10
111
(K10a
98
)
1
Arc Sequences
5 10 6 8 2 9 1 4 3 7
Solving Sequence
3,6 4,10
2 5 1 9 7 8
c
3
c
2
c
5
c
1
c
9
c
6
c
8
c
4
, c
7
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
6
u
5
+ 4u
2
4u + 1, 4u
5
+ 2u
4
+ u
3
+ a 16u + 9, 4u
5
2u
4
u
3
+ b + 16u 8i
I
u
2
= hu
14
+ u
13
+ u
12
+ u
11
+ 9u
10
+ 5u
9
+ 5u
8
+ 5u
7
+ 17u
6
+ 5u
5
+ 7u
4
4u
3
+ 8u
2
u + 1,
39480u
13
+ 1274u
12
+ ··· + 154933a + 163364, 39480u
13
1274u
12
+ ··· + 154933b 318297i
I
u
3
= hu
30
+ 5u
29
+ ··· 14u 1,
8.27934 × 10
37
u
29
3.87934 × 10
38
u
28
+ ··· + 1.76638 × 10
35
b + 2.60429 × 10
38
,
2.70329 × 10
43
u
29
1.27006 × 10
44
u
28
+ ··· + 1.82589 × 10
40
a + 8.84680 × 10
43
i
There are 3 irreducible components with 50 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
6
u
5
+4u
2
4u+1, 4u
5
+2u
4
+u
3
+a16u+9, 4u
5
2u
4
u
3
+b+16u8i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
4
=
1
u
2
a
10
=
4u
5
2u
4
u
3
+ 16u 9
4u
5
+ 2u
4
+ u
3
16u + 8
a
2
=
3u
5
+ 2u
4
+ u
3
12u + 8
u
5
4u + 1
a
5
=
5u
5
+ 3u
4
+ u
3
+ u
2
20u + 12
u 1
a
1
=
4u
5
2u
4
u
3
+ 15u 8
2u
5
+ u
4
7u + 3
a
9
=
1
4u
5
+ 2u
4
+ u
3
16u + 8
a
7
=
u
2u
5
+ u
4
7u + 4
a
8
=
4u
5
2u
4
u
3
u
2
+ 16u 9
3u
5
+ u
4
+ u
3
12u + 6
(ii) Obstruction class = 1
(iii) Cusp Shapes = 25u
5
+ 10u
4
+ 7u
3
+ 5u
2
96u + 32
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.02499 0.98915I
a = 1.124001 + 0.785288I
b = 0.124001 0.785288I
7.16447 + 3.17324I 9.24905 1.07022I
u = 1.02499 + 0.98915I
a = 1.124001 0.785288I
b = 0.124001 + 0.785288I
7.16447 3.17324I 9.24905 + 1.07022I
u = 0.449699
a = 1.90398
b = 0.903984
4.18532 9.57419
u = 0.584070
a = 0.185012
b = 1.18501
2.13209 21.5059
u = 1.008103 0.927438I
a = 0.516513 1.114980I
b = 0.483487 + 1.114980I
0.45357 + 5.18068I 8.71093 6.15331I
u = 1.008103 + 0.927438I
a = 0.516513 + 1.114980I
b = 0.483487 1.114980I
0.45357 5.18068I 8.71093 + 6.15331I
3
II. I
u
2
= hu
14
+ u
13
+ · · · u + 1, 3.95 × 10
4
u
13
+ 1274u
12
+ · · · + 1.55 ×
10
5
a + 1.63 × 10
5
, 39480u
13
1274u
12
+ · · · + 154933b 318297i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
4
=
1
u
2
a
10
=
0.254820u
13
0.00822291u
12
+ ··· + 3.57261u 1.05442
0.254820u
13
+ 0.00822291u
12
+ ··· 3.57261u + 2.05442
a
2
=
1.23639u
13
1.16081u
12
+ ··· 8.80020u + 1.50958
1.49121u
13
+ 1.15259u
12
+ ··· + 12.3728u 2.56399
a
5
=
1.23092u
13
1.01347u
12
+ ··· 9.83592u + 2.03566
1.69744u
13
+ 1.28920u
12
+ ··· + 14.8569u 3.30753
a
1
=
0.116528u
13
0.0612071u
12
+ ··· + 2.26064u 0.978836
0.445618u
13
0.321436u
12
+ ··· 4.37795u + 1.58789
a
9
=
1
0.254820u
13
+ 0.00822291u
12
+ ··· 3.57261u + 2.05442
a
7
=
u
0.263043u
13
0.260306u
12
+ ··· 0.799597u 0.254820
a
8
=
0.254820u
13
0.00822291u
12
+ ··· + 3.57261u 1.05442
0.252083u
13
+ 0.0818935u
12
+ ··· 4.09047u + 2.31746
(ii) Obstruction class = 1
(iii) Cusp Shapes =
319153
154933
u
13
+
874852
154933
u
12
+ ··· +
171234
154933
u +
3427848
154933
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.22366 1.15195I
a = 0.353292 1.290879I
b = 0.64671 + 1.29088I
7.5584 14.2650I 9.61306 + 7.45903I
u = 1.22366 + 1.15195I
a = 0.353292 + 1.290879I
b = 0.64671 1.29088I
7.5584 + 14.2650I 9.61306 7.45903I
u = 1.048938 0.652179I
a = 0.187430 + 0.232557I
b = 1.187430 0.232557I
10.88727 7.87015I 12.80126 + 5.10311I
u = 1.048938 + 0.652179I
a = 0.187430 0.232557I
b = 1.187430 + 0.232557I
10.88727 + 7.87015I 12.80126 5.10311I
u = 0.480433 0.861641I
a = 0.839666 1.114138I
b = 0.160334 + 1.114138I
3.38735 + 0.25005I 0.72285 2.46287I
u = 0.480433 + 0.861641I
a = 0.839666 + 1.114138I
b = 0.160334 1.114138I
3.38735 0.25005I 0.72285 + 2.46287I
u = 0.025861 0.375115I
a = 0.66482 1.30083I
b = 1.66482 + 1.30083I
3.88016 + 0.30866I 21.1851 + 1.2071I
u = 0.025861 + 0.375115I
a = 0.66482 + 1.30083I
b = 1.66482 1.30083I
3.88016 0.30866I 21.1851 1.2071I
u = 0.486006 0.497659I
a = 0.703941 0.328567I
b = 0.296059 + 0.328567I
0.46723 + 1.34385I 4.53324 5.44435I
u = 0.486006 + 0.497659I
a = 0.703941 + 0.328567I
b = 0.296059 0.328567I
0.46723 1.34385I 4.53324 + 5.44435I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.775295 1.049044I
a = 0.248822 + 1.259319I
b = 0.75118 1.25932I
1.57595 + 8.21733I 8.00559 6.90533I
u = 0.775295 + 1.049044I
a = 0.248822 1.259319I
b = 0.75118 + 1.25932I
1.57595 8.21733I 8.00559 + 6.90533I
u = 0.96587 1.05626I
a = 0.706527 + 1.007888I
b = 0.293473 1.007888I
1.24241 + 4.01184I 4.00908 1.51637I
u = 0.96587 + 1.05626I
a = 0.706527 1.007888I
b = 0.293473 + 1.007888I
1.24241 4.01184I 4.00908 + 1.51637I
6
III. I
u
3
=
hu
30
+5u
29
+· · ·14u1, 8.28×10
37
u
29
3.88×10
38
u
28
+· · ·+1.77×10
35
b+
2.60× 10
38
, 2.70× 10
43
u
29
1.27 ×10
44
u
28
+· · · +1.83 ×10
40
a + 8.85× 10
43
i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
4
=
1
u
2
a
10
=
1480.53u
29
+ 6955.82u
28
+ ··· 52108.0u 4845.20
468.718u
29
+ 2196.21u
28
+ ··· 16062.1u 1474.36
a
2
=
1085.37u
29
5070.52u
28
+ ··· + 36285.0u + 3322.75
350.689u
29
+ 1665.42u
28
+ ··· 13806.5u 1340.83
a
5
=
991.168u
29
4565.91u
28
+ ··· + 28334.6u + 2429.77
398.530u
29
+ 1884.97u
28
+ ··· 15091.1u 1445.96
a
1
=
1780.35u
29
8495.89u
28
+ ··· + 72515.7u + 7062.23
181.114u
29
833.946u
28
+ ··· + 5098.47u + 429.698
a
9
=
1949.25u
29
+ 9152.03u
28
+ ··· 68170.1u 6319.56
468.718u
29
+ 2196.21u
28
+ ··· 16062.1u 1474.36
a
7
=
1388.04u
29
+ 6649.23u
28
+ ··· 58366.7u 5729.66
268.310u
29
+ 1245.03u
28
+ ··· 8303.63u 734.682
a
8
=
1659.04u
29
+ 7795.68u
28
+ ··· 58477.9u 5439.43
437.990u
29
+ 2052.51u
28
+ ··· 15026.1u 1379.63
(ii) Obstruction class = 1
(iii) Cusp Shapes = 1720.55u
29
8153.31u
28
+ ··· + 66031.6u + 6330.33
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.86908 1.30976I
a = 0.215319 + 0.754139I
b = 0.339110 0.822375I
7.62983 + 4.35870I 12.9946 7.4101I
u = 1.86908 + 1.30976I
a = 0.215319 0.754139I
b = 0.339110 + 0.822375I
7.62983 4.35870I 12.9946 + 7.4101I
u = 1.44704 1.35792I
a = 0.183612 + 1.220713I
b = 0.455697 1.200152I
2.08637 7.22895I 8.76544 + 6.47803I
u = 1.44704 + 1.35792I
a = 0.183612 1.220713I
b = 0.455697 + 1.200152I
2.08637 + 7.22895I 8.76544 6.47803I
u = 1.204330 0.431350I
a = 0.000782 0.403066I
b = 0.766826
5.55785 2.82812I 12.02861 + 2.97945I
u = 1.204330 + 0.431350I
a = 0.000782 + 0.403066I
b = 0.766826
5.55785 + 2.82812I 12.02861 2.97945I
u = 1.108116 0.043065I
a = 0.244564 + 0.869058I
b = 0.339110 + 0.822375I
7.62983 + 1.29754I 12.99464 + 1.45120I
u = 1.108116 + 0.043065I
a = 0.244564 0.869058I
b = 0.339110 0.822375I
7.62983 1.29754I 12.99464 1.45120I
u = 0.711013 0.598106I
a = 0.891374 + 0.831846I
b = 0.455697 1.200152I
2.05122 4.40083I 2.23618 + 3.49859I
u = 0.711013 + 0.598106I
a = 0.891374 0.831846I
b = 0.455697 + 1.200152I
2.05122 + 4.40083I 2.23618 3.49859I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.57394 1.89944I
a = 0.156804 1.350681I
b = 0.339110 + 0.822375I
7.62983 + 1.29754I 12.99464 + 1.45120I
u = 0.57394 + 1.89944I
a = 0.156804 + 1.350681I
b = 0.339110 0.822375I
7.62983 1.29754I 12.99464 1.45120I
u = 0.320748 0.034597I
a = 5.85652 + 4.23753I
b = 0.339110 0.822375I
7.62983 + 4.35870I 12.9946 7.4101I
u = 0.320748 + 0.034597I
a = 5.85652 4.23753I
b = 0.339110 + 0.822375I
7.62983 4.35870I 12.9946 + 7.4101I
u = 0.291032 0.058576I
a = 1.59228 + 3.37585I
b = 0.455697 1.200152I
2.08637 1.57271I 8.76544 + 0.51914I
u = 0.291032 + 0.058576I
a = 1.59228 3.37585I
b = 0.455697 + 1.200152I
2.08637 + 1.57271I 8.76544 0.51914I
u = 0.289671
a = 1.58078
b = 0.766826
1.42027 5.49935
u = 0.375132 0.793055I
a = 0.26221 + 2.41312I
b = 0.339110 0.822375I
3.49225 + 1.53058I 6.46537 4.43065I
u = 0.375132 + 0.793055I
a = 0.26221 2.41312I
b = 0.339110 + 0.822375I
3.49225 1.53058I 6.46537 + 4.43065I
u = 0.680032
a = 0.340860
b = 0.766826
1.42027 5.49935
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.723473 0.513234I
a = 0.226305 + 0.956455I
b = 0.455697 1.200152I
2.08637 1.57271I 8.76544 + 0.51914I
u = 0.723473 + 0.513234I
a = 0.226305 0.956455I
b = 0.455697 + 1.200152I
2.08637 + 1.57271I 8.76544 0.51914I
u = 0.737499 1.067561I
a = 0.235020 1.389003I
b = 0.455697 + 1.200152I
2.05122 + 4.40083I 2.23618 3.49859I
u = 0.737499 + 1.067561I
a = 0.235020 + 1.389003I
b = 0.455697 1.200152I
2.05122 4.40083I 2.23618 + 3.49859I
u = 0.750590 0.816532I
a = 0.254359 0.536218I
b = 0.766826
5.55785 + 2.82812I 12.02861 2.97945I
u = 0.750590 + 0.816532I
a = 0.254359 + 0.536218I
b = 0.766826
5.55785 2.82812I 12.02861 + 2.97945I
u = 0.953025 0.838377I
a = 1.10687 1.14990I
b = 0.455697 + 1.200152I
2.08637 + 7.22895I 8.76544 6.47803I
u = 0.953025 + 0.838377I
a = 1.10687 + 1.14990I
b = 0.455697 1.200152I
2.08637 7.22895I 8.76544 + 6.47803I
u = 1.290395 0.535753I
a = 0.308791 0.195205I
b = 0.339110 + 0.822375I
3.49225 1.53058I 6.46537 + 4.43065I
u = 1.290395 + 0.535753I
a = 0.308791 + 0.195205I
b = 0.339110 0.822375I
3.49225 + 1.53058I 6.46537 4.43065I
10
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
7
(u
6
+ 2u
5
u
4
4u
3
u
2
+ u + 1)
(u
14
6u
12
+ 15u
10
u
9
15u
8
+ 2u
7
+ 3u
6
u
3
+ 4u
2
+ 2u + 1)
(u
30
+ u
29
+ ··· + 212u + 11)
c
2
(u
5
u
4
+ 2u
3
u
2
+ u 1)
6
(u
6
+ u
5
+ u
4
u
2
1)
(u
14
+ 10u
13
+ ··· + 60u + 8)
c
3
, c
6
(u
6
u
5
+ 4u
2
4u + 1)(u
14
+ u
13
+ ··· u + 1)
(u
30
+ 5u
29
+ ··· 14u 1)
c
4
(u
3
u
2
+ 2u 1)
10
(u
6
+ u
4
u
2
u 1)
(u
14
+ 11u
13
+ ··· + 224u + 32)
c
5
, c
10
(u
6
2u
5
u
4
+ 4u
3
u
2
u + 1)
(u
14
6u
12
+ 15u
10
u
9
15u
8
+ 2u
7
+ 3u
6
u
3
+ 4u
2
+ 2u + 1)
(u
30
+ u
29
+ ··· + 212u + 11)
c
8
(u
3
u
2
+ 2u 1)
10
(u
6
+ u
4
u
2
+ u 1)
(u
14
+ 11u
13
+ ··· + 224u + 32)
c
9
(u
5
u
4
+ 2u
3
u
2
+ u 1)
6
(u
6
u
5
+ u
4
u
2
1)
(u
14
+ 10u
13
+ ··· + 60u + 8)
11
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
, c
7
c
10
(y
6
6y
5
+ ··· 3y + 1)(y
14
12y
13
+ ··· + 4y + 1)
(y
30
25y
29
+ ··· 32360y + 121)
c
2
, c
9
(y
5
+ 3y
4
+ 4y
3
+ y
2
y 1)
6
(y
6
+ y
5
y
4
4y
3
y
2
+ 2y + 1)
(y
14
+ 4y
13
+ ··· + 240y + 64)
c
3
, c
6
(y
6
y
5
+ ··· 8y + 1)(y
14
+ y
13
+ ··· + 15y + 1)
(y
30
5y
29
+ ··· 56y + 1)
c
4
, c
8
(y
3
+ 3y
2
+ 2y 1)
10
(y
6
+ 2y
5
y
4
4y
3
y
2
+ y + 1)
(y
14
+ 5y
13
+ ··· + 512y + 1024)
12