10
144
(K10n
28
)
1
Arc Sequences
5 8 7 6 2 10 3 4 6 7
Solving Sequence
2,8 3,6
5 1 4 9 10 7
c
2
c
5
c
1
c
4
c
8
c
9
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
10
u
9
u
8
+ 2u
7
+ 3u
6
4u
5
+ 4u
3
u + 1, u
8
u
7
+ u
5
+ 3u
4
3u
3
+ u
2
+ 2a + u + 1,
u
8
+ u
7
u
5
u
4
+ 3u
3
3u
2
+ 2b u + 1i
I
u
4
= hu
12
u
11
2u
10
+ 4u
9
+ u
8
5u
7
u
6
+ 7u
5
u
4
9u
3
+ 6u
2
+ 2u 3,
22u
11
8u
10
+ 63u
9
32u
8
87u
7
+ 64u
6
+ 105u
5
92u
4
112u
3
+ 182u
2
+ 47b + 35u 86,
74u
11
+ 50u
10
+ ··· + 141a 520i
There are 4 irreducible components with 25 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
8
=
1
0
a
3
=
1
0
a
6
=
0
1
a
5
=
1
1
a
1
=
0
1
a
4
=
1
0
a
9
=
1
0
a
10
=
1
1
a
7
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
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
3.28987 12.0000
3
II. I
u
2
= hb
2
+ 2, u + 1, b + a + 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
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
9
=
1
b
a
10
=
1
b 1
a
7
=
1
b
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
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
8.22467 12.0000
u = 1.00000
a = 1.00000 1.41421I
b = 1.41421I
8.22467 12.0000
5
III. I
u
3
= hu
10
u
9
u
8
+ 2u
7
+ 3u
6
4u
5
+ 4u
3
u + 1, u
8
u
7
+ u
5
+
3u
4
3u
3
+ u
2
+ 2a + u + 1, u
8
+ u
7
u
5
u
4
+ 3u
3
3u
2
+ 2b u + 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
1
2
u
8
+
1
2
u
7
+ ···
1
2
u
1
2
1
2
u
8
1
2
u
7
+ ··· +
1
2
u
1
2
a
3
=
1
2
u
9
+
1
2
u
8
+ ···
1
2
u
2
+
3
2
u
1
2
u
9
u
8
+ ··· u +
1
2
a
6
=
0
u
a
5
=
u
u
a
1
=
u
2
+ 1
u
2
a
4
=
u
u
3
+ u
a
9
=
1
1
2
u
8
1
2
u
7
+ ··· +
1
2
u
1
2
a
10
=
1
1
2
u
8
1
2
u
7
+ ··· +
1
2
u
1
2
a
7
=
u
1
2
u
9
1
2
u
8
+ ··· +
1
2
u
2
+
1
2
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
9
+ u
8
4u
7
+ u
6
+ 9u
5
+ u
4
11u
3
+ 9u
2
+ 9u 6
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.014862 0.798709I
a = 0.986385 + 0.539596I
b = 0.880108 0.189454I
3.61170 6.23908I 1.40880 + 5.42921I
u = 1.014862 + 0.798709I
a = 0.986385 0.539596I
b = 0.880108 + 0.189454I
3.61170 + 6.23908I 1.40880 5.42921I
u = 0.773203 0.317670I
a = 0.53813 1.49598I
b = 0.02945 + 1.49900I
7.11290 1.33139I 5.94848 + 5.33149I
u = 0.773203 + 0.317670I
a = 0.53813 + 1.49598I
b = 0.02945 1.49900I
7.11290 + 1.33139I 5.94848 5.33149I
u = 0.351677 0.481849I
a = 0.758504 + 0.165558I
b = 0.246909 0.578012I
0.143235 + 1.179705I 1.77268 5.86187I
u = 0.351677 + 0.481849I
a = 0.758504 0.165558I
b = 0.246909 + 0.578012I
0.143235 1.179705I 1.77268 + 5.86187I
u = 0.794058 0.823254I
a = 0.206373 0.375553I
b = 0.453532 1.055339I
0.94791 + 1.45588I 3.02190 1.71983I
u = 0.794058 + 0.823254I
a = 0.206373 + 0.375553I
b = 0.453532 + 1.055339I
0.94791 1.45588I 3.02190 + 1.71983I
u = 1.142329 0.733576I
a = 1.60388 0.50530I
b = 0.38382 + 1.39954I
1.41581 + 10.79659I 5.84814 6.97307I
u = 1.142329 + 0.733576I
a = 1.60388 + 0.50530I
b = 0.38382 1.39954I
1.41581 10.79659I 5.84814 + 6.97307I
7
IV. I
u
4
= hu
12
u
11
+ · · · + 2u 3, 22u
11
8u
10
+ · · · + 47b
86, 74u
11
+ 50u
10
+ · · · + 141a 520i
(i) Arc colorings
a
2
=
1
0
a
8
=
0.524823u
11
0.354610u
10
+ ··· 0.531915u + 3.68794
0.468085u
11
+ 0.170213u
10
+ ··· 0.744681u + 1.82979
a
3
=
0.695035u
11
0.0709220u
10
+ ··· + 1.89362u 0.262411
0.0851064u
11
0.212766u
10
+ ··· + 1.68085u + 0.212766
a
6
=
0
u
a
5
=
u
u
a
1
=
u
2
+ 1
u
2
a
4
=
u
u
3
+ u
a
9
=
0.460993u
11
0.0141844u
10
+ ··· 1.02128u + 2.34752
0.0638298u
11
+ 0.340426u
10
+ ··· 0.489362u 0.340426
a
10
=
0.460993u
11
0.0141844u
10
+ ··· 1.02128u + 2.34752
1
a
7
=
0.780142u
11
+ 0.716312u
10
+ ··· 2.42553u 2.04965
0.446809u
11
+ 0.382979u
10
+ ··· 0.425532u 1.38298
(ii) Obstruction class = 1
(iii) Cusp Shapes =
84
47
u
11
+
72
47
u
10
+
232
47
u
9
276
47
u
8
204
47
u
7
+
364
47
u
6
+
324
47
u
5
488
47
u
4
120
47
u
3
+
712
47
u
2
268
47
u
354
47
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.298593 0.085372I
a = 0.001528 1.205411I
b = 0.138835 + 1.234445I
6.25011 + 1.97241I 7.42428 3.68478I
u = 1.298593 + 0.085372I
a = 0.001528 + 1.205411I
b = 0.138835 1.234445I
6.25011 1.97241I 7.42428 + 3.68478I
u = 0.769522 0.881187I
a = 1.049649 0.365296I
b = 0.873214
4.37022 0.269499
u = 0.769522 + 0.881187I
a = 1.049649 + 0.365296I
b = 0.873214
4.37022 0.269499
u = 0.666209
a = 2.04823
b = 0.413150
2.55102 1.41678
u = 0.547085 0.953523I
a = 0.493698 + 0.413265I
b = 0.408802 + 1.276377I
0.40571 4.59213I 3.41886 + 3.20482I
u = 0.547085 + 0.953523I
a = 0.493698 0.413265I
b = 0.408802 1.276377I
0.40571 + 4.59213I 3.41886 3.20482I
u = 0.805413 0.489916I
a = 1.80888 + 1.01497I
b = 0.138835 + 1.234445I
6.25011 + 1.97241I 7.42428 3.68478I
u = 0.805413 + 0.489916I
a = 1.80888 1.01497I
b = 0.138835 1.234445I
6.25011 1.97241I 7.42428 + 3.68478I
u = 0.973781 0.790428I
a = 1.46944 + 0.17813I
b = 0.408802 1.276377I
0.40571 + 4.59213I 3.41886 3.20482I
9
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.973781 + 0.790428I
a = 1.46944 0.17813I
b = 0.408802 + 1.276377I
0.40571 4.59213I 3.41886 + 3.20482I
u = 1.14988
a = 0.0232063
b = 0.413150
2.55102 1.41678
10
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
, c
6
c
9
, c
10
(u 1)
2
(u + 1)(u
10
+ u
9
u
8
2u
7
+ 3u
6
+ 4u
5
4u
3
+ u + 1)
(u
12
+ u
11
2u
10
4u
9
+ u
8
+ 5u
7
u
6
7u
5
u
4
+ 9u
3
+ 6u
2
2u 3)
c
2
, c
3
, c
7
u(u
2
+ 2)(u
6
u
5
+ 3u
4
2u
3
+ 2u
2
u 1)
2
(u
10
+ 3u
9
+ 9u
8
+ 16u
7
+ 24u
6
+ 27u
5
+ 23u
4
+ 16u
3
+ 8u
2
+ 4u + 2)
c
4
(u + 1)
3
(u
10
+ 3u
9
+ 11u
8
+ 18u
7
+ 33u
6
+ 32u
5
+ 34u
4
+ 18u
3
+ 8u
2
+ u + 1)
(u
12
+ 5u
11
+ ··· + 40u + 9)
c
8
u(u
2
+ 2)(u
6
u
5
3u
4
+ 2u
3
+ 2u
2
+ u 1)
2
(u
10
+ 3u
9
+ 3u
8
8u
6
17u
5
+ 17u
4
+ 58u
3
+ 48u
2
+ 16u + 10)
11
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
, c
6
c
9
, c
10
(y 1)
3
(y
10
3y
9
+ 11y
8
18y
7
+ 33y
6
32y
5
+ 34y
4
18y
3
+ 8y
2
y + 1)
(y
12
5y
11
+ ··· 40y + 9)
c
2
, c
3
, c
7
y(y + 2)
2
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
6y
2
5y + 1)
2
(y
10
+ 9y
9
+ ··· + 16y + 4)
c
4
(y 1)
3
(y
10
+ 13y
9
+ ··· + 15y + 1)(y
12
+ 3y
11
+ ··· 196y + 81)
c
8
y(y + 2)
2
(y
6
7y
5
+ 17y
4
16y
3
+ 6y
2
5y + 1)
2
(y
10
3y
9
+ ··· + 704y + 100)
12