11a
176
(K11a
176
)
1
Arc Sequences
6 1 10 9 11 2 3 4 5 8 7
Solving Sequence
2,6
7 1 3 8 11 5 10 9 4
c
6
c
1
c
2
c
7
c
11
c
5
c
10
c
9
c
4
c
3
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu 1i
I
u
2
= hu
54
+ 2u
53
+ ··· + u + 1i
There are 2 irreducible components with 55 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 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
1
a
7
=
1
1
a
1
=
1
1
a
3
=
0
1
a
8
=
1
2
a
11
=
1
1
a
5
=
1
2
a
10
=
0
1
a
9
=
1
1
a
4
=
0
1
a
4
=
0
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 = 1.00000
1.64493 6.00000
3
II. I
u
2
= hu
54
+ 2u
53
+ · · · + u + 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
7
=
u
u
a
1
=
1
u
2
a
3
=
u
2
+ 1
u
4
a
8
=
u
7
+ 2u
5
2u
3
u
9
u
7
+ u
5
+ u
a
11
=
u
4
u
2
+ 1
u
4
a
5
=
u
9
+ 2u
7
3u
5
+ 2u
3
u
u
9
u
7
+ u
5
+ u
a
10
=
u
20
+ 5u
18
13u
16
+ 20u
14
20u
12
+ 13u
10
7u
8
+ 4u
6
u
4
u
2
+ 1
u
22
4u
20
+ 9u
18
12u
16
+ 12u
14
10u
12
+ 9u
10
6u
8
+ 3u
6
2u
4
+ u
2
a
9
=
u
40
9u
38
+ ··· 3u
4
+ 1
u
40
+ 8u
38
+ ··· + 6u
6
2u
4
a
4
=
u
38
+ 9u
36
+ ··· 5u
4
+ 1
u
40
8u
38
+ ··· 6u
6
+ 2u
4
a
4
=
u
38
+ 9u
36
+ ··· 5u
4
+ 1
u
40
8u
38
+ ··· 6u
6
+ 2u
4
(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 = 1.103822 0.601081I
2.1336 14.1348I 5.42152 + 9.37737I
u = 1.103822 + 0.601081I
2.1336 + 14.1348I 5.42152 9.37737I
u = 1.097975 0.388274I
8.77151 + 0.63321I 12.03225 + 0.38631I
u = 1.097975 + 0.388274I
8.77151 0.63321I 12.03225 0.38631I
u = 1.080999 0.596600I
1.40844 6.51845I 3.38857 + 4.08750I
u = 1.080999 + 0.596600I
1.40844 + 6.51845I 3.38857 4.08750I
u = 1.077949 0.450954I
2.75425 5.16303I 6.62576 + 8.31738I
u = 1.077949 + 0.450954I
2.75425 + 5.16303I 6.62576 8.31738I
u = 1.073583 0.116171I
0.11288 + 3.24680I 3.97449 4.31964I
u = 1.073583 + 0.116171I
0.11288 3.24680I 3.97449 + 4.31964I
u = 1.064752 0.232168I
7.31911 1.48659I 11.69621 + 1.05685I
u = 1.064752 + 0.232168I
7.31911 + 1.48659I 11.69621 1.05685I
u = 1.063550 0.603687I
1.70219 6.38060I 2.45012 + 6.07310I
u = 1.063550 + 0.603687I
1.70219 + 6.38060I 2.45012 6.07310I
u = 1.026963 0.612535I
0.833592 + 0.995204I 3.58557 + 0.09789I
u = 1.026963 + 0.612535I
0.833592 0.995204I 3.58557 0.09789I
u = 0.813292 0.457492I
1.20313 1.95407I 2.79385 + 4.45291I
u = 0.813292 + 0.457492I
1.20313 + 1.95407I 2.79385 4.45291I
u = 0.545423 0.733742I
0.59736 6.12710I 1.42529 + 5.10753I
u = 0.545423 + 0.733742I
0.59736 + 6.12710I 1.42529 5.10753I
u = 0.484808 0.741490I
3.41733 + 1.25909I 0.259004 1.112987I
u = 0.484808 + 0.741490I
3.41733 1.25909I 0.259004 + 1.112987I
u = 0.453732 0.747475I
3.26830 + 1.40826I 0.268768 + 0.442670I
u = 0.453732 + 0.747475I
3.26830 1.40826I 0.268768 0.442670I
u = 0.417796 0.781578I
0.09729 + 8.92706I 2.38304 5.21667I
u = 0.417796 + 0.781578I
0.09729 8.92706I 2.38304 + 5.21667I
u = 0.110946 0.513974I
0.243608 + 1.371008I 2.52596 5.06044I
u = 0.110946 + 0.513974I
0.243608 1.371008I 2.52596 + 5.06044I
u = 0.090303 0.613474I
5.56771 4.07219I 6.96248 + 3.87127I
u = 0.090303 + 0.613474I
5.56771 + 4.07219I 6.96248 3.87127I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.386339 0.688925I
3.09355 0.76124I 4.72747 + 0.80623I
u = 0.386339 + 0.688925I
3.09355 + 0.76124I 4.72747 0.80623I
u = 0.431777 0.772015I
5.12841 5.22917I 2.37428 + 4.44764I
u = 0.431777 + 0.772015I
5.12841 + 5.22917I 2.37428 4.44764I
u = 0.522597 0.736334I
5.62286 + 2.45822I 3.43259 3.89075I
u = 0.522597 + 0.736334I
5.62286 2.45822I 3.43259 + 3.89075I
u = 0.632564 0.508807I
2.32233 0.58829I 2.37061 0.50602I
u = 0.632564 + 0.508807I
2.32233 + 0.58829I 2.37061 + 0.50602I
u = 0.889638 0.519400I
3.00640 + 4.83893I 3.57705 6.36658I
u = 0.889638 + 0.519400I
3.00640 4.83893I 3.57705 + 6.36658I
u = 0.966367 0.158023I
1.48527 + 0.15164I 7.75648 0.91671I
u = 0.966367 + 0.158023I
1.48527 0.15164I 7.75648 + 0.91671I
u = 1.042307 0.608486I
4.07878 + 2.66694I 1.14279 1.40015I
u = 1.042307 + 0.608486I
4.07878 2.66694I 1.14279 + 1.40015I
u = 1.062023 0.405530I
3.08098 + 1.83888I 8.29710 0.16550I
u = 1.062023 + 0.405530I
3.08098 1.83888I 8.29710 + 0.16550I
u = 1.089611 0.565989I
5.12761 + 5.61169I 8.17583 5.02124I
u = 1.089611 + 0.565989I
5.12761 5.61169I 8.17583 + 5.02124I
u = 1.095684 0.601552I
3.15875 + 10.41643I 0.77334 8.79469I
u = 1.095684 + 0.601552I
3.15875 10.41643I 0.77334 + 8.79469I
u = 1.101111 0.130732I
5.13575 6.76281I 8.82931 + 4.64431I
u = 1.101111 + 0.130732I
5.13575 + 6.76281I 8.82931 4.64431I
u = 1.105269 0.453209I
8.33472 + 8.06621I 10.75532 7.48177I
u = 1.105269 + 0.453209I
8.33472 8.06621I 10.75532 + 7.48177I
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
(u 1)(u
54
+ 2u
53
+ ··· + u + 1)
c
2
(u + 1)(u
54
+ 24u
53
+ ··· u + 1)
c
3
(u)(u
54
+ 3u
53
+ ··· + 13u + 5)
c
4
(u 1)(u
54
+ 2u
53
+ ··· u + 1)
c
5
, c
7
(u + 1)(u
54
20u
52
+ ··· 23u + 1)
c
8
, c
9
(u 1)(u
54
+ 2u
53
+ ··· u + 1)
c
10
(u + 1)(u
54
+ 12u
53
+ ··· + 297u + 23)
c
11
(u)(u
54
+ 3u
53
+ ··· 11u + 5)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y 1)(y
54
24y
53
+ ··· + y + 1)
c
2
(y 1)(y
54
+ 12y
53
+ ··· 11y + 1)
c
3
(y)(y
54
+ 3y
53
+ ··· 269y + 25)
c
4
(y 1)(y
54
48y
53
+ ··· + y + 1)
c
5
, c
7
(y 1)(y
54
40y
53
+ ··· 207y + 1)
c
8
, c
9
(y 1)(y
54
48y
53
+ ··· + y + 1)
c
10
(y 1)(y
54
+ 8y
53
+ ··· + 11749y + 529)
c
11
(y)(y
54
+ 3y
53
+ ··· + 1139y + 25)
8