11n
139
(K11n
139
)
1
Arc Sequences
7 1 8 11 10 2 4 3 1 5 4
Solving Sequence
1,7 2,4
8 3 9 6 11 5 10
c
1
c
7
c
3
c
8
c
6
c
11
c
4
c
10
c
2
, c
5
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
4
+ 3u
2
+ 1, u
2
+ a + 2, u
3
+ b 2ui
I
u
2
= hu
8
+ u
7
+ 8u
6
+ 4u
5
+ 18u
4
+ 11u
2
5u + 2, u
7
8u
5
18u
3
+ 2u
2
+ 4b 9u + 2,
u
7
6u
5
+ 2u
4
8u
3
+ 8u
2
+ 4a + u + 4i
There are 2 irreducible components with 12 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
4
+ 3u
2
+ 1, u
2
+ a + 2, u
3
+ b 2ui
(i) Arc colorings
a
1
=
1
0
a
7
=
u
2
2
u
3
+ 2u
a
2
=
u
3
+ 3u + 1
1
a
4
=
0
u
a
8
=
u
2
2
u
3
+ u
2
+ 2u + 1
a
3
=
u
3
+ 3u
1
a
9
=
0
u
2
+ 1
a
6
=
u
3
2u
0
a
11
=
1
u
2
a
5
=
u
u
3
+ u
a
10
=
u
2
+ 1
u
2
+ 1
a
10
=
u
2
+ 1
u
2
+ 1
(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 = 0.618034I
a = 1.61803
b = 1.00000I
0.986960 0
u = 0.618034I
a = 1.61803
b = 1.00000I
0.986960 0
u = 1.61803I
a = 0.618034
b = 1.00000I
8.88264 0
u = 1.61803I
a = 0.618034
b = 1.00000I
8.88264 0
3
II. I
u
2
= hu
8
+ u
7
+ 8u
6
+ 4u
5
+ 18u
4
+ 11u
2
5u + 2, u
7
8u
5
18u
3
+
2u
2
+ 4b 9u + 2, u
7
6u
5
+ 2u
4
8u
3
+ 8u
2
+ 4a + u + 4i
(i) Arc colorings
a
1
=
1
0
a
7
=
1
4
u
7
+
3
2
u
5
+ ···
1
4
u 1
1
4
u
7
+ 2u
5
+ ··· +
9
4
u
1
2
a
2
=
1
4
u
7
+
3
2
u
5
+ ··· u
2
+
11
4
u
1
2
u
6
1
2
u
5
+ ···
3
2
u
2
+
3
2
u
a
4
=
0
u
a
8
=
1
4
u
7
+
3
2
u
5
+ ···
1
4
u 1
1
2
u
6
+
1
2
u
5
+ ··· +
5
2
u
2
+
1
2
u
a
3
=
1
4
u
7
+
1
2
u
6
+ ··· +
1
2
u
2
+
5
4
u
1
2
u
6
1
2
u
5
+ ···
3
2
u
2
+
3
2
u
a
9
=
u
4
3u
2
1
u
4
+ 2u
2
a
6
=
u
3
2u
u
5
+ 3u
3
+ u
a
11
=
1
u
2
a
5
=
u
u
3
+ u
a
10
=
u
2
+ 1
u
4
2u
2
a
10
=
u
2
+ 1
u
4
2u
2
(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 = 0.36613 1.66771I
a = 1.005552 0.492957I
b = 1.52012 + 0.43970I
12.93019 1.89326I 1.23462 + 1.04722I
u = 0.36613 + 1.66771I
a = 1.005552 + 0.492957I
b = 1.52012 0.43970I
12.93019 + 1.89326I 1.23462 1.04722I
u = 0.234808 1.029494I
a = 0.740043 + 0.762517I
b = 0.776759 + 0.445071I
3.66920 + 1.06491I 1.31198 1.63429I
u = 0.234808 + 1.029494I
a = 0.740043 0.762517I
b = 0.776759 0.445071I
3.66920 1.06491I 1.31198 + 1.63429I
u = 0.15755 1.96154I
a = 1.131185 0.148871I
b = 1.39990 1.57125I
12.82706 5.56972I 0.47783 + 1.89693I
u = 0.15755 + 1.96154I
a = 1.131185 + 0.148871I
b = 1.39990 + 1.57125I
12.82706 + 5.56972I 0.47783 1.89693I
u = 0.258486 0.303432I
a = 1.115676 + 0.333931I
b = 0.156540 0.733537I
0.482455 + 0.984921I 7.02443 7.03211I
u = 0.258486 + 0.303432I
a = 1.115676 0.333931I
b = 0.156540 + 0.733537I
0.482455 0.984921I 7.02443 + 7.03211I
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
(u
2
+ 1)
2
(u
8
+ u
7
u
6
8u
5
+ 2u
4
+ 7u
3
+ 5u
2
+ 4u + 5)
c
2
(u + 1)
4
(u
8
+ 3u
7
+ ··· 34u + 25)
c
3
, c
7
, c
8
(u
2
+ 1)
2
(u
8
+ u
7
+ 9u
6
+ 2u
5
+ 22u
4
5u
3
+ 23u
2
+ 6u + 5)
c
4
, c
5
, c
10
c
11
(u
4
+ 3u
2
+ 1)(u
8
+ u
7
+ 8u
6
+ 4u
5
+ 18u
4
+ 11u
2
5u + 2)
c
9
(u
2
u 1)
2
(u
8
+ 11u
7
+ 44u
6
+ 62u
5
+ 426u
4
1920u
3
+ 1693u
2
389u + 136)
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y + 1)
4
(y
8
3y
7
+ ··· + 34y + 25)
c
2
(y 1)
4
(y
8
+ 33y
7
+ ··· 1706y + 625)
c
3
, c
7
, c
8
(y + 1)
4
(y
8
+ 17y
7
+ 121y
6
+ 448y
5
+ 916y
4
+ 1053y
3
+ 809y
2
+ 194y + 25)
c
4
, c
5
, c
10
c
11
(y
2
+ 3y + 1)
2
(y
8
+ 15y
7
+ 92y
6
+ 294y
5
+ 514y
4
+ 468y
3
+ 193y
2
+ 19y + 4)
c
9
(y
2
3y + 1)
2
(y
8
33y
7
+ ··· + 309175y + 18496)
7