10
145
(K10n
14
)
1
Arc Sequences
8 10 8 3 9 2 4 1 3 6
Solving Sequence
3,8
4
5,10
2 1 7 6 9
c
3
c
4
c
2
c
1
c
7
c
6
c
9
c
5
, c
8
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
4
2a
3
+ 5a
2
4a + 1, 2a
3
+ 3a
2
+ b 9a + 4, 2a
3
3a
2
+ 8a + u 3i
I
u
2
= hu
5
+ 2u
4
+ 15u
3
+ 14u
2
+ 17u + 4, 3u
4
9u
3
+ 31u
2
+ 118b 54u + 26,
17u
4
+ 8u
3
+ 215u
2
+ 236a 70u + 167i
There are 2 irreducible components with 9 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
= ha
4
2a
3
+ 5a
2
4a + 1, 2a
3
+ 3a
2
+ b 9a + 4, 2a
3
3a
2
+ 8a + u 3i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
2a
3
+ 3a
2
8a + 3
a
4
=
1
2a
3
+ 3a
2
8a + 4
a
5
=
2a
3
3a
2
+ 8a 3
2a
3
+ 3a
2
8a + 4
a
10
=
a
2a
3
3a
2
+ 9a 4
a
2
=
a
3
a
2
+ 4a 1
1
a
1
=
a
3
a
2
+ 4a 1
a
3
2a
2
+ 5a 4
a
7
=
2a
3
3a
2
+ 8a 3
2a
3
+ 3a
2
8a + 4
a
6
=
a
a
3
+ 2a
2
5a + 4
a
9
=
2a
3
+ 3a
2
8a + 4
2a
3
3a
2
+ 9a 4
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8a
3
+ 12a
2
32a + 16
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 0.500000 0.133975I
b = 1.00000I
3.28987 2.02988I 2.00000 + 3.46410I
u = 0.500000 0.866025I
a = 0.500000 + 0.133975I
b = 1.00000I
3.28987 + 2.02988I 2.00000 3.46410I
u = 0.500000 0.866025I
a = 0.50000 1.86603I
b = 1.00000I
3.28987 + 2.02988I 2.00000 3.46410I
u = 0.500000 + 0.866025I
a = 0.50000 + 1.86603I
b = 1.00000I
3.28987 2.02988I 2.00000 + 3.46410I
3
II. I
u
2
= hu
5
+ 2u
4
+ 15u
3
+ 14u
2
+ 17u + 4, 3u
4
9u
3
+ 31u
2
+ 118b
54u + 26, 17u
4
+ 8u
3
+ 215u
2
+ 236a 70u + 167i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
4
=
1
u
2
a
5
=
u
2
+ 1
u
2
a
10
=
0.0720339u
4
0.0338983u
3
+ ··· + 0.296610u 0.707627
0.0254237u
4
+ 0.0762712u
3
+ ··· + 0.457627u 0.220339
a
2
=
0.0550847u
4
+ 0.0847458u
3
+ ··· + 0.508475u + 1.39407
0.0169492u
4
+ 0.0508475u
3
+ ··· + 0.305085u + 0.186441
a
1
=
0.0550847u
4
+ 0.0847458u
3
+ ··· + 0.508475u + 1.39407
0.110169u
4
+ 0.169492u
3
+ ··· + 0.516949u + 0.288136
a
7
=
u
u
3
+ u
a
6
=
0.0720339u
4
+ 0.0338983u
3
+ ··· 0.296610u + 0.707627
0.110169u
4
0.169492u
3
+ ··· 0.516949u 0.288136
a
9
=
0.0466102u
4
0.110169u
3
+ ··· 0.161017u 0.487288
0.0254237u
4
+ 0.0762712u
3
+ ··· + 0.457627u 0.220339
(ii) Obstruction class = 1
(iii) Cusp Shapes =
33
59
u
4
+
78
59
u
3
+
518
59
u
2
+
645
59
u +
994
59
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.51390 3.52451I
a = 0.131172 + 0.610069I
b = 0.76439 + 2.80121I
15.2298 + 5.0449I 4.96361 1.80446I
u = 0.51390 + 3.52451I
a = 0.131172 0.610069I
b = 0.76439 2.80121I
15.2298 5.0449I 4.96361 + 1.80446I
u = 0.345349 1.000391I
a = 0.062320 0.860264I
b = 0.078472 0.559300I
1.59932 + 2.36167I 6.81651 4.70099I
u = 0.345349 + 1.000391I
a = 0.062320 + 0.860264I
b = 0.078472 + 0.559300I
1.59932 2.36167I 6.81651 + 4.70099I
u = 0.281507
a = 0.863015
b = 0.371844
0.702837 14.4397
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
2
, c
8
c
9
(u
2
+ 1)
2
(u
5
+ u
4
+ 8u
3
4u
2
+ 3u 1)
c
3
(u
2
+ u + 1)
2
(u
5
+ 2u
4
+ 15u
3
+ 14u
2
+ 17u + 4)
c
4
(u
2
+ u + 1)
2
(u
5
+ 26u
4
+ 203u
3
+ 298u
2
+ 177u 16)
c
5
(u
4
2u
3
+ 2u
2
4u + 4)(u
5
+ u
4
+ 26u
3
+ 18u
2
4u 4)
c
6
(u
4
+ 2u
3
+ 2u
2
+ 4u + 4)(u
5
+ u
4
+ 26u
3
+ 18u
2
4u 4)
c
7
(u
2
u + 1)
2
(u
5
+ 2u
4
+ 15u
3
+ 14u
2
+ 17u + 4)
c
10
(u
4
u
2
+ 1)(u
5
+ 4u
4
+ 7u
3
+ 4u
2
u 2)
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
2
, c
8
c
9
(y + 1)
4
(y
5
+ 15y
4
+ 78y
3
+ 34y
2
+ y 1)
c
3
, c
7
(y
2
+ y + 1)
2
(y
5
+ 26y
4
+ 203y
3
+ 298y
2
+ 177y 16)
c
4
(y
2
+ y + 1)
2
(y
5
270y
4
+ 26067y
3
16110y
2
+ 40865y 256)
c
5
, c
6
(y
4
4y
2
+ 16)(y
5
+ 51y
4
+ 632y
3
524y
2
+ 160y 16)
c
10
(y
2
y + 1)
2
(y
5
2y
4
+ 15y
3
14y
2
+ 17y 4)
7