10
48
(K10a
79
)
1
Arc Sequences
4 7 1 2 9 10 3 5 6 8
Solving Sequence
5,8
9 6
2,10
4 1 3 7
c
8
c
5
c
9
c
4
c
1
c
3
c
7
c
2
, c
6
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
2
+ a 1, b, u 1i
I
u
2
= hu
26
3u
25
+ ··· 9u
2
1, 2u
25
3u
24
+ ··· + b + 1, 3u
25
4u
24
+ ··· + 2a + 3i
There are 2 irreducible components with 28 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
2
+ a 1, b, u 1i
(i) Arc colorings
a
5
=
0
1
a
8
=
a
0
a
9
=
a
a
a
6
=
a + 1
a
a
2
=
1
0
a
10
=
a + 1
1
a
4
=
1
1
a
1
=
0
1
a
3
=
1
0
a
7
=
a
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.61803
b = 0
7.23771 5.00000
u = 1.00000
a = 0.618034
b = 0
0.657974 5.00000
3
II.
I
u
2
= hu
26
3u
25
+· · ·9u
2
1, 2u
25
3u
24
+· · ·+b+1, 3u
25
4u
24
+· · ·+2a+3i
(i) Arc colorings
a
5
=
0
u
a
8
=
3
2
u
25
+ 2u
24
+ ···
15
2
u
3
2
2u
25
+ 3u
24
+ ··· 2u 1
a
9
=
3
2
u
25
+ 2u
24
+ ···
15
2
u
3
2
3
2
u
25
3u
24
+ ···
1
2
u +
3
2
a
6
=
1
2
u
25
u
24
+ ···
7
2
u
1
2
1
2
u
25
+ u
24
+ ···
1
2
u
1
2
a
2
=
1
0
a
10
=
1
2
u
25
u
24
+ ···
7
2
u +
1
2
u
7
+ 3u
5
+ 2u
4
2u
3
4u
2
u
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
3
+ 2u
u
3
+ u
a
7
=
1
2
u
25
+ 6u
23
+ ···
15
2
u
3
2
3u
25
5u
24
+ ··· + 14u
2
+ 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
25
+ 3u
24
+ 47u
23
17u
22
248u
21
21u
20
+ 744u
19
+
432u
18
1274u
17
1555u
16
+ 894u
15
+ 2682u
14
+ 830u
13
2170u
12
2214u
11
+
190u
10
+ 1502u
9
+ 792u
8
125u
7
221u
6
54u
5
89u
4
118u
3
54u
2
+ 4u + 1
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.391236 0.188512I
a = 1.070413 0.235267I
b = 1.87529 2.10880I
3.48759 3.82064I 0.89607 + 2.40126I
u = 1.391236 + 0.188512I
a = 1.070413 + 0.235267I
b = 1.87529 + 2.10880I
3.48759 + 3.82064I 0.89607 2.40126I
u = 1.343902 0.071348I
a = 0.342162 + 0.379199I
b = 0.63975 + 2.10165I
3.27468 1.70414I 2.66466 + 3.89699I
u = 1.343902 + 0.071348I
a = 0.342162 0.379199I
b = 0.63975 2.10165I
3.27468 + 1.70414I 2.66466 3.89699I
u = 0.878676 0.575534I
a = 1.25305 + 0.72364I
b = 0.071717 + 0.625709I
4.66926 + 1.10360I 0.162083 0.384354I
u = 0.878676 + 0.575534I
a = 1.25305 0.72364I
b = 0.071717 0.625709I
4.66926 1.10360I 0.162083 + 0.384354I
u = 0.865819
a = 0.326651
b = 0.364658
1.22611 10.4965
u = 0.557648 0.540231I
a = 0.731221 + 0.115201I
b = 0.139839 + 0.169070I
1.59551 0.48344I 4.02832 0.08458I
u = 0.557648 + 0.540231I
a = 0.731221 0.115201I
b = 0.139839 0.169070I
1.59551 + 0.48344I 4.02832 + 0.08458I
u = 0.403777 0.726209I
a = 0.166289 0.900683I
b = 0.294786 0.784403I
0.89453 3.84444I 0.49259 + 7.28090I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.403777 + 0.726209I
a = 0.166289 + 0.900683I
b = 0.294786 + 0.784403I
0.89453 + 3.84444I 0.49259 7.28090I
u = 0.330593 0.851302I
a = 0.47819 + 1.65273I
b = 0.60804 + 1.35634I
6.35171 6.10679I 2.97747 + 5.14405I
u = 0.330593 + 0.851302I
a = 0.47819 1.65273I
b = 0.60804 1.35634I
6.35171 + 6.10679I 2.97747 5.14405I
u = 0.075738 0.290375I
a = 0.04394 + 2.15442I
b = 0.512456 + 0.395630I
1.166696 + 0.399409I 7.28789 1.42640I
u = 0.075738 + 0.290375I
a = 0.04394 2.15442I
b = 0.512456 0.395630I
1.166696 0.399409I 7.28789 + 1.42640I
u = 0.259688 0.494614I
a = 1.23552 2.55466I
b = 0.880046 0.886686I
8.75625 + 1.27302I 7.34031 0.88258I
u = 0.259688 + 0.494614I
a = 1.23552 + 2.55466I
b = 0.880046 + 0.886686I
8.75625 1.27302I 7.34031 + 0.88258I
u = 1.25836
a = 1.32966
b = 0.227753
6.22196 3.01543
u = 1.45576 0.04814I
a = 0.825848 0.069396I
b = 0.764431 0.391611I
3.39138 + 0.04941I 2.26185 + 0.23755I
u = 1.45576 + 0.04814I
a = 0.825848 + 0.069396I
b = 0.764431 + 0.391611I
3.39138 0.04941I 2.26185 0.23755I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.46140 0.33031I
a = 1.008094 + 0.354468I
b = 0.96069 + 2.36911I
0.59945 + 10.37890I 0.23306 5.66856I
u = 1.46140 + 0.33031I
a = 1.008094 0.354468I
b = 0.96069 2.36911I
0.59945 10.37890I 0.23306 + 5.66856I
u = 1.47480 0.26620I
a = 0.368266 0.472378I
b = 0.25171 2.13870I
6.97144 + 7.45946I 3.34661 6.43325I
u = 1.47480 + 0.26620I
a = 0.368266 + 0.472378I
b = 0.25171 + 2.13870I
6.97144 7.45946I 3.34661 + 6.43325I
u = 1.48218 0.19094I
a = 0.246657 + 0.429290I
b = 0.39153 + 1.65497I
8.11148 + 3.15061I 5.88075 0.93673I
u = 1.48218 + 0.19094I
a = 0.246657 0.429290I
b = 0.39153 1.65497I
8.11148 3.15061I 5.88075 + 0.93673I
7
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
2
(u
26
+ 3u
25
+ ··· 9u
2
1)
c
2
, c
7
u
2
(u
26
+ u
25
+ ··· + 17u
2
4)
c
3
, c
4
(u + 1)
2
(u
26
+ 3u
25
+ ··· 9u
2
1)
c
5
, c
6
(u
2
u 1)(u
26
+ 2u
25
+ ··· 3u + 1)
c
8
, c
9
(u
2
+ u 1)(u
26
+ 2u
25
+ ··· 3u + 1)
c
10
(u
2
+ u 1)(u
26
+ 6u
25
+ ··· + 57u 9)
8
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
4
(y 1)
2
(y
26
25y
25
+ ··· + 18y + 1)
c
2
, c
7
y
2
(y
26
15y
25
+ ··· 136y + 16)
c
5
, c
6
, c
8
c
9
(y
2
3y + 1)(y
26
30y
25
+ ··· 19y + 1)
c
10
(y
2
3y + 1)(y
26
+ 6y
25
+ ··· 3159y + 81)
9