11n
62
(K11n
62
)
1
Arc Sequences
4 1 9 2 9 11 10 3 1 7 6
Solving Sequence
1,4 2,6
11 7 10 8 9 3 5
c
1
c
11
c
6
c
10
c
7
c
9
c
3
c
5
c
2
, c
4
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
4
b
3
+ 3b
2
2b + 1, a + 1, u 1i
I
u
2
= hu
20
+ 5u
19
+ ··· + 4u + 1, u
4
+ u
2
+ a + 2u 1, u
19
+ 4u
18
+ ··· + 8b + 1i
There are 2 irreducible components with 24 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
4
b
3
+ 3b
2
2b + 1, a + 1, u 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
6
=
1
b
a
11
=
b + 1
b
2
a
7
=
b
2
+ b 1
b
3
+ b
a
10
=
b
3
+ b
2
2b + 1
b
3
b
2
+ 2b 1
a
8
=
0
b
3
b
2
+ 2b 1
a
9
=
0
b
3
b
2
+ 2b 1
a
3
=
0
1
a
5
=
1
0
a
5
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 0.10488 1.55249I
8.43568 3.16396I 5.67855 + 1.65351I
u = 1.00000
a = 1.00000
b = 0.10488 + 1.55249I
8.43568 + 3.16396I 5.67855 1.65351I
u = 1.00000
a = 1.00000
b = 0.395123 0.506844I
1.43393 1.41510I 0.82145 + 5.62908I
u = 1.00000
a = 1.00000
b = 0.395123 + 0.506844I
1.43393 + 1.41510I 0.82145 5.62908I
3
II.
I
u
2
= hu
20
+5u
19
+· · ·+ 4u + 1, u
4
+u
2
+a + 2u 1, u
19
+4u
18
+· · ·+ 8b + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
6
=
u
4
u
2
2u + 1
1
8
u
19
1
2
u
18
+ ···
3
8
u
1
8
a
11
=
3
8
u
19
3
2
u
18
+ ···
9
8
u +
5
8
9
8
u
19
19
4
u
18
+ ···
31
8
u
11
8
a
7
=
5
2
u
19
43
4
u
18
+ ··· 11u
9
4
1
2
u
19
3u
18
+ ··· 4u 2
a
10
=
3
2
u
19
29
4
u
18
+ ···
17
2
u
13
4
7
4
u
19
+
31
4
u
18
+ ··· +
29
4
u + 2
a
8
=
3
4
u
19
+
9
2
u
18
+ ··· +
13
4
u +
5
4
2u
19
37
4
u
18
+ ··· 7u
7
4
a
9
=
1
4
u
19
+
1
2
u
18
+ ···
5
4
u
5
4
7
4
u
19
+
31
4
u
18
+ ··· +
29
4
u + 2
a
3
=
u
2
+ 1
u
2
a
5
=
u
u
3
+ u
a
5
=
u
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.14360 0.97217I
a = 1.88812 + 1.33370I
b = 0.22806 1.54393I
2.29524 9.61446I 0.95211 + 4.92599I
u = 1.14360 + 0.97217I
a = 1.88812 1.33370I
b = 0.22806 + 1.54393I
2.29524 + 9.61446I 0.95211 4.92599I
u = 1.07938 1.03654I
a = 1.93071 + 0.24104I
b = 0.693187 0.560449I
9.21922 6.23574I 2.42777 + 5.05678I
u = 1.07938 + 1.03654I
a = 1.93071 0.24104I
b = 0.693187 + 0.560449I
9.21922 + 6.23574I 2.42777 5.05678I
u = 1.00029 1.08975I
a = 1.53048 0.81585I
b = 0.716949 + 0.471005I
9.48767 1.51858I 3.11046 0.47571I
u = 1.00029 + 1.08975I
a = 1.53048 + 0.81585I
b = 0.716949 0.471005I
9.48767 + 1.51858I 3.11046 + 0.47571I
u = 0.89119 1.13673I
a = 0.57676 1.77039I
b = 0.24160 + 1.48775I
3.14125 + 1.95377I 0.018441 0.726692I
u = 0.89119 + 1.13673I
a = 0.57676 + 1.77039I
b = 0.24160 1.48775I
3.14125 1.95377I 0.018441 + 0.726692I
u = 0.597208 0.324863I
a = 1.85580 + 0.45659I
b = 0.09851 + 1.58176I
7.51526 3.82239I 0.11541 + 4.60594I
u = 0.597208 + 0.324863I
a = 1.85580 0.45659I
b = 0.09851 1.58176I
7.51526 + 3.82239I 0.11541 4.60594I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.363060 0.401326I
a = 1.67131 + 0.49419I
b = 0.394382 + 0.630620I
0.02154 2.08472I 2.36846 + 5.36236I
u = 0.363060 + 0.401326I
a = 1.67131 0.49419I
b = 0.394382 0.630620I
0.02154 + 2.08472I 2.36846 5.36236I
u = 0.042499 0.536678I
a = 1.28106 + 1.14509I
b = 0.478289 0.318748I
0.995000 + 0.993446I 5.17867 4.04800I
u = 0.042499 + 0.536678I
a = 1.28106 1.14509I
b = 0.478289 + 0.318748I
0.995000 0.993446I 5.17867 + 4.04800I
u = 0.352937 0.780079I
a = 0.70910 + 2.64377I
b = 0.11570 1.45345I
4.77753 + 2.99094I 0.69176 3.46155I
u = 0.352937 + 0.780079I
a = 0.70910 2.64377I
b = 0.11570 + 1.45345I
4.77753 2.99094I 0.69176 + 3.46155I
u = 0.982345 0.153948I
a = 1.111425 + 0.040944I
b = 0.172981 + 0.339397I
1.76070 + 0.62769I 5.52555 + 1.68478I
u = 0.982345 + 0.153948I
a = 1.111425 0.040944I
b = 0.172981 0.339397I
1.76070 0.62769I 5.52555 1.68478I
u = 1.196942 0.306717I
a = 1.47976 0.61803I
b = 0.03411 + 1.47694I
7.81656 + 1.26535I 3.51291 0.02866I
u = 1.196942 + 0.306717I
a = 1.47976 + 0.61803I
b = 0.03411 1.47694I
7.81656 1.26535I 3.51291 + 0.02866I
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
(u 1)
4
(u
20
+ 5u
19
+ ··· + 4u + 1)
c
2
(u + 1)
4
(u
20
+ 3u
19
+ ··· 4u + 1)
c
3
, c
8
u
4
(u
20
+ u
19
+ ··· + 8u + 16)
c
5
(u
4
+ u
3
+ u
2
+ 1)(u
20
+ 2u
19
+ ··· + 154u + 445)
c
6
, c
7
, c
10
c
11
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
20
+ 2u
19
+ ··· + 2u + 1)
c
9
(u
4
+ u
3
+ u
2
+ 1)(u
20
+ 2u
19
+ ··· + 5u
2
+ 1)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
4
(y
20
3y
19
+ ··· + 4y + 1)
c
2
(y 1)
4
(y
20
+ 33y
19
+ ··· + 4y + 1)
c
3
, c
8
y
4
(y
20
27y
19
+ ··· 1344y + 256)
c
5
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
20
+ 38y
19
+ ··· + 4809874y + 198025)
c
6
, c
7
, c
10
c
11
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
20
+ 22y
19
+ ··· + 10y + 1)
c
9
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
20
+ 26y
19
+ ··· + 10y + 1)
8