10
87
(K10a
39
)
1
Arc Sequences
6 7 1 8 4 10 9 5 2 3
Solving Sequence
3,10
1
4,7
2 6 5 9 8
c
10
c
3
c
2
c
6
c
5
c
9
c
7
c
1
, c
4
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu 1, a + 1, b 1i
I
u
2
= hu
41
2u
40
+ ··· 97u 29,
2.36615 × 10
84
u
40
+ 1.00102 × 10
85
u
39
+ ··· + 2.02004 × 10
85
b 2.22392 × 10
86
,
1.45715 × 10
86
u
40
2.37849 × 10
86
u
39
+ ··· + 5.85812 × 10
86
a 2.22717 × 10
87
i
There are 2 irreducible components with 42 representations.
1
The knot diagram image is adapter from “C. Livingston and A. H. Moore, KnotInfo: Table of Knot
Invariants, http://www.indiana.edu/ knotinfo”
1
I. I
u
1
= hu 1, a + 1, b 1i
(i) Arc colorings
a
3
=
1
0
a
10
=
1
1
a
1
=
0
1
a
4
=
1
1
a
7
=
0
1
a
2
=
1
1
a
6
=
1
2
a
5
=
0
1
a
9
=
1
1
a
8
=
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 1.00000
0 0
3
II. I
u
2
=
hu
41
2u
40
+· · ·97u29, 2.37×10
84
u
40
+1.00×10
85
u
39
+· · ·+2.02×10
85
b
2.22 × 10
86
, 1.46 × 10
86
u
40
2.38 × 10
86
u
39
+ · · · + 5.86 × 10
86
a 2.23 × 10
87
i
(i) Arc colorings
a
3
=
1
0
a
10
=
0.248740u
40
+ 0.406016u
39
+ ··· + 16.6441u + 3.80185
0.117134u
40
0.495543u
39
+ ··· + 29.0009u + 11.0093
a
1
=
0.131606u
40
0.0895269u
39
+ ··· + 45.6450u + 14.8111
0.117134u
40
0.495543u
39
+ ··· + 29.0009u + 11.0093
a
4
=
0.292917u
40
+ 0.513744u
39
+ ··· + 23.4665u + 3.07201
0.480424u
40
+ 1.12971u
39
+ ··· 3.94821u 8.19114
a
7
=
0
u
a
2
=
1
u
2
a
6
=
0.309265u
40
+ 0.667402u
39
+ ··· + 12.5382u 3.04639
0.240957u
40
+ 0.384787u
39
+ ··· + 29.4513u + 5.43770
a
5
=
0.461677u
40
0.955241u
39
+ ··· 39.4863u 7.28846
0.488568u
40
0.687641u
39
+ ··· 84.2115u 18.9283
a
9
=
0.0853670u
40
0.467864u
39
+ ··· + 29.5596u + 12.1587
0.283335u
40
0.962531u
39
+ ··· + 39.2615u + 16.9736
a
8
=
0.0783018u
40
+ 0.335178u
39
+ ··· 27.0594u 10.5696
0.259788u
40
0.136631u
39
+ ··· 77.8718u 19.5569
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0.964869u
40
+ 2.21650u
39
+ ··· + 15.7583u 11.1379
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.84706 1.39224I
a = 0.155159 0.941463I
b = 1.45266 + 0.46932I
9.17940 7.97252I 3.27060 + 3.71618I
u = 0.84706 + 1.39224I
a = 0.155159 + 0.941463I
b = 1.45266 0.46932I
9.17940 + 7.97252I 3.27060 3.71618I
u = 0.817889 1.149023I
a = 0.493407 0.092769I
b = 0.335334 0.324976I
3.49362 + 1.78935I 0.87448 4.34492I
u = 0.817889 + 1.149023I
a = 0.493407 + 0.092769I
b = 0.335334 + 0.324976I
3.49362 1.78935I 0.87448 + 4.34492I
u = 0.803428 0.914535I
a = 0.020946 + 0.842025I
b = 0.163245 1.244783I
3.27367 8.18385I 0.29544 + 8.35233I
u = 0.803428 + 0.914535I
a = 0.020946 0.842025I
b = 0.163245 + 1.244783I
3.27367 + 8.18385I 0.29544 8.35233I
u = 0.676854 0.607503I
a = 0.007234 + 1.105057I
b = 0.096162 0.914015I
2.78156 3.84619I 6.97849 + 6.75687I
u = 0.676854 + 0.607503I
a = 0.007234 1.105057I
b = 0.096162 + 0.914015I
2.78156 + 3.84619I 6.97849 6.75687I
u = 0.616531 1.004935I
a = 0.178036 1.400954I
b = 1.284695 + 0.271028I
4.36449 4.62926I 4.68493 + 4.91932I
u = 0.616531 + 1.004935I
a = 0.178036 + 1.400954I
b = 1.284695 0.271028I
4.36449 + 4.62926I 4.68493 4.91932I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.552035 0.294297I
a = 0.842537 + 0.587546I
b = 1.275178 0.127224I
2.56290 0.10225I 4.38337 2.22967I
u = 0.552035 + 0.294297I
a = 0.842537 0.587546I
b = 1.275178 + 0.127224I
2.56290 + 0.10225I 4.38337 + 2.22967I
u = 0.348292 0.221145I
a = 0.34967 + 2.50510I
b = 0.408078 0.420450I
1.94008 + 1.34771I 7.61480 3.42502I
u = 0.348292 + 0.221145I
a = 0.34967 2.50510I
b = 0.408078 + 0.420450I
1.94008 1.34771I 7.61480 + 3.42502I
u = 0.112780 0.808744I
a = 1.80140 + 1.14585I
b = 1.277529 + 0.155370I
7.57848 5.87702I 4.98103 + 5.65225I
u = 0.112780 + 0.808744I
a = 1.80140 1.14585I
b = 1.277529 0.155370I
7.57848 + 5.87702I 4.98103 5.65225I
u = 0.052118 0.819445I
a = 0.328066 0.998011I
b = 1.55855 + 0.65970I
7.62795 + 5.14257I 6.43416 5.95767I
u = 0.052118 + 0.819445I
a = 0.328066 + 0.998011I
b = 1.55855 0.65970I
7.62795 5.14257I 6.43416 + 5.95767I
u = 0.016292 0.887375I
a = 1.42252 1.37633I
b = 1.310630 0.070763I
8.52020 0.56768I 6.81847 + 0.32338I
u = 0.016292 + 0.887375I
a = 1.42252 + 1.37633I
b = 1.310630 + 0.070763I
8.52020 + 0.56768I 6.81847 0.32338I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.00877 3.29065I
a = 0.157549 0.007328I
b = 1.134146 0.025263I
5.18095 2.90757I 15.8800 + 0.0973I
u = 0.00877 + 3.29065I
a = 0.157549 + 0.007328I
b = 1.134146 + 0.025263I
5.18095 + 2.90757I 15.8800 0.0973I
u = 0.083569 0.725138I
a = 0.386389 + 1.105088I
b = 1.60494 0.57469I
7.77351 + 0.87153I 6.95024 + 0.30904I
u = 0.083569 + 0.725138I
a = 0.386389 1.105088I
b = 1.60494 + 0.57469I
7.77351 0.87153I 6.95024 0.30904I
u = 0.350374 0.701645I
a = 0.404942 0.966845I
b = 0.142705 + 0.550416I
0.05414 + 1.50218I 0.18723 4.24532I
u = 0.350374 + 0.701645I
a = 0.404942 + 0.966845I
b = 0.142705 0.550416I
0.05414 1.50218I 0.18723 + 4.24532I
u = 0.358882 0.899587I
a = 0.025505 0.769488I
b = 1.072998 + 0.545709I
1.56831 + 2.54987I 1.57226 7.77175I
u = 0.358882 + 0.899587I
a = 0.025505 + 0.769488I
b = 1.072998 0.545709I
1.56831 2.54987I 1.57226 + 7.77175I
u = 0.519866 0.633729I
a = 0.18225 + 2.28261I
b = 1.116563 0.153454I
0.067811 + 0.953358I 2.02910 7.42558I
u = 0.519866 + 0.633729I
a = 0.18225 2.28261I
b = 1.116563 + 0.153454I
0.067811 0.953358I 2.02910 + 7.42558I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.662740
a = 0.845831
b = 0.0835204
1.43130 6.87204
u = 0.66708 2.46190I
a = 0.107052 0.134257I
b = 0.938830 0.044980I
0.431185 0.284475I 13.1815 15.1990I
u = 0.66708 + 2.46190I
a = 0.107052 + 0.134257I
b = 0.938830 + 0.044980I
0.431185 + 0.284475I 13.1815 + 15.1990I
u = 0.726138 0.952037I
a = 0.023496 0.831457I
b = 0.265202 + 1.192169I
3.76974 + 2.25598I 1.50138 3.41744I
u = 0.726138 + 0.952037I
a = 0.023496 + 0.831457I
b = 0.265202 1.192169I
3.76974 2.25598I 1.50138 + 3.41744I
u = 0.897003 1.048182I
a = 0.059395 + 1.136795I
b = 1.253449 0.430907I
0.85372 + 8.63849I 1.84811 8.19635I
u = 0.897003 + 1.048182I
a = 0.059395 1.136795I
b = 1.253449 + 0.430907I
0.85372 8.63849I 1.84811 + 8.19635I
u = 0.93078 1.40973I
a = 0.113769 + 0.904087I
b = 1.44670 0.51890I
8.3544 + 14.2736I 1.90848 8.33140I
u = 0.93078 + 1.40973I
a = 0.113769 0.904087I
b = 1.44670 + 0.51890I
8.3544 14.2736I 1.90848 + 8.33140I
u = 0.986991 0.919090I
a = 0.549234 + 0.027574I
b = 0.197100 + 0.379729I
3.15987 + 3.90045I 0.31532 1.57295I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.986991 + 0.919090I
a = 0.549234 0.027574I
b = 0.197100 0.379729I
3.15987 3.90045I 0.31532 + 1.57295I
8
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)(u
41
+ 8u
39
+ ··· + 687u 229)
c
2
(u 1)(u
41
+ 2u
40
+ ··· 97u + 29)
c
3
(u 1)(u
41
+ 2u
40
+ ··· + 9u + 1)
c
4
(u 1)(u
41
+ 2u
40
+ ··· u + 1)
c
5
(u + 1)(u
41
+ 12u
40
+ ··· + 5u + 1)
c
6
(u 1)(u
41
+ 4u
40
+ ··· + u + 1)
c
7
(u 1)(u
41
+ 12u
40
+ ··· + 5u + 1)
c
8
(u + 1)(u
41
+ 2u
40
+ ··· u + 1)
c
9
(u)(u
41
+ 7u
40
+ ··· + 6u + 2)
c
10
(u + 1)(u
41
+ 2u
40
+ ··· + 9u + 1)
9
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
(y 1)(y
41
+ 16y
40
+ ··· + 1271637y 52441)
c
2
(y 1)(y
41
+ 48y
40
+ ··· 7063y 841)
c
3
, c
10
(y 1)(y
41
32y
40
+ ··· + 141y 1)
c
4
, c
8
(y 1)(y
41
12y
40
+ ··· + 5y 1)
c
5
, c
7
(y 1)(y
41
+ 36y
40
+ ··· + 5y 1)
c
6
(y 1)(y
41
8y
40
+ ··· + 5y 1)
c
9
(y)(y
41
+ 9y
40
+ ··· 16y 4)
10