11n
80
(K11n
80
)
1
Arc Sequences
5 1 7 2 4 10 3 11 1 7 9
Solving Sequence
2,4
5
6,10
7 1 3 8 9 11
c
4
c
5
c
6
c
1
c
3
c
7
c
9
c
11
c
2
, c
8
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= ha
4
a
3
+ 2a
2
+ a + 1, a
3
2a
2
+ 2b + 2a + 1, a
3
+ 2a
2
2a + 2u + 1i
I
u
2
= hu
4
u
3
+ u
2
+ 1, b + u 1, u
3
+ u
2
+ a 1i
I
u
3
= hu
15
4u
14
+ ··· + 12u 1, 19u
14
+ 63u
13
+ ··· + 2098b 1803,
1482u
14
5963u
13
+ ··· + 2098a + 12656i
There are 3 irreducible components with 23 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
= ha
4
a
3
+ 2a
2
+ a + 1, a
3
2a
2
+ 2b + 2a + 1, a
3
+ 2a
2
2a + 2u + 1i
(i) Arc colorings
a
2
=
1
0
a
4
=
0
1
2
a
3
a
2
+ a
1
2
a
5
=
1
2
a
3
a
2
+ a
1
2
1
2
a
3
a
2
+ a
1
2
a
6
=
1
2
a
3
a
2
+ a
1
2
1
2
a
3
a
2
+ a +
1
2
a
10
=
a
1
2
a
3
+ a
2
a
1
2
a
7
=
0
3
2
a
3
2a
2
+ 3a +
3
2
a
1
=
1
2
a
3
+ a
2
a +
1
2
1
2
a
3
+ a
2
a
1
2
a
3
=
0
1
2
a
3
a
2
+ a
1
2
a
8
=
0
3
2
a
3
2a
2
+ 3a +
3
2
a
9
=
1
2
a
3
+ a
2
a +
1
2
3
2
a
3
+ 2a
2
3a
3
2
a
11
=
a
2a
3
2a
2
+ 4a + 2
a
11
=
a
2a
3
2a
2
+ 4a + 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 = 0.500000 0.866025I
a = 0.309017 0.535233I
b = 0.500000 + 0.866025I
8.88264 + 2.02988I 4.50000 + 2.34537I
u = 0.500000 + 0.866025I
a = 0.309017 + 0.535233I
b = 0.500000 0.866025I
8.88264 2.02988I 4.50000 2.34537I
u = 0.500000 + 0.866025I
a = 0.80902 1.40126I
b = 0.500000 0.866025I
0.98696 2.02988I 4.50000 + 9.27358I
u = 0.500000 0.866025I
a = 0.80902 + 1.40126I
b = 0.500000 + 0.866025I
0.98696 + 2.02988I 4.50000 9.27358I
3
II. I
u
2
= hu
4
u
3
+ u
2
+ 1, b + u 1, u
3
+ u
2
+ a 1i
(i) Arc colorings
a
2
=
1
0
a
4
=
0
u
a
5
=
u
u
a
6
=
u
u
3
+ u
a
10
=
u
3
u
2
+ 1
u + 1
a
7
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
2
a
3
=
u
3
u
3
u
2
1
a
8
=
u
2
1
u
2
a
9
=
u
3
2u
2
u
2
u + 1
a
11
=
u
3
u
2
+ 1
u + 1
a
11
=
u
3
u
2
+ 1
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 = 0.351808 0.720342I
a = 1.89923 0.40053I
b = 1.35181 + 0.72034I
1.43393 + 1.41510I 11.48794 2.21528I
u = 0.351808 + 0.720342I
a = 1.89923 + 0.40053I
b = 1.35181 0.72034I
1.43393 1.41510I 11.48794 + 2.21528I
u = 0.851808 0.911292I
a = 0.399232 + 0.325640I
b = 0.148192 + 0.911292I
8.43568 3.16396I 4.01206 + 4.08190I
u = 0.851808 + 0.911292I
a = 0.399232 0.325640I
b = 0.148192 0.911292I
8.43568 + 3.16396I 4.01206 4.08190I
5
III. I
u
3
= hu
15
4u
14
+ · · · + 12u 1, 19u
14
+ 63u
13
+ · · · + 2098b
1803, 1482u
14
5963u
13
+ · · · + 2098a + 12656i
(i) Arc colorings
a
2
=
1
0
a
4
=
0
u
a
5
=
u
u
a
6
=
u
u
3
+ u
a
10
=
0.706387u
14
+ 2.84223u
13
+ ··· + 6.91849u 6.03241
0.00905624u
14
0.0300286u
13
+ ··· 2.28742u + 0.859390
a
7
=
0.337941u
14
1.38370u
13
+ ··· 4.67255u + 3.64776
0.0896092u
14
+ 0.376072u
13
+ ··· + 2.02812u 0.477121
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
+ u
2
+ 1
u
4
a
8
=
0.447092u
14
1.95615u
13
+ ··· 8.29457u + 4.26883
0.163489u
14
+ 0.252622u
13
+ ··· 0.153480u 0.277407
a
9
=
0.859390u
14
+ 3.42850u
13
+ ··· + 9.95853u 7.02526
0.0104862u
14
+ 0.166349u
13
+ ··· 0.693518u + 0.715443
a
11
=
0.409438u
14
+ 1.69971u
13
+ ··· + 4.12583u 3.90610
0.0614871u
14
0.361773u
13
+ ··· 3.31983u + 0.782173
a
11
=
0.409438u
14
+ 1.69971u
13
+ ··· + 4.12583u 3.90610
0.0614871u
14
0.361773u
13
+ ··· 3.31983u + 0.782173
(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 = 0.443471 0.899923I
a = 2.62437 + 1.29922I
b = 0.36489 + 2.30827I
1.31612 + 1.82919I 23.9935 13.4254I
u = 0.443471 + 0.899923I
a = 2.62437 1.29922I
b = 0.36489 2.30827I
1.31612 1.82919I 23.9935 + 13.4254I
u = 0.416218 0.666363I
a = 0.385003 + 0.578192I
b = 0.427401 + 0.423054I
0.075833 + 1.377122I 0.42484 4.74084I
u = 0.416218 + 0.666363I
a = 0.385003 0.578192I
b = 0.427401 0.423054I
0.075833 1.377122I 0.42484 + 4.74084I
u = 0.136912 1.276844I
a = 0.595692 0.052604I
b = 0.10821 + 1.48209I
2.05262 + 0.52363I 2.28909 0.30141I
u = 0.136912 + 1.276844I
a = 0.595692 + 0.052604I
b = 0.10821 1.48209I
2.05262 0.52363I 2.28909 + 0.30141I
u = 0.0927870
a = 5.52238
b = 0.683573
1.10369 8.82437
u = 0.39863 1.51864I
a = 1.057418 + 0.254123I
b = 0.289448 + 1.141243I
11.10028 1.26356I 2.58190 + 0.63912I
u = 0.39863 + 1.51864I
a = 1.057418 0.254123I
b = 0.289448 1.141243I
11.10028 + 1.26356I 2.58190 0.63912I
u = 0.68964 1.30605I
a = 1.193831 0.355171I
b = 0.65950 1.93623I
8.47013 10.83431I 4.46568 + 4.98924I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.68964 + 1.30605I
a = 1.193831 + 0.355171I
b = 0.65950 + 1.93623I
8.47013 + 10.83431I 4.46568 4.98924I
u = 0.707815 0.947595I
a = 0.104702 + 0.150418I
b = 0.299642 0.759085I
9.44393 2.71266I 11.43593 + 3.34052I
u = 0.707815 + 0.947595I
a = 0.104702 0.150418I
b = 0.299642 + 0.759085I
9.44393 + 2.71266I 11.43593 3.34052I
u = 1.154123 0.257445I
a = 0.124420 1.403621I
b = 0.45790 1.43110I
5.23991 + 4.29122I 5.74651 1.92061I
u = 1.154123 + 0.257445I
a = 0.124420 + 1.403621I
b = 0.45790 + 1.43110I
5.23991 4.29122I 5.74651 + 1.92061I
8
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)
2
(u
4
u
3
+ u
2
+ 1)(u
15
+ 4u
14
+ ··· + 12u + 1)
c
2
, c
5
(u
2
+ u + 1)
2
(u
4
+ u
3
+ ··· + 2u + 1)(u
15
+ 10u
14
+ ··· + 112u 1)
c
3
u
4
(u
4
u
3
+ ··· 2u + 1)(u
15
+ 2u
14
+ ··· + 16u + 16)
c
4
(u
2
u + 1)
2
(u
4
+ u
3
+ u
2
+ 1)(u
15
+ 4u
14
+ ··· + 12u + 1)
c
6
u
4
(u
2
u 1)
2
(u
15
+ 3u
14
+ ··· 24u + 16)
c
7
u
4
(u
4
+ u
3
+ ··· + 2u + 1)(u
15
+ 2u
14
+ ··· + 16u + 16)
c
8
, c
9
(u + 1)
4
(u
2
u 1)
2
(u
15
+ 7u
14
+ ··· 16u 1)
c
10
u
4
(u
2
+ u 1)
2
(u
15
+ 3u
14
+ ··· 24u + 16)
c
11
(u 1)
4
(u
2
+ u 1)
2
(u
15
+ 7u
14
+ ··· 16u 1)
9
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
2
+ y + 1)
2
(y
4
+ y
3
+ ··· + 2y + 1)(y
15
+ 10y
14
+ ··· + 112y 1)
c
2
, c
5
(y
2
+ y + 1)
2
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
15
6y
14
+ ··· + 13488y 1)
c
3
, c
7
y
4
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
15
20y
14
+ ··· + 128y 256)
c
6
, c
10
y
4
(y
2
3y + 1)
2
(y
15
+ 21y
14
+ ··· 1984y 256)
c
8
, c
9
, c
11
(y 1)
4
(y
2
3y + 1)
2
(y
15
3y
14
+ ··· + 134y 1)
10