11n
87
(K11n
87
)
1
Arc Sequences
6 1 10 8 2 3 4 1 5 7 8
Solving Sequence
2,5
6 1
3,8
9 4 7 11 10
c
5
c
1
c
2
c
8
c
4
c
7
c
11
c
10
c
3
, c
6
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
\
I
v
1
I
u
1
= hu
4
+ 2u
2
+ 2, u
3
+ 2a 2u, u
3
+ u
2
+ b u + 1i
I
u
2
= ha
16
3a
15
+ 5a
14
6a
13
+ 4a
12
3a
11
6a
10
+ a
9
2a
8
11a
7
+ a
6
8a
5
+ 6a
4
8a
3
a
2
+ 4a 1,
118830a
15
+ 38487u + ··· 400157a + 270811,
308180a
15
+ 115461b + ··· + 1078224a 840233i
I
u
3
= hu
16
3u
15
+ ··· 6u + 2,
u
14
2u
13
+ 5u
12
7u
11
+ 10u
10
12u
9
+ 12u
8
11u
7
+ 9u
6
6u
5
+ 5u
4
3u
3
+ 3u
2
+ b 3u + 1,
u
15
+ 3u
14
+ ··· + 2a + 2i
I
v
1
= hv + 1, b 1, ai
There are 4 irreducible components with 37 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
+ 2u
2
+ 2, u
3
+ 2a 2u, u
3
+ u
2
+ b u + 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
2
1
2u
2
2
a
8
=
1
2
u
3
+ u
u
3
u
2
+ u 1
a
9
=
1
2
u
3
+ u
2
+ u + 1
u
3
+ u 1
a
4
=
1
2
u
3
+ u
u
3
u
2
+ 2u 1
a
7
=
0
u
a
11
=
1
2
u
3
+ u
2
+ u + 1
u
3
+ u 1
a
10
=
1
2
u
3
+ u
2
+ u + 1
u
3
+ u
2
+ 2u + 1
a
10
=
1
2
u
3
+ u
2
+ u + 1
u
3
+ u
2
+ 2u + 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 = 0.455090 1.098684I
a = 0.321797 0.776887I
b = 1.09868 1.45509I
0.82247 + 3.66386I 8.00000 4.00000I
u = 0.455090 + 1.098684I
a = 0.321797 + 0.776887I
b = 1.09868 + 1.45509I
0.82247 3.66386I 8.00000 + 4.00000I
u = 0.455090 1.098684I
a = 0.321797 0.776887I
b = 1.098684 + 0.544910I
0.82247 3.66386I 8.00000 + 4.00000I
u = 0.455090 + 1.098684I
a = 0.321797 + 0.776887I
b = 1.098684 0.544910I
0.82247 + 3.66386I 8.00000 4.00000I
3
II. I
u
2
= ha
16
3a
15
+ · · · + 4a 1, 118830a
15
+ 38487u + · · · 400157a +
270811, 1.15 × 10
5
b 3.08 × 10
5
a
15
+ · · · + 1.08 × 10
6
a 8.40 × 10
5
i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
3.08754a
15
+ 7.69158a
14
+ ··· + 10.3972a 7.03643
a
6
=
3.08754a
15
+ 7.69158a
14
+ ··· + 10.3972a 7.03643
3.08754a
15
+ 7.69158a
14
+ ··· + 10.3972a 7.03643
a
1
=
2.15121a
15
+ 6.00333a
14
+ ··· + 8.01260a 6.09977
2.15121a
15
+ 6.00333a
14
+ ··· + 8.01260a 7.09977
a
3
=
2.53984a
15
7.07201a
14
+ ··· 11.4886a + 7.80982
4.69105a
15
13.0753a
14
+ ··· 19.5012a + 13.9096
a
8
=
a
2.66913a
15
6.91429a
14
+ ··· 9.33843a + 7.27720
a
9
=
2.48913a
15
+ 6.31317a
14
+ ··· + 8.20519a 6.89545
0.727501a
15
2.26520a
14
+ ··· 5.48276a + 1.92159
a
4
=
0.854574a
15
2.42887a
14
+ ··· 3.19656a + 1.57102
1.63455a
15
+ 3.88430a
14
+ ··· + 5.00863a 5.32442
a
7
=
0.359489a
15
0.562770a
14
+ ··· + 0.457583a + 0.248179
3.10164a
15
7.30795a
14
+ ··· 9.44519a + 6.44568
a
11
=
2.12964a
15
+ 5.75040a
14
+ ··· + 8.66277a 6.64727
1.46709a
15
3.42365a
14
+ ··· 4.43656a + 1.12126
a
10
=
2.48913a
15
+ 6.31317a
14
+ ··· + 8.20519a 6.89545
1.63455a
15
+ 3.88430a
14
+ ··· + 5.00863a 5.32442
a
10
=
2.48913a
15
+ 6.31317a
14
+ ··· + 8.20519a 6.89545
1.63455a
15
+ 3.88430a
14
+ ··· + 5.00863a 5.32442
(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 = 0.473514 + 1.273022I
a = 0.982629 0.371709I
b = 2.67280 0.68277I
10.78258 4.93524I 1.01557 + 2.99422I
u = 0.473514 1.273022I
a = 0.982629 + 0.371709I
b = 2.67280 + 0.68277I
10.78258 + 4.93524I 1.01557 2.99422I
u = 0.578577
a = 0.617583
b = 0.801052
1.93558 4.99681
u = 0.394459 1.112499I
a = 0.465116 0.844545I
b = 1.33580 0.60094I
1.05533 3.63283I 1.57760 + 4.51802I
u = 0.394459 + 1.112499I
a = 0.465116 + 0.844545I
b = 1.33580 + 0.60094I
1.05533 + 3.63283I 1.57760 4.51802I
u = 0.252896 + 0.819281I
a = 0.245877 1.350008I
b = 0.050903 + 0.938748I
2.79859 1.27532I 6.81947 + 5.08518I
u = 0.252896 0.819281I
a = 0.245877 + 1.350008I
b = 0.050903 0.938748I
2.79859 + 1.27532I 6.81947 5.08518I
u = 0.394459 1.112499I
a = 0.383061 0.033126I
b = 0.724062 + 0.960570I
1.05533 3.63283I 1.57760 + 4.51802I
u = 0.394459 + 1.112499I
a = 0.383061 + 0.033126I
b = 0.724062 0.960570I
1.05533 + 3.63283I 1.57760 4.51802I
u = 0.473514 + 1.273022I
a = 0.586259 0.833880I
b = 1.68418 1.18701I
10.78258 4.93524I 1.01557 + 2.99422I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.473514 1.273022I
a = 0.586259 + 0.833880I
b = 1.68418 + 1.18701I
10.78258 + 4.93524I 1.01557 2.99422I
u = 0.252896 0.819281I
a = 0.638611 0.935621I
b = 2.26358 1.10055I
2.79859 + 1.27532I 6.81947 5.08518I
u = 0.252896 + 0.819281I
a = 0.638611 + 0.935621I
b = 2.26358 + 1.10055I
2.79859 1.27532I 6.81947 + 5.08518I
u = 0.914675
a = 0.91831 1.15526I
b = 0.194501 + 0.440277I
6.88602 4.17790
u = 0.914675
a = 0.91831 + 1.15526I
b = 0.194501 0.440277I
6.88602 4.17790
u = 0.578577
a = 1.95233
b = 0.585296
1.93558 4.99681
6
III.
I
u
3
= hu
16
3u
15
+· · ·6u+2, u
14
2u
13
+· · ·+b+1, u
15
+3u
14
+· · ·+2a+2i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
+ u
2
+ 1
u
4
a
8
=
1
2
u
15
3
2
u
14
+ ··· + 3u 1
u
14
+ 2u
13
+ ··· + 3u 1
a
9
=
1
2
u
15
+
3
2
u
14
+ ··· 3u + 2
u
13
+ u
12
+ ··· u + 1
a
4
=
1
2
u
15
1
2
u
14
+ ··· + u 1
u
14
2u
13
+ ··· 2u + 1
a
7
=
u
7
2u
5
2u
3
u
7
u
5
+ u
a
11
=
1
2
u
15
3
2
u
14
+ ··· 3u
2
+ 3u
u
13
u
12
+ ··· + u 1
a
10
=
1
2
u
15
+
3
2
u
14
+ ··· 3u + 2
u
14
2u
13
+ ··· 2u + 1
a
10
=
1
2
u
15
+
3
2
u
14
+ ··· 3u + 2
u
14
2u
13
+ ··· 2u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.641580 0.478671I
a = 0.261902 + 1.077326I
b = 0.629700 0.297007I
0.96609 2.28706I 6.88422 + 4.18311I
u = 0.641580 + 0.478671I
a = 0.261902 1.077326I
b = 0.629700 + 0.297007I
0.96609 + 2.28706I 6.88422 4.18311I
u = 0.569839 0.991415I
a = 0.847460 0.348051I
b = 1.85072 + 0.46336I
0.48607 + 7.00413I 4.93065 8.89860I
u = 0.569839 + 0.991415I
a = 0.847460 + 0.348051I
b = 1.85072 0.46336I
0.48607 7.00413I 4.93065 + 8.89860I
u = 0.142689 1.132381I
a = 0.703529 0.463706I
b = 1.41732 0.31663I
3.75188 0.61754I 1.35608 + 1.57553I
u = 0.142689 + 1.132381I
a = 0.703529 + 0.463706I
b = 1.41732 + 0.31663I
3.75188 + 0.61754I 1.35608 1.57553I
u = 0.409686 1.284695I
a = 0.460627 + 0.987088I
b = 1.60185 + 1.06559I
10.32592 + 2.59855I 1.50083 1.34763I
u = 0.409686 + 1.284695I
a = 0.460627 0.987088I
b = 1.60185 1.06559I
10.32592 2.59855I 1.50083 + 1.34763I
u = 0.482015 1.060217I
a = 0.113993 0.507268I
b = 0.700496 0.004789I
0.74617 3.29967I 2.58175 + 1.95258I
u = 0.482015 + 1.060217I
a = 0.113993 + 0.507268I
b = 0.700496 + 0.004789I
0.74617 + 3.29967I 2.58175 1.95258I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.522071 1.247144I
a = 0.951591 + 0.542924I
b = 2.96708 + 0.68151I
9.4923 12.3434I 2.79056 + 7.18778I
u = 0.522071 + 1.247144I
a = 0.951591 0.542924I
b = 2.96708 0.68151I
9.4923 + 12.3434I 2.79056 7.18778I
u = 0.531551 0.451405I
a = 0.835102 + 0.156457I
b = 0.484866 0.424635I
1.066483 0.823485I 7.17691 + 4.58909I
u = 0.531551 + 0.451405I
a = 0.835102 0.156457I
b = 0.484866 + 0.424635I
1.066483 + 0.823485I 7.17691 4.58909I
u = 0.908785 0.099623I
a = 1.09410 1.12213I
b = 0.453304 + 0.669187I
6.01654 + 7.18776I 5.49115 4.28840I
u = 0.908785 + 0.099623I
a = 1.09410 + 1.12213I
b = 0.453304 0.669187I
6.01654 7.18776I 5.49115 + 4.28840I
9
IV. I
v
1
= hv + 1, b 1, ai
(i) Arc colorings
a
2
=
1
0
a
5
=
1
0
a
6
=
1
0
a
1
=
1
0
a
3
=
1
0
a
8
=
0
1
a
9
=
1
1
a
4
=
1
1
a
7
=
1
0
a
11
=
1
1
a
10
=
2
1
a
10
=
2
1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
10
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
3.28987 12.0000
11
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
u(u
4
+ 2u
2
+ 2)(u
8
u
7
+ 3u
6
2u
5
+ 3u
4
2u
3
1)
2
(u
16
+ 3u
15
+ ··· + 6u + 2)
c
2
u(u
2
+ 2u + 2)
2
(u
8
+ 5u
7
+ 11u
6
+ 10u
5
u
4
10u
3
6u
2
+ 1)
2
(u
16
+ 9u
15
+ ··· + 4u + 4)
c
3
, c
7
(u 1)(u + 1)
4
(u
16
+ u
15
+ ··· + 2u 1)(u
16
+ u
15
+ ··· 4u
2
+ 1)
c
4
(u 1)
4
(u + 1)(u
16
+ u
15
+ ··· + 2u 1)(u
16
+ u
15
+ ··· 4u
2
+ 1)
c
6
u(u
4
2u
2
+ 2)(u
8
u
7
5u
6
+ 4u
5
+ 7u
4
4u
3
2u
2
+ 2u 1)
2
(u
16
+ 3u
15
+ ··· 22u + 10)
c
8
(u + 1)
5
(u
16
+ 3u
15
+ ··· + 8u + 1)(u
16
+ 5u
15
+ ··· + 4u + 1)
c
9
(u 1)(u
4
+ 4u
3
+ 4u
2
+ 1)(u
16
+ u
15
+ ··· + 376u + 419)
(u
16
+ u
15
+ ··· + 2u
2
+ 1)
c
10
(u 1)(u + 1)
16
(u
4
4u
3
+ 4u
2
+ 1)(u
16
14u
15
+ ··· 1024u + 256)
c
11
(u 1)
5
(u
16
+ 3u
15
+ ··· + 8u + 1)(u
16
+ 5u
15
+ ··· + 4u + 1)
12
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
y(y
2
+ 2y + 2)
2
(y
8
+ 5y
7
+ 11y
6
+ 10y
5
y
4
10y
3
6y
2
+ 1)
2
(y
16
+ 9y
15
+ ··· + 4y + 4)
c
2
y(y
2
+ 4)
2
(y
8
3y
7
+ 19y
6
34y
5
+ 71y
4
66y
3
+ 34y
2
12y + 1)
2
(y
16
3y
15
+ ··· + 144y + 16)
c
3
, c
4
, c
7
(y 1)
5
(y
16
5y
15
+ ··· 4y + 1)(y
16
3y
15
+ ··· 8y + 1)
c
6
y(y
2
2y + 2)
2
(y
8
11y
7
+ 47y
6
98y
5
+ 103y
4
50y
3
+ 6y
2
+ 1)
2
(y
16
15y
15
+ ··· 364y + 100)
c
8
, c
11
(y 1)
5
(y
16
+ 11y
15
+ ··· 88y + 1)(y
16
+ 29y
15
+ ··· + 4y + 1)
c
9
(y 1)(y
4
8y
3
+ 18y
2
+ 8y + 1)
(y
16
+ 15y
15
+ ··· 846972y + 175561)(y
16
+ 33y
15
+ ··· + 4y + 1)
c
10
(y 1)
17
(y
4
8y
3
+ 18y
2
+ 8y + 1)
(y
16
12y
15
+ ··· + 65536y + 65536)
13