10
71
(K10a
10
)
1
Arc Sequences
3 5 7 2 10 8 4 1 6 9
Solving Sequence
2,5 3,10
6 4 9 1 8 7
c
2
c
5
c
4
c
9
c
10
c
8
c
7
c
1
, c
3
, c
6
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
2
+ 3, u + 1, b + 2a 1i
I
u
2
= hu
40
+ 3u
39
+ ··· + 3u + 1, u
39
2u
38
+ ··· + 2a 1, u
39
+ 2u
38
+ ··· + b + 1i
There are 2 irreducible components with 42 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
= hb
2
+ 3, u + 1, b + 2a 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
1
a
3
=
1
1
a
10
=
1
2
b +
1
2
b
a
6
=
1
2
b
1
2
1
2
b +
1
2
a
4
=
1
1
a
9
=
1
2
b
1
2
b + 1
a
1
=
0
1
a
8
=
1
2
b
1
2
1
2
b +
1
2
a
7
=
1
2
b
1
2
1
2
b +
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2b + 3
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 0.500000 + 0.866025I
b = 1.73205I
1.64493 2.02988I 3.00000 + 3.46410I
u = 1.00000
a = 0.500000 0.866025I
b = 1.73205I
1.64493 + 2.02988I 3.00000 3.46410I
3
II.
I
u
2
= hu
40
+3u
39
+· · ·+3u+1, u
39
2u
38
+· · ·+2a1, u
39
+2u
38
+· · ·+b+1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
u
2
a
10
=
1
2
u
39
+ u
38
+ ··· + u +
1
2
u
39
2u
38
+ ··· 3u 1
a
6
=
5
2
u
39
6u
38
+ ··· 6u
7
2
5
2
u
39
+ 7u
38
+ ··· + 8u +
9
2
a
4
=
u
u
a
9
=
7
2
u
39
8u
38
+ ··· 8u
7
2
2u
39
+ 5u
38
+ ··· + 5u + 3
a
1
=
u
2
+ 1
u
4
a
8
=
11
2
u
39
12u
38
+ ··· 12u
11
2
9
2
u
39
+ 9u
38
+ ··· + 9u +
9
2
a
7
=
9
2
u
39
10u
38
+ ··· 11u
11
2
7
2
u
39
+ 7u
38
+ ··· + 8u +
9
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 11u
39
+ 27u
38
+ ··· + 31u + 8
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.182532 0.574885I
a = 0.134690 + 0.947345I
b = 0.33275 3.29714I
0.05370 + 13.38523I 0.42075 9.35928I
u = 1.182532 + 0.574885I
a = 0.134690 0.947345I
b = 0.33275 + 3.29714I
0.05370 13.38523I 0.42075 + 9.35928I
u = 1.153696 0.496328I
a = 0.281375 0.846655I
b = 0.18379 + 3.15325I
5.21580 + 6.90989I 5.24227 6.39245I
u = 1.153696 + 0.496328I
a = 0.281375 + 0.846655I
b = 0.18379 3.15325I
5.21580 6.90989I 5.24227 + 6.39245I
u = 1.152910 0.576867I
a = 0.789619 0.148475I
b = 1.040260 + 0.012851I
1.20323 + 7.65538I 1.63964 4.86252I
u = 1.152910 + 0.576867I
a = 0.789619 + 0.148475I
b = 1.040260 0.012851I
1.20323 7.65538I 1.63964 + 4.86252I
u = 1.070513 0.405845I
a = 0.780179 + 0.679209I
b = 0.34963 2.03038I
2.37466 + 0.03317I 2.30074 1.92960I
u = 1.070513 + 0.405845I
a = 0.780179 0.679209I
b = 0.34963 + 2.03038I
2.37466 0.03317I 2.30074 + 1.92960I
u = 1.060109 0.493550I
a = 0.428684 0.518252I
b = 0.468283 + 0.760309I
0.39800 + 4.72692I 1.63267 6.05913I
u = 1.060109 + 0.493550I
a = 0.428684 + 0.518252I
b = 0.468283 0.760309I
0.39800 4.72692I 1.63267 + 6.05913I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.823561 0.701523I
a = 0.002358 + 1.090112I
b = 0.47944 1.65555I
6.02457 + 5.56367I 5.18066 6.01609I
u = 0.823561 + 0.701523I
a = 0.002358 1.090112I
b = 0.47944 + 1.65555I
6.02457 5.56367I 5.18066 + 6.01609I
u = 0.759310 0.715307I
a = 1.093061 0.073821I
b = 0.007252 0.353745I
6.21108 0.22925I 5.84725 0.24543I
u = 0.759310 + 0.715307I
a = 1.093061 + 0.073821I
b = 0.007252 + 0.353745I
6.21108 + 0.22925I 5.84725 + 0.24543I
u = 0.695195 0.270902I
a = 0.012548 1.393236I
b = 0.28495 + 1.73971I
0.74845 + 2.81821I 1.95524 6.55211I
u = 0.695195 + 0.270902I
a = 0.012548 + 1.393236I
b = 0.28495 1.73971I
0.74845 2.81821I 1.95524 + 6.55211I
u = 0.412657 0.535700I
a = 0.718698 0.502951I
b = 0.148875 + 0.060490I
1.47568 0.52119I 6.28438 + 0.91978I
u = 0.412657 + 0.535700I
a = 0.718698 + 0.502951I
b = 0.148875 0.060490I
1.47568 + 0.52119I 6.28438 0.91978I
u = 0.308532 0.828965I
a = 0.404339 + 1.102772I
b = 0.138835 0.360604I
3.72005 2.44717I 4.96365 + 1.04542I
u = 0.308532 + 0.828965I
a = 0.404339 1.102772I
b = 0.138835 + 0.360604I
3.72005 + 2.44717I 4.96365 1.04542I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.261280 0.867465I
a = 1.292565 + 0.423506I
b = 0.928329 0.594777I
2.70648 8.09252I 2.94350 + 6.08172I
u = 0.261280 + 0.867465I
a = 1.292565 0.423506I
b = 0.928329 + 0.594777I
2.70648 + 8.09252I 2.94350 6.08172I
u = 0.129708 0.704300I
a = 1.43620 + 0.12766I
b = 0.960054 0.315198I
2.32493 2.41163I 2.33571 + 3.34704I
u = 0.129708 + 0.704300I
a = 1.43620 0.12766I
b = 0.960054 + 0.315198I
2.32493 + 2.41163I 2.33571 3.34704I
u = 0.283071 0.547471I
a = 1.82389 0.22572I
b = 0.674246 + 0.886942I
0.60920 + 2.86826I 1.22261 1.95241I
u = 0.283071 + 0.547471I
a = 1.82389 + 0.22572I
b = 0.674246 0.886942I
0.60920 2.86826I 1.22261 + 1.95241I
u = 0.467744 0.454035I
a = 0.11763 1.56846I
b = 0.956048 + 0.303178I
1.07354 2.17702I 2.16670 + 4.43587I
u = 0.467744 + 0.454035I
a = 0.11763 + 1.56846I
b = 0.956048 0.303178I
1.07354 + 2.17702I 2.16670 4.43587I
u = 0.978018 0.193176I
a = 0.148963 + 0.397208I
b = 0.574759 0.833284I
1.75548 0.68997I 4.17661 0.16492I
u = 0.978018 + 0.193176I
a = 0.148963 0.397208I
b = 0.574759 + 0.833284I
1.75548 + 0.68997I 4.17661 + 0.16492I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.045402 0.468373I
a = 0.811525 + 0.261280I
b = 1.115032 + 0.433573I
0.63968 1.74616I 0.044303 + 1.257582I
u = 1.045402 + 0.468373I
a = 0.811525 0.261280I
b = 1.115032 0.433573I
0.63968 + 1.74616I 0.044303 1.257582I
u = 1.098051 0.496688I
a = 0.166755 0.989201I
b = 0.92580 + 3.22127I
1.68055 7.12390I 1.84913 + 6.13601I
u = 1.098051 + 0.496688I
a = 0.166755 + 0.989201I
b = 0.92580 3.22127I
1.68055 + 7.12390I 1.84913 6.13601I
u = 1.169535 0.383434I
a = 0.245456 + 0.810272I
b = 0.31764 3.17797I
6.00686 1.32070I 7.28134 + 0.72610I
u = 1.169535 + 0.383434I
a = 0.245456 0.810272I
b = 0.31764 + 3.17797I
6.00686 + 1.32070I 7.28134 0.72610I
u = 1.211641 0.200716I
a = 0.433230 + 0.626819I
b = 0.35003 1.51378I
1.25887 0.68759I 0.543601 0.759704I
u = 1.211641 + 0.200716I
a = 0.433230 0.626819I
b = 0.35003 + 1.51378I
1.25887 + 0.68759I 0.543601 + 0.759704I
u = 1.256542 0.267461I
a = 0.656103 0.646617I
b = 0.02126 + 2.16902I
2.21178 + 4.43619I 1.72906 5.48285I
u = 1.256542 + 0.267461I
a = 0.656103 + 0.646617I
b = 0.02126 2.16902I
2.21178 4.43619I 1.72906 + 5.48285I
8
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u + 1)
2
(u
40
+ 21u
39
+ ··· + 3u + 1)
c
2
(u + 1)
2
(u
40
+ 3u
39
+ ··· + 3u + 1)
c
3
, c
7
u
2
(u
40
+ u
39
+ ··· 8u + 4)
c
4
(u 1)
2
(u
40
+ 3u
39
+ ··· + 3u + 1)
c
5
(u
2
u + 1)(u
40
+ 2u
39
+ ··· + 4u
2
+ 1)
c
6
u
2
(u
40
+ 15u
39
+ ··· + 120u + 16)
c
8
(u
2
+ u + 1)(u
40
+ 14u
39
+ ··· + 8u + 1)
c
9
(u
2
+ u + 1)(u
40
+ 2u
39
+ ··· + 4u
2
+ 1)
c
10
(u
2
+ 1)(u
2
u + 1)(u
38
+ 14u
37
+ ··· + 8u + 1)
9
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
(y 1)
2
(y
40
y
39
+ ··· + 17y + 1)
c
2
, c
4
(y 1)
2
(y
40
21y
39
+ ··· 3y + 1)
c
3
, c
7
y
2
(y
40
15y
39
+ ··· 120y + 16)
c
5
(y
2
+ y + 1)(y
40
+ 14y
39
+ ··· + 8y + 1)
c
6
y
2
(y
40
+ 17y
39
+ ··· + 2016y + 256)
c
8
(y
2
+ y + 1)(y
40
+ 26y
39
+ ··· + 44y + 1)
c
9
(y
2
+ y + 1)(y
40
+ 14y
39
+ ··· + 8y + 1)
c
10
(y
2
+ y + 1)(y
40
+ 26y
39
+ ··· + 44y + 1)
10