11n
103
(K11n
103
)
1
Arc Sequences
6 1 8 11 2 4 3 7 1 4 10
Solving Sequence
4,10
11 5
1,8
3 2 7 6 9
c
10
c
4
c
11
c
3
c
2
c
7
c
6
c
9
c
1
, c
5
, c
8
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hu
12
4u
10
+ 7u
8
+ u
7
4u
6
3u
5
2u
4
+ 3u
3
+ 3u
2
1, b u,
u
11
+ u
10
4u
9
3u
8
+ 6u
7
+ 4u
6
u
4
7u
3
u
2
+ 2a + ui
I
u
2
= hu
28
u
27
+ ··· u
2
+ 1, 735606923u
27
133880069u
26
+ ··· + 373862627b 628531014,
3300346385u
27
+ 1114715184u
26
+ ··· + 373862627a + 4122165546i
I
u
3
= hu
4
u
2
+ 1, b + u, 2u
3
+ a + u 1i
I
u
4
= hu
4
u
2
+ 1, u
2
+ a, u
3
+ b ui
There are 4 irreducible components with 48 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
12
4u
10
+ · · · + 3u
2
1, b u, u
11
+ u
10
+ · · · + 2a + ui
(i) Arc colorings
a
4
=
1
0
a
10
=
1
2
u
11
1
2
u
10
+ ··· +
1
2
u
2
1
2
u
u
a
11
=
1
2
u
11
1
2
u
10
+ ··· +
1
2
u
2
+
1
2
u
u
a
5
=
1
2
u
11
3
2
u
9
+ ··· 2u
2
+
3
2
u
2
a
1
=
1
2
u
11
+
3
2
u
9
+ ··· + u +
1
2
u
3
+ u
a
8
=
0
u
a
3
=
1
u
2
a
2
=
u
11
1
2
u
10
+ ··· +
1
2
u +
1
2
1
2
u
10
1
2
u
9
+ ··· +
1
2
u
1
2
a
7
=
u
u
3
+ u
a
6
=
u
3
u
3
+ u
a
9
=
u
3
u
5
u
3
+ u
a
9
=
u
3
u
5
u
3
+ u
(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 = 1.278420 0.477758I
a = 2.05428 + 1.02363I
b = 1.278420 0.477758I
9.66215 + 6.36480I 11.90320 3.80747I
u = 1.278420 + 0.477758I
a = 2.05428 1.02363I
b = 1.278420 + 0.477758I
9.66215 6.36480I 11.90320 + 3.80747I
u = 1.18703
a = 2.12208
b = 1.18703
5.86925 15.2180
u = 0.583435 0.389720I
a = 0.567285 0.807318I
b = 0.583435 0.389720I
2.15137 + 2.12179I 2.80850 2.85099I
u = 0.583435 + 0.389720I
a = 0.567285 + 0.807318I
b = 0.583435 + 0.389720I
2.15137 2.12179I 2.80850 + 2.85099I
u = 0.123324 0.852412I
a = 0.114185 + 0.560956I
b = 0.123324 0.852412I
1.59916 2.93187I 4.52833 + 2.34392I
u = 0.123324 + 0.852412I
a = 0.114185 0.560956I
b = 0.123324 + 0.852412I
1.59916 + 2.93187I 4.52833 2.34392I
u = 0.541203
a = 0.399849
b = 0.541203
0.811000 12.4599
u = 1.070851 0.293331I
a = 2.81627 + 0.02424I
b = 1.070851 0.293331I
0.88385 3.90933I 11.22062 + 4.69963I
u = 1.070851 + 0.293331I
a = 2.81627 0.02424I
b = 1.070851 + 0.293331I
0.88385 + 3.90933I 11.22062 4.69963I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.237241 0.570642I
a = 1.91766 + 1.40726I
b = 1.237241 0.570642I
8.0502 13.4162I 9.70040 + 8.08040I
u = 1.237241 + 0.570642I
a = 1.91766 1.40726I
b = 1.237241 + 0.570642I
8.0502 + 13.4162I 9.70040 8.08040I
4
II.
I
u
2
= hu
28
u
27
+· · ·u
2
+1, 7.36×10
8
u
27
1.34×10
8
u
26
+· · ·+3.74×10
8
b
6.29 × 10
8
, 3.30 × 10
9
u
27
+ 1.11 × 10
9
u
26
+ · · · + 3.74 × 10
8
a + 4.12 × 10
9
i
(i) Arc colorings
a
4
=
1
0
a
10
=
8.82770u
27
2.98162u
26
+ ··· 18.8225u 11.0259
1.96759u
27
+ 0.358100u
26
+ ··· + 2.54082u + 1.68118
a
11
=
6.86011u
27
2.62352u
26
+ ··· 16.2817u 9.34470
1.96759u
27
+ 0.358100u
26
+ ··· + 2.54082u + 1.68118
a
5
=
0.604072u
27
0.672673u
26
+ ··· 2.58371u + 1.13808
4.82738u
27
+ 1.39404u
26
+ ··· + 6.80258u + 5.67111
a
1
=
9.69214u
27
2.16126u
26
+ ··· 21.1172u 12.6487
3.54018u
27
1.27699u
26
+ ··· 6.39747u 5.90810
a
8
=
0
u
a
3
=
1
u
2
a
2
=
11.1683u
27
2.69831u
26
+ ··· 24.1271u 14.6661
2.00633u
27
0.821463u
26
+ ··· 3.57032u 3.52639
a
7
=
u
u
3
+ u
a
6
=
u
3
u
3
+ u
a
9
=
u
3
u
5
u
3
+ u
a
9
=
u
3
u
5
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
5
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.307490 0.340934I
a = 1.73157 0.35436I
b = 1.283467 0.468890I
9.73046 3.61450I 11.98207 + 2.36533I
u = 1.307490 + 0.340934I
a = 1.73157 + 0.35436I
b = 1.283467 + 0.468890I
9.73046 + 3.61450I 11.98207 2.36533I
u = 1.223102 0.521275I
a = 0.689321 0.331934I
b = 0.195025 0.938101I
4.87993 + 7.94699I 7.43585 5.15066I
u = 1.223102 + 0.521275I
a = 0.689321 + 0.331934I
b = 0.195025 + 0.938101I
4.87993 7.94699I 7.43585 + 5.15066I
u = 1.056903 0.376008I
a = 1.13106 1.12117I
b = 0.852569 + 0.695292I
0.44424 + 5.41715I 9.45457 7.85187I
u = 1.056903 + 0.376008I
a = 1.13106 + 1.12117I
b = 0.852569 0.695292I
0.44424 5.41715I 9.45457 + 7.85187I
u = 0.961456 0.569453I
a = 0.57202 1.76941I
b = 0.733151 + 0.047660I
1.13610 + 2.11470I 7.32861 1.96391I
u = 0.961456 + 0.569453I
a = 0.57202 + 1.76941I
b = 0.733151 0.047660I
1.13610 2.11470I 7.32861 + 1.96391I
u = 0.772763 0.566548I
a = 0.212171 0.821160I
b = 0.200841 + 0.299347I
1.79770 + 2.20081I 2.89649 4.56299I
u = 0.772763 + 0.566548I
a = 0.212171 + 0.821160I
b = 0.200841 0.299347I
1.79770 2.20081I 2.89649 + 4.56299I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.200841 0.299347I
a = 2.15734 0.65443I
b = 0.772763 + 0.566548I
1.79770 2.20081I 2.89649 + 4.56299I
u = 0.200841 + 0.299347I
a = 2.15734 + 0.65443I
b = 0.772763 0.566548I
1.79770 + 2.20081I 2.89649 4.56299I
u = 0.006608 0.931399I
a = 0.722066 + 0.536584I
b = 1.245824 0.402354I
5.74815 1.35811I 8.91559 + 0.70318I
u = 0.006608 + 0.931399I
a = 0.722066 0.536584I
b = 1.245824 + 0.402354I
5.74815 + 1.35811I 8.91559 0.70318I
u = 0.195025 0.938101I
a = 0.639670 + 0.847278I
b = 1.223102 0.521275I
4.87993 + 7.94699I 7.43585 5.15066I
u = 0.195025 + 0.938101I
a = 0.639670 0.847278I
b = 1.223102 + 0.521275I
4.87993 7.94699I 7.43585 + 5.15066I
u = 0.733151 0.047660I
a = 1.99409 2.00575I
b = 0.961456 + 0.569453I
1.13610 2.11470I 7.32861 + 1.96391I
u = 0.733151 + 0.047660I
a = 1.99409 + 2.00575I
b = 0.961456 0.569453I
1.13610 + 2.11470I 7.32861 1.96391I
u = 0.772691 0.621111I
a = 0.120289 0.111843I
b = 0.946438 + 0.337794I
0.225291 + 0.283453I 9.98682 1.62016I
u = 0.772691 + 0.621111I
a = 0.120289 + 0.111843I
b = 0.946438 0.337794I
0.225291 0.283453I 9.98682 + 1.62016I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.852569 0.695292I
a = 0.70263 1.46407I
b = 1.056903 + 0.376008I
0.44424 5.41715I 9.45457 + 7.85187I
u = 0.852569 + 0.695292I
a = 0.70263 + 1.46407I
b = 1.056903 0.376008I
0.44424 + 5.41715I 9.45457 7.85187I
u = 0.946438 0.337794I
a = 0.031893 + 0.158868I
b = 0.772691 + 0.621111I
0.225291 0.283453I 9.98682 + 1.62016I
u = 0.946438 + 0.337794I
a = 0.031893 0.158868I
b = 0.772691 0.621111I
0.225291 + 0.283453I 9.98682 1.62016I
u = 1.245824 0.402354I
a = 0.518510 0.375216I
b = 0.006608 0.931399I
5.74815 1.35811I 8.91559 + 0.70318I
u = 1.245824 + 0.402354I
a = 0.518510 + 0.375216I
b = 0.006608 + 0.931399I
5.74815 + 1.35811I 8.91559 0.70318I
u = 1.283467 0.468890I
a = 1.60742 0.68620I
b = 1.307490 0.340934I
9.73046 3.61450I 11.98207 + 2.36533I
u = 1.283467 + 0.468890I
a = 1.60742 + 0.68620I
b = 1.307490 + 0.340934I
9.73046 + 3.61450I 11.98207 2.36533I
8
III. I
u
3
= hu
4
u
2
+ 1, b + u, 2u
3
+ a + u 1i
(i) Arc colorings
a
4
=
1
0
a
10
=
2u
3
u + 1
u
a
11
=
2u
3
2u + 1
u
a
5
=
u 1
u
2
a
1
=
u
3
u
2
+ 1
u
3
u
a
8
=
0
u
a
3
=
1
u
2
a
2
=
u
3
u
2
+ 2
u
3
+ u
2
u
a
7
=
u
u
3
+ u
a
6
=
u
3
u
3
+ u
a
9
=
u
3
0
a
9
=
u
3
0
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
9
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.866025 0.500000I
a = 1.86603 1.50000I
b = 0.866025 + 0.500000I
1.64493 + 4.05977I 4.00000 6.92820I
u = 0.866025 + 0.500000I
a = 1.86603 + 1.50000I
b = 0.866025 0.500000I
1.64493 4.05977I 4.00000 + 6.92820I
u = 0.866025 0.500000I
a = 0.13397 1.50000I
b = 0.866025 + 0.500000I
1.64493 4.05977I 4.00000 + 6.92820I
u = 0.866025 + 0.500000I
a = 0.13397 + 1.50000I
b = 0.866025 0.500000I
1.64493 + 4.05977I 4.00000 6.92820I
10
IV. I
u
4
= hu
4
u
2
+ 1, u
2
+ a, u
3
+ b ui
(i) Arc colorings
a
4
=
1
0
a
10
=
u
2
u
3
+ u
a
11
=
u
3
u
2
+ u
u
3
+ u
a
5
=
u
2
+ u
u
2
1
a
1
=
u
3
u
2
+ u + 1
u
a
8
=
0
u
a
3
=
1
u
2
a
2
=
u
3
u
2
+ u + 2
u
2
+ u
a
7
=
u
u
3
+ u
a
6
=
u
3
u
3
+ u
a
9
=
u
3
0
a
9
=
u
3
0
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
11
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.866025 0.500000I
a = 0.500000 0.866025I
b = 0.866025 + 0.500000I
1.64493 4.00000
u = 0.866025 + 0.500000I
a = 0.500000 + 0.866025I
b = 0.866025 0.500000I
1.64493 4.00000
u = 0.866025 0.500000I
a = 0.500000 + 0.866025I
b = 0.866025 + 0.500000I
1.64493 4.00000
u = 0.866025 + 0.500000I
a = 0.500000 0.866025I
b = 0.866025 0.500000I
1.64493 4.00000
12
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
(u
2
+ 1)
4
(u
12
+ 5u
11
+ ··· 20u 4)
(1 + 2u
2
2u
3
+ 9u
4
10u
5
+ 18u
6
16u
7
+ 17u
8
10u
9
+ 8u
10
4u
11
+ 4u
12
2u
13
+ u
14
)
2
c
2
(u + 1)
8
(u
12
+ 5u
11
+ ··· 88u + 16)
(1 + 4u + 22u
2
+ 68u
3
+ 147u
4
+ 244u
5
+ 310u
6
+ 302u
7
+ 245u
8
+ 166u
9
+ 92u
10
+ 42u
11
+ 16u
12
+ 4u
13
+ u
14
)
2
c
3
, c
4
, c
7
c
10
(u
4
u
2
+ 1)
2
(u
12
4u
10
+ ··· + 3u
2
1)
(u
28
+ u
27
+ ··· u
2
+ 1)
c
6
(u
4
u
2
+ 1)
2
(u
12
+ 8u
10
+ ··· + 2u + 1)(u
28
+ 3u
27
+ ··· + 64u + 17)
c
8
, c
11
(u
2
u + 1)
4
(u
12
+ 8u
11
+ ··· + 6u + 1)(u
28
+ 15u
27
+ ··· + 2u + 1)
c
9
(u
2
+ u + 1)
4
(u
12
+ 8u
11
+ ··· + 6u + 1)(u
28
+ 15u
27
+ ··· + 2u + 1)
13
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y + 1)
8
(y
12
+ 5y
11
+ ··· 88y + 16)
(1 + 4y + 22y
2
+ 68y
3
+ 147y
4
+ 244y
5
+ 310y
6
+ 302y
7
+ 245y
8
+ 166y
9
+ 92y
10
+ 42y
11
+ 16y
12
+ 4y
13
+ y
14
)
2
c
2
(y 1)
8
(y
12
+ 5y
11
+ ··· 13088y + 256)
(1 + 28y + 234y
2
+ 512y
3
+ 139y
4
+ 168y
5
+ 1922y
6
+ 1698y
7
+ 509y
8
+ 418y
9
+ 564y
10
+ 342y
11
+ 104y
12
+ 16y
13
+ y
14
)
2
c
3
, c
4
, c
7
c
10
(y
2
y + 1)
4
(y
12
8y
11
+ ··· 6y + 1)(y
28
15y
27
+ ··· 2y + 1)
c
6
(y
2
y + 1)
4
(y
12
+ 16y
11
+ ··· 10y + 1)
(y
28
+ 21y
27
+ ··· 3178y + 289)
c
8
, c
9
, c
11
(y
2
+ y + 1)
4
(y
12
4y
11
+ ··· 10y + 1)(y
28
3y
27
+ ··· + 78y + 1)
14