11a
175
(K11a
175
)
1
Arc Sequences
6 1 8 11 10 2 3 4 5 9 7
Solving Sequence
6,10
5 9 11 4 8 3 7 1 2
c
5
c
9
c
10
c
4
c
8
c
3
c
7
c
11
c
2
c
1
, c
6
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu + 1i
I
u
2
= hu
11
u
10
2u
9
+ 2u
8
+ 3u
7
2u
6
2u
5
+ 2u
3
u + 1i
I
u
3
= hu
40
u
39
+ ··· 3u
3
+ 1i
There are 3 irreducible components with 52 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
6
=
1
0
a
10
=
0
1
a
5
=
1
1
a
9
=
1
0
a
11
=
1
1
a
4
=
1
1
a
8
=
0
1
a
3
=
1
0
a
7
=
1
1
a
1
=
1
1
a
2
=
2
1
a
2
=
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 = 1.00000
1.64493 6.00000
3
II. I
u
2
= hu
11
u
10
2u
9
+ 2u
8
+ 3u
7
2u
6
2u
5
+ 2u
3
u + 1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
5
=
1
u
2
a
9
=
u
u
3
+ u
a
11
=
u
3
u
5
u
3
+ u
a
4
=
u
6
u
4
+ 1
u
8
+ 2u
6
2u
4
a
8
=
u
10
+ 2u
8
+ u
7
2u
6
2u
5
+ 3u
3
u + 1
u
8
+ 2u
6
2u
4
a
3
=
u
10
u
8
u
7
+ u
6
+ 2u
5
+ u
4
2u
3
+ u
u
3
+ u
a
7
=
u
10
+ 3u
8
+ u
7
5u
6
2u
5
+ 3u
4
+ 3u
3
u + 1
u
2
a
1
=
2u
10
u
9
4u
8
+ u
7
+ 5u
6
+ u
5
2u
4
3u
3
+ u
2
+ u 1
u
a
2
=
2u
10
u
9
4u
8
+ u
7
+ 5u
6
+ u
5
2u
4
3u
3
+ u
2
1
u
a
2
=
2u
10
u
9
4u
8
+ u
7
+ 5u
6
+ u
5
2u
4
3u
3
+ u
2
1
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 = 1.081803 0.517146I
2.76698 9.75515I 4.05162 + 10.29185I
u = 1.081803 + 0.517146I
2.76698 + 9.75515I 4.05162 10.29185I
u = 0.912079
1.65611 5.73715
u = 0.472789 0.800775I
9.12060 + 3.24476I 5.98156 0.51441I
u = 0.472789 + 0.800775I
9.12060 3.24476I 5.98156 + 0.51441I
u = 0.361975 0.559972I
1.36102 0.98826I 4.35867 + 1.84291I
u = 0.361975 + 0.559972I
1.36102 + 0.98826I 4.35867 1.84291I
u = 1.054491 0.371149I
4.87523 + 4.09967I 8.95070 5.15592I
u = 1.054491 + 0.371149I
4.87523 4.09967I 8.95070 + 5.15592I
u = 1.094166 0.624458I
5.3908 + 13.9605I 0.53068 9.48051I
u = 1.094166 + 0.624458I
5.3908 13.9605I 0.53068 + 9.48051I
5
III. I
u
3
= hu
40
u
39
+ · · · 3u
3
+ 1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
5
=
1
u
2
a
9
=
u
u
3
+ u
a
11
=
u
3
u
5
u
3
+ u
a
4
=
u
6
u
4
+ 1
u
8
+ 2u
6
2u
4
a
8
=
u
11
+ 2u
9
2u
7
+ u
3
u
13
3u
11
+ 5u
9
4u
7
+ 2u
5
u
3
+ u
a
3
=
u
16
+ 3u
14
5u
12
+ 4u
10
u
8
+ 1
u
18
4u
16
+ 9u
14
12u
12
+ 11u
10
8u
8
+ 6u
6
4u
4
+ u
2
a
7
=
u
21
4u
19
+ 9u
17
12u
15
+ 10u
13
6u
11
+ 3u
9
2u
7
u
5
+ 2u
3
u
u
23
+ 5u
21
+ ··· 2u
3
+ u
a
1
=
u
39
+ 8u
37
+ ··· + 15u
7
6u
5
u
38
+ 9u
36
+ ··· + u
3
1
a
2
=
u
39
+ u
38
+ ··· u
3
+ 1
u
38
+ 9u
36
+ ··· + u
3
1
a
2
=
u
39
+ u
38
+ ··· u
3
+ 1
u
38
+ 9u
36
+ ··· + u
3
1
(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 = 1.119575 0.049168I
1.78732 + 6.69475I 2.60998 4.97701I
u = 1.119575 + 0.049168I
1.78732 6.69475I 2.60998 + 4.97701I
u = 1.087959 0.626575I
7.28190 8.60190I 3.29856 + 5.07396I
u = 1.087959 + 0.626575I
7.28190 + 8.60190I 3.29856 5.07396I
u = 1.071009 0.632590I
7.58837 5.35722I 3.80298 + 4.77693I
u = 1.071009 + 0.632590I
7.58837 + 5.35722I 3.80298 4.77693I
u = 1.065390 0.469454I
4.22715 2.78049I 7.53200 + 3.56896I
u = 1.065390 + 0.469454I
4.22715 + 2.78049I 7.53200 3.56896I
u = 0.989179 0.332673I
1.87696 1.08776I 3.66948 + 0.80831I
u = 0.989179 + 0.332673I
1.87696 + 1.08776I 3.66948 0.80831I
u = 0.912778 0.528712I
0.112919 1.81750
u = 0.912778 + 0.528712I
0.112919 1.81750
u = 0.674204 0.548152I
0.79488 4.38017I 2.87668 + 6.69250I
u = 0.674204 + 0.548152I
0.79488 + 4.38017I 2.87668 6.69250I
u = 0.502219 0.792060I
9.28815 6.23474
u = 0.502219 + 0.792060I
9.28815 6.23474
u = 0.289073 0.622325I
0.55874 + 5.32051I 0.06135 6.50240I
u = 0.289073 + 0.622325I
0.55874 5.32051I 0.06135 + 6.50240I
u = 0.138437 0.513103I
1.87696 1.08776I 3.66948 + 0.80831I
u = 0.138437 + 0.513103I
1.87696 + 1.08776I 3.66948 0.80831I
u = 0.461488 0.804643I
7.28190 8.60190I 3.29856 + 5.07396I
u = 0.461488 + 0.804643I
7.28190 + 8.60190I 3.29856 5.07396I
u = 0.475874 0.769365I
3.57846 1.46542I 0.189647 + 0.302471I
u = 0.475874 + 0.769365I
3.57846 + 1.46542I 0.189647 0.302471I
u = 0.515254 0.788495I
7.58837 + 5.35722I 3.80298 4.77693I
u = 0.515254 + 0.788495I
7.58837 5.35722I 3.80298 + 4.77693I
u = 0.565990 0.536897I
1.96889 5.82360
u = 0.565990 + 0.536897I
1.96889 5.82360
u = 0.976421 0.536361I
0.79488 + 4.38017I 2.87668 6.69250I
u = 0.976421 + 0.536361I
0.79488 4.38017I 2.87668 + 6.69250I
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.047754 0.294823I
4.22715 2.78049I 7.53200 + 3.56896I
u = 1.047754 + 0.294823I
4.22715 + 2.78049I 7.53200 3.56896I
u = 1.053774 0.517468I
0.55874 + 5.32051I 0.06135 6.50240I
u = 1.053774 + 0.517468I
0.55874 5.32051I 0.06135 + 6.50240I
u = 1.062888 0.635226I
5.95204 1.53406
u = 1.062888 + 0.635226I
5.95204 1.53406
u = 1.077536 0.613425I
1.78732 + 6.69475I 2.60998 4.97701I
u = 1.077536 + 0.613425I
1.78732 6.69475I 2.60998 + 4.97701I
u = 1.112843 0.027837I
3.57846 1.46542I 0.189647 + 0.302471I
u = 1.112843 + 0.027837I
3.57846 + 1.46542I 0.189647 0.302471I
7
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
, c
6
c
9
(u 1)(u
11
+ u
10
2u
9
2u
8
+ 3u
7
+ 2u
6
2u
5
+ 2u
3
u 1)
(u
40
+ u
39
+ ··· + 3u
3
+ 1)
c
2
, c
10
(u + 1)(u
11
+ 5u
10
+ ··· + u + 1)(u
40
+ 17u
39
+ ··· + 2u
2
+ 1)
c
3
, c
7
, c
8
(u + 1)(u
11
+ 4u
10
+ ··· 5u
2
+ 4)
(1 2u 13u
2
4u
3
+ 35u
4
+ 38u
5
16u
6
78u
7
+ 15u
8
+ 90u
9
47u
10
114u
11
+ 72u
12
+ 114u
13
64u
14
64u
15
+ 33u
16
+ 18u
17
9u
18
2u
19
+ u
20
)
2
c
4
, c
11
u(u
11
+ u
9
2u
8
+ 7u
7
u
6
+ 4u
5
+ 3u
4
+ 12u
3
+ 8u
2
+ 5u 1)
(u
40
+ 3u
39
+ ··· 6u + 3)
8
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
, c
6
c
9
(y 1)(y
11
5y
10
+ ··· + y 1)(y
40
17y
39
+ ··· + 2y
2
+ 1)
c
2
(y 1)(y
11
+ 3y
10
+ ··· 7y 1)(y
40
+ 11y
39
+ ··· + 4y + 1)
c
3
, c
7
, c
8
(y 1)(y
11
10y
10
+ ··· + 40y 16)
(1 30y + 223y
2
806y
3
+ 1663y
4
3312y
5
+ 8864y
6
1.92 × 10
4
y
7
+ 3.19 × 10
4
y
8
4.41 × 10
4
y
9
+ 5.23 × 10
4
y
10
5.53 × 10
4
y
11
+ 5.30 × 10
4
y
12
4.38 × 10
4
y
13
+ 2.88 × 10
4
y
14
1.43 × 10
4
y
15
+ 5145y
16
1302y
17
+ 219y
18
22y
19
+ y
20
)
2
c
4
, c
11
(y)(y
11
+ 2y
10
+ ··· + 41y 1)(y
40
5y
39
+ ··· 282y + 9)
c
10
(y 1)(y
11
+ 3y
10
+ ··· 7y 1)(y
40
+ 11y
39
+ ··· + 4y + 1)
9