9
, c
10
u
2
+ u − 1
c
4
, c
6
, c
11
c
12
u
2
− u − 1
c
5
, c
8
u
2
24
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
9
, c
10
, c
11
c
12
y
2
− 3y + 1
c
5
, c
8
y
2
25
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 2.00000
b = −0.618034
−1.97392 −20.0000
u = −1.61803
a = 2.00000
b = 1.61803
−17.7653 −20.0000
26
VI. I
u
6
= hb − u − 1, a + u + 1, u
2
+ u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
−u + 1
a
5
=
−u
−u + 1
a
10
=
−u − 1
u + 1
a
11
=
−2u − 2
u + 1
a
12
=
u + 3
−u − 2
a
9
=
2u + 3
−u − 2
a
8
=
2u + 3
−u − 2
a
6
=
−u
−u + 1
a
7
=
−1
0
a
3
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −65
27
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
7
, c
9
, c
10
u
2
+ u − 1
c
4
, c
6
, c
11
c
12
u
2
− u − 1
c
5
, c
8
u
2
28
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
9
, c
10
, c
11
c
12
y
2
− 3y + 1
c
5
, c
8
y
2
29
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −1.61803
b = 1.61803
−9.86960 −65.0000
u = −1.61803
a = 0.618034
b = −0.618034
−9.86960 −65.0000
30
VII. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
9
c
10
(u − 1)
3
(u
2
+ u − 1)
2
(u
3
+ u
2
− 1)
· (u
13
− 3u
12
− u
11
+ 10u
10
− 4u
9
− 9u
8
+ 3u
7
+ 8u
5
− 7u
3
+ u
2
+ u + 1)
· (u
44
− 4u
43
+ ··· − 116u − 1)
c
3
, c
7
u
3
(u
2
+ u − 1)
2
(u
3
− u
2
+ 2u − 1)(u
13
+ u
12
+ ··· + 5u + 1)
· (u
44
+ 3u
43
+ ··· − 44u + 8)
c
4
, c
12
(u + 1)
3
(u
2
− u − 1)
2
(u
3
− u
2
+ 1)
· (u
13
− 3u
12
− u
11
+ 10u
10
− 4u
9
− 9u
8
+ 3u
7
+ 8u
5
− 7u
3
+ u
2
+ u + 1)
· (u
44
− 4u
43
+ ··· − 116u − 1)
c
5
u
4
(u
3
+ 3u
2
+ 2u − 1)
2
(u
13
+ 5u
12
+ ··· − 8u − 4)
· (u
22
− u
21
+ ··· − 9u − 2)
2
c
6
, c
11
u
3
(u
2
− u − 1)
2
(u
3
+ u
2
+ 2u + 1)(u
13
+ u
12
+ ··· + 5u + 1)
· (u
44
+ 3u
43
+ ··· − 44u + 8)
c
8
u
4
(u
3
− 3u
2
+ 2u + 1)
2
(u
13
+ 5u
12
+ ··· − 8u − 4)
· (u
22
− u
21
+ ··· − 9u − 2)
2
31
VIII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
9
, c
10
, c
12
((y −1)
3
)(y
2
− 3y + 1)
2
(y
3
− y
2
+ 2y −1)(y
13
− 11y
12
+ ··· − y −1)
· (y
44
− 40y
43
+ ··· − 12428y + 1)
c
3
, c
6
, c
7
c
11
y
3
(y
2
− 3y + 1)
2
(y
3
+ 3y
2
+ 2y −1)(y
13
− 3y
12
+ ··· + 7y −1)
· (y
44
− 21y
43
+ ··· − 7760y + 64)
c
5
, c
8
y
4
(y
3
− 5y
2
+ 10y −1)
2
(y
13
− 5y
12
+ ··· + 96y −16)
· (y
22
− 15y
21
+ ··· − 113y + 4)
2
32
