11n
38
(K11n
38
)
1
Arc Sequences
4 1 7 2 10 4 1 11 6 9 8
Solving Sequence
9,11
8 1
4,7
3 2 5 10 6
c
8
c
11
c
7
c
3
c
2
c
4
c
10
c
5
c
1
, c
6
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
4
+ u
2
u + 1, u
3
+ b 1, u
2
+ a u 1i
I
u
2
= hu
5
+ u
4
+ 21u
3
+ 8u
2
+ 6u + 1, 64u
4
+ 63u
3
+ 1308u
2
+ 249b + 476u + 361,
212u
4
+ 162u
3
+ 4395u
2
+ 249a + 643u + 869i
There are 2 irreducible components with 9 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
4
+ u
2
u + 1, u
3
+ b 1, u
2
+ a u 1i
(i) Arc colorings
a
9
=
0
u
a
11
=
u
2
+ u + 1
u
3
+ 1
a
8
=
2u
3
u
2
2u + 2
u
3
u
2
u
a
1
=
u
2
u 2
1
a
4
=
1
0
a
7
=
u
3
+ u
2
+ u
0
a
3
=
1
0
a
2
=
u
2
u 1
1
a
5
=
u
2
+ u + 2
1
a
10
=
u
2
+ u + 1
u
a
6
=
u
3
+ u
2
+ u
0
a
6
=
u
3
+ u
2
+ u
0
(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 = 0.547424 1.120873I
a = 0.504108 + 0.106312I
b = 0.899232 0.400532I
1.43393 + 1.41510I 1.48175 2.96122I
u = 0.547424 + 1.120873I
a = 0.504108 0.106312I
b = 0.899232 + 0.400532I
1.43393 1.41510I 1.48175 + 2.96122I
u = 0.547424 0.585652I
a = 1.50411 1.22685I
b = 1.39923 + 0.32564I
8.43568 3.16396I 3.01825 + 2.83489I
u = 0.547424 + 0.585652I
a = 1.50411 + 1.22685I
b = 1.39923 0.32564I
8.43568 + 3.16396I 3.01825 2.83489I
3
II. I
u
2
= hu
5
+ u
4
+ 21u
3
+ 8u
2
+ 6u + 1, 64u
4
+ 63u
3
+ 1308u
2
+ 249b +
476u + 361, 212u
4
+ 162u
3
+ · · · + 249a + 869i
(i) Arc colorings
a
9
=
0
u
a
11
=
0.851406u
4
0.650602u
3
+ ··· 2.58233u 3.48996
0.257028u
4
0.253012u
3
+ ··· 1.91165u 1.44980
a
8
=
0.397590u
4
+ 0.313253u
3
+ ··· + 1.14458u + 2.44578
0.200803u
4
+ 0.228916u
3
+ ··· + 1.61847u + 0.851406
a
1
=
1.19277u
4
0.939759u
3
+ ··· 4.43373u 5.33735
0.341365u
4
0.289157u
3
+ ··· 1.85141u 1.84739
a
4
=
1
0
a
7
=
1.04819u
4
+ 0.734940u
3
+ ··· + 3.10843u + 6.08434
0.453815u
4
+ 0.337349u
3
+ ··· + 2.43775u + 2.04418
a
3
=
2.26908u
4
+ 1.68675u
3
+ ··· + 10.1888u + 13.2209
0.618474u
4
+ 0.265060u
3
+ ··· + 2.22490u + 3.58233
a
2
=
0.851406u
4
0.650602u
3
+ ··· 2.58233u 3.48996
0.341365u
4
0.289157u
3
+ ··· 1.85141u 1.84739
a
5
=
1.19277u
4
+ 0.939759u
3
+ ··· + 4.43373u + 7.33735
0.626506u
4
+ 0.554217u
3
+ ··· + 2.40964u + 3.09639
a
10
=
0.851406u
4
0.650602u
3
+ ··· 2.58233u 3.48996
0.228916u
4
0.240964u
3
+ ··· 2.26506u 1.65060
a
6
=
0.594378u
4
+ 0.397590u
3
+ ··· + 0.670683u + 4.04016
0.453815u
4
+ 0.337349u
3
+ ··· + 2.43775u + 2.04418
a
6
=
0.594378u
4
+ 0.397590u
3
+ ··· + 0.670683u + 4.04016
0.453815u
4
+ 0.337349u
3
+ ··· + 2.43775u + 2.04418
(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 = 0.31023 4.51405I
a = 0.546751 0.052231I
b = 2.81245 + 0.17314I
10.26498 4.12490I 3.22285 + 1.83437I
u = 0.31023 + 4.51405I
a = 0.546751 + 0.052231I
b = 2.81245 0.17314I
10.26498 + 4.12490I 3.22285 1.83437I
u = 0.191115
a = 3.63772
b = 1.27490
2.84858 4.39074
u = 0.094211 0.496693I
a = 0.865610 0.402477I
b = 0.050105 + 0.441667I
0.206446 + 1.108915I 2.91822 5.88873I
u = 0.094211 + 0.496693I
a = 0.865610 + 0.402477I
b = 0.050105 0.441667I
0.206446 1.108915I 2.91822 + 5.88873I
5
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
4
(u
5
+ 5u
4
20u
2
u 1)
c
2
(u + 1)
4
(u
5
+ 25u
4
+ 198u
3
+ 390u
2
39u + 1)
c
3
, c
6
u
4
(u
5
+ 4u
4
+ 38u
3
+ 40u
2
40u + 16)
c
4
(u + 1)
4
(u
5
+ 5u
4
20u
2
u 1)
c
5
(u
4
+ u
3
+ u
2
+ 1)(u
5
+ 2u
4
+ 2u
3
u
2
u 1)
c
7
, c
8
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
5
+ 6u
3
+ u
2
u + 1)
c
9
(u
4
u
3
+ u
2
+ 1)(u
5
+ 2u
4
+ 2u
3
u
2
u 1)
c
10
, c
11
(u
4
u
3
+ 3u
2
2u + 1)(u
5
+ 6u
3
+ u
2
u + 1)
6
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
4
(y
5
25y
4
+ 198y
3
390y
2
39y 1)
c
2
(y 1)
4
(y
5
229y
4
+ 19626y
3
167594y
2
+ 741y 1)
c
3
, c
6
y
4
(y
5
+ 60y
4
+ 1044y
3
4768y
2
+ 320y 256)
c
5
, c
9
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
5
+ 6y
3
y
2
y 1)
c
7
, c
8
, c
10
c
11
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)(y
5
+ 12y
4
+ 34y
3
13y
2
y 1)
7