11a
186
(K11a
186
)
1
Arc Sequences
6 1 8 10 7 2 3 4 11 5 9
Solving Sequence
2,7
6 1 3 8 4 9 5 11 10
c
6
c
1
c
2
c
7
c
3
c
8
c
5
c
11
c
10
c
4
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
11
2u
9
+ 4u
7
+ u
6
4u
5
u
4
+ 3u
3
+ 2u
2
2u 1i
I
u
2
= hu
36
u
35
+ ··· u
3
+ 1i
There are 2 irreducible components with 47 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
11
2u
9
+ 4u
7
+ u
6
4u
5
u
4
+ 3u
3
+ 2u
2
2u 1i
(i) Arc colorings
a
2
=
1
0
a
7
=
0
u
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
u
2
+ 1
u
4
a
8
=
u
9
+ 2u
7
3u
5
+ 2u
3
u
u
9
+ u
7
u
5
+ u
a
4
=
u
9
u
8
2u
7
+ u
6
+ 3u
5
u
4
3u
3
+ u + 1
u
10
+ u
9
2u
8
u
7
+ 3u
6
+ 2u
5
2u
4
u
3
+ u
2
a
9
=
u
9
u
8
+ 2u
7
+ u
6
3u
5
3u
4
+ 2u
3
+ 2u
2
u
u
10
u
9
2u
8
+ u
7
+ 3u
6
u
5
4u
4
+ u
2
+ u
a
5
=
u
u
3
+ u
a
11
=
u
10
+ u
8
+ u
7
2u
6
2u
5
+ u
4
+ 2u
3
+ u
2
2u
u
9
u
8
+ 2u
7
+ u
6
3u
5
3u
4
+ 2u
3
+ 3u
2
u 1
a
10
=
u
10
+ u
8
+ u
7
2u
6
2u
5
+ u
4
+ 2u
3
2u
u
9
u
8
+ 2u
7
+ u
6
3u
5
2u
4
+ 2u
3
+ 2u
2
u 1
a
10
=
u
10
+ u
8
+ u
7
2u
6
2u
5
+ u
4
+ 2u
3
2u
u
9
u
8
+ 2u
7
+ u
6
3u
5
2u
4
+ 2u
3
+ 2u
2
u 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 = 1.13447
11.8669 21.5187
u = 0.937682 0.702007I
4.20048 8.65870I 8.03545 + 9.01618I
u = 0.937682 + 0.702007I
4.20048 + 8.65870I 8.03545 9.01618I
u = 0.574057 0.778762I
0.32700 + 2.62828I 9.00950 0.39606I
u = 0.574057 + 0.778762I
0.32700 2.62828I 9.00950 + 0.39606I
u = 0.424792
0.633212 15.6941
u = 0.786275 0.725485I
5.13423 + 2.26440I 5.35075 2.78673I
u = 0.786275 + 0.725485I
5.13423 2.26440I 5.35075 + 2.78673I
u = 0.903688
4.12325 21.6837
u = 1.053251 0.672906I
3.16344 + 13.64352I 13.1560 9.4873I
u = 1.053251 + 0.672906I
3.16344 13.64352I 13.1560 + 9.4873I
3
II. I
u
2
= hu
36
u
35
+ · · · u
3
+ 1i
(i) Arc colorings
a
2
=
1
0
a
7
=
0
u
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
u
2
+ 1
u
4
a
8
=
u
9
+ 2u
7
3u
5
+ 2u
3
u
u
9
+ u
7
u
5
+ u
a
4
=
u
14
+ 3u
12
6u
10
+ 7u
8
6u
6
+ 4u
4
2u
2
+ 1
u
14
+ 2u
12
3u
10
+ 2u
8
+ u
2
a
9
=
u
19
4u
17
+ 10u
15
16u
13
+ 19u
11
18u
9
+ 14u
7
10u
5
+ 5u
3
2u
u
19
3u
17
+ 6u
15
7u
13
+ 5u
11
3u
9
u
3
+ u
a
5
=
u
u
3
+ u
a
11
=
u
35
+ u
34
+ ··· + u
2
+ 2
u
35
+ 7u
33
+ ··· 2u
2
+ 1
a
10
=
2u
35
+ u
34
+ ··· u + 2
2u
35
u
34
+ ··· 3u
2
2u
a
10
=
2u
35
+ u
34
+ ··· u + 2
2u
35
u
34
+ ··· 3u
2
2u
(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 = 1.129814 0.032613I
7.69896 6.87816I 17.6593 + 5.1131I
u = 1.129814 + 0.032613I
7.69896 + 6.87816I 17.6593 5.1131I
u = 1.045367 0.669009I
1.72161 8.10595I 11.08535 + 5.00657I
u = 1.045367 + 0.669009I
1.72161 + 8.10595I 11.08535 5.00657I
u = 1.038668 0.636561I
2.30993 5.17624I 11.82231 + 5.02355I
u = 1.038668 + 0.636561I
2.30993 + 5.17624I 11.82231 5.02355I
u = 0.936753 0.611605I
0.88834 4.72205I 15.5195 + 7.2621I
u = 0.936753 + 0.611605I
0.88834 + 4.72205I 15.5195 7.2621I
u = 0.772239 0.333861I
0.218096 0.036628I 13.43748 + 0.95651I
u = 0.772239 + 0.333861I
0.218096 + 0.036628I 13.43748 0.95651I
u = 0.759891 0.733182I
4.73704 + 3.18642I 6.45994 3.31717I
u = 0.759891 + 0.733182I
4.73704 3.18642I 6.45994 + 3.31717I
u = 0.720307 0.524101I
0.218096 + 0.036628I 13.43748 0.95651I
u = 0.720307 + 0.524101I
0.218096 0.036628I 13.43748 + 0.95651I
u = 0.510565 0.712216I
0.822851 9.60076
u = 0.510565 + 0.712216I
0.822851 9.60076
u = 0.049508 0.478803I
1.83259 2.50180I 6.41929 + 3.81694I
u = 0.049508 + 0.478803I
1.83259 + 2.50180I 6.41929 3.81694I
u = 0.475172 0.740129I
2.30993 + 5.17624I 11.82231 5.02355I
u = 0.475172 + 0.740129I
2.30993 5.17624I 11.82231 + 5.02355I
u = 0.527375 0.775874I
6.14948 1.48503I 15.5689 + 0.3788I
u = 0.527375 + 0.775874I
6.14948 + 1.48503I 15.5689 0.3788I
u = 0.568398 0.797612I
1.72161 8.10595I 11.08535 + 5.00657I
u = 0.568398 + 0.797612I
1.72161 + 8.10595I 11.08535 5.00657I
u = 0.853258 0.641261I
1.83259 + 2.50180I 6.41929 3.81694I
u = 0.853258 + 0.641261I
1.83259 2.50180I 6.41929 + 3.81694I
u = 0.898798 0.229050I
0.88834 + 4.72205I 15.5195 7.2621I
u = 0.898798 + 0.229050I
0.88834 4.72205I 15.5195 + 7.2621I
u = 0.917289 0.702643I
4.73704 + 3.18642I 6.45994 3.31717I
u = 0.917289 + 0.702643I
4.73704 3.18642I 6.45994 + 3.31717I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.051517 0.626704I
3.95239 14.4550
u = 1.051517 + 0.626704I
3.95239 14.4550
u = 1.056179 0.652350I
7.69896 + 6.87816I 17.6593 5.1131I
u = 1.056179 + 0.652350I
7.69896 6.87816I 17.6593 + 5.1131I
u = 1.115127 0.024468I
6.14948 + 1.48503I 15.5689 0.3788I
u = 1.115127 + 0.024468I
6.14948 1.48503I 15.5689 + 0.3788I
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
, c
6
c
10
(u
11
2u
9
+ 4u
7
u
6
4u
5
+ u
4
+ 3u
3
2u
2
2u + 1)
(u
36
+ u
35
+ ··· + u
3
+ 1)
c
2
, c
5
, c
9
c
11
(u
11
+ 4u
10
+ ··· + 8u + 1)(u
36
+ 13u
35
+ ··· 10u
2
+ 1)
c
3
, c
7
, c
8
(u
11
+ 5u
10
+ 8u
9
+ 5u
8
+ 9u
7
+ 19u
6
+ 8u
5
2u
4
+ 9u
3
+ u
2
12u 4)
(1 + 2u 3u
2
6u
3
+ 6u
4
+ 2u
5
+ 36u
6
12u
7
77u
8
+ 6u
9
+ 63u
10
+ 28u
11
37u
12
34u
13
+ 20u
14
+ 14u
15
7u
16
2u
17
+ u
18
)
2
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
, c
6
c
10
(y
11
4y
10
+ ··· + 8y 1)(y
36
13y
35
+ ··· 10y
2
+ 1)
c
2
, c
5
, c
9
c
11
(y
11
+ 8y
10
+ ··· + 28y 1)(y
36
+ 19y
35
+ ··· 20y + 1)
c
3
, c
7
, c
8
(y
11
9y
10
+ ··· + 152y 16)
(1 10y + 45y
2
8y
3
262y
4
+ 848y
5
72y
6
4222y
7
+ 9455y
8
1.11 × 10
4
y
9
+ 9785y
10
8266y
11
+ 6727y
12
4432y
13
+ 2108y
14
686y
15
+ 145y
16
18y
17
+ y
18
)
2
7