10
150
(K10n
9
)
1
Arc Sequences
9 4 1 7 4 10 5 2 7 3
Solving Sequence
4,7
5
1,8
3 2 10 6 9
c
4
c
7
c
3
c
2
c
10
c
6
c
9
c
1
, c
5
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
3
+ u
2
1, b u 1, u
2
+ a + ui
I
u
2
= hu
17
+ 2u
16
+ ··· 3u + 1, 6376u
16
5959u
15
+ ··· + 24209b 14691,
23873u
16
52964u
15
+ ··· + 24209a + 86827i
There are 2 irreducible components with 20 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
3
+ u
2
1, b u 1, u
2
+ a + ui
(i) Arc colorings
a
4
=
0
u
a
7
=
u
2
u
u + 1
a
5
=
u
2
u
2u + 1
a
1
=
1
0
a
8
=
0
u
a
3
=
u
u
a
2
=
u
u
2
+ u 1
a
10
=
u
2
+ 1
u
2
a
6
=
u
2
u
u + 1
a
9
=
u
2
+ 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
8u 16
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.877439 0.744862I
a = 0.662359 0.562280I
b = 0.122561 0.744862I
1.37919 2.82812I 9.19557 + 4.65175I
u = 0.877439 + 0.744862I
a = 0.662359 + 0.562280I
b = 0.122561 + 0.744862I
1.37919 + 2.82812I 9.19557 4.65175I
u = 0.754878
a = 1.32472
b = 1.75488
2.75839 22.6089
3
II. I
u
2
= hu
17
+ 2u
16
+ · · · 3u + 1, 6376u
16
5959u
15
+ · · · + 24209b
14691, 23873u
16
52964u
15
+ · · · + 24209a + 86827i
(i) Arc colorings
a
4
=
0
u
a
7
=
0.986121u
16
+ 2.18778u
15
+ ··· + 0.310670u 3.58656
0.263373u
16
+ 0.246148u
15
+ ··· + 0.747532u + 0.606840
a
5
=
1.05552u
16
+ 2.24887u
15
+ ··· 1.24268u 3.65377
0.194308u
16
0.371060u
15
+ ··· + 1.34937u + 0.788178
a
1
=
1
0
a
8
=
0.861209u
16
0.877814u
15
+ ··· + 3.89330u 0.134413
0.943823u
16
+ 1.37912u
15
+ ··· 0.456814u 0.421785
a
3
=
u
u
a
2
=
u
u
3
+ u
a
10
=
u
2
+ 1
u
2
a
6
=
1.05552u
16
+ 2.24887u
15
+ ··· 1.24268u 3.65377
0.242431u
16
+ 0.0830683u
15
+ ··· + 0.707382u + 0.650337
a
9
=
0.667892u
16
+ 1.06477u
15
+ ··· 2.49469u 0.464084
0.598496u
16
1.00368u
15
+ ··· 0.0586559u + 0.396877
(ii) Obstruction class = 1
(iii) Cusp Shapes =
76049
24209
u
16
104431
24209
u
15
+ ··· +
330360
24209
u
115800
24209
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.39748 0.52974I
a = 1.174478 + 0.164845I
b = 0.57010 + 1.79791I
12.6337 10.0814I 9.96961 + 5.13034I
u = 1.39748 + 0.52974I
a = 1.174478 0.164845I
b = 0.57010 1.79791I
12.6337 + 10.0814I 9.96961 5.13034I
u = 1.26347
a = 1.48097
b = 0.336658
6.78936 15.0236
u = 1.229706 0.222583I
a = 0.824417 + 0.821497I
b = 0.456483 1.192643I
4.39628 4.11745I 11.29745 + 5.99012I
u = 1.229706 + 0.222583I
a = 0.824417 0.821497I
b = 0.456483 + 1.192643I
4.39628 + 4.11745I 11.29745 5.99012I
u = 0.876782 0.644726I
a = 0.320354 + 0.445756I
b = 0.160900 + 0.049329I
2.13008 2.53959I 0.76560 + 1.98769I
u = 0.876782 + 0.644726I
a = 0.320354 0.445756I
b = 0.160900 0.049329I
2.13008 + 2.53959I 0.76560 1.98769I
u = 0.026050 1.128115I
a = 0.19304 + 1.44626I
b = 0.27826 + 1.53513I
8.13487 + 4.20505I 7.98094 2.47792I
u = 0.026050 + 1.128115I
a = 0.19304 1.44626I
b = 0.27826 1.53513I
8.13487 4.20505I 7.98094 + 2.47792I
u = 0.057966 0.464686I
a = 1.22085 0.93247I
b = 0.333227 0.938302I
0.61170 + 1.48793I 4.64409 4.66231I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.057966 + 0.464686I
a = 1.22085 + 0.93247I
b = 0.333227 + 0.938302I
0.61170 1.48793I 4.64409 + 4.66231I
u = 0.306131
a = 3.73776
b = 1.04293
2.29521 1.20574
u = 0.819663
a = 0.0776955
b = 0.608393
1.19406 8.42607
u = 1.089064 0.132960I
a = 0.780793 + 0.121123I
b = 0.29863 2.21266I
3.18058 + 0.67411I 10.63151 + 5.49435I
u = 1.089064 + 0.132960I
a = 0.780793 0.121123I
b = 0.29863 + 2.21266I
3.18058 0.67411I 10.63151 5.49435I
u = 1.39973 0.55866I
a = 1.068562 + 0.386504I
b = 0.13413 + 1.53307I
12.44686 + 1.83083I 10.41430 0.85064I
u = 1.39973 + 0.55866I
a = 1.068562 0.386504I
b = 0.13413 1.53307I
12.44686 1.83083I 10.41430 + 0.85064I
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
3
+ u
2
+ 2u + 1)(u
17
+ 2u
16
+ ··· + u + 1)
c
2
(u
3
u
2
+ 2u 1)(u
17
+ 12u
16
+ ··· + 7u + 1)
c
3
(u
3
u
2
+ 1)(u
17
+ 2u
16
+ ··· 3u + 1)
c
4
(u 1)
3
(u
17
+ 4u
16
+ ··· + 16u + 1)
c
5
(u + 1)
3
(u
17
+ 22u
16
+ ··· + 256u + 1)
c
6
, c
9
u
3
(u
17
+ 3u
16
+ ··· + 20u + 8)
c
7
(u + 1)
3
(u
17
+ 4u
16
+ ··· + 16u + 1)
c
8
(u
3
u
2
+ 2u 1)(u
17
+ 2u
16
+ ··· + u + 1)
c
10
(u
3
+ u
2
1)(u
17
+ 2u
16
+ ··· 3u + 1)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
8
(y
3
+ 3y
2
+ 2y 1)(y
17
+ 18y
15
+ ··· + 7y 1)
c
2
(y
3
+ 3y
2
+ 2y 1)(y
17
12y
16
+ ··· + 155y 1)
c
3
, c
10
(y
3
y
2
+ 2y 1)(y
17
12y
16
+ ··· + 7y 1)
c
4
, c
7
(y 1)
3
(y
17
22y
16
+ ··· + 256y 1)
c
5
(y 1)
3
(y
17
50y
16
+ ··· + 60796y 1)
c
6
, c
9
y
3
(y
17
+ 21y
16
+ ··· + 976y 64)
8