11n
101
(K11n
101
)
1
Arc Sequences
7 1 8 7 10 2 4 11 6 9 8
Solving Sequence
5,10 6,7
4 9 11 8 1 3 2
c
5
c
4
c
9
c
10
c
8
c
11
c
3
c
2
c
1
, c
6
, c
7
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hb
6
+ b
4
+ 2b
2
+ 1, b
5
+ b + u, b
5
+ b
4
+ b
3
+ 2b + a + 1i
I
u
2
= hu
12
+ u
11
+ 2u
10
+ 2u
9
+ 5u
8
+ 5u
7
+ 13u
6
+ 11u
5
+ 15u
4
+ 11u
3
+ 8u
2
+ 4u + 1,
402u
11
264u
10
+ ··· + 281a 157, 290u
11
224u
10
+ ··· + 281b 576i
I
u
3
= hu
13
+ u
11
u
10
+ 7u
9
+ 3u
7
5u
6
+ 8u
5
+ u
4
+ 4u
3
2u
2
1,
3u
12
+ 3u
11
+ u
9
+ 14u
8
+ 24u
7
13u
6
2u
5
2u
4
+ 37u
3
7u
2
+ 8b + u 11,
3u
12
5u
11
+ 2u
10
3u
9
14u
8
38u
7
+ 21u
6
2u
5
6u
4
53u
3
+ 9u
2
+ 8a 3u + 7i
There are 3 irreducible components with 31 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
= hb
6
+ b
4
+ 2b
2
+ 1, b
5
+ b + u, b
5
+ b
4
+ b
3
+ 2b + a + 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
b
5
b
4
b
3
2b 1
b
a
6
=
b
5
+ b
b
2
a
7
=
0
b
5
b
a
4
=
1
1
a
9
=
b
5
b
3
+ b
2
2b
b
3
+ b
a
11
=
b
5
b
3
b 1
b
5
+ b
3
+ b
a
8
=
b
5
b
0
a
1
=
1
b
5
+ b
3
+ b
a
3
=
0
1
a
2
=
1
b
5
+ b
3
+ b 1
a
2
=
1
b
5
+ b
3
+ b 1
(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 = 1.00000I
a = 0.100079 + 1.224638I
b = 0.744862 0.877439I
0.26574 + 2.82812I 0.49024 2.97945I
u = 1.00000I
a = 0.100079 1.224638I
b = 0.744862 + 0.877439I
0.26574 2.82812I 0.49024 + 2.97945I
u = 1.00000I
a = 1.32472 + 1.32472I
b = 0.754878I
4.40332 7.01951
u = 1.00000I
a = 1.32472 1.32472I
b = 0.754878I
4.40332 7.01951
u = 1.00000I
a = 1.224638 + 0.100079I
b = 0.744862 0.877439I
0.26574 2.82812I 0.49024 + 2.97945I
u = 1.00000I
a = 1.224638 0.100079I
b = 0.744862 + 0.877439I
0.26574 + 2.82812I 0.49024 2.97945I
3
II. I
u
2
= hu
12
+ u
11
+ · · · + 4u + 1, 402u
11
264u
10
+ · · · + 281a
157, 290u
11
224u
10
+ · · · + 281b 576i
(i) Arc colorings
a
5
=
1
0
a
10
=
1.43060u
11
+ 0.939502u
10
+ ··· + 4.47687u + 0.558719
1.03203u
11
+ 0.797153u
10
+ ··· + 5.01068u + 2.04982
a
6
=
0.220641u
11
+ 0.935943u
10
+ ··· + 4.74021u + 3.12100
0.779359u
11
0.0640569u
10
+ ··· 3.25979u 0.879004
a
7
=
0
u
a
4
=
1
u
2
a
9
=
0.754448u
11
0.555160u
10
+ ··· 4.91815u 3.28470
0.896797u
11
+ 0.320285u
10
+ ··· + 5.29893u + 1.39502
a
11
=
0.163701u
11
+ 0.370107u
10
+ ··· 2.38790u 0.476868
0.199288u
11
0.0711744u
10
+ ··· + 0.266904u 0.754448
a
8
=
u
u
3
+ u
a
1
=
0.879004u
11
0.0996441u
10
+ ··· 4.62633u 1.25623
0.715302u
11
0.469751u
10
+ ··· 2.23843u 1.77936
a
3
=
u
2
+ 1
u
4
+ 2u
2
a
2
=
0.879004u
11
0.0996441u
10
+ ··· 4.62633u 1.25623
1
a
2
=
0.879004u
11
0.0996441u
10
+ ··· 4.62633u 1.25623
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 = 1.18584 0.84722I
a = 0.951052 + 0.505357I
b = 0.932789 + 0.951611I
12.06461 3.42721I 5.95500 + 2.25224I
u = 1.18584 + 0.84722I
a = 0.951052 0.505357I
b = 0.932789 0.951611I
12.06461 + 3.42721I 5.95500 2.25224I
u = 0.419110 0.222236I
a = 0.496924 0.802130I
b = 0.716019 0.809696I
1.35295 2.65597I 5.58115 + 3.39809I
u = 0.419110 + 0.222236I
a = 0.496924 + 0.802130I
b = 0.716019 + 0.809696I
1.35295 + 2.65597I 5.58115 3.39809I
u = 0.247920 0.814674I
a = 2.14725 0.42856I
b = 0.283231 0.633899I
3.54796 + 1.10871I 0.46385 6.18117I
u = 0.247920 + 0.814674I
a = 2.14725 + 0.42856I
b = 0.283231 + 0.633899I
3.54796 1.10871I 0.46385 + 6.18117I
u = 0.073688 1.173753I
a = 0.276467 1.109671I
b = 0.283231 + 0.633899I
3.54796 1.10871I 0.46385 + 6.18117I
u = 0.073688 + 1.173753I
a = 0.276467 + 1.109671I
b = 0.283231 0.633899I
3.54796 + 1.10871I 0.46385 6.18117I
u = 0.276186 0.937280I
a = 0.825821 + 0.685494I
b = 0.716019 0.809696I
1.35295 2.65597I 5.58115 + 3.39809I
u = 0.276186 + 0.937280I
a = 0.825821 0.685494I
b = 0.716019 + 0.809696I
1.35295 + 2.65597I 5.58115 3.39809I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.15037 0.92808I
a = 1.50706 0.08967I
b = 0.932789 + 0.951611I
12.06461 3.42721I 5.95500 + 2.25224I
u = 1.15037 + 0.92808I
a = 1.50706 + 0.08967I
b = 0.932789 0.951611I
12.06461 + 3.42721I 5.95500 2.25224I
6
III. I
u
3
=
hu
13
+u
11
+· · ·2u
2
1, 3u
12
+3u
11
+· · ·+8b11, 3u
12
5u
11
+· · ·+8a+7i
(i) Arc colorings
a
5
=
1
0
a
10
=
3
8
u
12
+
5
8
u
11
+ ··· +
3
8
u
7
8
3
8
u
12
3
8
u
11
+ ···
1
8
u +
11
8
a
6
=
u
1
8
u
12
+
1
8
u
11
+ ···
9
8
u
1
8
a
7
=
0
u
a
4
=
1
u
2
a
9
=
3
4
u
12
+
1
4
u
11
+ ··· u
1
2
5
8
u
12
1
8
u
11
+ ··· +
15
8
u +
3
8
a
11
=
1
8
u
12
1
8
u
11
+ ···
1
8
u +
9
8
1
8
u
12
+
1
8
u
11
+ ··· +
1
8
u
1
8
a
8
=
u
u
3
+ u
a
1
=
1
1
8
u
12
+
1
8
u
11
+ ··· +
1
8
u
1
8
a
3
=
u
2
+ 1
u
4
+ 2u
2
a
2
=
1
1
8
u
12
+
1
8
u
11
+ ··· +
1
8
u
1
8
a
2
=
1
1
8
u
12
+
1
8
u
11
+ ··· +
1
8
u
1
8
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.95082 1.19131I
a = 1.63050 0.50532I
b = 0.885010 + 1.004921I
10.8970 + 11.1670I 4.44754 6.34112I
u = 0.95082 + 1.19131I
a = 1.63050 + 0.50532I
b = 0.885010 1.004921I
10.8970 11.1670I 4.44754 + 6.34112I
u = 0.745925 0.860258I
a = 1.37950 + 0.99779I
b = 0.534793 1.010413I
1.03858 + 7.07395I 2.58380 8.11816I
u = 0.745925 + 0.860258I
a = 1.37950 0.99779I
b = 0.534793 + 1.010413I
1.03858 7.07395I 2.58380 + 8.11816I
u = 0.438163 0.579645I
a = 0.62882 1.49528I
b = 0.014294 + 0.955793I
2.29540 + 1.46021I 0.76105 4.77537I
u = 0.438163 + 0.579645I
a = 0.62882 + 1.49528I
b = 0.014294 0.955793I
2.29540 1.46021I 0.76105 + 4.77537I
u = 0.052177 0.598239I
a = 0.165174 + 1.066639I
b = 0.696907 0.878523I
1.24085 2.67797I 5.53095 + 2.23117I
u = 0.052177 + 0.598239I
a = 0.165174 1.066639I
b = 0.696907 + 0.878523I
1.24085 + 2.67797I 5.53095 2.23117I
u = 0.622206
a = 0.720555
b = 0.524309
0.957360 10.3808
u = 0.871545 0.665952I
a = 0.501025 0.543340I
b = 0.780232 0.483051I
2.81429 2.20167I 6.81300 + 2.37182I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.871545 + 0.665952I
a = 0.501025 + 0.543340I
b = 0.780232 + 0.483051I
2.81429 + 2.20167I 6.81300 2.37182I
u = 1.00444 1.14917I
a = 0.595760 + 0.655816I
b = 0.956659 + 0.875818I
11.32248 4.40088I 5.19535 + 1.84237I
u = 1.00444 + 1.14917I
a = 0.595760 0.655816I
b = 0.956659 0.875818I
11.32248 + 4.40088I 5.19535 1.84237I
9
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
3
, c
4
c
6
, c
7
(u
2
+ 1)
3
(u
12
+ u
11
+ ··· + 4u + 1)
(u
13
+ u
11
u
10
+ 7u
9
+ 3u
7
5u
6
+ 8u
5
+ u
4
+ 4u
3
2u
2
1)
c
2
(u + 1)
6
(u
12
+ 3u
11
+ ··· + 6u
2
+ 1)(u
13
+ 2u
12
+ ··· 4u 1)
c
5
, c
9
(u
6
+ u
4
+ 2u
2
+ 1)(u
6
u
5
+ u
4
+ 2u
2
u + 1)
2
(u
13
+ 3u
12
+ ··· + 5u + 2)
c
8
, c
10
, c
11
(u
3
+ u
2
+ 2u + 1)
2
(u
6
+ u
5
+ 5u
4
+ 4u
3
+ 6u
2
+ 3u + 1)
2
(u
13
+ 3u
12
+ ··· + 5u 4)
10
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
4
c
6
, c
7
(y + 1)
6
(y
12
+ 3y
11
+ ··· + 6y
2
+ 1)(y
13
+ 2y
12
+ ··· 4y 1)
c
2
(y 1)
6
(y
12
+ 11y
11
+ ··· + 12y + 1)(y
13
+ 26y
12
+ ··· + 12y 1)
c
5
, c
9
(y
3
+ y
2
+ 2y + 1)
2
(y
6
+ y
5
+ 5y
4
+ 4y
3
+ 6y
2
+ 3y + 1)
2
(y
13
+ 3y
12
+ ··· + 5y 4)
c
8
, c
10
, c
11
(y
3
+ 3y
2
+ 2y 1)
2
(y
6
+ 9y
5
+ 29y
4
+ 40y
3
+ 22y
2
+ 3y + 1)
2
(y
13
+ 15y
12
+ ··· + 177y 16)
11