11n
135
(K11n
135
)
1
Arc Sequences
7 1 9 11 10 2 3 11 5 3 5
Solving Sequence
5,10 3,6
11 1 2 4 9 8 7
c
5
c
10
c
11
c
2
c
4
c
9
c
8
c
7
c
1
, c
3
, c
6
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
8
6u
7
+ 11u
6
u
5
14u
4
+ 4u
3
+ 11u
2
6u + 1, u
7
+ 6u
6
10u
5
2u
4
+ 14u
3
+ u
2
+ b 11u + 3,
3u
7
+ 17u
6
27u
5
7u
4
+ 40u
3
+ 2u
2
+ a 32u + 7i
I
u
2
= hu
10
+ 3u
9
4u
8
16u
7
+ 6u
6
+ 37u
5
+ 11u
4
21u
3
16u
2
5u 1,
7u
9
+ 14u
8
40u
7
71u
6
+ 100u
5
+ 158u
4
51u
3
95u
2
+ b 32u 9,
9u
9
+ 20u
8
50u
7
104u
6
+ 125u
5
+ 233u
4
59u
3
138u
2
+ a 49u 13i
There are 2 irreducible components with 18 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
8
6u
7
+· · ·6u+1, u
7
+6u
6
+· · ·+b+3, 3u
7
+17u
6
+· · ·+a+7i
(i) Arc colorings
a
5
=
0
u
a
10
=
3u
7
17u
6
+ 27u
5
+ 7u
4
40u
3
2u
2
+ 32u 7
u
7
6u
6
+ 10u
5
+ 2u
4
14u
3
u
2
+ 11u 3
a
3
=
1
0
a
6
=
u
6
5u
5
+ 6u
4
+ 5u
3
9u
2
5u + 6
u
7
5u
6
+ 6u
5
+ 5u
4
9u
3
5u
2
+ 7u
a
11
=
2u
7
11u
6
+ 17u
5
+ 5u
4
26u
3
u
2
+ 21u 4
u
7
6u
6
+ 10u
5
+ 2u
4
14u
3
u
2
+ 11u 3
a
1
=
2u
7
11u
6
+ 17u
5
+ 5u
4
26u
3
u
2
+ 21u 4
2u
7
10u
6
+ 13u
5
+ 7u
4
19u
3
4u
2
+ 15u 4
a
2
=
u
7
+ 6u
6
11u
5
+ u
4
+ 14u
3
4u
2
12u + 8
u
7
5u
6
+ 6u
5
+ 5u
4
10u
3
4u
2
+ 8u
a
4
=
u
7
6u
6
+ 11u
5
u
4
14u
3
+ 4u
2
+ 11u 5
u 1
a
9
=
3u
7
17u
6
+ 27u
5
+ 7u
4
40u
3
2u
2
+ 32u 7
u
7
5u
6
+ 7u
5
+ 2u
4
9u
3
u
2
+ 8u 2
a
8
=
2u
7
+ 11u
6
17u
5
5u
4
+ 26u
3
+ u
2
21u + 4
2u
6
8u
5
+ 5u
4
+ 12u
3
7u
2
11u + 4
a
7
=
2u
7
+ 13u
6
25u
5
+ 38u
3
6u
2
32u + 8
2u
6
8u
5
+ 5u
4
+ 12u
3
7u
2
11u + 4
a
7
=
2u
7
+ 13u
6
25u
5
+ 38u
3
6u
2
32u + 8
2u
6
8u
5
+ 5u
4
+ 12u
3
7u
2
11u + 4
(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 = 0.880687 0.411861I
a = 0.384446 + 0.633801I
b = 0.077538 0.716519I
2.76707 + 1.04226I 7.14108 + 0.01449I
u = 0.880687 + 0.411861I
a = 0.384446 0.633801I
b = 0.077538 + 0.716519I
2.76707 1.04226I 7.14108 0.01449I
u = 0.272591 0.146873I
a = 1.44122 3.46089I
b = 0.115448 1.155084I
6.36547 2.93267I 4.14670 + 1.68828I
u = 0.272591 + 0.146873I
a = 1.44122 + 3.46089I
b = 0.115448 + 1.155084I
6.36547 + 2.93267I 4.14670 1.68828I
u = 1.60890 0.18698I
a = 0.746630 0.202358I
b = 1.239094 0.185966I
3.52853 0.48963I 9.23600 1.05814I
u = 1.60890 + 0.18698I
a = 0.746630 + 0.202358I
b = 1.239094 + 0.185966I
3.52853 + 0.48963I 9.23600 + 1.05814I
u = 1.99919 0.45726I
a = 0.420965 + 0.197898I
b = 0.932081 + 0.203144I
5.60402 + 3.77609I 14.7696 2.3802I
u = 1.99919 + 0.45726I
a = 0.420965 0.197898I
b = 0.932081 0.203144I
5.60402 3.77609I 14.7696 + 2.3802I
3
II.
I
u
2
= hu
10
+3u
9
+· · ·5u1, 7u
9
+14u
8
+· · ·+b9, 9u
9
+20u
8
+· · ·+a13i
(i) Arc colorings
a
5
=
0
u
a
10
=
9u
9
20u
8
+ ··· + 49u + 13
7u
9
14u
8
+ ··· + 32u + 9
a
3
=
1
0
a
6
=
2u
9
3u
8
+ 12u
7
+ 14u
6
30u
5
30u
4
+ 17u
3
+ 18u
2
+ 7u + 1
3u
9
4u
8
+ 18u
7
+ 18u
6
44u
5
39u
4
+ 24u
3
+ 25u
2
+ 10u + 2
a
11
=
2u
9
6u
8
+ 10u
7
+ 33u
6
25u
5
75u
4
+ 8u
3
+ 43u
2
+ 17u + 4
7u
9
14u
8
+ ··· + 32u + 9
a
1
=
2u
9
6u
8
+ 10u
7
+ 33u
6
25u
5
75u
4
+ 8u
3
+ 43u
2
+ 17u + 4
5u
9
13u
8
+ ··· + 30u + 9
a
2
=
u
9
+ u
8
6u
7
4u
6
+ 14u
5
+ 9u
4
7u
3
7u
2
3u + 1
3u
9
+ 4u
8
18u
7
18u
6
+ 44u
5
+ 39u
4
25u
3
24u
2
9u 2
a
4
=
u
9
+ u
8
6u
7
4u
6
+ 14u
5
+ 9u
4
7u
3
7u
2
2u
2u
9
2u
8
+ 12u
7
+ 8u
6
28u
5
18u
4
+ 14u
3
+ 14u
2
+ 5u + 1
a
9
=
9u
9
20u
8
+ ··· + 49u + 13
2u
8
u
7
+ 13u
6
+ u
5
30u
4
u
3
+ 15u
2
+ 6u + 2
a
8
=
2u
9
6u
8
+ 10u
7
+ 33u
6
25u
5
75u
4
+ 8u
3
+ 43u
2
+ 17u + 4
6u
9
+ 17u
8
+ ··· 37u 11
a
7
=
4u
9
+ 11u
8
22u
7
60u
6
+ 60u
5
+ 134u
4
38u
3
75u
2
20u 7
6u
9
+ 17u
8
+ ··· 37u 11
a
7
=
4u
9
+ 11u
8
22u
7
60u
6
+ 60u
5
+ 134u
4
38u
3
75u
2
20u 7
6u
9
+ 17u
8
+ ··· 37u 11
(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 = 1.91166 0.13941I
a = 1.71182 0.70510I
b = 3.17411 1.58657I
17.7391 8.0399I 7.30663 + 2.83159I
u = 1.91166 + 0.13941I
a = 1.71182 + 0.70510I
b = 3.17411 + 1.58657I
17.7391 + 8.0399I 7.30663 2.83159I
u = 1.85920
a = 2.10858
b = 3.92027
13.1860 6.15586
u = 0.672782 0.035203I
a = 0.07501 1.84158I
b = 0.115292 1.236343I
5.72141 + 3.10928I 8.09311 4.32692I
u = 0.672782 + 0.035203I
a = 0.07501 + 1.84158I
b = 0.115292 + 1.236343I
5.72141 3.10928I 8.09311 + 4.32692I
u = 0.125563 0.292930I
a = 0.451377 1.260813I
b = 0.312653 0.290533I
0.422559 + 0.990373I 6.71540 6.78739I
u = 0.125563 + 0.292930I
a = 0.451377 + 1.260813I
b = 0.312653 + 0.290533I
0.422559 0.990373I 6.71540 + 6.78739I
u = 1.07196
a = 0.382834
b = 0.410385
1.55504 5.61011
u = 1.60362 0.62517I
a = 0.225319 + 0.092507I
b = 0.419159 0.007483I
4.86323 + 4.26845I 7.00188 6.39401I
u = 1.60362 + 0.62517I
a = 0.225319 0.092507I
b = 0.419159 + 0.007483I
4.86323 4.26845I 7.00188 + 6.39401I
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
8
+ 2u
6
+ ··· u + 1)(u
10
+ 6u
9
+ ··· + 14u + 4)
c
2
(u
8
+ 4u
7
+ 10u
6
+ 16u
5
+ 19u
4
+ 15u
3
+ 8u
2
+ 3u + 1)
(u
10
+ 4u
9
+ ··· + 36u + 16)
c
3
, c
11
(u
8
+ u
6
+ u
5
2u
4
u + 1)
(u
10
+ u
9
10u
8
30u
7
+ 42u
6
+ 25u
5
18u
4
7u
3
10u
2
2u 1)
c
4
(u
8
+ u
6
u
5
2u
4
+ u + 1)
(u
10
+ u
9
10u
8
30u
7
+ 42u
6
+ 25u
5
18u
4
7u
3
10u
2
2u 1)
c
5
, c
10
(u
8
u
7
2u
4
+ u
3
+ u
2
+ 1)
(u
10
+ 2u
9
12u
8
41u
7
+ 4u
6
65u
5
+ 12u
4
+ 18u
3
8u
2
u + 1)
c
6
(u
8
+ 2u
6
+ ··· + u + 1)(u
10
+ 6u
9
+ ··· + 14u + 4)
c
7
(u
8
+ 2u
6
+ ··· + 3u + 1)(u
10
+ 6u
9
+ ··· + 42u + 180)
c
8
(u
8
+ 7u
7
+ 17u
6
+ 15u
5
+ u
4
+ 5u
2
+ 2u + 1)
(u
10
+ 4u
9
3u
8
29u
7
21u
6
+ 48u
5
+ 80u
4
+ 47u
3
+ 16u
2
+ 5u + 1)
c
9
(u
8
+ u
7
2u
4
u
3
+ u
2
+ 1)
(u
10
+ 2u
9
12u
8
41u
7
+ 4u
6
65u
5
+ 12u
4
+ 18u
3
8u
2
u + 1)
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y
8
+ 4y
7
+ 10y
6
+ 16y
5
+ 19y
4
+ 15y
3
+ 8y
2
+ 3y + 1)
(y
10
+ 4y
9
+ ··· + 36y + 16)
c
2
(y
8
+ 4y
7
+ 10y
6
+ 20y
5
+ 19y
4
+ 3y
3
+ 12y
2
+ 7y + 1)
(y
10
+ 4y
9
+ ··· 1648y + 256)
c
3
, c
4
, c
11
(y
8
+ 2y
7
3y
6
5y
5
+ 6y
4
+ 4y
3
4y
2
y + 1)
(y
10
21y
9
+ ··· + 16y + 1)
c
5
, c
9
, c
10
(y
8
y
7
4y
6
+ 4y
5
+ 6y
4
5y
3
3y
2
+ 2y + 1)
(y
10
28y
9
+ ··· 17y + 1)
c
7
(y
8
+ 4y
7
+ 6y
6
11y
5
+ 33y
4
+ 43y
3
+ 21y
2
+ y + 1)
(y
10
56y
9
+ ··· + 244836y + 32400)
c
8
(y
8
15y
7
+ 81y
6
181y
5
+ 145y
4
16y
3
+ 27y
2
+ 6y + 1)
(y
10
22y
9
+ ··· + 7y + 1)
7