10
68
(K10a
67
)
1
Arc Sequences
6 9 8 10 7 2 1 3 5 4
Solving Sequence
2,9 3,7
6 1 5 10 4 8
c
2
c
6
c
1
c
5
c
9
c
4
c
8
c
3
, c
7
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
4
u
2
+ 1, u
3
+ b, u
3
u
2
+ a + u + 1i
I
u
2
= hb
18
+ b
17
+ ··· 6b + 1, 389b
17
3339b
16
+ ··· + 12107a + 6552,
4802b
17
+ 12107u + ··· + 19536b 16224i
I
u
3
= hu
14
+ 3u
13
+ u
12
8u
11
12u
10
+ u
9
+ 19u
8
+ 18u
7
2u
6
18u
5
13u
4
+ 3u
3
+ 11u
2
+ 7u + 2,
u
13
2u
12
+ u
11
+ 7u
10
+ 5u
9
6u
8
13u
7
5u
6
+ 8u
5
+ 11u
4
+ u
3
6u
2
+ b 5u 1,
5u
13
11u
12
+ 5u
11
+ 38u
10
+ 28u
9
35u
8
71u
7
26u
6
+ 44u
5
+ 58u
4
+ 9u
3
33u
2
+ 2a 27u 7i
There are 3 irreducible components with 36 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
4
u
2
+ 1, u
3
+ b, u
3
u
2
+ a + u + 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
u
3
+ u
2
u 1
u
3
a
3
=
u
2
+ u + 1
1
a
7
=
0
u
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
5
=
u
u
3
+ u
a
10
=
u
3
u
u
3
u
2
a
4
=
u
2
+ u
1
a
8
=
u
3
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
4
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.866025 0.500000I
a = 0.366025 + 0.366025I
b = 1.00000I
3.28987 2.02988I 2.00000 + 3.46410I
u = 0.866025 + 0.500000I
a = 0.366025 0.366025I
b = 1.00000I
3.28987 + 2.02988I 2.00000 3.46410I
u = 0.866025 0.500000I
a = 1.36603 1.36603I
b = 1.00000I
3.28987 + 2.02988I 2.00000 3.46410I
u = 0.866025 + 0.500000I
a = 1.36603 + 1.36603I
b = 1.00000I
3.28987 2.02988I 2.00000 + 3.46410I
3
II. I
u
2
= hb
18
+ b
17
+ · · · 6b + 1, 389b
17
3339b
16
+ · · · + 12107a +
6552, 4802b
17
+ 12107u + · · · + 19536b 16224i
(i) Arc colorings
a
2
=
1
0
a
9
=
0.0321302b
17
+ 0.275791b
16
+ ··· + 3.26704b 0.541175
b
a
3
=
0.307921b
17
1.01247b
16
+ ··· + 0.733956b + 0.967870
b
2
a
7
=
0
0.396630b
17
0.682911b
16
+ ··· 1.61361b + 1.34005
a
6
=
0.396630b
17
0.682911b
16
+ ··· 1.61361b + 1.34005
0.396630b
17
0.682911b
16
+ ··· 1.61361b + 1.34005
a
1
=
0.579830b
17
+ 0.616833b
16
+ ··· + 4.42265b 0.550012
0.579830b
17
+ 0.616833b
16
+ ··· + 4.42265b 1.55001
a
5
=
0.396630b
17
0.682911b
16
+ ··· 1.61361b + 1.34005
0.275791b
17
+ 0.288263b
16
+ ··· + 1.53308b + 0.490956
a
10
=
0.184439b
17
+ 0.0818535b
16
+ ··· 0.333443b 0.0474106
0.521516b
17
0.492690b
16
+ ··· 4.87387b + 1.79698
a
4
=
0.606674b
17
0.941687b
16
+ ··· 2.45147b + 1.64029
b
4
2b
2
a
8
=
0.672421b
17
+ 0.971174b
16
+ ··· + 4.14669b 0.849096
b
3
+ b
(ii) Obstruction class = 1
(iii) Cusp Shapes =
31900
12107
b
17
61944
12107
b
16
+ ···
168364
12107
b +
112870
12107
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.172473 0.500383I
a = 1.30737 + 0.64095I
b = 0.912264 0.491243I
6.88799 + 7.08493I 1.57680 5.91335I
u = 1.172473 + 0.500383I
a = 1.30737 0.64095I
b = 0.912264 + 0.491243I
6.88799 7.08493I 1.57680 + 5.91335I
u = 1.173911 0.391555I
a = 0.295022 0.527057I
b = 0.792965 0.741615I
7.66122 1.33617I 3.28409 + 0.70175I
u = 1.173911 + 0.391555I
a = 0.295022 + 0.527057I
b = 0.792965 + 0.741615I
7.66122 + 1.33617I 3.28409 0.70175I
u = 0.141484 + 0.739668I
a = 0.580256 0.436812I
b = 0.118400 1.390976I
3.90681 + 2.45442I 1.67208 2.91298I
u = 0.141484 0.739668I
a = 0.580256 + 0.436812I
b = 0.118400 + 1.390976I
3.90681 2.45442I 1.67208 + 2.91298I
u = 0.772920 0.510351I
a = 1.134685 0.621833I
b = 0.103396 1.069764I
1.50643 + 2.09337I 4.51499 4.16283I
u = 0.772920 + 0.510351I
a = 1.134685 + 0.621833I
b = 0.103396 + 1.069764I
1.50643 2.09337I 4.51499 + 4.16283I
u = 1.173911 + 0.391555I
a = 0.93744 + 1.09998I
b = 0.00304 1.47476I
7.66122 + 1.33617I 3.28409 0.70175I
u = 1.173911 0.391555I
a = 0.93744 1.09998I
b = 0.00304 + 1.47476I
7.66122 1.33617I 3.28409 + 0.70175I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.172473 + 0.500383I
a = 1.71906 + 0.77036I
b = 0.18330 1.47754I
6.88799 7.08493I 1.57680 + 5.91335I
u = 1.172473 0.500383I
a = 1.71906 0.77036I
b = 0.18330 + 1.47754I
6.88799 + 7.08493I 1.57680 5.91335I
u = 0.772920 + 0.510351I
a = 0.651119 0.926889I
b = 0.243739 0.102909I
1.50643 2.09337I 4.51499 + 4.16283I
u = 0.772920 0.510351I
a = 0.651119 + 0.926889I
b = 0.243739 + 0.102909I
1.50643 + 2.09337I 4.51499 4.16283I
u = 0.825933
a = 1.78189 + 0.72543I
b = 0.256179 1.094021I
4.48831 4.65235
u = 0.825933
a = 1.78189 0.72543I
b = 0.256179 + 1.094021I
4.48831 4.65235
u = 0.141484 0.739668I
a = 1.60270 0.68359I
b = 0.746849 0.515863I
3.90681 2.45442I 1.67208 + 2.91298I
u = 0.141484 + 0.739668I
a = 1.60270 + 0.68359I
b = 0.746849 + 0.515863I
3.90681 + 2.45442I 1.67208 2.91298I
6
III. I
u
3
=
hu
14
+3u
13
+· · ·+7u+2, u
13
2u
12
+· · ·+b1, 5u
13
11u
12
+· · ·+2a7i
(i) Arc colorings
a
2
=
1
0
a
9
=
5
2
u
13
+
11
2
u
12
+ ··· +
27
2
u +
7
2
u
13
+ 2u
12
+ ··· + 5u + 1
a
3
=
1
2
u
13
1
2
u
12
+ ···
1
2
u +
1
2
u
12
+ u
11
+ ··· + 3u + 1
a
7
=
0
u
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
5
=
u
u
3
+ u
a
10
=
3
2
u
13
+
7
2
u
12
+ ··· +
15
2
u +
3
2
u
13
+ 2u
12
+ ··· + 4u + 1
a
4
=
1
2
u
13
1
2
u
12
+ ··· +
1
2
u +
1
2
u
13
+ 3u
12
+ ··· + 9u + 3
a
8
=
u
5
2u
3
+ u
u
5
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 2u
13
4u
12
+4u
11
+18u
10
+6u
9
26u
8
30u
7
+8u
6
+32u
5
+20u
4
16u
3
20u
2
6u+10
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.211205 0.579083I
a = 1.90400 0.21727I
b = 0.33038 + 1.55103I
13.5268 11.6370I 3.43423 + 6.31221I
u = 1.211205 + 0.579083I
a = 1.90400 + 0.21727I
b = 0.33038 1.55103I
13.5268 + 11.6370I 3.43423 6.31221I
u = 1.041837 0.481714I
a = 0.984465 + 0.702849I
b = 0.552436 0.381452I
0.78724 4.41668I 3.49417 + 7.88625I
u = 1.041837 + 0.481714I
a = 0.984465 0.702849I
b = 0.552436 + 0.381452I
0.78724 + 4.41668I 3.49417 7.88625I
u = 0.830389 0.784414I
a = 0.92687 1.26515I
b = 0.04509 1.43706I
6.78342 2.90589I 2.10855 + 2.91897I
u = 0.830389 + 0.784414I
a = 0.92687 + 1.26515I
b = 0.04509 + 1.43706I
6.78342 + 2.90589I 2.10855 2.91897I
u = 0.400528 0.482833I
a = 1.076986 0.113831I
b = 0.498731 0.157320I
1.035524 + 0.368514I 9.33320 2.06000I
u = 0.400528 + 0.482833I
a = 1.076986 + 0.113831I
b = 0.498731 + 0.157320I
1.035524 0.368514I 9.33320 + 2.06000I
u = 0.243278 0.917020I
a = 0.780413 + 1.135089I
b = 0.26550 + 1.53094I
10.58651 + 6.18900I 1.00936 2.90508I
u = 0.243278 + 0.917020I
a = 0.780413 1.135089I
b = 0.26550 1.53094I
10.58651 6.18900I 1.00936 + 2.90508I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.941064 0.407114I
a = 0.198597 + 0.140678I
b = 0.164790 0.466680I
1.42730 + 1.54478I 1.163355 0.228482I
u = 0.941064 + 0.407114I
a = 0.198597 0.140678I
b = 0.164790 + 0.466680I
1.42730 1.54478I 1.163355 + 0.228482I
u = 1.286173 0.280982I
a = 0.237831 0.480033I
b = 0.19870 + 1.61232I
15.6273 2.2414I 5.43859 + 0.46441I
u = 1.286173 + 0.280982I
a = 0.237831 + 0.480033I
b = 0.19870 1.61232I
15.6273 + 2.2414I 5.43859 0.46441I
9
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
(u
4
u
2
+ 1)(u
9
u
8
2u
7
+ 3u
6
+ u
5
3u
4
+ 2u
3
u + 1)
2
(u
14
+ 3u
13
+ ··· + 7u + 2)
c
2
, c
3
, c
4
c
8
, c
9
, c
10
(u
2
+ 1)
2
(u
14
+ 9u
12
+ ··· + u
2
+ 1)(u
18
+ u
17
+ ··· 6u + 1)
c
5
(u
2
u + 1)
2
(u
9
+ 5u
8
+ 12u
7
+ 15u
6
+ 9u
5
u
4
4u
3
2u
2
+ u + 1)
2
(u
14
+ 7u
13
+ ··· + 5u + 4)
c
7
(u
4
u
2
+ 1)
(u
9
3u
8
+ 8u
7
13u
6
+ 17u
5
17u
4
+ 12u
3
6u
2
+ u + 1)
2
(u
14
+ 9u
13
+ ··· + 115u + 26)
10
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y
2
y + 1)
2
(y
9
5y
8
+ 12y
7
15y
6
+ 9y
5
+ y
4
4y
3
+ 2y
2
+ y 1)
2
(y
14
7y
13
+ ··· 5y + 4)
c
2
, c
3
, c
4
c
8
, c
9
, c
10
(y + 1)
4
(y
14
+ 18y
13
+ ··· + 2y + 1)(y
18
+ 15y
17
+ ··· 16y + 1)
c
5
(y
2
+ y + 1)
2
(y
9
y
8
+ 12y
7
7y
6
+ 37y
5
+ y
4
10y
2
+ 5y 1)
2
(y
14
+ y
13
+ ··· + 191y + 16)
c
7
(y
2
y + 1)
2
(y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
15y
4
+ 22y
2
+ 13y 1)
2
(y
14
+ 5y
13
+ ··· 69y + 676)
11