9
30
(K9a
1
)
1
Arc Sequences
5 9 1 7 2 4 6 3 8
Solving Sequence
4,7 1,5
2 3 6 8 9
c
4
c
1
c
3
c
6
c
7
c
9
c
2
, c
5
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
2
u + 1, b + u, a + u 1i
I
u
2
= hu
28
2u
27
+ ··· 22u + 17,
1.10786 × 10
30
u
27
+ 1.56975 × 10
30
u
26
+ ··· + 5.64156 × 10
30
b + 1.84737 × 10
31
,
9.86099 × 10
30
u
27
+ 1.53334 × 10
31
u
26
+ ··· + 9.59066 × 10
31
a + 1.00203 × 10
32
i
There are 2 irreducible components with 30 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
2
u + 1, b + u, a + u 1i
(i) Arc colorings
a
4
=
0
u
a
7
=
u + 1
u
a
1
=
1
0
a
5
=
u + 1
0
a
2
=
1
0
a
3
=
u
u
a
6
=
u + 1
0
a
8
=
0
u
a
9
=
1
u + 1
a
9
=
1
u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u + 1
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 0.500000 + 0.866025I
b = 0.500000 + 0.866025I
1.64493 + 2.02988I 3.00000 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 0.866025I
b = 0.500000 0.866025I
1.64493 2.02988I 3.00000 + 3.46410I
3
II. I
u
2
=
hu
28
2u
27
+· · ·22u+17, 1.11×10
30
u
27
+1.57×10
30
u
26
+· · ·+5.64×10
30
b+
1.85× 10
31
, 9.86×10
30
u
27
+1.53 ×10
31
u
26
+· · · +9.59 ×10
31
a + 1.00× 10
32
i
(i) Arc colorings
a
4
=
0
u
a
7
=
0.102819u
27
0.159878u
26
+ ··· 1.72157u 1.04480
0.196374u
27
0.278248u
26
+ ··· + 4.36028u 3.27457
a
1
=
1
0
a
5
=
0.0289468u
27
+ 0.0511954u
26
+ ··· 0.946887u + 3.10177
0.0660702u
27
+ 0.00650029u
26
+ ··· + 1.41339u + 0.622499
a
2
=
0.247232u
27
0.289355u
26
+ ··· 0.803406u 3.36355
0.0701438u
27
0.0601191u
26
+ ··· + 1.45892u 0.856112
a
3
=
u
u
a
6
=
0.102819u
27
0.159878u
26
+ ··· 1.72157u 1.04480
0.149697u
27
0.238519u
26
+ ··· + 3.61906u 4.05247
a
8
=
0.0503595u
27
+ 0.0305753u
26
+ ··· 1.67757u 0.351015
0.124940u
27
0.197271u
26
+ ··· + 1.38849u 3.01050
a
9
=
0.0657848u
27
0.0912517u
26
+ ··· 1.95710u 2.43782
0.241085u
27
0.319098u
26
+ ··· + 1.10897u 5.09730
a
9
=
0.0657848u
27
0.0912517u
26
+ ··· 1.95710u 2.43782
0.241085u
27
0.319098u
26
+ ··· + 1.10897u 5.09730
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0.0691432u
27
0.164535u
26
+ ··· + 0.898606u 12.0931
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.191128 0.619206I
a = 0.446676 + 0.694630I
b = 0.11783 + 1.66482I
0.41268 5.75423I 0.10698 + 5.96655I
u = 1.191128 + 0.619206I
a = 0.446676 0.694630I
b = 0.11783 1.66482I
0.41268 + 5.75423I 0.10698 5.96655I
u = 1.13985 0.88919I
a = 0.336232 0.805130I
b = 1.02430 1.92233I
0.59978 6.77427I 1.77406 + 4.95962I
u = 1.13985 + 0.88919I
a = 0.336232 + 0.805130I
b = 1.02430 + 1.92233I
0.59978 + 6.77427I 1.77406 4.95962I
u = 1.105364 0.510425I
a = 0.609049 + 0.428795I
b = 0.138597 0.133720I
2.52313 1.47542I 1.29345 + 0.59666I
u = 1.105364 + 0.510425I
a = 0.609049 0.428795I
b = 0.138597 + 0.133720I
2.52313 + 1.47542I 1.29345 0.59666I
u = 0.847077 0.345927I
a = 0.893222 0.255924I
b = 0.319452 1.172605I
1.32210 1.34593I 1.91932 + 0.66126I
u = 0.847077 + 0.345927I
a = 0.893222 + 0.255924I
b = 0.319452 + 1.172605I
1.32210 + 1.34593I 1.91932 0.66126I
u = 0.66840 1.28739I
a = 0.499324 0.405264I
b = 0.52286 1.56384I
0.967687 0.906276I 0.59768 1.67094I
u = 0.66840 + 1.28739I
a = 0.499324 + 0.405264I
b = 0.52286 + 1.56384I
0.967687 + 0.906276I 0.59768 + 1.67094I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.236722 0.655524I
a = 0.82244 + 1.42154I
b = 0.901799 + 0.826328I
2.40233 + 0.64414I 4.35398 + 1.30683I
u = 0.236722 + 0.655524I
a = 0.82244 1.42154I
b = 0.901799 0.826328I
2.40233 0.64414I 4.35398 1.30683I
u = 0.191038 0.606129I
a = 0.482592 + 0.293625I
b = 0.387987 0.437755I
0.22315 1.43304I 1.58225 + 4.97603I
u = 0.191038 + 0.606129I
a = 0.482592 0.293625I
b = 0.387987 + 0.437755I
0.22315 + 1.43304I 1.58225 4.97603I
u = 0.015157 1.395582I
a = 0.334699 0.686844I
b = 0.03901 1.87739I
1.56772 4.24816I 1.88645 + 6.97904I
u = 0.015157 + 1.395582I
a = 0.334699 + 0.686844I
b = 0.03901 + 1.87739I
1.56772 + 4.24816I 1.88645 6.97904I
u = 0.623667 0.562813I
a = 0.526785 + 1.121544I
b = 0.309747 + 1.323956I
1.85217 + 1.40144I 4.69947 1.74630I
u = 0.623667 + 0.562813I
a = 0.526785 1.121544I
b = 0.309747 1.323956I
1.85217 1.40144I 4.69947 + 1.74630I
u = 0.763781 0.287418I
a = 0.87066 1.23875I
b = 1.03249 1.42256I
5.39487 + 3.62399I 4.20871 2.76186I
u = 0.763781 + 0.287418I
a = 0.87066 + 1.23875I
b = 1.03249 + 1.42256I
5.39487 3.62399I 4.20871 + 2.76186I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.991759 0.593054I
a = 1.002081 + 0.480415I
b = 1.53289 + 0.52019I
0.65193 + 3.28147I 1.23266 4.99392I
u = 0.991759 + 0.593054I
a = 1.002081 0.480415I
b = 1.53289 0.52019I
0.65193 3.28147I 1.23266 + 4.99392I
u = 1.019474 0.068324I
a = 0.659580 + 0.779415I
b = 0.250962 0.164029I
6.61232 2.08114I 5.79595 + 2.78862I
u = 1.019474 + 0.068324I
a = 0.659580 0.779415I
b = 0.250962 + 0.164029I
6.61232 + 2.08114I 5.79595 2.78862I
u = 1.43260 0.55257I
a = 0.468429 + 0.450927I
b = 0.155652 + 0.114978I
5.32101 + 6.23266I 4.14975 4.30079I
u = 1.43260 + 0.55257I
a = 0.468429 0.450927I
b = 0.155652 0.114978I
5.32101 6.23266I 4.14975 + 4.30079I
u = 1.53314 0.75996I
a = 0.430947 0.639054I
b = 1.29528 1.97686I
3.12706 + 11.95453I 1.04116 8.32221I
u = 1.53314 + 0.75996I
a = 0.430947 + 0.639054I
b = 1.29528 + 1.97686I
3.12706 11.95453I 1.04116 + 8.32221I
7
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
u
2
(u
28
+ u
27
+ ··· + 8u + 4)
c
2
(u
2
u + 1)(u
28
+ 2u
27
+ ··· + 2u + 1)
c
3
(u
2
+ u + 1)(u
28
+ 2u
27
+ ··· + 22u + 17)
c
4
(u + 1)
2
(u
28
+ 3u
27
+ ··· + u + 1)
c
6
(u 1)
2
(u
28
+ 3u
27
+ ··· + u + 1)
c
7
(u 1)
2
(u
28
+ 13u
27
+ ··· 7u + 1)
c
8
(u
2
+ u + 1)(u
28
+ 2u
27
+ ··· + 2u + 1)
c
9
(u
2
+ u + 1)(u
28
+ 14u
27
+ ··· + 2u + 1)
8
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
y
2
(y
28
15y
27
+ ··· 88y + 16)
c
2
, c
8
(y
2
+ y + 1)(y
28
+ 14y
27
+ ··· + 2y + 1)
c
3
(y
2
+ y + 1)(y
28
10y
27
+ ··· 246y + 289)
c
4
, c
6
(y 1)
2
(y
28
13y
27
+ ··· + 7y + 1)
c
7
(y 1)
2
(y
28
+ 7y
27
+ ··· 61y + 1)
c
9
(y
2
+ y + 1)(y
28
+ 2y
27
+ ··· + 14y + 1)
9