10
148
(K10n
12
)
1
Arc Sequences
9 5 10 7 2 8 5 1 3 8
Solving Sequence
1,9 2,5
3 6 8 7 10 4
c
1
c
2
c
5
c
8
c
7
c
10
c
3
c
4
, c
6
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
2
+ b 1, u + 1, b + a 1i
I
u
2
= hu
17
+ 3u
16
+ ··· + 8u + 1, 4.47697 × 10
22
u
16
1.36976 × 10
23
u
15
+ ··· + 1.73292 × 10
23
b 9.48772 × 10
22
,
2.20248 × 10
23
u
16
+ 6.62167 × 10
23
u
15
+ ··· + 1.73292 × 10
23
a + 2.22549 × 10
24
i
There are 2 irreducible components with 19 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
+ b 1, u + 1, b + a 1i
(i) Arc colorings
a
1
=
1
0
a
9
=
b + 1
b
a
2
=
0
b 1
a
5
=
0
1
a
3
=
0
b 1
a
6
=
0
1
a
8
=
1
b
a
7
=
1
b 1
a
10
=
b + 1
b + 1
a
4
=
1
b
(ii) Obstruction class = 1
(iii) Cusp Shapes = 1
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 0.618034
b = 1.61803
7.23771 1.00000
u = 1.00000
a = 1.61803
b = 0.618034
0.657974 1.00000
3
II. I
u
2
=
hu
17
+3u
16
+· · ·+8u+1, 4.48×10
22
u
16
1.37×10
23
u
15
+· · ·+1.73×10
23
b
9.49 × 10
22
, 2.20 × 10
23
u
16
+ 6.62 × 10
23
u
15
+ · · · + 1.73 × 10
23
a + 2.23 × 10
24
i
(i) Arc colorings
a
1
=
1
0
a
9
=
1.27097u
16
3.82111u
15
+ ··· 39.1307u 12.8425
0.258349u
16
+ 0.790438u
15
+ ··· + 5.91867u + 0.547500
a
2
=
1.41268u
16
+ 3.93142u
15
+ ··· + 73.4194u + 0.763566
0.143938u
16
+ 0.488408u
15
+ ··· 2.88260u + 1.01570
a
5
=
0
u
a
3
=
1.41268u
16
+ 3.93142u
15
+ ··· + 73.4194u + 0.763566
0.118767u
16
+ 0.413875u
15
+ ··· 3.92298u + 0.709070
a
6
=
1.42225u
16
+ 4.40850u
15
+ ··· + 34.3881u + 21.9681
0.292173u
16
0.817353u
15
+ ··· 14.8546u 1.93156
a
8
=
1.52932u
16
4.61155u
15
+ ··· 45.0494u 13.3900
0.258349u
16
+ 0.790438u
15
+ ··· + 5.91867u + 0.547500
a
7
=
1.52932u
16
4.61155u
15
+ ··· 45.0494u 13.3900
0.235051u
16
+ 0.733571u
15
+ ··· + 4.20051u + 0.523894
a
10
=
1.26875u
16
3.44301u
15
+ ··· 76.3020u + 2.25214
0.143938u
16
0.488408u
15
+ ··· + 2.88260u 1.01570
a
4
=
2.12393u
16
+ 6.41496u
15
+ ··· + 61.1583u + 21.8492
0.333639u
16
1.00042u
15
+ ··· 9.93282u 1.22558
(ii) Obstruction class = 1
(iii) Cusp Shapes =
18146848036291991028487
21661450370363269586357
u
16
57256859957671720832173
21661450370363269586357
u
15
+ ···
57664361149698647803319
21661450370363269586357
u +
2588816807740110800204
21661450370363269586357
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 2.57956
a = 1.30094
b = 0.886101
0.349343 37.8910
u = 1.63712 0.76206I
a = 0.630527 0.262244I
b = 1.40843 0.58705I
5.29829 1.87159I 2.27258 + 1.08933I
u = 1.63712 + 0.76206I
a = 0.630527 + 0.262244I
b = 1.40843 + 0.58705I
5.29829 + 1.87159I 2.27258 1.08933I
u = 1.017964 0.809310I
a = 0.323399 0.187083I
b = 0.753939 + 0.337936I
2.60021 1.61334I 0.56805 + 3.90220I
u = 1.017964 + 0.809310I
a = 0.323399 + 0.187083I
b = 0.753939 0.337936I
2.60021 + 1.61334I 0.56805 3.90220I
u = 0.66721 1.87186I
a = 0.473849 + 0.370899I
b = 1.086973 + 0.371718I
0.79066 4.66548I 5.23718 + 7.00226I
u = 0.66721 + 1.87186I
a = 0.473849 0.370899I
b = 1.086973 0.371718I
0.79066 + 4.66548I 5.23718 7.00226I
u = 0.175480 0.518586I
a = 1.25553 1.48760I
b = 0.152522 0.439635I
1.71156 1.29590I 0.73837 + 2.68816I
u = 0.175480 + 0.518586I
a = 1.25553 + 1.48760I
b = 0.152522 + 0.439635I
1.71156 + 1.29590I 0.73837 2.68816I
u = 0.0980350 0.1012124I
a = 8.95152 + 6.17792I
b = 0.028909 1.130359I
9.74801 4.21913I 0.41910 + 2.45985I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.0980350 + 0.1012124I
a = 8.95152 6.17792I
b = 0.028909 + 1.130359I
9.74801 + 4.21913I 0.41910 2.45985I
u = 0.404986
a = 0.272501
b = 1.56075
7.69334 17.1878
u = 0.508295
a = 0.709514
b = 0.490972
0.859712 11.9142
u = 1.27497 1.10919I
a = 0.791728 + 0.167842I
b = 1.060613 + 0.200554I
1.19150 + 0.82781I 6.36851 + 0.89620I
u = 1.27497 + 1.10919I
a = 0.791728 0.167842I
b = 1.060613 0.200554I
1.19150 0.82781I 6.36851 0.89620I
u = 1.65397 1.53374I
a = 0.717289 0.244886I
b = 1.35486 0.58404I
5.64268 + 10.26024I 3.51255 5.70568I
u = 1.65397 + 1.53374I
a = 0.717289 + 0.244886I
b = 1.35486 + 0.58404I
5.64268 10.26024I 3.51255 + 5.70568I
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
8
, c
10
(u
2
+ u 1)(u
17
+ 2u
16
+ ··· + u + 1)
c
2
, c
5
u
2
(u
17
+ 3u
16
+ ··· + 20u 4)
c
3
, c
9
(u
2
u 1)(u
17
+ 2u
16
+ ··· + u + 1)
c
4
(u + 1)
2
(u
17
+ 3u
16
+ ··· 6u 1)
c
6
(u + 1)
2
(u
17
+ 19u
16
+ ··· + 90u + 1)
c
7
(u 1)
2
(u
17
+ 3u
16
+ ··· 6u 1)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
8
, c
10
(y
2
3y + 1)(y
17
12y
16
+ ··· + 3y 1)
c
2
, c
5
y
2
(y
17
+ 15y
16
+ ··· + 168y 16)
c
3
, c
9
(y
2
3y + 1)(y
17
+ 18y
15
+ ··· + 3y 1)
c
4
, c
7
(y 1)
2
(y
17
19y
16
+ ··· + 90y 1)
c
6
(y 1)
2
(y
17
39y
16
+ ··· + 6538y 1)
8