11a
85
(K11a
85
)
1
Arc Sequences
5 1 11 9 2 3 10 4 8 7 6
Solving Sequence
1,5
2 3 6 7 11 4 10 8 9
c
1
c
2
c
5
c
6
c
11
c
3
c
10
c
7
c
9
c
4
, c
8
Representation Ideals
I = I
u
1
I
u
1
= hu
53
+ u
52
+ ··· + 3u + 1i
There are 1 irreducible components with 53 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
53
+ u
52
+ · · · + 3u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
6
=
u
u
3
+ u
a
7
=
u
7
+ 2u
5
2u
3
u
7
u
5
+ u
a
11
=
u
4
u
2
+ 1
u
6
2u
4
+ u
2
a
4
=
u
12
3u
10
+ 5u
8
4u
6
+ 2u
4
u
2
+ 1
u
14
4u
12
+ 7u
10
6u
8
+ 2u
6
+ u
2
a
10
=
u
20
5u
18
+ 13u
16
20u
14
+ 20u
12
13u
10
+ 7u
8
4u
6
+ 3u
4
u
2
+ 1
u
20
+ 4u
18
8u
16
+ 8u
14
3u
12
2u
10
+ 2u
8
u
4
a
8
=
u
33
+ 8u
31
+ ··· 4u
5
u
u
33
7u
31
+ ··· + 2u
13
+ u
a
9
=
u
46
11u
44
+ ··· + 2u
4
+ 1
u
46
+ 10u
44
+ ··· 2u
6
u
2
a
9
=
u
46
11u
44
+ ··· + 2u
4
+ 1
u
46
+ 10u
44
+ ··· 2u
6
u
2
(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 = 1.119277 0.581631I
5.0695 14.0551I 3.27946 + 9.53540I
u = 1.119277 + 0.581631I
5.0695 + 14.0551I 3.27946 9.53540I
u = 1.114576 0.556654I
2.57269 10.01430I 8.16672 + 10.13225I
u = 1.114576 + 0.556654I
2.57269 + 10.01430I 8.16672 10.13225I
u = 1.109655 0.186764I
2.94416 + 0.20510I 5.58587 + 0.61243I
u = 1.109655 + 0.186764I
2.94416 0.20510I 5.58587 0.61243I
u = 1.108378 0.429410I
0.525380 0.893417I 6.66537 + 1.06487I
u = 1.108378 + 0.429410I
0.525380 + 0.893417I 6.66537 1.06487I
u = 1.102860 0.521380I
3.95438 4.57454I 11.60058 + 3.32502I
u = 1.102860 + 0.521380I
3.95438 + 4.57454I 11.60058 3.32502I
u = 1.044498 0.281512I
2.20764 0.64085I 5.73049 + 0.77381I
u = 1.044498 + 0.281512I
2.20764 + 0.64085I 5.73049 0.77381I
u = 0.974822 0.492221I
0.225106 0.871265I 4.08839 0.80386I
u = 0.974822 + 0.492221I
0.225106 + 0.871265I 4.08839 + 0.80386I
u = 0.971931 0.595508I
7.41049 + 1.56089I 0.452720 + 0.072646I
u = 0.971931 + 0.595508I
7.41049 1.56089I 0.452720 0.072646I
u = 0.601217 0.686706I
8.50588 6.50884I 1.27031 + 5.73425I
u = 0.601217 + 0.686706I
8.50588 + 6.50884I 1.27031 5.73425I
u = 0.592887 0.582181I
0.88261 3.44954I 2.55100 + 7.40984I
u = 0.592887 + 0.582181I
0.88261 + 3.44954I 2.55100 7.40984I
u = 0.559370
1.01608 9.59731
u = 0.366014 0.773541I
7.29360 + 8.95041I 0.09124 5.62138I
u = 0.366014 + 0.773541I
7.29360 8.95041I 0.09124 + 5.62138I
u = 0.332464 0.722196I
0.29834 + 5.13851I 4.80621 6.56976I
u = 0.332464 + 0.722196I
0.29834 5.13851I 4.80621 + 6.56976I
u = 0.262528 0.616219I
1.62999 + 0.09542I 8.49556 + 0.70141I
u = 0.262528 + 0.616219I
1.62999 0.09542I 8.49556 0.70141I
u = 0.022954 0.641795I
3.51831 2.96655I 2.71846 + 2.72944I
u = 0.022954 + 0.641795I
3.51831 + 2.96655I 2.71846 2.72944I
u = 0.376178 0.768520I
7.69936 2.64631I 0.774092 + 0.704677I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.376178 + 0.768520I
7.69936 + 2.64631I 0.774092 0.704677I
u = 0.378665 0.682475I
1.76473 1.60253I 1.33859 + 1.09595I
u = 0.378665 + 0.682475I
1.76473 + 1.60253I 1.33859 1.09595I
u = 0.491709 0.618554I
2.27338 0.13836I 2.37846 + 0.39315I
u = 0.491709 + 0.618554I
2.27338 + 0.13836I 2.37846 0.39315I
u = 0.586774 0.690510I
8.78157 + 0.18779I 1.88251 0.64861I
u = 0.586774 + 0.690510I
8.78157 0.18779I 1.88251 + 0.64861I
u = 0.983871 0.595556I
7.60812 + 4.77001I 0.11119 5.11123I
u = 0.983871 + 0.595556I
7.60812 4.77001I 0.11119 + 5.11123I
u = 1.038830 0.540274I
0.66178 + 4.72185I 1.41180 6.22617I
u = 1.038830 + 0.540274I
0.66178 4.72185I 1.41180 + 6.22617I
u = 1.093004 0.554934I
0.31357 + 6.39150I 2.37542 5.36487I
u = 1.093004 + 0.554934I
0.31357 6.39150I 2.37542 + 5.36487I
u = 1.098216 0.328717I
5.25608 + 2.82044I 13.7464 4.6104I
u = 1.098216 + 0.328717I
5.25608 2.82044I 13.7464 + 4.6104I
u = 1.102420 0.258443I
4.56669 2.49112I 11.89781 + 4.18392I
u = 1.102420 + 0.258443I
4.56669 + 2.49112I 11.89781 4.18392I
u = 1.114173 0.582898I
5.52079 + 7.74501I 2.36979 4.72622I
u = 1.114173 + 0.582898I
5.52079 7.74501I 2.36979 + 4.72622I
u = 1.117789 0.404581I
0.34604 + 6.68644I 7.28464 6.64124I
u = 1.117789 + 0.404581I
0.34604 6.68644I 7.28464 + 6.64124I
u = 1.122118 0.196280I
2.54225 6.40452I 6.41624 + 4.33917I
u = 1.122118 + 0.196280I
2.54225 + 6.40452I 6.41624 4.33917I
4
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
(u
53
+ u
52
+ ··· + 3u + 1)
c
2
(u
53
+ 25u
52
+ ··· + 3u + 1)
c
3
(u
53
+ 7u
52
+ ··· + 41u + 5)
c
4
(u
53
+ u
52
+ ··· + u 1)
c
6
(u
53
+ u
52
+ ··· + 149u 97)
c
7
, c
9
, c
10
(u
53
+ 13u
52
+ ··· + 3u + 1)
c
8
(u
53
+ u
52
+ ··· + u 1)
c
11
(u
53
+ 3u
52
+ ··· + 213u + 39)
5
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
53
25y
52
+ ··· + 3y 1)
c
2
(y
53
+ 7y
52
+ ··· + 3y 1)
c
3
(y
53
5y
52
+ ··· + 911y 25)
c
4
(y
53
13y
52
+ ··· + 3y 1)
c
6
(y
53
17y
52
+ ··· + 199323y 9409)
c
7
, c
9
, c
10
(y
53
+ 55y
52
+ ··· 5y 1)
c
8
(y
53
13y
52
+ ··· + 3y 1)
c
11
(y
53
+ 11y
52
+ ··· 28185y 1521)
6