9
39
(K9a
32
)
1
Arc Sequences
7 9 6 1 8 2 4 3 5
Solving Sequence
1,7 2,5
4 8 6 3 9
c
1
c
4
c
7
c
6
c
3
c
9
c
2
, c
5
, c
8
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
4
+ 2u
2
+ u + 1, b + u, u
3
u
2
+ a + ui
I
u
2
= hu
11
+ 4u
9
u
8
+ 7u
7
3u
6
+ 4u
5
3u
4
+ u
3
u
2
+ u + 1, b u,
17u
10
+ 18u
9
+ 65u
8
+ 43u
7
+ 91u
6
+ 38u
5
+ 20u
4
21u
3
+ 8u
2
+ 25a + 15u + 27i
I
u
3
= hu
20
+ u
19
+ ··· 8u + 7, 63269332u
19
195765489u
18
+ ··· + 599392561b 259471427,
5187188287u
19
+ 10884118778u
18
+ ··· + 20978739635a + 61432716820i
There are 3 irreducible components with 35 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
4
+ 2u
2
+ u + 1, b + u, u
3
u
2
+ a + ui
(i) Arc colorings
a
1
=
1
0
a
7
=
u
3
+ u
2
u
u
a
2
=
u
3
+ u
2
+ u + 2
u
2
a
5
=
0
u
a
4
=
u
u
a
8
=
u
3
u 1
u
2
u + 1
a
6
=
1
u
3
u
a
3
=
u
2
u + 1
u
2
+ u
a
9
=
1
u
2
a
9
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 7u
3
2u
2
+ 11u + 8
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.343815 0.625358I
a = 0.291777 + 1.032580I
b = 0.343815 + 0.625358I
1.13814 + 3.38562I 7.30286 7.57942I
u = 0.343815 + 0.625358I
a = 0.291777 1.032580I
b = 0.343815 0.625358I
1.13814 3.38562I 7.30286 + 7.57942I
u = 0.343815 1.358435I
a = 0.20822 1.60071I
b = 0.343815 + 1.358435I
4.42801 2.37936I 2.19714 + 1.10073I
u = 0.343815 + 1.358435I
a = 0.20822 + 1.60071I
b = 0.343815 1.358435I
4.42801 + 2.37936I 2.19714 1.10073I
3
II. I
u
2
= hu
11
+ 4u
9
+ · · · + u + 1, b u, 17u
10
+ 18u
9
+ · · · + 25a + 27i
(i) Arc colorings
a
1
=
1
0
a
7
=
0.680000u
10
0.720000u
9
+ ··· 0.600000u 1.08000
u
a
2
=
18
25
u
10
3
25
u
9
+ ··· +
2
5
u +
8
25
u
2
a
5
=
0
u
a
4
=
u
u
a
8
=
0.560000u
10
0.240000u
9
+ ··· + 0.800000u 0.360000
0.120000u
10
0.480000u
9
+ ··· 0.400000u 0.720000
a
6
=
0.560000u
10
0.240000u
9
+ ··· 0.200000u 0.360000
u
3
+ u
a
3
=
6
25
u
10
1
25
u
9
+ ···
1
5
u +
11
25
8
25
u
10
+
7
25
u
9
+ ··· +
2
5
u
2
25
a
9
=
1
u
2
a
9
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
31
25
u
10
51
25
u
9
+
24
5
u
8
201
25
u
7
+
238
25
u
6
341
25
u
5
+
47
5
u
4
103
25
u
3
+
144
25
u
2
6
5
u +
211
25
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.56939 1.41435I
a = 0.51246 + 1.61981I
b = 0.56939 1.41435I
3.64137 + 12.81031I 2.99547 7.42806I
u = 0.56939 + 1.41435I
a = 0.51246 1.61981I
b = 0.56939 + 1.41435I
3.64137 12.81031I 2.99547 + 7.42806I
u = 0.483399 0.706724I
a = 0.621005 + 0.007669I
b = 0.483399 0.706724I
1.11176 + 2.13095I 7.34122 2.95650I
u = 0.483399 + 0.706724I
a = 0.621005 0.007669I
b = 0.483399 + 0.706724I
1.11176 2.13095I 7.34122 + 2.95650I
u = 0.450687
a = 0.984340
b = 0.450687
0.895812 11.2897
u = 0.127465 1.057022I
a = 0.39223 + 2.89250I
b = 0.127465 1.057022I
0.55548 3.69188I 3.27466 + 4.59532I
u = 0.127465 + 1.057022I
a = 0.39223 2.89250I
b = 0.127465 + 1.057022I
0.55548 + 3.69188I 3.27466 4.59532I
u = 0.424463 1.293844I
a = 0.90958 + 1.56370I
b = 0.424463 1.293844I
6.69869 6.38540I 0.12486 + 5.46357I
u = 0.424463 + 1.293844I
a = 0.90958 1.56370I
b = 0.424463 + 1.293844I
6.69869 + 6.38540I 0.12486 5.46357I
u = 0.726207 0.303425I
a = 0.108291 + 1.300027I
b = 0.726207 0.303425I
3.57861 + 2.27941I 10.11894 1.15857I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.726207 + 0.303425I
a = 0.108291 1.300027I
b = 0.726207 + 0.303425I
3.57861 2.27941I 10.11894 + 1.15857I
5
III.
I
u
3
= hu
20
+u
19
+· · ·8u+7, 6.33×10
7
u
19
1.96×10
8
u
18
+· · ·+5.99×10
8
b
2.59 × 10
8
, 5.19 × 10
9
u
19
+ 1.09 × 10
10
u
18
+ · · · + 2.10 × 10
10
a + 6.14 × 10
10
i
(i) Arc colorings
a
1
=
1
0
a
7
=
0.247259u
19
0.518817u
18
+ ··· + 0.335520u 2.92833
0.105556u
19
+ 0.326606u
18
+ ··· + 1.69754u + 0.432891
a
2
=
0.349552u
19
0.0796380u
18
+ ··· 1.25873u + 1.73518
0.551898u
19
+ 0.446663u
18
+ ··· + 6.28274u 0.316957
a
5
=
0
u
a
4
=
u
u
a
8
=
0.0543867u
19
0.364665u
18
+ ··· + 0.923392u 2.57479
0.0873168u
19
+ 0.172455u
18
+ ··· + 1.10967u + 0.0793444
a
6
=
0.120650u
19
+ 0.527823u
18
+ ··· 1.39873u + 2.23494
0.135717u
19
0.241519u
18
+ ··· + 6.24336u 2.65232
a
3
=
0.392610u
19
0.353345u
18
+ ··· 2.93530u + 0.162741
0.478230u
19
+ 0.643378u
18
+ ··· + 4.73897u + 1.29758
a
9
=
1
u
2
a
9
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes =
1402773924
2996962805
u
19
100220208
2996962805
u
18
+ ···
22381283924
2996962805
u +
26005127826
2996962805
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.286373 0.028870I
a = 0.016417 0.483818I
b = 0.387179 + 1.147990I
0.93776 + 6.43072I 4.74431 6.96269I
u = 1.286373 + 0.028870I
a = 0.016417 + 0.483818I
b = 0.387179 1.147990I
0.93776 6.43072I 4.74431 + 6.96269I
u = 0.590252 0.825819I
a = 0.478334 0.384375I
b = 0.070663 0.512466I
0.93776 + 2.37095I 4.74431 0.03448I
u = 0.590252 + 0.825819I
a = 0.478334 + 0.384375I
b = 0.070663 + 0.512466I
0.93776 2.37095I 4.74431 + 0.03448I
u = 0.38849 1.61565I
a = 0.509858 1.243014I
b = 0.067213 + 1.072302I
4.60570 + 0.49930I 0.515115 + 0.966547I
u = 0.38849 + 1.61565I
a = 0.509858 + 1.243014I
b = 0.067213 1.072302I
4.60570 0.49930I 0.515115 0.966547I
u = 0.133857 1.341626I
a = 0.08909 1.65343I
b = 0.76505 + 1.34819I
4.60570 + 3.56046I 0.51511 7.89475I
u = 0.133857 + 1.341626I
a = 0.08909 + 1.65343I
b = 0.76505 1.34819I
4.60570 3.56046I 0.51511 + 7.89475I
u = 0.130820 1.153334I
a = 0.050211 0.442673I
b = 0.739688 + 0.098744I
2.53372 + 2.02988I 1.48114 3.46410I
u = 0.130820 + 1.153334I
a = 0.050211 + 0.442673I
b = 0.739688 0.098744I
2.53372 2.02988I 1.48114 + 3.46410I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.067213 1.072302I
a = 0.44509 2.02969I
b = 0.38849 + 1.61565I
4.60570 0.49930I 0.515115 0.966547I
u = 0.067213 + 1.072302I
a = 0.44509 + 2.02969I
b = 0.38849 1.61565I
4.60570 + 0.49930I 0.515115 + 0.966547I
u = 0.070663 0.512466I
a = 1.200159 + 0.097022I
b = 0.590252 0.825819I
0.93776 + 2.37095I 4.74431 0.03448I
u = 0.070663 + 0.512466I
a = 1.200159 0.097022I
b = 0.590252 + 0.825819I
0.93776 2.37095I 4.74431 + 0.03448I
u = 0.387179 1.147990I
a = 0.477144 0.191488I
b = 1.286373 + 0.028870I
0.93776 6.43072I 4.74431 + 6.96269I
u = 0.387179 + 1.147990I
a = 0.477144 + 0.191488I
b = 1.286373 0.028870I
0.93776 + 6.43072I 4.74431 6.96269I
u = 0.739688 0.098744I
a = 0.686864 0.091692I
b = 0.130820 + 1.153334I
2.53372 2.02988I 1.48114 + 3.46410I
u = 0.739688 + 0.098744I
a = 0.686864 + 0.091692I
b = 0.130820 1.153334I
2.53372 + 2.02988I 1.48114 3.46410I
u = 0.76505 1.34819I
a = 0.51123 1.34643I
b = 0.133857 + 1.341626I
4.60570 3.56046I 0.51511 + 7.89475I
u = 0.76505 + 1.34819I
a = 0.51123 + 1.34643I
b = 0.133857 1.341626I
4.60570 + 3.56046I 0.51511 7.89475I
8
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
(u
4
+ 2u
2
u + 1)
(u
11
+ 4u
9
u
8
+ 7u
7
3u
6
+ 4u
5
3u
4
+ u
3
u
2
+ u + 1)
(u
20
+ u
19
+ ··· 8u + 7)
c
2
(u
4
+ u
3
+ 2u
2
+ 1)(u
5
u
4
+ 2u
3
u
2
+ u 1)
4
(u
11
+ 6u
10
+ ··· + 26u + 4)
c
3
, c
5
(u
4
+ u + 1)(u
11
2u
9
+ ··· u 1)
(u
20
+ 5u
19
+ ··· + 2u + 1)
c
6
, c
9
(u
4
+ 2u
2
+ u + 1)
(u
11
+ 4u
9
u
8
+ 7u
7
3u
6
+ 4u
5
3u
4
+ u
3
u
2
+ u + 1)
(u
20
+ u
19
+ ··· 8u + 7)
c
7
(u
2
u + 1)
10
(u
4
u
3
+ 1)(u
11
+ 10u
10
+ ··· + 176u + 32)
c
8
(u
4
u
3
+ 2u
2
+ 1)(u
5
u
4
+ 2u
3
u
2
+ u 1)
4
(u
11
+ 6u
10
+ ··· + 26u + 4)
9
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
, c
6
c
9
(y
4
+ 4y
3
+ ··· + 3y + 1)(y
11
+ 8y
10
+ ··· + 3y 1)
(y
20
+ 15y
19
+ ··· + 468y + 49)
c
2
, c
8
(y
4
+ 3y
3
+ 6y
2
+ 4y + 1)(y
5
+ 3y
4
+ 4y
3
+ y
2
y 1)
4
(y
11
+ 6y
10
+ ··· + 124y 16)
c
3
, c
5
(y
4
+ 2y
2
y + 1)(y
11
4y
10
+ ··· + 11y 1)
(y
20
+ 3y
19
+ ··· + 12y + 1)
c
7
(y
2
+ y + 1)
10
(y
4
y
3
+ 2y
2
+ 1)(y
11
+ 2y
9
+ ··· + 1792y 1024)
10