10
154
(K10n
7
)
1
Arc Sequences
9 4 1 7 4 9 5 2 6 3
Solving Sequence
3,10
1 4
2,6
5 9 7 8
c
10
c
3
c
2
c
5
c
9
c
6
c
7
c
1
, c
4
, c
8
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hb
6
2b
5
+ 4b
4
b
3
+ 2b
2
+ 3b + 1, b
5
+ 3b
4
3b
3
+ 2b + 4u 1, b
5
+ b
4
3b
3
2b
2
4b + 4a 7i
I
u
2
= hu
6
u
5
13u
4
2u
3
+ 16u
2
17u + 5, 46u
5
+ 78u
4
+ 485u
3
44u
2
+ 193b 311u + 176,
56u
5
+ 246u
4
+ 238u
3
1178u
2
+ 965a u 1078i
I
u
3
= hu
3
3u
2
+ 2u 1, a, u
2
+ b 2ui
I
u
4
= hu
3
3u
2
+ 2u 1, b + 1, u
2
+ a + 2u + 1i
There are 4 irreducible components with 18 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
= hb
6
2b
5
+ 4b
4
b
3
+ 2b
2
+ 3b + 1, b
5
+ 3b
4
3b
3
+ 2b + 4u
1, b
5
+ b
4
3b
3
2b
2
4b + 4a 7i
(i) Arc colorings
a
3
=
1
0
a
10
=
1
4
b
5
1
4
b
4
+ ··· + b +
7
4
b
a
1
=
1
4
b
5
1
4
b
4
+ ··· +
1
2
b
2
+
7
4
b
a
4
=
1
4
b
5
1
4
b
4
+ ··· + b +
3
4
b
2
a
2
=
1
4
b
5
+
5
4
b
4
+ ··· + b +
5
4
b
4
a
6
=
0
1
4
b
5
3
4
b
4
+ ···
1
2
b +
1
4
a
5
=
b
3
1
4
b
5
3
4
b
4
+ ··· b +
1
4
a
9
=
1
4
b
5
1
4
b
4
+ ··· + b +
7
4
1
4
b
5
3
4
b
4
+ ···
1
2
b
3
4
a
7
=
b
3
+ b
2
1
1
2
b
5
1
2
b
4
+
3
2
b
3
b +
1
2
a
8
=
1
4
b
5
+
1
4
b
4
+ ··· + b
3
4
5
4
b
5
11
4
b
4
+ ···
9
2
b
3
4
(ii) Obstruction class = 1
(iii) Cusp Shapes =
1
2
b
5
+ b
4
3
2
b
3
1
2
b
2
+
1
2
b 9
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.445042
a = 1.40097 0.19310I
b = 0.400969 0.193096I
0.939962 9.24698
u = 0.445042
a = 1.40097 + 0.19310I
b = 0.400969 + 0.193096I
0.939962 9.24698
u = 1.24698
a = 0.72252 1.21572I
b = 0.277479 1.215715I
4.69981 6.19806
u = 1.24698
a = 0.72252 + 1.21572I
b = 0.277479 + 1.215715I
4.69981 6.19806
u = 1.80194
a = 0.12349 1.40881I
b = 1.12349 1.40881I
15.9794 7.55496
u = 1.80194
a = 0.12349 + 1.40881I
b = 1.12349 + 1.40881I
15.9794 7.55496
3
II. I
u
2
= hu
6
u
5
13u
4
2u
3
+ 16u
2
17u + 5, 46u
5
+ 78u
4
+ · · · +
193b + 176, 56u
5
+ 246u
4
+ · · · + 965a 1078i
(i) Arc colorings
a
3
=
1
0
a
10
=
0.0580311u
5
0.254922u
4
+ ··· + 0.00103627u + 1.11710
0.238342u
5
0.404145u
4
+ ··· + 1.61140u 0.911917
a
1
=
0.180311u
5
+ 0.149223u
4
+ ··· 1.61036u + 2.02902
0.238342u
5
0.404145u
4
+ ··· + 1.61140u 0.911917
a
4
=
0.00725389u
5
+ 0.281865u
4
+ ··· + 2.62487u 0.389637
0.274611u
5
0.813472u
4
+ ··· 0.512953u + 0.0362694
a
2
=
0.0186528u
5
0.153368u
4
+ ··· 2.17824u + 1.85907
0.134715u
5
+ 0.663212u
4
+ ··· + 1.17617u 0.0932642
a
6
=
0
u
a
5
=
0.0186528u
5
+ 0.153368u
4
+ ··· + 2.17824u 0.859067
0.134715u
5
0.663212u
4
+ ··· 0.176166u + 0.0932642
a
9
=
0.0580311u
5
0.254922u
4
+ ··· + 0.00103627u + 1.11710
0.0725389u
5
+ 0.818653u
4
+ ··· + 5.24870u 1.89637
a
7
=
0.198964u
5
0.302591u
4
+ ··· 0.567876u 0.169948
0.373057u
5
+ 1.06736u
4
+ ··· + 0.564767u + 0.818653
a
8
=
0.433161u
5
1.11710u
4
+ ··· 2.02798u 0.161658
1.11917u
5
+ 3.20207u
4
+ ··· + 7.69430u 1.54404
(ii) Obstruction class = 1
(iii) Cusp Shapes =
168
193
u
5
352
193
u
4
1872
193
u
3
+
832
193
u
2
+
2512
193
u
2942
193
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 2.25137 0.09770I
a = 0.320342 0.957140I
b = 1.33572 1.10504I
14.9943 + 9.2499I 8.40611 3.97593I
u = 2.25137 + 0.09770I
a = 0.320342 + 0.957140I
b = 1.33572 + 1.10504I
14.9943 9.2499I 8.40611 + 3.97593I
u = 0.434265
a = 1.31939
b = 0.385643
0.943503 9.62413
u = 0.480327 0.566769I
a = 1.162623 0.626285I
b = 0.822978 0.498752I
1.78286 + 4.10821I 5.51333 7.68125I
u = 0.480327 + 0.566769I
a = 1.162623 + 0.626285I
b = 0.822978 + 0.498752I
1.78286 4.10821I 5.51333 + 7.68125I
u = 4.10782
a = 0.0853230
b = 0.931750
3.00199 62.5370
5
III. I
u
3
= hu
3
3u
2
+ 2u 1, a, u
2
+ b 2ui
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
2
+ 2u
a
1
=
u
2
2u
u
2
+ 2u
a
4
=
u
2
3u + 2
u
2
+ 3u 1
a
2
=
u
2
2u
u + 2
a
6
=
0
u
a
5
=
u
2
3u + 2
u
2
+ 4u 1
a
9
=
0
u
2
+ 2u
a
7
=
0
u
a
8
=
u
2
+ 3u 2
u
2
3u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
+ 3u 9
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.337641 0.562280I
a = 0
b = 0.877439 0.744862I
1.37919 + 2.82812I 7.78492 1.30714I
u = 0.337641 + 0.562280I
a = 0
b = 0.877439 + 0.744862I
1.37919 2.82812I 7.78492 + 1.30714I
u = 2.32472
a = 0
b = 0.754878
2.75839 7.43016
7
IV. I
u
4
= hu
3
3u
2
+ 2u 1, b + 1, u
2
+ a + 2u + 1i
(i) Arc colorings
a
3
=
1
0
a
10
=
u
2
2u 1
1
a
1
=
u
2
2u
1
a
4
=
u
2
+ 2u + 1
1
a
2
=
u
2
2u
1
a
6
=
0
u
a
5
=
2u
2
+ 6u 3
u
2
+ 4u 1
a
9
=
u
2
2u 1
u 2
a
7
=
2u
2
6u + 3
2u
2
4u + 1
a
8
=
u
2
2u 1
u 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
+ 3u 9
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.337641 0.562280I
a = 1.87744 + 0.74486I
b = 1.00000
1.37919 + 2.82812I 7.78492 1.30714I
u = 0.337641 + 0.562280I
a = 1.87744 0.74486I
b = 1.00000
1.37919 2.82812I 7.78492 + 1.30714I
u = 2.32472
a = 0.245122
b = 1.00000
2.75839 7.43016
9
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
u
3
(u
3
u
2
+ 2u 1)(u
6
+ u
5
+ 5u
4
2u
3
+ 4u
2
u 1)
(u
6
+ u
5
+ 8u
4
+ u
3
+ 8u
2
20u + 8)
c
2
(u 1)
3
(u
3
+ u
2
+ 2u + 1)(u
6
4u
5
+ ··· + 5u + 1)
(u
6
+ 3u
5
+ 13u
4
+ 20u
3
+ 18u
2
+ 9u + 1)
c
3
(u + 1)
3
(u
3
+ u
2
1)(u
6
+ 2u
5
+ 4u
4
+ u
3
+ 2u
2
3u + 1)
(u
6
+ 3u
5
+ 3u
4
2u
3
4u
2
u + 1)
c
4
(u 1)
3
(u
3
+ u
2
1)(u
6
+ 2u
5
+ 4u
4
+ u
3
+ 2u
2
3u + 1)
(u
6
+ 3u
5
+ 3u
4
2u
3
4u
2
u + 1)
c
5
(u + 1)
3
(u
3
+ u
2
+ 2u + 1)(u
6
4u
5
+ ··· + 5u + 1)
(u
6
+ 3u
5
+ 13u
4
+ 20u
3
+ 18u
2
+ 9u + 1)
c
7
(u 1)
3
(u
3
u
2
+ 1)(u
6
+ 2u
5
+ 4u
4
+ u
3
+ 2u
2
3u + 1)
(u
6
+ 3u
5
+ 3u
4
2u
3
4u
2
u + 1)
c
8
, c
9
u
3
(u
3
+ u
2
+ 2u + 1)(u
6
+ u
5
+ 5u
4
2u
3
+ 4u
2
u 1)
(u
6
+ u
5
+ 8u
4
+ u
3
+ 8u
2
20u + 8)
c
10
(u 1)
3
(u
3
+ u
2
1)(u
6
+ 2u
5
+ 4u
4
+ u
3
+ 2u
2
3u + 1)
(u
6
+ 3u
5
+ 3u
4
2u
3
4u
2
u + 1)
10
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
, c
8
c
9
y
3
(y
3
+ 3y
2
+ 2y 1)(y
6
+ 9y
5
+ 37y
4
+ 36y
3
+ 2y
2
9y + 1)
(y
6
+ 15y
5
+ 78y
4
+ 183y
3
+ 232y
2
272y + 64)
c
2
, c
5
(y 1)
3
(y
3
+ 3y
2
+ 2y 1)
(y
6
+ 16y
5
+ 60y
4
223y
3
+ 646y
2
+ 11y + 1)
(y
6
+ 17y
5
+ 85y
4
+ 16y
3
10y
2
45y + 1)
c
3
, c
4
, c
7
c
10
(y 1)
3
(y
3
y
2
+ 2y 1)(y
6
3y
5
+ ··· 9y + 1)
(y
6
+ 4y
5
+ 16y
4
+ 29y
3
+ 18y
2
5y + 1)
11