10
65
(K10a
42
)
1
Arc Sequences
5 9 8 6 2 10 1 3 4 7
Solving Sequence
2,9 3,6
5 1 4 10 8 7
c
2
c
5
c
1
c
4
c
9
c
8
c
7
c
3
, c
6
, c
10
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hu 1, b, a 1i
I
u
2
= hb
2
+ 2, u + 1, b + a 1i
I
u
3
= hu
9
3u
7
+ u
6
+ 3u
5
2u
4
u
3
+ u
2
1, u
4
+ u
2
+ a + 1, u
6
2u
4
+ u
2
+ bi
I
u
4
= hu
25
2u
24
+ ··· u + 3, 11610u
24
+ 12849u
23
+ ··· + 85658b + 129131,
109744u
24
56921u
23
+ ··· + 256974a + 486119i
There are 4 irreducible components with 37 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 1, b, a 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
1
0
a
3
=
1
0
a
6
=
0
1
a
5
=
1
1
a
1
=
0
1
a
4
=
1
0
a
10
=
1
0
a
8
=
1
0
a
7
=
1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 0
0 0
3
II. I
u
2
= hb
2
+ 2, u + 1, b + a 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
b + 1
b
a
3
=
b 1
2
a
6
=
0
1
a
5
=
1
1
a
1
=
0
1
a
4
=
1
0
a
10
=
1
b
a
8
=
1
b
a
7
=
1
b 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.00000 + 1.41421I
b = 1.41421I
4.93480 0
u = 1.00000
a = 1.00000 1.41421I
b = 1.41421I
4.93480 0
5
III.
I
u
3
= hu
9
3u
7
+u
6
+3u
5
2u
4
u
3
+u
2
1, u
4
+u
2
+a+1, u
6
2u
4
+u
2
+bi
(i) Arc colorings
a
2
=
1
0
a
9
=
u
4
u
2
1
u
6
+ 2u
4
u
2
a
3
=
u
7
u
6
+ 2u
5
+ 2u
4
u
3
u
2
+ u + 1
u
6
+ 2u
4
u
3
u
2
+ u + 1
a
6
=
0
u
a
5
=
u
u
a
1
=
u
2
+ 1
u
2
a
4
=
u
u
3
+ u
a
10
=
1
0
a
8
=
u
3
+ 2u
u
3
+ u
a
7
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
3
4u + 6
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.324816 0.175904I
a = 0.031420 + 1.141150I
b = 0.215080 1.307141I
3.02413 + 2.82812I 2.49024 2.97945I
u = 1.324816 + 0.175904I
a = 0.031420 1.141150I
b = 0.215080 + 1.307141I
3.02413 2.82812I 2.49024 + 2.97945I
u = 0.631920 0.444935I
a = 1.47702 0.33587I
b = 0.569840
1.11345 9.01951
u = 0.631920 + 0.444935I
a = 1.47702 + 0.33587I
b = 0.569840
1.11345 9.01951
u = 0.376870 0.700062I
a = 0.809231 + 0.894977I
b = 0.215080 + 1.307141I
3.02413 2.82812I 2.49024 + 2.97945I
u = 0.376870 + 0.700062I
a = 0.809231 0.894977I
b = 0.215080 1.307141I
3.02413 + 2.82812I 2.49024 2.97945I
u = 0.947946 0.524157I
a = 2.22219 0.24617I
b = 0.215080 1.307141I
3.02413 + 2.82812I 2.49024 2.97945I
u = 0.947946 + 0.524157I
a = 2.22219 + 0.24617I
b = 0.215080 + 1.307141I
3.02413 2.82812I 2.49024 + 2.97945I
u = 1.26384
a = 0.0459561
b = 0.569840
1.11345 9.01951
7
IV. I
u
4
= hu
25
2u
24
+ · · · u + 3, 11610u
24
+ 12849u
23
+ · · · + 85658b +
129131, 1.10 × 10
5
u
24
5.69 × 10
4
u
23
+ · · · + 2.57 × 10
5
a + 4.86 × 10
5
i
(i) Arc colorings
a
2
=
1
0
a
9
=
0.427063u
24
+ 0.221505u
23
+ ··· + 0.444166u 1.89170
0.135539u
24
0.150004u
23
+ ··· 1.43626u 1.50752
a
3
=
0.919723u
24
1.09561u
23
+ ··· + 1.84439u 0.787628
0.0326765u
24
+ 0.283838u
23
+ ··· + 2.14742u 0.527645
a
6
=
0
u
a
5
=
u
u
a
1
=
u
2
+ 1
u
2
a
4
=
u
u
3
+ u
a
10
=
0.393767u
24
+ 0.711325u
23
+ ··· + 1.22050u 1.57814
0.579304u
24
+ 0.514429u
23
+ ··· 1.11645u + 0.475274
a
8
=
0.130114u
24
+ 0.460451u
23
+ ··· + 4.45388u + 0.657576
0.104520u
24
+ 0.242336u
23
+ ··· + 1.98843u + 0.880852
a
7
=
0.409543u
24
+ 0.711006u
23
+ ··· + 2.36142u + 1.51464
0.567968u
24
+ 0.448551u
23
+ ··· + 1.86788u + 0.556609
(ii) Obstruction class = 1
(iii) Cusp Shapes =
144607
42829
u
24
194866
42829
u
23
+ ···
750725
42829
u
15780
42829
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.121399 0.226598I
a = 0.162911 1.241721I
b = 0.11000 + 1.41509I
7.43417 + 0.37131I 4.72924 + 0.01538I
u = 1.121399 + 0.226598I
a = 0.162911 + 1.241721I
b = 0.11000 1.41509I
7.43417 0.37131I 4.72924 0.01538I
u = 1.096786 0.679709I
a = 0.986127 0.662737I
b = 0.814894 + 0.282583I
5.88761 7.92352I 7.71863 + 6.25521I
u = 1.096786 + 0.679709I
a = 0.986127 + 0.662737I
b = 0.814894 0.282583I
5.88761 + 7.92352I 7.71863 6.25521I
u = 0.976768 0.540770I
a = 1.186827 + 0.711118I
b = 0.678633 0.221561I
0.03499 4.24383I 4.60496 + 6.78537I
u = 0.976768 + 0.540770I
a = 1.186827 0.711118I
b = 0.678633 + 0.221561I
0.03499 + 4.24383I 4.60496 6.78537I
u = 0.881284 0.447818I
a = 0.515096 + 1.241640I
b = 0.06236 1.52702I
3.72172 1.83282I 3.75932 + 4.01286I
u = 0.881284 + 0.447818I
a = 0.515096 1.241640I
b = 0.06236 + 1.52702I
3.72172 + 1.83282I 3.75932 4.01286I
u = 0.516228 0.881834I
a = 1.094163 + 0.211837I
b = 0.809668 + 0.163514I
7.64625 + 2.15851I 10.42476 1.29245I
u = 0.516228 + 0.881834I
a = 1.094163 0.211837I
b = 0.809668 0.163514I
7.64625 2.15851I 10.42476 + 1.29245I
9
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.435793
a = 1.82975
b = 0.403977
0.909052 12.0937
u = 0.378354 0.934639I
a = 0.707965 0.423297I
b = 0.339400 1.358960I
2.85055 6.29490I 6.20266 + 3.49250I
u = 0.378354 + 0.934639I
a = 0.707965 + 0.423297I
b = 0.339400 + 1.358960I
2.85055 + 6.29490I 6.20266 3.49250I
u = 0.611097 0.519026I
a = 0.381470 1.359473I
b = 0.045104 1.169878I
2.09053 + 1.42730I 3.69318 4.01748I
u = 0.611097 + 0.519026I
a = 0.381470 + 1.359473I
b = 0.045104 + 1.169878I
2.09053 1.42730I 3.69318 + 4.01748I
u = 0.686093 0.799024I
a = 1.127655 + 0.082339I
b = 0.404191 + 1.026875I
4.98459 + 2.21818I 7.23817 3.39990I
u = 0.686093 + 0.799024I
a = 1.127655 0.082339I
b = 0.404191 1.026875I
4.98459 2.21818I 7.23817 + 3.39990I
u = 0.901860 0.293308I
a = 0.0736169 0.0690093I
b = 0.245363 0.498558I
1.52585 + 1.04428I 1.27127 1.42914I
u = 0.901860 + 0.293308I
a = 0.0736169 + 0.0690093I
b = 0.245363 + 0.498558I
1.52585 1.04428I 1.27127 + 1.42914I
u = 0.969881 0.673526I
a = 0.078511 + 0.246322I
b = 0.498082 + 0.831864I
4.11705 + 3.30443I 6.15585 1.80924I
10
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.969881 + 0.673526I
a = 0.078511 0.246322I
b = 0.498082 0.831864I
4.11705 3.30443I 6.15585 + 1.80924I
u = 1.090161 0.576724I
a = 2.08235 0.43233I
b = 0.276880 + 1.384381I
5.06500 + 7.73599I 0.26723 6.67404I
u = 1.090161 + 0.576724I
a = 2.08235 + 0.43233I
b = 0.276880 1.384381I
5.06500 7.73599I 0.26723 + 6.67404I
u = 1.172915 0.648513I
a = 1.78321 + 0.63873I
b = 0.33191 1.42709I
0.44144 + 12.07646I 3.42339 7.22441I
u = 1.172915 + 0.648513I
a = 1.78321 0.63873I
b = 0.33191 + 1.42709I
0.44144 12.07646I 3.42339 + 7.22441I
11
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
(u 1)
2
(u + 1)(u
9
3u
7
u
6
+ 3u
5
+ 2u
4
u
3
u
2
+ 1)
(u
25
+ 2u
24
+ ··· u 3)
c
2
, c
3
, c
8
(u)(u
2
+ 2)(1 + 2u + u
2
+ u
3
)
3
(u
25
2u
24
+ ··· 4u + 2)
c
4
(u + 1)
3
(u
9
+ 6u
8
+ ··· + 2u + 1)
(u
25
+ 10u
24
+ ··· + 97u + 9)
c
6
, c
10
(u 1)
2
(u + 1)(u
9
3u
7
+ u
6
+ 3u
5
2u
4
u
3
+ u
2
1)
(u
25
+ 2u
24
+ ··· 5u + 3)
c
7
(u 1)(u + 1)
2
(u
9
3u
7
+ u
6
+ 3u
5
2u
4
u
3
+ u
2
1)
(u
25
+ 2u
24
+ ··· 5u + 3)
c
9
(u)(u
2
+ 2)(1 + u
2
+ u
3
)
3
(u
25
2u
24
+ ··· + 56u 16)
12
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y 1)
3
(y
9
6y
8
+ ··· + 2y 1)
(y
25
10y
24
+ ··· + 97y 9)
c
2
, c
3
, c
8
(y)(y + 2)
2
(1 + 2y + 3y
2
+ y
3
)
3
(y
25
+ 22y
24
+ ··· + 8y 4)
c
4
(y 1)
3
(y
9
6y
8
+ ··· 6y 1)
(y
25
+ 14y
24
+ ··· + 1561y 81)
c
6
, c
7
, c
10
(y 1)
3
(y
9
6y
8
+ ··· + 2y 1)
(y
25
26y
24
+ ··· 47y 9)
c
9
(y)(y + 2)
2
(1 + 2y y
2
+ y
3
)
3
(y
25
2y
24
+ ··· 2624y 256)
13