10
98
(K10a
96
)
1
Arc Sequences
6 9 7 10 8 1 4 2 5 3
Solving Sequence
2,9 3,6
1 7 4 8 5 10
c
2
c
1
c
6
c
3
c
8
c
5
c
10
c
4
, c
7
, c
9
Representation Ideals
I =
6
\
i=1
I
u
i
\
I
v
1
I
u
1
= hb + 1, u
3
a u
3
+ au + 1i
I
u
2
= hb, a 1, u
3
+ u + 1i
I
u
3
= hb + 1, a u 1, u
3
+ u + 1i
I
u
4
= hb u, 2a + u 1, u
2
+ 1i
I
u
5
= hb
19
+ 32u + ··· + 44b 14, b
20
2b
19
+ ··· 58b + 31, 15b
19
+ 14b
18
+ ··· + 32a + 402i
I
u
6
= h3u
12
+ 9u
11
+ 28u
10
+ 60u
9
+ 108u
8
+ 168u
7
+ 205u
6
+ 211u
5
+ 181u
4
+ 114u
3
+ 66u
2
+ 26u + 5,
16107147u
11
24322986u
10
+ ··· + 49091480a + 70838891,
5477571u
11
15482940u
10
+ ··· + 9818296b 26892483i
I
v
1
= ha, v + 1, b + 1i
There are 7 irreducible components with 41 representations.
There are 1 irreducible components of dim
C
= 1 for 10
98
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
= hb + 1, u
3
a u
3
+ au + 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
a
1
a
3
=
a + 1
1
a
6
=
0
u
a
1
=
1
u
2
a
7
=
u
u
3
+ u
a
4
=
a + u + 1
u
3
+ u + 1
a
8
=
a 1
1
a
5
=
a
2
u 2au + u
au + 2u
a
10
=
a
2
u
2
2u
2
a + u
2
+ a
u
2
a + 2u
2
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = ···
a = ···
b = ···
3.28987 12.0000
3
II. I
u
2
= hb, a 1, u
3
+ u + 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
1
0
a
3
=
1
0
a
6
=
0
u
a
1
=
1
u
2
a
7
=
u
1
a
4
=
u + 1
1
a
8
=
1
0
a
5
=
u
u
a
10
=
u
2
+ 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.682328
a = 1.00000
b = 0
1.64493 6.00000
u = 0.341164 1.161541I
a = 1.00000
b = 0
1.64493 6.00000
u = 0.341164 + 1.161541I
a = 1.00000
b = 0
1.64493 6.00000
5
III. I
u
3
= hb + 1, a u 1, u
3
+ u + 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
u + 1
1
a
3
=
u
1
a
6
=
0
u
a
1
=
1
u
2
a
7
=
u
1
a
4
=
u
1
a
8
=
u
1
a
5
=
u 1
u
2
+ u
a
10
=
u
2
+ 1
u
2
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.682328
a = 0.317672
b = 1.00000
1.64493 6.00000
u = 0.341164 1.161541I
a = 1.34116 1.16154I
b = 1.00000
1.64493 6.00000
u = 0.341164 + 1.161541I
a = 1.34116 + 1.16154I
b = 1.00000
1.64493 6.00000
7
IV. I
u
4
= hb u, 2a + u 1, u
2
+ 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
1
2
u +
1
2
u
a
3
=
1
2
u +
3
2
1
a
6
=
0
u
a
1
=
1
1
a
7
=
u
0
a
4
=
1
2
u +
1
2
1
a
8
=
1
2
u +
1
2
u
a
5
=
0.5
1
2
u
1
2
a
10
=
u +
1
2
1
2
u
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4.
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.00000I
a = 0.500000 + 0.500000I
b = 1.00000I
1.64493 4.00000
u = 1.00000I
a = 0.500000 0.500000I
b = 1.00000I
1.64493 4.00000
9
V. I
u
5
= hb
19
+ 32u + · · · + 44b 14, b
20
2b
19
+ · · · 58b + 31, 15b
19
+
14b
18
+ · · · + 32a + 402i
(i) Arc colorings
a
2
=
1
0
a
9
=
0.468750b
19
0.437500b
18
+ ··· + 13.8125b 12.5625
b
a
3
=
0.500000b
19
0.468750b
18
+ ··· + 14.6250b 13.5313
b
2
a
6
=
0
0.0312500b
19
+ 0.187500b
17
+ ··· 1.37500b + 0.437500
a
1
=
1
0.0312500b
18
0.187500b
16
+ ··· 0.218750b + 0.312500
a
7
=
0.0312500b
19
+ 0.187500b
17
+ ··· 1.37500b + 0.437500
1
16
b
18
+
3
8
b
16
+ ··· +
1
2
b 1
a
4
=
0.437500b
19
0.406250b
18
+ ··· + 12.2813b 11.0625
1
a
8
=
0.468750b
19
0.437500b
18
+ ··· + 14.8125b 12.5625
b
a
5
=
1.03125b
19
+ 0.906250b
18
+ ··· 31.3438b + 25.6250
0.156250b
19
+ 0.0937500b
18
+ ··· 5.53125b + 2.37500
a
10
=
0.187500b
19
+ 0.375000b
18
+ ··· 5.96875b + 11.9688
0.218750b
19
+ 0.218750b
18
+ ··· 5.28125b + 6.12500
(ii) Obstruction class = 1
(iii) Cusp Shapes =
1
2
b
19
1
2
b
18
3b
17
+ 2b
16
+
35
4
b
15
7
2
b
14
39
4
b
13
2b
12
5b
11
+
15b
10
+
59
2
b
9
47
2
b
8
101
4
b
7
+ 5b
6
23
4
b
5
+ 24b
4
+
31
4
b
3
19
2
b
2
+
37
4
b
43
2
10
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.347624 1.331993I
a = 1.76349 0.46038I
b = 1.39510 0.62944I
2.90872 8.28632I 7.82440 + 6.14881I
u = 0.347624 + 1.331993I
a = 1.76349 + 0.46038I
b = 1.39510 + 0.62944I
2.90872 + 8.28632I 7.82440 6.14881I
u = 0.057928 + 1.351674I
a = 1.41449 0.30537I
b = 1.24454 0.70845I
8.22706 2.31006I 12.86369 + 3.52133I
u = 0.057928 1.351674I
a = 1.41449 + 0.30537I
b = 1.24454 + 0.70845I
8.22706 + 2.31006I 12.86369 3.52133I
u = 0.309318 0.396943I
a = 0.38841 1.74306I
b = 1.157779 0.163121I
2.84181 + 1.23169I 7.09823 5.44908I
u = 0.309318 + 0.396943I
a = 0.38841 + 1.74306I
b = 1.157779 + 0.163121I
2.84181 1.23169I 7.09823 + 5.44908I
u = 0.280829 + 1.292557I
a = 0.161068 + 0.000633I
b = 0.255380 0.856092I
5.70347 3.47839I 11.19503 + 2.79515I
u = 0.280829 1.292557I
a = 0.161068 0.000633I
b = 0.255380 + 0.856092I
5.70347 + 3.47839I 11.19503 2.79515I
u = 0.347624 + 1.331993I
a = 0.215749 + 0.421066I
b = 0.065535 1.177787I
2.90872 + 8.28632I 7.82440 6.14881I
u = 0.347624 1.331993I
a = 0.215749 0.421066I
b = 0.065535 + 1.177787I
2.90872 8.28632I 7.82440 + 6.14881I
11
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.800451 0.099834I
a = 0.575091 0.621305I
b = 0.193027 0.767853I
1.58679 4.14585I 3.01866 + 3.97600I
u = 0.800451 + 0.099834I
a = 0.575091 + 0.621305I
b = 0.193027 + 0.767853I
1.58679 + 4.14585I 3.01866 3.97600I
u = 0.309318 0.396943I
a = 1.69749 2.41922I
b = 0.965077 0.285214I
2.84181 + 1.23169I 7.09823 5.44908I
u = 0.309318 + 0.396943I
a = 1.69749 + 2.41922I
b = 0.965077 + 0.285214I
2.84181 1.23169I 7.09823 + 5.44908I
u = 0.800451 + 0.099834I
a = 0.100538 0.188636I
b = 1.096356 0.477116I
1.58679 + 4.14585I 3.01866 3.97600I
u = 0.800451 0.099834I
a = 0.100538 + 0.188636I
b = 1.096356 + 0.477116I
1.58679 4.14585I 3.01866 + 3.97600I
u = 0.280829 1.292557I
a = 1.88115 0.56080I
b = 1.359949 0.294980I
5.70347 + 3.47839I 11.19503 2.79515I
u = 0.280829 + 1.292557I
a = 1.88115 + 0.56080I
b = 1.359949 + 0.294980I
5.70347 3.47839I 11.19503 + 2.79515I
u = 0.057928 + 1.351674I
a = 1.87561 0.05942I
b = 1.50393 0.39581I
8.22706 2.31006I 12.86369 + 3.52133I
u = 0.057928 1.351674I
a = 1.87561 + 0.05942I
b = 1.50393 + 0.39581I
8.22706 + 2.31006I 12.86369 3.52133I
12
VI.
I
u
6
= h3u
12
+9u
11
+· · · +26u + 5, 1.61× 10
7
u
11
2.43 × 10
7
u
10
+· · · +4.91 ×
10
7
a+7.08×10
7
, 5.48×10
6
u
11
1.55×10
7
u
10
+· · ·+9.82×10
6
b2.69×10
7
i
(i) Arc colorings
a
2
=
1
0
a
9
=
0.328105u
11
+ 0.495462u
10
+ ··· + 0.0367006u 1.44300
0.557894u
11
+ 1.57695u
10
+ ··· + 9.27371u + 2.73902
a
3
=
1.05782u
11
2.97033u
10
+ ··· 13.5027u 3.49251
0.796686u
11
+ 2.01762u
10
+ ··· + 6.38513u + 1.13810
a
6
=
0
u
a
1
=
1
u
2
a
7
=
u
u
3
+ u
a
4
=
1.15456u
11
3.11361u
10
+ ··· 15.5988u 4.42234
0.430950u
11
+ 1.13948u
10
+ ··· + 3.17696u 0.0365961
a
8
=
0.885999u
11
+ 2.07241u
10
+ ··· + 9.31041u + 1.29602
0.557894u
11
+ 1.57695u
10
+ ··· + 9.27371u + 2.73902
a
5
=
1.00247u
11
+ 2.75725u
10
+ ··· + 14.4353u + 4.34408
0.169298u
11
0.421204u
10
+ ··· 0.0912175u + 0.435230
a
10
=
0.0109788u
11
0.248412u
10
+ ··· 2.77357u 0.683631
0.326845u
11
+ 1.00760u
10
+ ··· + 6.22974u + 2.02991
(ii) Obstruction class = 1
(iii) Cusp Shapes =
6140907
2454574
u
11
+
11367045
1227287
u
10
+ ··· +
103749145
2454574
u +
23836473
2454574
13
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
6
1(vol +
1CS) Cusp shape
u = 1.159520 0.322882I
a = 0.626412 0.425218I
b = 1.170888 0.448059I
1.38664 + 8.65525I 8.05360 7.75821I
u = 1.159520 + 0.322882I
a = 0.626412 + 0.425218I
b = 1.170888 + 0.448059I
1.38664 8.65525I 8.05360 + 7.75821I
u = 0.47591 1.48268I
a = 1.70879 + 0.61964I
b = 1.36378 + 0.57208I
7.0244 + 14.4129I 10.01184 7.82077I
u = 0.47591 + 1.48268I
a = 1.70879 0.61964I
b = 1.36378 0.57208I
7.0244 14.4129I 10.01184 + 7.82077I
u = 0.388433 0.220879I
a = 1.80496 0.48863I
b = 0.435810 0.817904I
3.72744 + 0.65506I 0.96539 2.36054I
u = 0.388433 + 0.220879I
a = 1.80496 + 0.48863I
b = 0.435810 + 0.817904I
3.72744 0.65506I 0.96539 + 2.36054I
u = 0.121457 1.180507I
a = 0.144294 + 0.714956I
b = 0.312276 1.374797I
0.93953 + 1.28267I 9.97159 5.49540I
u = 0.121457 + 1.180507I
a = 0.144294 0.714956I
b = 0.312276 + 1.374797I
0.93953 1.28267I 9.97159 + 5.49540I
u = 0.153074 0.769500I
a = 0.519469 0.259632I
b = 0.255755 0.338417I
0.534161 1.008594I 7.65789 + 6.71362I
u = 0.153074 + 0.769500I
a = 0.519469 + 0.259632I
b = 0.255755 + 0.338417I
0.534161 + 1.008594I 7.65789 6.71362I
14
Solution to I
u
6
1(vol +
1CS) Cusp shape
u = 0.49224 1.58068I
a = 1.54117 + 0.48377I
b = 1.325114 + 0.371579I
10.52615 7.73722I 13.2705 + 5.1580I
u = 0.49224 + 1.58068I
a = 1.54117 0.48377I
b = 1.325114 0.371579I
10.52615 + 7.73722I 13.2705 5.1580I
15
VII. I
v
1
= ha, v + 1, b + 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
1
a
3
=
1
1
a
6
=
1
0
a
1
=
1
0
a
7
=
1
0
a
4
=
0
1
a
8
=
1
1
a
5
=
0
1
a
10
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
16
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
3.28987 12.0000
17
VIII. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
u(u
2
+ 1)(u
3
+ u + 1)
2
(u
10
u
9
+ 5u
8
5u
7
+ 9u
6
9u
5
+ 6u
4
6u
3
+ u
2
+ 1)
2
(3u
12
+ 9u
11
+ ··· + 26u + 5)
c
2
, c
8
u
3
(u 1)
4
(u
2
+ 1)
(u
12
+ 2u
11
u
9
3u
7
+ u
6
+ 15u
5
+ 21u
4
+ 14u
3
+ 3u
2
u + 2)
(u
20
+ 2u
19
+ ··· + 58u + 31)
c
3
, c
7
u
3
(u 1)
3
(u + 1)(u
2
+ 1)
(u
12
+ 2u
11
u
9
3u
7
+ u
6
+ 15u
5
+ 21u
4
+ 14u
3
+ 3u
2
u + 2)
(u
20
+ 2u
19
+ ··· + 58u + 31)
c
4
, c
9
u(u
2
+ 1)(u
3
+ u + 1)
2
(u
10
u
9
+ 3u
8
3u
7
+ 5u
6
5u
5
+ 4u
4
4u
3
+ 3u
2
2u + 1)
2
(3u
12
+ 9u
11
+ ··· + 20u + 5)
c
5
, c
10
(u + 1)(2u
2
2u + 1)(u
3
+ u + 1)(u
3
+ 2u
2
+ u 1)
(2u
12
4u
11
+ ··· + 12u + 3)(2u
20
13u
18
+ ··· 518u + 121)
18
IX. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
y(y + 1)
2
(y
3
+ 2y
2
+ y 1)
2
(1 + 2y + 13y
2
6y
3
44y
4
21y
5
+ 41y
6
+ 59y
7
+ 33y
8
+ 9y
9
+ y
10
)
2
(9y
12
+ 87y
11
+ ··· 16y + 25)
c
2
, c
3
, c
7
c
8
y
3
(y 1)
4
(y + 1)
2
(y
12
4y
11
+ ··· + 11y + 4)
(y
20
14y
19
+ ··· 388y + 961)
c
4
, c
9
y(y + 1)
2
(y
3
+ 2y
2
+ y 1)
2
(y
10
+ 5y
9
+ 13y
8
+ 19y
7
+ 17y
6
+ 7y
5
2y
3
+ y
2
+ 2y + 1)
2
(9y
12
+ 51y
11
+ ··· + 240y + 25)
c
5
, c
10
(y 1)(4y
2
+ 1)(y
3
2y
2
+ 5y 1)(y
3
+ 2y
2
+ y 1)
(4y
12
20y
11
+ ··· 30y + 9)(4y
20
52y
19
+ ··· 56574y + 14641)
19