11n
19
(K11n
19
)
1
Arc Sequences
5 1 9 2 3 10 11 1 3 7 8
Solving Sequence
1,5
2 3
4,9
8 11 7 10 6
c
1
c
2
c
4
c
8
c
11
c
7
c
10
c
6
c
3
, c
5
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
4
a
3
+ 2a
2
+ a + 1, a
3
+ 2b 1, a
3
2a
2
+ 2a + 2u + 1i
I
u
2
= hu
6
3u
5
+ 4u
4
u
3
3u + 1, u
5
+ 2u
4
3u
3
+ 2b u + 2, 2u
5
5u
4
+ 6u
3
+ u
2
+ 2a 5i
There are 2 irreducible components with 10 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
a
3
+ 2a
2
+ a + 1, a
3
+ 2b 1, a
3
2a
2
+ 2a + 2u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
2
a
3
+ a
2
a
1
2
a
2
=
1
1
2
a
3
+ a
2
a +
1
2
a
3
=
1
2
a
3
a
2
+ a +
1
2
1
2
a
3
+ a
2
a +
1
2
a
4
=
1
2
a
3
a
2
+ a +
1
2
1
2
a
3
+ a
2
a +
1
2
a
9
=
a
1
2
a
3
+
1
2
a
8
=
1
2
a
3
+ a
1
2
1
2
a
3
+
1
2
a
11
=
a
3
a
2
1
2
a
3
+
1
2
a
7
=
1
2
a
3
a
2
3
2
1
2
a
3
+
1
2
a
10
=
a
1
2
a
3
+
1
2
a
6
=
1
0
a
6
=
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 = 0.500000 + 0.866025I
a = 0.309017 0.535233I
b = 0.618034
0.98696 2.02988I 6.50000 + 5.40059I
u = 0.500000 0.866025I
a = 0.309017 + 0.535233I
b = 0.618034
0.98696 + 2.02988I 6.50000 5.40059I
u = 0.500000 0.866025I
a = 0.80902 1.40126I
b = 1.61803
8.88264 + 2.02988I 6.50000 1.52761I
u = 0.500000 + 0.866025I
a = 0.80902 + 1.40126I
b = 1.61803
8.88264 2.02988I 6.50000 + 1.52761I
3
II. I
u
2
= hu
6
3u
5
+ 4u
4
u
3
3u + 1, u
5
+ 2u
4
3u
3
+ 2b u +
2, 2u
5
5u
4
+ 6u
3
+ u
2
+ 2a 5i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
4
=
u
u
3
+ u
a
9
=
u
5
+
5
2
u
4
3u
3
1
2
u
2
+
5
2
1
2
u
5
u
4
+
3
2
u
3
+
1
2
u 1
a
8
=
3
2
u
5
+
7
2
u
4
+ ···
1
2
u +
7
2
1
2
u
5
u
4
+
3
2
u
3
+
1
2
u 1
a
11
=
1
2
u
5
+
1
2
u
4
+ ···
5
2
u
1
2
1
2
u
5
u
4
+
1
2
u
3
+ u
2
+
3
2
u
a
7
=
3
2
u
4
u
3
+
3
2
u
2
+ u +
3
2
1
2
u
5
1
2
u
3
u
2
1
2
u
a
10
=
3u
5
15
2
u
4
+ 7u
3
+
5
2
u
2
+ 3u
9
2
3
2
u
5
+ 4u
4
7
2
u
3
2u
2
5
2
u + 2
a
6
=
u
5
2u
3
u
u
5
+ u
3
+ u
a
6
=
u
5
2u
3
u
u
5
+ 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 = 0.538017 0.647846I
a = 0.307113 0.332011I
b = 0.049860 0.377590I
0.088081 + 1.387967I 1.39961 3.44965I
u = 0.538017 + 0.647846I
a = 0.307113 + 0.332011I
b = 0.049860 + 0.377590I
0.088081 1.387967I 1.39961 + 3.44965I
u = 0.333806
a = 2.35960
b = 0.787648
1.36678 7.32047
u = 1.22230 1.32718I
a = 0.871194 + 0.796010I
b = 2.12131 + 0.18327I
11.30140 4.76989I 7.67813 + 1.77109I
u = 1.22230 + 1.32718I
a = 0.871194 0.796010I
b = 2.12131 0.18327I
11.30140 + 4.76989I 7.67813 1.77109I
u = 1.29762
a = 1.48776
b = 1.93055
11.1902 8.52403
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)
2
(u
6
+ 3u
5
+ 4u
4
+ u
3
+ 3u + 1)
c
2
(u
2
+ u + 1)
2
(u
6
+ u
5
+ 10u
4
+ 17u
3
+ 2u
2
+ 9u + 1)
c
3
, c
9
u
4
(u
6
+ 6u
5
+ 28u
4
+ 60u
3
+ 20u
2
16u 16)
c
4
(u
2
u + 1)
2
(u
6
+ 3u
5
+ 4u
4
+ u
3
+ 3u + 1)
c
5
(u
2
+ u + 1)
2
(u
6
+ 3u
5
+ 12u
4
127u
3
+ 52u
2
113u + 41)
c
6
, c
7
, c
8
(u
2
u 1)
2
(u
6
+ 3u
5
2u
4
11u
3
6u
2
u 1)
c
10
, c
11
(u
2
+ u 1)
2
(u
6
+ 3u
5
2u
4
11u
3
6u
2
u 1)
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
2
+ y + 1)
2
(y
6
y
5
+ 10y
4
17y
3
+ 2y
2
9y + 1)
c
2
(y
2
+ y + 1)
2
(y
6
+ 19y
5
+ 70y
4
265y
3
282y
2
77y + 1)
c
3
, c
9
y
4
(y
6
+ 20y
5
+ 104y
4
2320y
3
+ 1424y
2
896y + 256)
c
5
(y
2
+ y + 1)
2
(y
6
+ 15y
5
+ 1010y
4
14121y
3
25014y
2
8505y + 1681)
c
6
, c
7
, c
8
c
10
, c
11
(y
2
3y + 1)
2
(y
6
13y
5
+ 58y
4
93y
3
+ 18y
2
+ 11y + 1)
7