11n
152
(K11n
152
)
1
Arc Sequences
4 8 1 2 10 11 3 6 1 8 9
Solving Sequence
2,4 5,10
6 1 3 9 8 11 7
c
4
c
5
c
1
c
3
c
9
c
8
c
11
c
6
c
2
, c
7
, c
10
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hb
5
+ 4b
4
+ 4b
3
b
2
2b + 1, a 1, u 1i
I
u
2
= hu
5
+ u
4
2u
3
u
2
+ u 1, b + u + 1, u
4
u
3
+ 2u
2
+ a + u 1i
I
u
3
= hu
10
+ 2u
9
+ 5u
8
+ u
7
+ 3u
6
8u
5
2u
4
19u
3
6u
2
8u 1,
u
9
2u
8
5u
7
u
6
3u
5
+ 8u
4
+ 2u
3
+ 19u
2
+ a + 6u + 8,
u
9
u
8
4u
7
+ 3u
6
6u
5
+ 14u
4
12u
3
+ 31u
2
+ 16b 9u + 17i
I
u
4
= hu
11
+ 5u
10
+ 8u
9
3u
8
22u
7
14u
6
+ 18u
5
+ 19u
4
7u
3
7u
2
+ 2u 1, a u 2, u
3
+ 2u
2
+ b + ui
There are 4 irreducible components with 31 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
5
+ 4b
4
+ 4b
3
b
2
2b + 1, a 1, u 1i
(i) Arc colorings
a
2
=
1
0
a
4
=
0
1
a
5
=
1
1
a
10
=
1
b
a
6
=
b 2
b
2
b + 1
a
1
=
1
1
a
3
=
1
0
a
9
=
b + 2
1
a
8
=
b
4
5b
3
8b
2
3b + 2
0
a
11
=
b
2
3b 1
b
a
7
=
b
4
5b
3
8b
2
3b + 2
0
a
7
=
b
4
5b
3
8b
2
3b + 2
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 = 2.21774
4.04602 15.9649
u = 1.00000
a = 1.00000
b = 1.30992 0.54991I
1.97403 1.53058I 3.57269 + 4.45807I
u = 1.00000
a = 1.00000
b = 1.30992 + 0.54991I
1.97403 + 1.53058I 3.57269 4.45807I
u = 1.00000
a = 1.00000
b = 0.418784 0.219165I
7.51750 + 4.40083I 3.44484 1.78781I
u = 1.00000
a = 1.00000
b = 0.418784 + 0.219165I
7.51750 4.40083I 3.44484 + 1.78781I
3
II. I
u
2
= hu
5
+ u
4
2u
3
u
2
+ u 1, b + u + 1, u
4
u
3
+ 2u
2
+ a + u 1i
(i) Arc colorings
a
2
=
1
0
a
4
=
0
u
a
5
=
u
u
a
10
=
u
4
+ u
3
2u
2
u + 1
u 1
a
6
=
u
4
u
3
+ 2u
2
+ u
u
3
2u
2
+ u + 1
a
1
=
1
u
2
a
3
=
u
u
3
+ u
a
9
=
u
4
+ u
3
2u
2
u
u
2
u 1
a
8
=
u
4
u
2
1
u
4
+ 2u
2
a
11
=
u
4
+ u
3
2u
2
u + 1
u 1
a
7
=
u
u
a
7
=
u
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 = 1.41878 0.21917I
a = 0.688402 + 0.106340I
b = 0.418784 + 0.219165I
7.51750 4.40083I 3.44484 + 1.78781I
u = 1.41878 + 0.21917I
a = 0.688402 0.106340I
b = 0.418784 0.219165I
7.51750 + 4.40083I 3.44484 1.78781I
u = 0.309916 0.549911I
a = 0.77780 + 1.38013I
b = 1.30992 + 0.54991I
1.97403 + 1.53058I 3.57269 4.45807I
u = 0.309916 + 0.549911I
a = 0.77780 1.38013I
b = 1.30992 0.54991I
1.97403 1.53058I 3.57269 + 4.45807I
u = 1.21774
a = 0.821196
b = 2.21774
4.04602 15.9649
5
III.
I
u
3
= hu
10
+2u
9
+· · ·8u1, u
9
2u
8
+· · ·+a+8, u
9
u
8
+· · ·+16b+17i
(i) Arc colorings
a
2
=
1
0
a
4
=
0
u
a
5
=
u
u
a
10
=
u
9
+ 2u
8
+ 5u
7
+ u
6
+ 3u
5
8u
4
2u
3
19u
2
6u 8
0.0625000u
9
+ 0.0625000u
8
+ ··· + 0.562500u 1.06250
a
6
=
1.06250u
9
2.06250u
8
+ ··· + 4.43750u + 9.06250
0.187500u
9
0.312500u
8
+ ··· + 1.06250u + 1.31250
a
1
=
1
u
2
a
3
=
u
u
3
+ u
a
9
=
1.06250u
9
+ 2.06250u
8
+ ··· 4.43750u 9.06250
u
2
+ u 1
a
8
=
1
4
u
9
+
1
2
u
8
+ ···
3
2
u
13
4
1
4
u
8
1
2
u
7
+ ··· + u
1
4
a
11
=
1.43750u
9
+ 2.56250u
8
+ ··· 3.81250u 10.5625
0.187500u
9
+ 0.187500u
8
+ ··· + 1.68750u 1.18750
a
7
=
1
2
u
9
u
8
+ ··· +
5
4
u +
13
4
u
9
3
4
u
8
+ ··· u +
1
4
a
7
=
1
2
u
9
u
8
+ ··· +
5
4
u +
13
4
u
9
3
4
u
8
+ ··· u +
1
4
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.93349 1.31744I
a = 0.358064 + 0.505338I
b = 1.85704 0.19680I
9.31336 3.33174I 3.28666 + 2.53508I
u = 0.93349 + 1.31744I
a = 0.358064 0.505338I
b = 1.85704 + 0.19680I
9.31336 + 3.33174I 3.28666 2.53508I
u = 0.92355 1.51424I
a = 0.293576 + 0.481342I
b = 1.85704 + 0.19680I
9.31336 + 3.33174I 3.28666 2.53508I
u = 0.92355 + 1.51424I
a = 0.293576 0.481342I
b = 1.85704 0.19680I
9.31336 3.33174I 3.28666 + 2.53508I
u = 0.132691
a = 7.53631
b = 1.17329
2.52712 3.66494
u = 0.124008 0.699342I
a = 0.245824 + 1.386326I
b = 0.270394 0.413868I
0.17487 2.21397I 2.88087 + 4.04855I
u = 0.124008 + 0.699342I
a = 0.245824 1.386326I
b = 0.270394 + 0.413868I
0.17487 + 2.21397I 2.88087 4.04855I
u = 0.394402 1.113210I
a = 0.282768 + 0.798120I
b = 0.270394 + 0.413868I
0.17487 + 2.21397I 2.88087 4.04855I
u = 0.394402 + 1.113210I
a = 0.282768 0.798120I
b = 0.270394 0.413868I
0.17487 2.21397I 2.88087 + 4.04855I
u = 1.30598
a = 0.765707
b = 1.17329
2.52712 3.66494
7
IV. I
u
4
= hu
11
+ 5u
10
+ · · · + 2u 1, a u 2, u
3
+ 2u
2
+ b + ui
(i) Arc colorings
a
2
=
1
0
a
4
=
0
u
a
5
=
u
u
a
10
=
u + 2
u
3
2u
2
u
a
6
=
u
5
+ 4u
4
+ 4u
3
2u
2
5u
u
7
4u
6
5u
5
+ 4u
3
+ 2u
2
+ u
a
1
=
1
u
2
a
3
=
u
u
3
+ u
a
9
=
2
2u
2
u
a
8
=
u
10
4u
9
5u
8
+ 4u
7
+ 14u
6
+ 6u
5
11u
4
8u
3
+ 3u
2
+ 2u + 1
u
10
+ 4u
9
+ 4u
8
6u
7
13u
6
+ 12u
4
+ 2u
3
5u
2
a
11
=
2u + 1
2u
3
2u
2
a
7
=
2u
8
+ 5u
7
12u
5
6u
4
+ 10u
3
+ 4u
2
4u
2u
10
6u
9
2u
8
+ 13u
7
+ 12u
6
9u
5
10u
4
+ 2u
3
+ 2u
2
+ u
a
7
=
2u
8
+ 5u
7
12u
5
6u
4
+ 10u
3
+ 4u
2
4u
2u
10
6u
9
2u
8
+ 13u
7
+ 12u
6
9u
5
10u
4
+ 2u
3
+ 2u
2
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.40586 1.00997I
a = 0.594138 1.009967I
b = 2.03043 + 0.28871I
7.4453 12.4339I 4.94880 + 5.95992I
u = 1.40586 + 1.00997I
a = 0.594138 + 1.009967I
b = 2.03043 0.28871I
7.4453 + 12.4339I 4.94880 5.95992I
u = 1.31469 0.95832I
a = 0.685307 0.958324I
b = 1.65522 + 0.00776I
7.84139 5.06071I 4.48302 + 2.40182I
u = 1.31469 + 0.95832I
a = 0.685307 + 0.958324I
b = 1.65522 0.00776I
7.84139 + 5.06071I 4.48302 2.40182I
u = 1.263213 0.139301I
a = 0.736787 0.139301I
b = 0.0527895 + 0.0995813I
8.16883 4.71969I 15.8344 + 7.6612I
u = 1.263213 + 0.139301I
a = 0.736787 + 0.139301I
b = 0.0527895 0.0995813I
8.16883 + 4.71969I 15.8344 7.6612I
u = 0.113634 0.293281I
a = 2.11363 0.29328I
b = 0.060423 + 0.412722I
0.003691 + 1.266704I 0.27668 5.30833I
u = 0.113634 + 0.293281I
a = 2.11363 + 0.29328I
b = 0.060423 0.412722I
0.003691 1.266704I 0.27668 + 5.30833I
u = 0.832306 0.202239I
a = 2.83231 0.20224I
b = 2.61041 + 1.28756I
2.76312 + 1.08944I 13.75530 + 1.30535I
u = 0.832306 + 0.202239I
a = 2.83231 + 0.20224I
b = 2.61041 1.28756I
2.76312 1.08944I 13.75530 1.30535I
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.07566
a = 3.07566
b = 4.63430
3.78211 32.5964
9
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
9
(u 1)
5
(u
5
+ u
4
2u
3
u
2
+ u 1)
(u
10
+ 2u
9
+ 5u
8
+ u
7
+ 3u
6
8u
5
2u
4
19u
3
6u
2
8u 1)
(u
11
+ 5u
10
+ ··· + 2u 1)
c
2
u
5
(u
5
u
4
+ ··· + u 1)(u
10
+ u
9
+ ··· + 160u + 32)
(u
11
+ u
10
+ ··· + 2u + 1)
c
3
, c
4
, c
11
(u + 1)
5
(u
5
u
4
2u
3
+ u
2
+ u + 1)
(u
10
+ 2u
9
+ 5u
8
+ u
7
+ 3u
6
8u
5
2u
4
19u
3
6u
2
8u 1)
(u
11
+ 5u
10
+ ··· + 2u 1)
c
5
(u
5
+ u
4
2u
3
u
2
+ u 1)(u
5
+ u
4
u
3
4u
2
3u 1)
(u
10
+ 2u
9
+ ··· 100u 43)(u
11
+ u
10
+ ··· 33u
2
27)
c
6
(u
5
u
4
+ 2u
3
u
2
+ u 1)(u
5
+ u
4
u
3
4u
2
3u 1)
(u
10
10u
8
+ 43u
6
17u
5
35u
4
46u
3
+ 64u
2
+ 38u 29)
(u
11
+ u
10
3u
8
+ 12u
7
10u
6
6u
5
+ 33u
4
31u
3
+ 33u
2
10u + 11)
c
7
, c
10
u
5
(u
5
+ u
4
+ ··· + u + 1)(u
10
+ u
9
+ ··· + 160u + 32)
(u
11
+ u
10
+ ··· + 2u + 1)
c
8
(u
5
+ u
4
u
2
+ u + 1)
2
(u
5
+ 3u
4
+ 4u
3
+ u
2
u 1)
2
(u
11
+ u
10
u
8
+ 8u
7
+ 12u
6
+ 8u
5
3u
4
+ 3u
3
+ 3u
2
1)
10
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
4
c
9
, c
11
(y 1)
5
(y
5
5y
4
+ ··· y 1)(y
10
+ 6y
9
+ ··· 52y + 1)
(y
11
9y
10
+ ··· 10y 1)
c
2
, c
7
, c
10
y
5
(y
5
+ 3y
4
+ ··· y 1)(y
10
21y
9
+ ··· 9728y + 1024)
(y
11
9y
10
+ ··· 2y 1)
c
5
(y
5
5y
4
+ 8y
3
3y
2
y 1)(y
5
3y
4
+ 3y
3
8y
2
+ y 1)
(y
10
+ 20y
9
+ ··· 13440y + 1849)
(y
11
+ 15y
10
+ ··· 1782y 729)
c
6
(y
5
3y
4
+ 3y
3
8y
2
+ y 1)(y
5
+ 3y
4
+ 4y
3
+ y
2
y 1)
(y
10
20y
9
+ ··· 5156y + 841)(y
11
y
10
+ ··· 626y 121)
c
8
(y
5
y
4
+ 4y
3
3y
2
+ 3y 1)
2
(y
5
y
4
+ 8y
3
3y
2
+ 3y 1)
2
(y
11
y
10
+ ··· + 6y 1)
11