11n
45
(K11n
45
)
1
Arc Sequences
4 1 8 2 11 10 3 6 1 8 9
Solving Sequence
1,4
2
5,9
10 11 6 7 8 3
c
1
c
4
c
9
c
11
c
5
c
6
c
8
c
3
c
2
, c
7
, c
10
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hu
6
+ u
5
u
4
2u
3
+ u + 1, b + 1, u
5
+ u
4
u
3
2u
2
+ a + 1i
I
u
2
= hb
6
b
5
b
4
+ 2b
3
b + 1, a 1, u 1i
I
u
3
= hu
12
+ 5u
11
+ 9u
10
21u
8
22u
7
+ 10u
6
+ 26u
5
+ 4u
4
11u
3
3u
2
+ 2u + 1, b + u, a u 2i
I
u
4
= hu
12
+ 3u
11
+ 8u
10
+ 7u
9
+ 8u
8
8u
7
u
6
14u
5
+ 22u
4
+ 9u
3
+ 25u
2
+ 3u + 1,
u
11
+ 3u
10
+ 8u
9
+ 7u
8
+ 8u
7
8u
6
u
5
14u
4
+ 22u
3
+ 9u
2
+ a + 25u + 3,
u
11
2u
10
6u
9
u
8
7u
7
+ 15u
6
14u
5
+ 28u
4
50u
3
+ 41u
2
+ 32b 66u + 31i
There are 4 irreducible components with 36 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
6
+ u
5
u
4
2u
3
+ u + 1, b + 1, u
5
+ u
4
u
3
2u
2
+ a + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
u
u
3
+ u
a
9
=
u
5
u
4
+ u
3
+ 2u
2
1
1
a
10
=
u
5
u
4
+ u
3
+ 2u
2
1
a
11
=
u
5
u
4
+ u
3
+ 2u
2
1
a
6
=
u
5
+ u
4
2u
3
u
2
+ u + 1
a
7
=
u
u
3
+ u
a
8
=
u
4
+ u
2
1
u
5
+ u
4
2u
3
u
2
+ u + 1
a
3
=
u
2
+ 1
u
2
a
3
=
u
2
+ 1
u
2
(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 = 1.073950 0.558752I
a = 0.732786 + 0.381252I
b = 1.00000
1.64493 5.69302I 0.29418 + 2.69056I
u = 1.073950 + 0.558752I
a = 0.732786 0.381252I
b = 1.00000
1.64493 + 5.69302I 0.29418 2.69056I
u = 0.428243 0.664531I
a = 0.685196 + 1.063260I
b = 1.00000
0.245672 + 0.924305I 1.12292 1.33143I
u = 0.428243 + 0.664531I
a = 0.685196 1.063260I
b = 1.00000
0.245672 0.924305I 1.12292 + 1.33143I
u = 1.002193 0.295542I
a = 0.917982 + 0.270708I
b = 1.00000
3.53554 + 0.92430I 6.82874 7.13914I
u = 1.002193 + 0.295542I
a = 0.917982 0.270708I
b = 1.00000
3.53554 0.92430I 6.82874 + 7.13914I
3
II. I
u
2
= hb
6
b
5
b
4
+ 2b
3
b + 1, a 1, u 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
1
0
a
9
=
1
b
a
10
=
b + 1
b
a
11
=
b + 1
b
2
a
6
=
b
3
+ b
2
1
b
4
a
7
=
0
b
5
b
4
2b
3
+ b
2
+ b 1
a
8
=
0
b
5
b
4
2b
3
+ b
2
+ b 1
a
3
=
0
1
a
3
=
0
1
(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 = 1.00000
a = 1.00000
b = 1.002193 0.295542I
3.53554 0.92430I 6.82874 + 7.13914I
u = 1.00000
a = 1.00000
b = 1.002193 + 0.295542I
3.53554 + 0.92430I 6.82874 7.13914I
u = 1.00000
a = 1.00000
b = 0.428243 0.664531I
0.245672 0.924305I 1.12292 + 1.33143I
u = 1.00000
a = 1.00000
b = 0.428243 + 0.664531I
0.245672 + 0.924305I 1.12292 1.33143I
u = 1.00000
a = 1.00000
b = 1.073950 0.558752I
1.64493 + 5.69302I 0.29418 2.69056I
u = 1.00000
a = 1.00000
b = 1.073950 + 0.558752I
1.64493 5.69302I 0.29418 + 2.69056I
5
III. I
u
3
= hu
12
+ 5u
11
+ · · · + 2u + 1, b + u, a u 2i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
u
u
3
+ u
a
9
=
u + 2
u
a
10
=
2u + 2
u
a
11
=
u
2
+ 2u + 1
u
2
a
6
=
u
7
+ 4u
6
+ 6u
5
+ 2u
4
4u
3
4u
2
2u
u
7
2u
6
u
5
+ 2u
4
+ u
a
7
=
2u
9
6u
8
3u
7
+ 12u
6
+ 14u
5
6u
4
12u
3
+ 2u
u
9
+ 2u
8
u
7
6u
6
u
5
+ 6u
4
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
11
5u
10
+ ··· 2u 1
a
3
=
u
2
+ 1
u
2
a
3
=
u
2
+ 1
u
2
(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 = 1.37328 1.07803I
a = 0.626724 1.078028I
b = 1.37328 + 1.07803I
12.4026 12.7511I 0.69002 + 5.94531I
u = 1.37328 + 1.07803I
a = 0.626724 + 1.078028I
b = 1.37328 1.07803I
12.4026 + 12.7511I 0.69002 5.94531I
u = 1.29679 1.06566I
a = 0.703205 1.065657I
b = 1.29679 + 1.06566I
12.72394 5.46645I 0.22295 + 2.11548I
u = 1.29679 + 1.06566I
a = 0.703205 + 1.065657I
b = 1.29679 1.06566I
12.72394 + 5.46645I 0.22295 2.11548I
u = 0.997809 0.382742I
a = 1.002191 0.382742I
b = 0.997809 + 0.382742I
1.70690 6.65526I 0.69156 + 12.28500I
u = 0.997809 + 0.382742I
a = 1.002191 + 0.382742I
b = 0.997809 0.382742I
1.70690 + 6.65526I 0.69156 12.28500I
u = 0.417930 0.278210I
a = 1.58207 0.27821I
b = 0.417930 + 0.278210I
1.46216 + 0.16286I 7.96188 + 1.03516I
u = 0.417930 + 0.278210I
a = 1.58207 + 0.27821I
b = 0.417930 0.278210I
1.46216 0.16286I 7.96188 1.03516I
u = 0.568808 0.252332I
a = 2.56881 0.25233I
b = 0.568808 + 0.252332I
1.61529 + 1.35793I 3.64822 4.51645I
u = 0.568808 + 0.252332I
a = 2.56881 + 0.25233I
b = 0.568808 0.252332I
1.61529 1.35793I 3.64822 + 4.51645I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.017003 0.101771I
a = 3.01700 0.10177I
b = 1.017003 + 0.101771I
3.52730 + 0.57280I 2.7091 + 26.6989I
u = 1.017003 + 0.101771I
a = 3.01700 + 0.10177I
b = 1.017003 0.101771I
3.52730 0.57280I 2.7091 26.6989I
8
IV.
I
u
4
= hu
12
+3u
11
+· · ·+3u+1, u
11
+3u
10
+· · ·+a+3, u
11
2u
10
+· · ·+32b+31i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
u
u
3
+ u
a
9
=
u
11
3u
10
+ ··· 25u 3
0.0312500u
11
+ 0.0625000u
10
+ ··· + 2.06250u 0.968750
a
10
=
1.03125u
11
3.06250u
10
+ ··· 27.0625u 2.03125
0.0312500u
11
+ 0.0625000u
10
+ ··· + 2.06250u 0.968750
a
11
=
0.968750u
11
2.93750u
10
+ ··· 22.9375u 3.96875
7
32
u
11
+
1
2
u
10
+ ··· +
9
2
u
23
32
a
6
=
0.531250u
11
+ 1.75000u
10
+ ··· + 13.7500u + 5.09375
0.156250u
11
0.562500u
10
+ ··· 5.31250u 0.0312500
a
7
=
1
4
u
10
+
3
2
u
9
+ ··· +
35
4
u + 2
0.218750u
11
+ 0.437500u
10
+ ··· 1.81250u + 0.218750
a
8
=
1
4
u
11
3
4
u
10
+ ···
29
4
u
9
4
0.0312500u
11
+ 0.0625000u
10
+ ··· + 1.81250u 0.218750
a
3
=
u
2
+ 1
u
2
a
3
=
u
2
+ 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
9
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.07857 1.47659I
a = 0.322576 + 0.441612I
b = 1.07187 1.35065I
13.70954 + 3.42721I 0.48765 2.36550I
u = 1.07857 + 1.47659I
a = 0.322576 0.441612I
b = 1.07187 + 1.35065I
13.70954 3.42721I 0.48765 + 2.36550I
u = 1.07187 1.35065I
a = 0.360515 + 0.454280I
b = 1.07857 1.47659I
13.70954 3.42721I 0.48765 + 2.36550I
u = 1.07187 + 1.35065I
a = 0.360515 0.454280I
b = 1.07857 + 1.47659I
13.70954 + 3.42721I 0.48765 2.36550I
u = 0.282006 0.991713I
a = 0.265287 + 0.932918I
b = 0.042043 1.323158I
2.99789 2.65597I 1.54637 + 3.55162I
u = 0.282006 + 0.991713I
a = 0.265287 0.932918I
b = 0.042043 + 1.323158I
2.99789 + 2.65597I 1.54637 3.55162I
u = 0.058341 0.199318I
a = 1.35265 + 4.62119I
b = 1.032836 0.430283I
1.90302 1.10871I 2.03402 + 2.13465I
u = 0.058341 + 0.199318I
a = 1.35265 4.62119I
b = 1.032836 + 0.430283I
1.90302 + 1.10871I 2.03402 2.13465I
u = 0.042043 1.323158I
a = 0.023990 + 0.755006I
b = 0.282006 0.991713I
2.99789 + 2.65597I 1.54637 3.55162I
u = 0.042043 + 1.323158I
a = 0.023990 0.755006I
b = 0.282006 + 0.991713I
2.99789 2.65597I 1.54637 + 3.55162I
10
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.032836 0.430283I
a = 0.825019 + 0.343706I
b = 0.058341 0.199318I
1.90302 + 1.10871I 2.03402 2.13465I
u = 1.032836 + 0.430283I
a = 0.825019 0.343706I
b = 0.058341 + 0.199318I
1.90302 1.10871I 2.03402 + 2.13465I
11
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
6
(u
6
+ u
5
+ ··· + u + 1)(u
12
+ 3u
11
+ ··· + 3u + 1)
(u
12
+ 5u
11
+ ··· + 2u + 1)
c
2
(u + 1)
6
(u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)
(u
12
7u
11
+ ··· 41u + 1)(u
12
+ 7u
11
+ ··· + 10u + 1)
c
3
u
6
(u
6
u
5
+ ··· u + 1)(u
12
+ u
11
+ ··· + 320u + 64)
(u
12
+ u
11
+ ··· + 2u + 1)
c
4
(u 1)
6
(u
6
u
5
+ ··· u + 1)(u
12
+ 3u
11
+ ··· + 3u + 1)
(u
12
+ 5u
11
+ ··· + 2u + 1)
c
5
(u
6
+ u
5
+ ··· + 3u + 1)(u
6
+ 3u
5
+ ··· + u + 1)
(u
12
+ 2u
11
+ ··· 144u + 121)(u
12
+ 3u
11
+ ··· + 14u + 4)
c
6
(u
6
+ u
5
u
4
2u
3
+ u + 1)(u
6
+ u
5
+ 2u
4
+ 4u
3
+ 5u
2
+ 3u + 1)
(u
12
14u
10
+ ··· 120u + 77)(u
12
+ u
11
+ ··· 44u + 23)
c
7
, c
10
u
6
(u
6
+ u
5
+ ··· + u + 1)(u
12
+ u
11
+ ··· + 320u + 64)
(u
12
+ u
11
+ ··· + 2u + 1)
c
8
(u
6
3u
5
+ 5u
4
4u
3
+ 2u
2
u + 1)(u
6
+ u
5
+ u
4
+ 2u
2
+ u + 1)
2
(u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)
(u
12
+ u
11
+ u
10
+ 5u
8
4u
6
8u
5
+ 6u
4
+ 3u
3
+ 3u
2
+ 1)
c
9
(u 1)
6
(u
6
+ u
5
+ ··· + u + 1)(u
12
+ 3u
11
+ ··· + 3u + 1)
(u
12
+ 5u
11
+ ··· + 2u + 1)
c
11
(u + 1)
6
(u
6
+ u
5
+ ··· + u + 1)(u
12
+ 3u
11
+ ··· + 3u + 1)
(u
12
+ 5u
11
+ ··· + 2u + 1)
12
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
, c
9
c
11
(y 1)
6
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
(y
12
7y
11
+ ··· 10y + 1)(y
12
+ 7y
11
+ ··· + 41y + 1)
c
2
(y 1)
6
(y
6
+ y
5
+ ··· + 3y + 1)(y
12
+ 27y
11
+ ··· 451y + 1)
(y
12
+ 29y
11
+ ··· + 22y + 1)
c
3
, c
7
, c
10
y
6
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
(y
12
27y
11
+ ··· 12288y + 4096)(y
12
15y
11
+ ··· 2y + 1)
c
5
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)(y
6
+ 3y
5
+ 6y
4
+ 5y
2
+ y + 1)
(y
12
+ 5y
11
+ ··· + 68y + 16)(y
12
+ 24y
11
+ ··· + 28148y + 14641)
c
6
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)(y
6
+ 3y
5
+ 6y
4
+ 5y
2
+ y + 1)
(y
12
28y
11
+ ··· + 53360y + 5929)
(y
12
23y
11
+ ··· 4098y + 529)
c
8
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
(y
6
+ y
5
+ 5y
4
+ 4y
3
+ 6y
2
+ 3y + 1)
2
(y
12
+ y
11
+ ··· + 6y + 1)
13