11a
306
(K11a
306
)
1
Arc Sequences
8 7 1 11 10 2 3 4 5 6 9
Solving Sequence
3,7
8 2 1 4 9 6 11 10 5
c
7
c
2
c
1
c
3
c
8
c
6
c
11
c
10
c
5
c
4
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu + 1i
I
u
2
= hu
9
u
8
4u
7
+ 4u
6
+ 4u
5
4u
4
+ u
3
u 1i
I
u
3
= hu
42
u
41
+ ··· 2u
4
+ 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
3
=
1
0
a
7
=
0
1
a
8
=
1
1
a
2
=
1
1
a
1
=
1
1
a
4
=
2
1
a
9
=
1
0
a
6
=
1
0
a
11
=
2
1
a
10
=
1
1
a
5
=
2
1
a
5
=
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
9
u
8
4u
7
+ 4u
6
+ 4u
5
4u
4
+ u
3
u 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
8
=
u
u
a
2
=
1
u
2
a
1
=
u
4
+ u
2
+ 1
u
4
+ 2u
2
a
4
=
u
8
3u
6
+ u
4
+ 2u
2
+ 1
u
8
4u
6
+ 4u
4
a
9
=
u
8
3u
6
+ u
5
+ u
4
2u
3
+ 2u
2
+ u
u
8
+ u
7
4u
6
3u
5
+ 4u
4
+ 2u
3
+ u
a
6
=
u
u
3
+ u
a
11
=
u
6
2u
4
+ u
3
u
u
8
3u
6
+ u
5
+ 2u
4
2u
3
+ u + 1
a
10
=
u
6
2u
4
+ u
3
u
2
u
u
8
3u
6
+ u
5
+ u
4
2u
3
+ u
2
+ u + 1
a
5
=
u
7
+ 2u
5
u
4
+ u
3
+ u
2
u
u
8
u
7
+ 3u
6
+ 3u
5
2u
4
u
3
u
2
u 1
a
5
=
u
7
+ 2u
5
u
4
+ u
3
+ u
2
u
u
8
u
7
+ 3u
6
+ 3u
5
2u
4
u
3
u
2
u 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 = 1.44029 0.16872I
12.84676 + 4.88120I 15.4409 3.5107I
u = 1.44029 + 0.16872I
12.84676 4.88120I 15.4409 + 3.5107I
u = 0.423257 0.356395I
0.980950 + 0.551491I 9.15793 4.50455I
u = 0.423257 + 0.356395I
0.980950 0.551491I 9.15793 + 4.50455I
u = 0.287064 0.695105I
1.39752 6.41727I 2.65899 + 8.21479I
u = 0.287064 + 0.695105I
1.39752 + 6.41727I 2.65899 8.21479I
u = 1.30640
7.01397 12.1823
u = 1.42328 0.27641I
9.5593 13.5238I 11.6511 + 8.3193I
u = 1.42328 + 0.27641I
9.5593 + 13.5238I 11.6511 8.3193I
5
III. I
u
3
= hu
42
u
41
+ · · · 2u
4
+ 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
8
=
u
u
a
2
=
1
u
2
a
1
=
u
4
+ u
2
+ 1
u
4
+ 2u
2
a
4
=
u
8
3u
6
+ u
4
+ 2u
2
+ 1
u
8
4u
6
+ 4u
4
a
9
=
u
15
+ 6u
13
12u
11
+ 6u
9
+ 6u
7
2u
5
4u
3
u
15
+ 7u
13
18u
11
+ 19u
9
4u
7
4u
5
+ u
a
6
=
u
u
3
+ u
a
11
=
u
26
+ 11u
24
+ ··· + u
2
+ 1
u
26
+ 12u
24
+ ··· 2u
4
+ u
2
a
10
=
u
30
+ 13u
28
+ ··· + 2u
4
+ 1
u
32
+ 14u
30
+ ··· 2u
4
+ 2u
2
a
5
=
u
41
u
40
+ ··· u + 1
u
39
+ 18u
37
+ ··· u + 1
a
5
=
u
41
u
40
+ ··· u + 1
u
39
+ 18u
37
+ ··· u + 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 = 1.42800 0.24722I
11.72577 + 4.35170I 14.2357 2.9721I
u = 1.42800 + 0.24722I
11.72577 4.35170I 14.2357 + 2.9721I
u = 1.42429 0.12838I
6.08429 1.09840I 10.14786 + 3.17531I
u = 1.42429 + 0.12838I
6.08429 + 1.09840I 10.14786 3.17531I
u = 1.41609 0.27243I
4.04389 + 9.94224I 7.31059 8.24169I
u = 1.41609 + 0.27243I
4.04389 9.94224I 7.31059 + 8.24169I
u = 1.354035 0.243767I
1.62697 + 3.23317I 3.55215 1.92093I
u = 1.354035 + 0.243767I
1.62697 3.23317I 3.55215 + 1.92093I
u = 1.112717 0.206888I
4.64745 + 6.55351I 8.17560 6.03047I
u = 1.112717 + 0.206888I
4.64745 6.55351I 8.17560 + 6.03047I
u = 1.08927
1.57667 7.15487
u = 0.644973
1.57667 7.15487
u = 0.619519 0.389305I
5.30545 6.06326I 10.03226 + 2.92445I
u = 0.619519 + 0.389305I
5.30545 + 6.06326I 10.03226 2.92445I
u = 0.301718 0.707163I
4.04389 + 9.94224I 7.31059 8.24169I
u = 0.301718 + 0.707163I
4.04389 9.94224I 7.31059 + 8.24169I
u = 0.274697 0.655623I
0.07785 + 2.71696I 5.48517 3.12164I
u = 0.274697 + 0.655623I
0.07785 2.71696I 5.48517 + 3.12164I
u = 0.211792 0.670835I
0.40568 + 3.16875I 2.95224 5.22442I
u = 0.211792 + 0.670835I
0.40568 3.16875I 2.95224 + 5.22442I
u = 0.096884 0.668841I
1.62697 3.23317I 3.55215 + 1.92093I
u = 0.096884 + 0.668841I
1.62697 + 3.23317I 3.55215 1.92093I
u = 0.147288 0.653126I
3.11833 1.91795
u = 0.147288 + 0.653126I
3.11833 1.91795
u = 0.335269 0.641117I
6.08429 1.09840I 10.14786 + 3.17531I
u = 0.335269 + 0.641117I
6.08429 + 1.09840I 10.14786 3.17531I
u = 0.481440 0.468716I
6.75483 2.56601I 12.00469 + 3.90900I
u = 0.481440 + 0.468716I
6.75483 + 2.56601I 12.00469 3.90900I
u = 0.594417 0.333320I
0.07785 + 2.71696I 5.48517 3.12164I
u = 0.594417 + 0.333320I
0.07785 2.71696I 5.48517 + 3.12164I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.082919 0.161904I
0.40568 3.16875I 2.95224 + 5.22442I
u = 1.082919 + 0.161904I
0.40568 + 3.16875I 2.95224 5.22442I
u = 1.317379 0.229558I
6.02305 8.97052
u = 1.317379 + 0.229558I
6.02305 8.97052
u = 1.379259 0.261235I
4.64745 6.55351I 8.17560 + 6.03047I
u = 1.379259 + 0.261235I
4.64745 + 6.55351I 8.17560 6.03047I
u = 1.40967 0.25849I
5.30545 6.06326I 10.03226 + 2.92445I
u = 1.40967 + 0.25849I
5.30545 + 6.06326I 10.03226 2.92445I
u = 1.41719 0.15750I
6.75483 2.56601I 12.00469 + 3.90900I
u = 1.41719 + 0.15750I
6.75483 + 2.56601I 12.00469 3.90900I
u = 1.44204 0.12357I
11.72577 + 4.35170I 14.2357 2.9721I
u = 1.44204 + 0.12357I
11.72577 4.35170I 14.2357 + 2.9721I
8
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
u(u
9
u
7
2u
6
+ 8u
5
+ 5u
4
5u
3
+ 5u
2
+ 9u + 3)
(u
42
+ 3u
41
+ ··· + 2u 1)
c
2
, c
6
, c
7
c
9
, c
10
(u 1)(u
9
+ u
8
4u
7
4u
6
+ 4u
5
+ 4u
4
+ u
3
u + 1)
(u
42
+ u
41
+ ··· 2u
4
+ 1)
c
3
, c
11
(u + 1)(u
9
+ u
8
+ 4u
7
+ 2u
6
+ 8u
5
+ 6u
4
+ 9u
3
+ 6u
2
+ 3u + 1)
(u
42
+ 9u
41
+ ··· + 920u + 113)
c
5
(u 1)(u
9
+ u
8
4u
7
4u
6
+ 4u
5
+ 4u
4
+ u
3
u + 1)
(u
42
+ u
41
+ ··· 2u
4
+ 1)
c
8
(u + 1)(u
9
+ 6u
8
+ ··· 20u 8)
(1 + 4u 18u
2
2u
3
+ 67u
4
36u
5
83u
6
+ 79u
7
+ 47u
8
84u
9
+ 56u
11
16u
12
26u
13
+ 10u
14
+ 21u
15
15u
16
+ 2u
18
+ 3u
19
3u
20
+ u
21
)
2
9
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
y
(y
9
2y
8
+ 17y
7
30y
6
+ 112y
5
103y
4
+ 131y
3
145y
2
+ 51y 9)
(y
42
y
41
+ ··· 24y + 1)
c
2
, c
6
, c
7
c
10
(y 1)(y
9
9y
8
+ 32y
7
54y
6
+ 38y
5
2y
4
+ y
3
10y
2
+ y 1)
(y
42
37y
41
+ ··· 4y
2
+ 1)
c
3
, c
11
(y 1)(y
9
+ 7y
8
+ ··· 3y 1)
(y
42
+ 11y
41
+ ··· + 84720y + 12769)
c
5
(y 1)(y
9
9y
8
+ 32y
7
54y
6
+ 38y
5
2y
4
+ y
3
10y
2
+ y 1)
(y
42
37y
41
+ ··· 4y
2
+ 1)
c
8
(y 1)(y
9
+ 6y
7
+ ··· 16y 64)
(1 + 52y 474y
2
+ 2294y
3
6795y
4
+ 1.31 × 10
4
y
5
1.81 × 10
4
y
6
+ 1.91 × 10
4
y
7
1.67 × 10
4
y
8
+ 1.32 × 10
4
y
9
9752y
10
+ 7012y
11
4772y
12
+ 3064y
13
1736y
14
+ 973y
15
405y
16
+ 194y
17
52y
18
+ 21y
19
3y
20
+ y
21
)
2
c
9
(y 1)(y
9
9y
8
+ 32y
7
54y
6
+ 38y
5
2y
4
+ y
3
10y
2
+ y 1)
(y
42
37y
41
+ ··· 4y
2
+ 1)
10