11a
10
(K11a
10
)
1
Arc Sequences
5 1 9 2 3 11 10 4 6 7 8
Solving Sequence
1,5
2 3 6
4,8
9 11 7 10
c
1
c
2
c
5
c
4
c
8
c
11
c
6
c
10
c
3
, c
7
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
6
+ a
5
+ a
4
+ 2a
3
+ a
2
+ 1, a
4
a
3
a
2
a + u, a
5
a
4
a
3
a
2
+ bi
I
u
2
= hu
59
4u
58
+ ··· 4u + 1, 5u
58
+ 8u
57
+ ··· + 4a 13, 4u
58
17u
57
+ ··· + 4b 5i
There are 2 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
6
+a
5
+a
4
+2a
3
+a
2
+1, a
4
a
3
a
2
a +u, a
5
a
4
a
3
a
2
+bi
(i) Arc colorings
a
1
=
1
0
a
5
=
0
a
4
+ a
3
+ a
2
+ a
a
2
=
1
a
4
+ a
3
+ a
2
+ a + 1
a
3
=
a
4
a
3
a
2
a
a
4
+ a
3
+ a
2
+ a + 1
a
6
=
1
0
a
4
=
a
4
a
3
a
2
a
a
4
+ a
3
+ a
2
+ a + 1
a
8
=
a
a
5
+ a
4
+ a
3
+ a
2
a
9
=
a
a
5
+ a
4
+ a
3
+ a
2
a
11
=
a
3
+ a
2
+ 2
a
3
1
a
7
=
a
4
+ a
3
a
5
+ a
4
+ a
2
a
10
=
a
5
a
4
a
3
a
2
+ a
a
5
+ a
4
+ a
3
+ a
2
a
10
=
a
5
a
4
a
3
a
2
+ a
a
5
+ a
4
+ a
3
+ a
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3a
5
+ 6a
4
+ 7a
3
+ 7a
2
+ 2a 1
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 1.083789 0.387453I
b = 0.877439 0.744862I
3.02413 + 0.79824I 1.45566 + 0.28364I
u = 0.500000 0.866025I
a = 1.083789 + 0.387453I
b = 0.877439 + 0.744862I
3.02413 0.79824I 1.45566 0.28364I
u = 0.500000 + 0.866025I
a = 0.206350 1.132315I
b = 0.877439 + 0.744862I
3.02413 4.85801I 2.09851 + 6.80481I
u = 0.500000 0.866025I
a = 0.206350 + 1.132315I
b = 0.877439 0.744862I
3.02413 + 4.85801I 2.09851 6.80481I
u = 0.500000 0.866025I
a = 0.377439 0.653743I
b = 0.754878
1.11345 + 2.02988I 5.85715 2.43783I
u = 0.500000 + 0.866025I
a = 0.377439 + 0.653743I
b = 0.754878
1.11345 2.02988I 5.85715 + 2.43783I
3
II. I
u
2
=
hu
59
4u
58
+· · ·4u+1, 5u
58
+8u
57
+· · ·+4a13, 4u
58
17u
57
+· · ·+4b5i
(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
4
=
u
u
3
+ u
a
8
=
5
4
u
58
2u
57
+ ··· 7u +
13
4
u
58
+
17
4
u
57
+ ···
29
4
u +
5
4
a
9
=
7
4
u
58
8u
57
+ ··· + 8u
5
4
u
58
17
4
u
57
+ ···
23
4
u +
7
4
a
11
=
1
4
u
58
3
4
u
57
+ ···
3
4
u
2
3
4
u
1
4
u
58
+ u
57
+ ··· +
9
4
u
2
+
1
4
a
7
=
15
4
u
58
+
55
4
u
57
+ ···
45
4
u + 2
5
4
u
58
29
4
u
57
+ ··· +
53
4
u
7
2
a
10
=
7
4
u
58
31
4
u
57
+ ···
67
4
u
2
+
19
4
u
1
4
u
57
7
4
u
56
+ ···
15
4
u +
3
4
a
10
=
7
4
u
58
31
4
u
57
+ ···
67
4
u
2
+
19
4
u
1
4
u
57
7
4
u
56
+ ···
15
4
u +
3
4
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
58
+
53
2
u
57
+ ··· 43u + 5
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.734997 0.760157I
a = 0.96868 1.08226I
b = 1.267217 + 0.458315I
1.62049 + 6.58252I 3.09082 7.01397I
u = 0.734997 + 0.760157I
a = 0.96868 + 1.08226I
b = 1.267217 0.458315I
1.62049 6.58252I 3.09082 + 7.01397I
u = 0.710107 0.805974I
a = 1.069903 + 0.899033I
b = 1.182883 0.071664I
2.49974 + 2.68394I 8.69632 3.80104I
u = 0.710107 + 0.805974I
a = 1.069903 0.899033I
b = 1.182883 + 0.071664I
2.49974 2.68394I 8.69632 + 3.80104I
u = 0.702513 0.862278I
a = 1.25255 0.70988I
b = 1.146583 0.371799I
1.31708 1.15944I 4.48579 + 0.70163I
u = 0.702513 + 0.862278I
a = 1.25255 + 0.70988I
b = 1.146583 + 0.371799I
1.31708 + 1.15944I 4.48579 0.70163I
u = 0.633394 0.535575I
a = 0.410686 + 1.039013I
b = 0.033364 0.989055I
5.58829 + 1.44801I 3.47358 3.14593I
u = 0.633394 + 0.535575I
a = 0.410686 1.039013I
b = 0.033364 + 0.989055I
5.58829 1.44801I 3.47358 + 3.14593I
u = 0.614062 0.138877I
a = 0.206090 0.383859I
b = 1.190829 + 0.467468I
2.08956 3.75051I 0.10887 + 2.99236I
u = 0.614062 + 0.138877I
a = 0.206090 + 0.383859I
b = 1.190829 0.467468I
2.08956 + 3.75051I 0.10887 2.99236I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.576429 1.002578I
a = 1.302074 0.378195I
b = 0.173802 + 0.876561I
4.23548 + 3.29913I 0.76928 1.96138I
u = 0.576429 + 1.002578I
a = 1.302074 + 0.378195I
b = 0.173802 0.876561I
4.23548 3.29913I 0.76928 + 1.96138I
u = 0.557034
a = 0.357919
b = 1.12008
1.92594 4.78134
u = 0.482895 1.132475I
a = 1.33963 + 1.32642I
b = 1.37680 0.41824I
0.64183 + 8.01411I 4.18661 6.44361I
u = 0.482895 + 1.132475I
a = 1.33963 1.32642I
b = 1.37680 + 0.41824I
0.64183 8.01411I 4.18661 + 6.44361I
u = 0.473168 0.941406I
a = 0.600703 + 0.586424I
b = 0.213892 0.219473I
0.26736 + 2.47932I 1.42598 4.73162I
u = 0.473168 + 0.941406I
a = 0.600703 0.586424I
b = 0.213892 + 0.219473I
0.26736 2.47932I 1.42598 + 4.73162I
u = 0.449340 1.128282I
a = 1.16666 1.39879I
b = 1.297415 + 0.128016I
4.83655 + 3.88458I 8.90529 3.78678I
u = 0.449340 + 1.128282I
a = 1.16666 + 1.39879I
b = 1.297415 0.128016I
4.83655 3.88458I 8.90529 + 3.78678I
u = 0.439059 0.667015I
a = 0.348002 0.932721I
b = 0.004691 + 0.314983I
0.56495 + 1.37410I 1.41861 4.46189I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.439059 + 0.667015I
a = 0.348002 + 0.932721I
b = 0.004691 0.314983I
0.56495 1.37410I 1.41861 + 4.46189I
u = 0.406347 1.125174I
a = 0.96483 + 1.48708I
b = 1.173456 + 0.198614I
1.186548 0.224277I 5.26908 0.20287I
u = 0.406347 + 1.125174I
a = 0.96483 1.48708I
b = 1.173456 0.198614I
1.186548 + 0.224277I 5.26908 + 0.20287I
u = 0.101753 0.896841I
a = 0.014506 1.391838I
b = 0.412235 0.387167I
1.39618 + 2.09190I 4.96276 3.48066I
u = 0.101753 + 0.896841I
a = 0.014506 + 1.391838I
b = 0.412235 + 0.387167I
1.39618 2.09190I 4.96276 + 3.48066I
u = 0.129172 0.636296I
a = 0.21256 + 1.78664I
b = 0.816846 0.286892I
1.53739 0.64054I 8.33837 + 0.19638I
u = 0.129172 + 0.636296I
a = 0.21256 1.78664I
b = 0.816846 + 0.286892I
1.53739 + 0.64054I 8.33837 0.19638I
u = 0.253732 1.230484I
a = 0.604452 + 0.443681I
b = 1.57362 + 0.39515I
6.10402 + 5.89251I 7.50107 3.28658I
u = 0.253732 + 1.230484I
a = 0.604452 0.443681I
b = 1.57362 0.39515I
6.10402 5.89251I 7.50107 + 3.28658I
u = 0.284018 1.231301I
a = 0.578680 0.248538I
b = 1.46942 0.23463I
10.46571 + 1.43686I 11.55835 0.61012I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.284018 + 1.231301I
a = 0.578680 + 0.248538I
b = 1.46942 + 0.23463I
10.46571 1.43686I 11.55835 + 0.61012I
u = 0.318877 1.227114I
a = 0.502506 + 0.016091I
b = 1.273346 + 0.090577I
7.01962 3.10038I 8.30335 + 3.14724I
u = 0.318877 + 1.227114I
a = 0.502506 0.016091I
b = 1.273346 0.090577I
7.01962 + 3.10038I 8.30335 3.14724I
u = 0.323680 0.615536I
a = 0.27588 2.22662I
b = 1.074398 + 0.826755I
3.12629 3.79302I 1.44992 0.03738I
u = 0.323680 + 0.615536I
a = 0.27588 + 2.22662I
b = 1.074398 0.826755I
3.12629 + 3.79302I 1.44992 + 0.03738I
u = 0.396294 1.047504I
a = 1.052876 0.081007I
b = 0.91159 1.15307I
1.68472 + 0.64264I 5.71511 + 0.56548I
u = 0.396294 + 1.047504I
a = 1.052876 + 0.081007I
b = 0.91159 + 1.15307I
1.68472 0.64264I 5.71511 0.56548I
u = 0.416542 1.121066I
a = 0.549700 0.433591I
b = 0.476504 + 0.740501I
3.99025 1.99015I 8.92828 + 0.31986I
u = 0.416542 + 1.121066I
a = 0.549700 + 0.433591I
b = 0.476504 0.740501I
3.99025 + 1.99015I 8.92828 0.31986I
u = 0.478060 1.135302I
a = 0.630530 + 1.018974I
b = 0.159744 0.774161I
3.53532 5.79141I 7.16103 + 7.27058I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.478060 + 1.135302I
a = 0.630530 1.018974I
b = 0.159744 + 0.774161I
3.53532 + 5.79141I 7.16103 7.27058I
u = 0.523234 1.099604I
a = 1.09237 1.34895I
b = 0.54356 + 1.38063I
2.74503 7.63920I 2.14137 + 7.41850I
u = 0.523234 + 1.099604I
a = 1.09237 + 1.34895I
b = 0.54356 1.38063I
2.74503 + 7.63920I 2.14137 7.41850I
u = 0.537801 1.187015I
a = 0.31862 + 1.69270I
b = 1.073287 0.272661I
5.50561 5.74082I 6.55357 + 3.14507I
u = 0.537801 + 1.187015I
a = 0.31862 1.69270I
b = 1.073287 + 0.272661I
5.50561 + 5.74082I 6.55357 3.14507I
u = 0.561307 1.183047I
a = 0.41017 1.89331I
b = 1.41266 + 0.40428I
8.54972 10.26435I 9.15841 + 6.91772I
u = 0.561307 + 1.183047I
a = 0.41017 + 1.89331I
b = 1.41266 0.40428I
8.54972 + 10.26435I 9.15841 6.91772I
u = 0.576605 1.176116I
a = 0.50757 + 2.01290I
b = 1.62071 0.56998I
3.8828 14.6473I 4.76475 + 8.96217I
u = 0.576605 + 1.176116I
a = 0.50757 2.01290I
b = 1.62071 + 0.56998I
3.8828 + 14.6473I 4.76475 8.96217I
u = 0.616367 0.142359I
a = 0.37836 1.50406I
b = 0.232880 + 0.577229I
0.77600 + 1.53921I 3.61158 4.49048I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.616367 + 0.142359I
a = 0.37836 + 1.50406I
b = 0.232880 0.577229I
0.77600 1.53921I 3.61158 + 4.49048I
u = 0.641679 0.347844I
a = 0.88865 + 1.68574I
b = 0.424600 1.202053I
4.93456 + 3.07605I 2.05439 3.74419I
u = 0.641679 + 0.347844I
a = 0.88865 1.68574I
b = 0.424600 + 1.202053I
4.93456 3.07605I 2.05439 + 3.74419I
u = 0.836671 0.195187I
a = 0.754258 0.871830I
b = 1.074742 + 0.123170I
2.55192 + 0.70368I 3.76254 + 0.28369I
u = 0.836671 + 0.195187I
a = 0.754258 + 0.871830I
b = 1.074742 0.123170I
2.55192 0.70368I 3.76254 0.28369I
u = 0.851161 0.244308I
a = 0.959489 + 0.892501I
b = 1.370478 0.328698I
5.74133 + 5.06975I 6.65758 3.49400I
u = 0.851161 + 0.244308I
a = 0.959489 0.892501I
b = 1.370478 + 0.328698I
5.74133 5.06975I 6.65758 + 3.49400I
u = 0.857380 0.279212I
a = 1.10012 0.93353I
b = 1.54812 + 0.53383I
1.19374 + 9.36497I 2.15808 5.62672I
u = 0.857380 + 0.279212I
a = 1.10012 + 0.93353I
b = 1.54812 0.53383I
1.19374 9.36497I 2.15808 + 5.62672I
10
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)
3
(u
59
+ 4u
58
+ ··· 4u 1)
c
2
(u
2
+ u + 1)
3
(u
59
+ 30u
58
+ ··· 4u 1)
c
3
, c
8
u
6
(u
59
+ u
58
+ ··· + 32u + 64)
c
4
(u
2
u + 1)
3
(u
59
+ 4u
58
+ ··· 4u 1)
c
5
(u
2
+ u + 1)
3
(u
59
+ 4u
58
+ ··· 22u + 137)
c
6
, c
7
(u
3
u
2
+ 2u 1)
2
(u
59
+ 3u
58
+ ··· 3u 1)
c
9
, c
11
(u
3
u
2
+ 1)
2
(u
59
+ 3u
58
+ ··· 67u + 73)
c
10
(u
3
+ u
2
+ 2u + 1)
2
(u
59
+ 3u
58
+ ··· 3u 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
2
+ y + 1)
3
(y
59
+ 30y
58
+ ··· 4y 1)
c
2
(y
2
+ y + 1)
3
(y
59
+ 2y
58
+ ··· + 24y 1)
c
3
, c
8
y
6
(y
59
+ 35y
58
+ ··· 35840y 4096)
c
5
(y
2
+ y + 1)
3
(y
59
26y
58
+ ··· 288860y 18769)
c
6
, c
7
, c
10
(y
3
+ 3y
2
+ 2y 1)
2
(y
59
+ 49y
58
+ ··· 9y 1)
c
9
(y
3
y
2
+ 2y 1)
2
(y
59
43y
58
+ ··· 79169y 5329)
c
11
(y
3
y
2
+ 2y 1)
2
(y
59
43y
58
+ ··· 79169y 5329)
12