11a
102
(K11a
102
)
1
Arc Sequences
6 1 7 10 2 4 3 11 5 8 9
Solving Sequence
8,11
9
1,3
2 7 4 6 10 5
c
8
c
11
c
2
c
7
c
3
c
6
c
10
c
4
c
1
, c
5
, c
9
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= ha
4
+ 3a
2
+ 1, a
3
2a + u, a
3
+ a
2
+ b 2a + 2i
I
u
2
= hu
4
u
3
+ 3u
2
2u + 1, u
3
+ 2b + 3u + 1, u
3
+ u
2
+ a 3u + 2i
I
u
3
= hu
15
+ 5u
13
+ 3u
12
+ 10u
11
+ 12u
10
+ 14u
9
+ 18u
8
+ 17u
7
+ 13u
6
+ 13u
5
+ 5u
4
+ 3u
3
+ u
2
u 1,
u
11
+ 3u
9
+ 2u
8
+ 4u
7
+ 4u
6
+ 5u
5
+ 4u
4
+ 3u
3
+ 2u
2
+ b + u,
u
11
+ 4u
9
+ 2u
8
+ 6u
7
+ 6u
6
+ 6u
5
+ 6u
4
+ 5u
3
+ 2u
2
+ a + 2ui
I
u
4
= hu
42
2u
41
+ ··· 48u + 9,
7.99709 × 10
21
u
41
+ 1.02860 × 10
22
u
40
+ ··· + 1.31722 × 10
22
a + 1.40009 × 10
23
,
1.01341 × 10
22
u
41
+ 1.66129 × 10
22
u
40
+ ··· + 2.63445 × 10
22
b + 9.20064 × 10
22
i
There are 4 irreducible components with 65 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
2
+ 1, a
3
2a + u, a
3
+ a
2
+ b 2a + 2i
(i) Arc colorings
a
8
=
0
a
3
+ 2a
a
11
=
a
a
3
a
2
+ 2a 2
a
9
=
a
3
+ a
a
3
+ 3a 1
a
1
=
a
3
+ a
1
a
3
=
1
0
a
2
=
a
3
+ a + 1
1
a
7
=
a
3
+ 2a
a
3
+ 2a
a
4
=
0
1
a
6
=
a
3
+ 2a
0
a
10
=
a
a
3
a
2
+ 3a 2
a
5
=
a
2
a
3
+ 2a
a
5
=
a
2
a
3
+ 2a
(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 = 0.618034I
b = 1.61803 1.00000I
0.986960 8.00000
u = 1.00000I
a = 0.618034I
b = 1.61803 + 1.00000I
0.986960 8.00000
u = 1.00000I
a = 1.61803I
b = 0.618034 + 1.000000I
8.88264 8.00000
u = 1.00000I
a = 1.61803I
b = 0.618034 1.000000I
8.88264 8.00000
3
II. I
u
2
= hu
4
u
3
+ 3u
2
2u + 1, u
3
+ 2b + 3u + 1, u
3
+ u
2
+ a 3u + 2i
(i) Arc colorings
a
8
=
0
u
a
11
=
u
3
u
2
+ 3u 2
1
2
u
3
3
2
u
1
2
a
9
=
u
3
u
2
+ 3u 2
1
2
u
3
1
2
u
1
2
a
1
=
0
u
a
3
=
1
0
a
2
=
1
u
2
a
7
=
u
u
a
4
=
u
2
+ 1
u
2
a
6
=
u
3
+ 2u
u
3
+ u
a
10
=
u
3
u
2
+ 3u 2
1
2
u
3
1
2
u
1
2
a
5
=
u
2
+ 1
u
2
a
5
=
u
2
+ 1
u
2
(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 = 0.10488 1.55249I
a = 0.043315 0.641200I
b = 0.278726 + 0.483420I
5.14581 + 3.16396I 2.13894 + 0.11292I
u = 0.10488 + 1.55249I
a = 0.043315 + 0.641200I
b = 0.278726 0.483420I
5.14581 3.16396I 2.13894 0.11292I
u = 0.395123 0.506844I
a = 0.95668 1.22719I
b = 0.971274 + 0.813859I
1.85594 + 1.41510I 3.26394 5.88934I
u = 0.395123 + 0.506844I
a = 0.95668 + 1.22719I
b = 0.971274 0.813859I
1.85594 1.41510I 3.26394 + 5.88934I
5
III.
I
u
3
= hu
15
+ 5u
13
+ · · · u 1, u
11
+ 3u
9
+ · · · + b + u, u
11
+ 4u
9
+ · · · + a + 2ui
(i) Arc colorings
a
8
=
0
u
a
11
=
u
11
4u
9
2u
8
6u
7
6u
6
6u
5
6u
4
5u
3
2u
2
2u
u
11
3u
9
2u
8
4u
7
4u
6
5u
5
4u
4
3u
3
2u
2
u
a
9
=
u
8
3u
6
2u
5
3u
4
4u
3
2u
2
2u 1
u
10
4u
8
2u
7
5u
6
6u
5
4u
4
4u
3
3u
2
a
1
=
u
2
+ 1
u
2
a
3
=
1
0
a
2
=
u
4
u
2
+ 1
u
4
a
7
=
u
u
a
4
=
u
2
+ 1
u
2
a
6
=
u
3
+ 2u
u
3
+ u
a
10
=
u
11
4u
9
2u
8
6u
7
6u
6
6u
5
6u
4
5u
3
2u
2
2u
u
13
+ 3u
11
+ 2u
10
+ 3u
9
+ 4u
8
+ 2u
7
+ 2u
6
2u
4
u
3
2u
2
u
a
5
=
u
5
2u
3
u
u
5
u
3
u
a
5
=
u
5
2u
3
u
u
5
u
3
u
(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 = 0.874669 0.344338I
a = 1.48983 + 0.33594I
b = 1.93202 0.08403I
5.87256 4.40083I 6.74431 + 3.49859I
u = 0.874669 + 0.344338I
a = 1.48983 0.33594I
b = 1.93202 + 0.08403I
5.87256 + 4.40083I 6.74431 3.49859I
u = 0.550495 0.307358I
a = 0.854377 + 0.521914I
b = 0.171819 0.000842I
0.32910 1.53058I 2.51511 + 4.43065I
u = 0.550495 + 0.307358I
a = 0.854377 0.521914I
b = 0.171819 + 0.000842I
0.32910 + 1.53058I 2.51511 4.43065I
u = 0.239239 1.082454I
a = 0.237058 1.072584I
b = 2.91128 + 1.65716I
2.40108 3.48114
u = 0.239239 + 1.082454I
a = 0.237058 + 1.072584I
b = 2.91128 1.65716I
2.40108 3.48114
u = 0.157939 1.235434I
a = 0.469512 0.190833I
b = 1.82301 0.19779I
0.32910 + 1.53058I 2.51511 4.43065I
u = 0.157939 + 1.235434I
a = 0.469512 + 0.190833I
b = 1.82301 + 0.19779I
0.32910 1.53058I 2.51511 + 4.43065I
u = 0.25402 1.40443I
a = 0.328044 + 0.950888I
b = 2.06537 0.05212I
5.87256 + 4.40083I 6.74431 3.49859I
u = 0.25402 + 1.40443I
a = 0.328044 0.950888I
b = 2.06537 + 0.05212I
5.87256 4.40083I 6.74431 + 3.49859I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.392556 0.928076I
a = 0.382795 0.495847I
b = 0.115873 0.630746I
0.32910 1.53058I 2.51511 + 4.43065I
u = 0.392556 + 0.928076I
a = 0.382795 + 0.495847I
b = 0.115873 + 0.630746I
0.32910 + 1.53058I 2.51511 4.43065I
u = 0.478478
a = 2.54502
b = 1.68075
2.40108 3.48114
u = 0.620645 1.060089I
a = 0.429576 + 1.086861I
b = 1.33107 0.91192I
5.87256 4.40083I 6.74431 + 3.49859I
u = 0.620645 + 1.060089I
a = 0.429576 1.086861I
b = 1.33107 + 0.91192I
5.87256 + 4.40083I 6.74431 3.49859I
8
IV. I
u
4
=
hu
42
2u
41
+· · ·48u+9, 8.00×10
21
u
41
+1.03×10
22
u
40
+· · ·+1.32×10
22
a+
1.40 × 10
23
, 1.01× 10
22
u
41
+ 1.66 × 10
22
u
40
+ · · · + 2.63× 10
22
b + 9.20 ×10
22
i
(i) Arc colorings
a
8
=
0
u
a
11
=
0.607117u
41
0.780881u
40
+ ··· + 39.2356u 10.6291
0.384676u
41
0.630601u
40
+ ··· + 11.3534u 3.49244
a
9
=
0.156368u
41
+ 0.202779u
40
+ ··· 5.83398u + 0.985881
0.0795184u
41
+ 0.213521u
40
+ ··· 15.5452u + 3.30729
a
1
=
0.0179531u
41
0.298526u
40
+ ··· + 7.15707u 1.16257
0.321754u
41
+ 0.449573u
40
+ ··· 9.45646u + 1.29102
a
3
=
1
0
a
2
=
0.478299u
41
+ 0.451049u
40
+ ··· 18.1689u + 5.16489
0.496252u
41
+ 0.749576u
40
+ ··· 25.3260u + 5.32746
a
7
=
u
u
a
4
=
u
2
+ 1
u
2
a
6
=
u
3
+ 2u
u
3
+ u
a
10
=
0.607117u
41
0.780881u
40
+ ··· + 39.2356u 10.6291
0.155857u
41
0.142101u
40
+ ··· 3.98351u + 0.407741
a
5
=
0.0863918u
41
+ 0.0143476u
40
+ ··· 0.349456u 1.21758
0.205565u
41
+ 0.367169u
40
+ ··· 6.26652u + 1.19185
a
5
=
0.0863918u
41
+ 0.0143476u
40
+ ··· 0.349456u 1.21758
0.205565u
41
+ 0.367169u
40
+ ··· 6.26652u + 1.19185
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
9
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.721971 0.167290I
a = 1.85361 0.25337I
b = 2.40555 + 0.48631I
5.07603 3.61628I 8.76999 + 3.97464I
u = 0.721971 + 0.167290I
a = 1.85361 + 0.25337I
b = 2.40555 0.48631I
5.07603 + 3.61628I 8.76999 3.97464I
u = 0.685316 0.861119I
a = 0.715702 + 1.021237I
b = 1.34241 0.61978I
4.33907 0.92313I 6.58218 + 1.96577I
u = 0.685316 + 0.861119I
a = 0.715702 1.021237I
b = 1.34241 + 0.61978I
4.33907 + 0.92313I 6.58218 1.96577I
u = 0.547526 0.434925I
a = 0.447661 0.232192I
b = 0.478555 0.505398I
0.14555 1.89662I 0.46549 + 4.15100I
u = 0.547526 + 0.434925I
a = 0.447661 + 0.232192I
b = 0.478555 + 0.505398I
0.14555 + 1.89662I 0.46549 4.15100I
u = 0.34592 1.45551I
a = 0.442549 + 0.865413I
b = 2.02717 0.86796I
0.13244 8.79986I 2.66885 + 5.03818I
u = 0.34592 + 1.45551I
a = 0.442549 0.865413I
b = 2.02717 + 0.86796I
0.13244 + 8.79986I 2.66885 5.03818I
u = 0.292799 1.363811I
a = 0.374793 1.004435I
b = 1.93193 + 1.82849I
0.22800 7.29302I 3.85885 + 5.40090I
u = 0.292799 + 1.363811I
a = 0.374793 + 1.004435I
b = 1.93193 1.82849I
0.22800 + 7.29302I 3.85885 5.40090I
10
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.233928 0.613039I
a = 0.422347 + 0.000410I
b = 0.110077 0.379155I
0.165616 1.199761I 1.09413 + 6.46841I
u = 0.233928 + 0.613039I
a = 0.422347 0.000410I
b = 0.110077 + 0.379155I
0.165616 + 1.199761I 1.09413 6.46841I
u = 0.23140 1.40351I
a = 0.623929 0.059962I
b = 0.965902 + 0.293545I
5.09584 4.47116I 1.35353 + 3.51083I
u = 0.23140 + 1.40351I
a = 0.623929 + 0.059962I
b = 0.965902 0.293545I
5.09584 + 4.47116I 1.35353 3.51083I
u = 0.16943 1.45640I
a = 0.386896 + 0.252041I
b = 0.667170 0.867135I
5.99098 4.47238I 2.83567 + 4.51985I
u = 0.16943 + 1.45640I
a = 0.386896 0.252041I
b = 0.667170 + 0.867135I
5.99098 + 4.47238I 2.83567 4.51985I
u = 0.088530 1.329414I
a = 0.356281 0.818610I
b = 0.691424 + 0.486857I
2.61565 + 0.48442I 1.18826 1.33056I
u = 0.088530 + 1.329414I
a = 0.356281 + 0.818610I
b = 0.691424 0.486857I
2.61565 0.48442I 1.18826 + 1.33056I
u = 0.07935 1.63033I
a = 0.070287 + 0.733138I
b = 0.326444 0.381680I
4.54070 3.45793I 8.56940 + 4.77216I
u = 0.07935 + 1.63033I
a = 0.070287 0.733138I
b = 0.326444 + 0.381680I
4.54070 + 3.45793I 8.56940 4.77216I
11
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.03966 1.45077I
a = 0.457785 + 0.143329I
b = 0.664565 0.643345I
6.85342 1.08907I 3.57208 + 1.87970I
u = 0.03966 + 1.45077I
a = 0.457785 0.143329I
b = 0.664565 + 0.643345I
6.85342 + 1.08907I 3.57208 1.87970I
u = 0.136661 0.369266I
a = 1.33208 2.23399I
b = 0.68810 + 1.33005I
2.24911 + 0.80040I 10.34390 + 2.30566I
u = 0.136661 + 0.369266I
a = 1.33208 + 2.23399I
b = 0.68810 1.33005I
2.24911 0.80040I 10.34390 2.30566I
u = 0.145595 1.190175I
a = 0.106269 + 1.407179I
b = 0.38127 1.45641I
8.34619 + 1.31277I 3.80780 5.75825I
u = 0.145595 + 1.190175I
a = 0.106269 1.407179I
b = 0.38127 + 1.45641I
8.34619 1.31277I 3.80780 + 5.75825I
u = 0.183767 1.330836I
a = 0.345054 0.964778I
b = 1.77648 + 1.39586I
1.90495 + 2.38439I 0.748912 0.739188I
u = 0.183767 + 1.330836I
a = 0.345054 + 0.964778I
b = 1.77648 1.39586I
1.90495 2.38439I 0.748912 + 0.739188I
u = 0.238530 1.347953I
a = 0.409461 0.757225I
b = 0.090813 + 0.281522I
1.07465 + 4.16567I 3.63331 3.63134I
u = 0.238530 + 1.347953I
a = 0.409461 + 0.757225I
b = 0.090813 0.281522I
1.07465 4.16567I 3.63331 + 3.63134I
12
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.32163 1.40565I
a = 0.696116 0.089376I
b = 0.950880 + 0.691902I
2.78819 + 9.89486I 2.06286 7.44629I
u = 0.32163 + 1.40565I
a = 0.696116 + 0.089376I
b = 0.950880 0.691902I
2.78819 9.89486I 2.06286 + 7.44629I
u = 0.39883 1.43016I
a = 0.513637 + 0.893993I
b = 2.32607 1.14325I
3.0695 + 14.6861I 5.13652 8.10029I
u = 0.39883 + 1.43016I
a = 0.513637 0.893993I
b = 2.32607 + 1.14325I
3.0695 14.6861I 5.13652 + 8.10029I
u = 0.595387 0.224674I
a = 2.33450 + 0.67070I
b = 1.58096 + 0.51334I
11.12093 + 1.23717I 12.55029 0.91395I
u = 0.595387 + 0.224674I
a = 2.33450 0.67070I
b = 1.58096 0.51334I
11.12093 1.23717I 12.55029 + 0.91395I
u = 0.596491 0.127774I
a = 0.96038 1.18419I
b = 0.897982 + 0.287975I
3.60913 + 1.10388I 10.34002 1.20607I
u = 0.596491 + 0.127774I
a = 0.96038 + 1.18419I
b = 0.897982 0.287975I
3.60913 1.10388I 10.34002 + 1.20607I
u = 0.792407 0.241239I
a = 0.708585 + 0.788677I
b = 0.371841 + 0.410028I
2.45148 + 5.86761I 6.25811 7.21816I
u = 0.792407 + 0.241239I
a = 0.708585 0.788677I
b = 0.371841 0.410028I
2.45148 5.86761I 6.25811 + 7.21816I
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.947212 0.236763I
a = 1.53192 + 0.06227I
b = 2.23148 0.05406I
8.35687 + 9.85804I 8.74565 6.87807I
u = 0.947212 + 0.236763I
a = 1.53192 0.06227I
b = 2.23148 + 0.05406I
8.35687 9.85804I 8.74565 + 6.87807I
13
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ 1)
2
(u
4
u
3
+ u
2
+ 1)(u
15
+ 5u
13
+ ··· u + 1)
(u
42
+ 2u
41
+ ··· + 36u + 9)
c
2
(u + 1)
4
(u
4
+ u
3
+ ··· + 2u + 1)(u
15
+ 10u
14
+ ··· + 3u 1)
(u
42
+ 18u
41
+ ··· + 936u + 81)
c
3
(u
2
+ 1)
2
(u
4
u
3
+ ··· 2u + 1)(u
15
+ 5u
13
+ ··· u + 1)
(u
42
+ 2u
41
+ ··· + 48u + 9)
c
4
, c
9
u
4
(u
4
+ 3u
2
+ 1)(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
3
(u
42
2u
41
+ ··· 48u + 64)
c
5
(u
2
+ 1)
2
(u
4
+ u
3
+ u
2
+ 1)(u
15
+ 5u
13
+ ··· u + 1)
(u
42
+ 2u
41
+ ··· + 36u + 9)
c
6
, c
7
(u
2
+ 1)
2
(u
4
+ u
3
+ ··· + 2u + 1)(u
15
+ 5u
13
+ ··· u + 1)
(u
42
+ 2u
41
+ ··· + 48u + 9)
c
8
(u 1)
4
(u
2
+ u 1)
2
(u
5
+ u
4
2u
3
u
2
+ u 1)
3
(u
42
+ 4u
41
+ ··· 3u + 4)
c
10
, c
11
(u + 1)
4
(u
2
u 1)
2
(u
5
+ u
4
2u
3
u
2
+ u 1)
3
(u
42
+ 4u
41
+ ··· 3u + 4)
14
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y + 1)
4
(y
4
+ y
3
+ ··· + 2y + 1)(y
15
+ 10y
14
+ ··· + 3y 1)
(y
42
+ 18y
41
+ ··· + 936y + 81)
c
2
(y 1)
4
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
15
10y
14
+ ··· + 15y 1)
(y
42
+ 18y
41
+ ··· + 43092y + 6561)
c
3
, c
6
, c
7
(y + 1)
4
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
15
+ 10y
14
+ ··· + 3y 1)
(y
42
+ 42y
41
+ ··· + 648y + 81)
c
4
, c
9
y
4
(y
2
+ 3y + 1)
2
(y
5
+ 3y
4
+ 4y
3
+ y
2
y 1)
3
(y
42
+ 24y
41
+ ··· + 37632y + 4096)
c
8
, c
10
, c
11
(y 1)
4
(y
2
3y + 1)
2
(y
5
5y
4
+ 8y
3
3y
2
y 1)
3
(y
42
40y
41
+ ··· + 431y + 16)
15