11n
11
(K11n
11
)
1
Arc Sequences
5 1 8 2 3 10 3 11 1 7 9
Solving Sequence
1,5
2 3
6,9
10 7 4 11 8
c
1
c
2
c
5
c
9
c
6
c
4
c
11
c
8
c
3
, c
7
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
4
3a
3
+ 5a
2
6a + 4, a
3
+ a
2
a + 2u + 2, a
3
+ 3a
2
+ 2b 3a + 2i
I
u
2
= hu
31
3u
30
+ ··· + 8u + 1,
3907187141u
30
+ 9907766282u
29
+ ··· + 11106194846b 5307442484,
4690596084u
30
+ 17967804193u
29
+ ··· + 11106194846a 56611875743i
There are 2 irreducible components with 35 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
3a
3
+ 5a
2
6a + 4, a
3
+ a
2
a + 2u + 2, a
3
+ 3a
2
+ 2b 3a + 2i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
2
a
3
1
2
a
2
+
1
2
a 1
a
2
=
1
1
2
a
3
1
2
a
2
+
1
2
a
a
3
=
1
2
a
3
+
1
2
a
2
1
2
a + 1
1
2
a
3
1
2
a
2
+
1
2
a
a
6
=
1
0
a
9
=
a
1
2
a
3
3
2
a
2
+
3
2
a 1
a
10
=
1
2
a
3
+
3
2
a
2
1
2
a + 1
1
2
a
3
3
2
a
2
+
3
2
a 1
a
7
=
1
2
a
3
1
2
a
2
1
2
a + 1
1
2
a
3
+
3
2
a
2
3
2
a
a
4
=
1
2
a
3
+
1
2
a
2
1
2
a + 1
1
2
a
3
1
2
a
2
+
1
2
a
a
11
=
a
2
2a + 3
1
2
a
3
+
3
2
a
2
3
2
a
a
8
=
1
2
a
3
1
2
a
2
1
2
a + 1
1
2
a
3
+
3
2
a
2
3
2
a
a
8
=
1
2
a
3
1
2
a
2
1
2
a + 1
1
2
a
3
+
3
2
a
2
3
2
a
(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 = 0.19098 1.40126I
b = 1.61803
8.88264 2.02988I 4.50000 + 9.27358I
u = 0.500000 0.866025I
a = 0.19098 + 1.40126I
b = 1.61803
8.88264 + 2.02988I 4.50000 9.27358I
u = 0.500000 0.866025I
a = 1.30902 0.53523I
b = 0.618034
0.98696 + 2.02988I 4.50000 + 2.34537I
u = 0.500000 + 0.866025I
a = 1.30902 + 0.53523I
b = 0.618034
0.98696 2.02988I 4.50000 2.34537I
3
II. I
u
2
=
hu
31
3u
30
+ · · · + 8u + 1, 3.91 ×10
9
u
30
+ 9.91 × 10
9
u
29
+ · · · + 1.11× 10
10
b
5.31 × 10
9
, 4.69 × 10
9
u
30
+ 1.80 × 10
10
u
29
+ · · · + 1.11 × 10
10
a 5.66 × 10
10
i
(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
5
2u
3
u
u
5
+ u
3
+ u
a
9
=
0.422341u
30
1.61782u
29
+ ··· + 7 .34852u + 5.09732
0.351803u
30
0.892094u
29
+ ··· 2.21764u + 0.477881
a
10
=
0.0705380u
30
0.725725u
29
+ ··· + 9 .56615u + 4.61944
0.351803u
30
0.892094u
29
+ ··· 2.21764u + 0.477881
a
7
=
0.0298066u
30
+ 0.369831u
29
+ ··· + 11 .0518u + 3.33706
0.462869u
30
+ 1.23400u
29
+ ··· + 2 .39823u + 0.640946
a
4
=
u
u
3
+ u
a
11
=
1.04116u
30
1.86337u
29
+ ··· 10.3236u 3.63282
1.09279u
30
+ 3.43163u
29
+ ··· + 12 .6294u + 0.911525
a
8
=
2.22119u
30
4.64554u
29
+ ··· 1.83225u + 2.09311
2.01803u
30
+ 5.92094u
29
+ ··· + 15 .6764u + 2.22119
a
8
=
2.22119u
30
4.64554u
29
+ ··· 1.83225u + 2.09311
2.01803u
30
+ 5.92094u
29
+ ··· + 15 .6764u + 2.22119
(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 = 0.899829 0.326300I
a = 0.009918 0.301415I
b = 0.679882 + 0.287551I
1.58742 + 1.54591I 2.94722 4.18501I
u = 0.899829 + 0.326300I
a = 0.009918 + 0 .301415I
b = 0.679882 0.287551I
1.58742 1.54591I 2.94722 + 4.18501I
u = 0.809239 1.066072I
a = 0.106100 1.009315I
b = 0.931324 + 0.285581I
0.02246 + 4.44381I 1.53825 8.61147I
u = 0.809239 + 1.066072I
a = 0.106100 + 1.009315I
b = 0.931324 0.285581I
0.02246 4.44381I 1.53825 + 8.61147I
u = 0.764060 0.674237I
a = 0.127848 + 0.428082I
b = 0.391432 0.273361I
1.30059 + 1.62044I 1.54713 2.13328I
u = 0.764060 + 0.674237I
a = 0.127848 0.428082I
b = 0.391432 + 0.273361I
1.30059 1.62044I 1.54713 + 2.13328I
u = 0.546636 0.848867I
a = 2.58814 + 2.04636I
b = 0.912773 0.075536I
1.30863 + 2.18648I 32.4747 + 4.2586I
u = 0.546636 + 0.848867I
a = 2.58814 2.04636I
b = 0.912773 + 0 .075536I
1.30863 2.18648I 32.4747 4.2586I
u = 0.385132 0.990364I
a = 0.356417 + 0.108036I
b = 0.111818 0.363270I
0.54852 + 2.74241I 0.76165 6.33975I
u = 0.385132 + 0.990364I
a = 0.356417 0.108036I
b = 0.111818 + 0 .363270I
0.54852 2.74241I 0.76165 + 6.33975I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.223764 1.287054I
a = 0.939284 0.416238I
b = 1.168088 + 0.283606I
4.01618 + 5.37811I 8.38800 7.62748I
u = 0.223764 + 1.287054I
a = 0.939284 + 0.416238I
b = 1.168088 0.283606I
4.01618 5.37811I 8.38800 + 7.62748I
u = 0.218572 0.836243I
a = 2.58492 0.35963I
b = 1.193907 0.091335I
2.79337 + 1.66318I 5.72575 2.19283I
u = 0.218572 + 0.836243I
a = 2.58492 + 0 .35963I
b = 1.193907 + 0 .091335I
2.79337 1.66318I 5.72575 + 2.19283I
u = 0.118473
a = 4.59349
b = 0.779230
1.12597 9.35891
u = 0.065762 0.590377I
a = 0.46658 1.70189I
b = 0.623364 + 0 .404541I
1.47821 0.10102I 8.24537 + 0.31125I
u = 0.065762 + 0.590377I
a = 0.46658 + 1 .70189I
b = 0.623364 0.404541I
1.47821 + 0.10102I 8.24537 0.31125I
u = 0.395500 0.773342I
a = 0.23219 + 1.75077I
b = 1.60218 0.05522I
9.04764 1.61419I 11.37069 6.82904I
u = 0.395500 + 0 .773342I
a = 0.23219 1.75077I
b = 1.60218 + 0.05522I
9.04764 + 1.61419I 11.37069 + 6.82904I
u = 0.780157 1.088848I
a = 0.36111 + 1.73816I
b = 1.37851 0.54460I
2.96987 13.05181I 4.52638 + 7.60952I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.780157 + 1 .088848I
a = 0.36111 1.73816I
b = 1.37851 + 0.54460I
2.96987 + 13.05181I 4.52638 7.60952I
u = 0.794359 0.925045I
a = 0.22991 1.78336I
b = 1.42534 + 0 .52079I
3.19514 4.83034I 2.76361 + 3.70838I
u = 0.794359 + 0 .925045I
a = 0.22991 + 1 .78336I
b = 1.42534 0.52079I
3.19514 + 4.83034I 2.76361 3.70838I
u = 0.813553 0.859384I
a = 0.604357 + 0.213999I
b = 1.33715 0.61035I
3.39903 1.19447I 2.39419 + 1.64836I
u = 0.813553 + 0 .859384I
a = 0.604357 0.213999I
b = 1.33715 + 0 .61035I
3.39903 + 1.19447I 2.39419 1.64836I
u = 0.821771 0.994451I
a = 0.641022 1.066048I
b = 0.039656 + 1 .102600I
7.38206 7.22461I 1.02588 + 4.93399I
u = 0.821771 + 0 .994451I
a = 0.641022 + 1 .066048I
b = 0.039656 1.102600I
7.38206 + 7.22461I 1.02588 4.93399I
u = 0.900705 0.813465I
a = 0.536594 + 1.207935I
b = 0.094951 1.070965I
7.95570 + 0.85651I 0.056335 + 0.135364I
u = 0.900705 + 0 .813465I
a = 0.536594 1.207935I
b = 0.094951 + 1.070965I
7.95570 0.85651I 0.056335 0.135364I
u = 0.966186 0.671432I
a = 0.622938 0.524513I
b = 1.287254 + 0.574250I
4.28029 + 6.65397I 2.65676 3.57953I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.966186 + 0 .671432I
a = 0.622938 + 0 .524513I
b = 1.287254 0.574250I
4.28029 6.65397I 2.65676 + 3.57953I
7
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)
2
(u
31
+ 3u
30
+ ··· + 8u 1)
c
2
(u
2
+ u + 1)
2
(u
31
+ 9u
30
+ ··· + 60 u 1)
c
3
, c
7
u
4
(u
31
+ 3u
30
+ ··· + 112 u + 16)
c
4
(u
2
u + 1)
2
(u
31
+ 3u
30
+ ··· + 8u 1)
c
5
(u
2
+ u + 1)
2
(u
31
+ 3u
30
+ ··· + 4454u + 977)
c
6
(u
2
+ u 1)
2
(u
31
+ 3u
30
+ ··· 2u
2
+ 1)
c
8
, c
9
(u
2
+ u 1)
2
(u
31
+ 3u
30
+ ··· 2u 1)
c
10
(u
2
u 1)
2
(u
31
+ 3u
30
+ ··· 2u
2
+ 1)
c
11
(u
2
u 1)
2
(u
31
+ 3u
30
+ ··· 2u 1)
8
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
2
+ y + 1)
2
(y
31
+ 9y
30
+ ··· + 60 y 1)
c
2
(y
2
+ y + 1)
2
(y
31
+ 29y
30
+ ··· + 5084 y 1)
c
3
, c
7
y
4
(y
31
25y
30
+ ··· + 1152y 256)
c
5
(y
2
+ y + 1)
2
(y
31
+ 49y
30
+ ··· + 4 .45523 × 10
7
y 954529)
c
6
, c
10
(y
2
3y + 1)
2
(y
31
3y
30
+ ··· + 4 y 1)
c
8
, c
9
, c
11
(y
2
3y + 1)
2
(y
31
23y
30
+ ··· + 4y 1)
9