10
149
(K10n
11
)
1
Arc Sequences
9 5 10 7 2 8 5 1 2 4
Solving Sequence
1,8
9
2,5
3 7 4 6 10
c
8
c
1
c
2
c
7
c
4
c
6
c
10
c
3
, c
5
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
2
b 1, u + 1, a 1i
I
u
2
= hu
22
+ u
21
+ ··· + 104u
2
9,
7.35788 × 10
30
u
21
3.30448 × 10
30
u
20
+ ··· + 1.82577 × 10
31
a + 2.82744 × 10
31
,
1.28108 × 10
31
u
21
4.84463 × 10
30
u
20
+ ··· + 3.65154 × 10
31
b + 1.82963 × 10
32
i
There are 2 irreducible components with 24 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, a 1i
(i) Arc colorings
a
1
=
1
0
a
8
=
1
b
a
9
=
b + 1
b
a
2
=
0
b + 1
a
5
=
0
1
a
3
=
0
b + 1
a
7
=
1
b 1
a
4
=
1
b
a
6
=
0
1
a
10
=
b + 1
b 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 21
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.00000
b = 0.618034
2.63189 21.0000
u = 1.00000
a = 1.00000
b = 1.61803
10.5276 21.0000
3
II. I
u
2
=
hu
22
+u
21
+· · ·+104u
2
9, 7.36×10
30
u
21
3.30×10
30
u
20
+· · ·+1.83×10
31
a+
2.83 × 10
31
, 1.28× 10
31
u
21
4.84 × 10
30
u
20
+ · · · + 3.65 × 10
31
b + 1.83 × 10
32
i
(i) Arc colorings
a
1
=
1
0
a
8
=
0.403001u
21
+ 0.180991u
20
+ ··· + 11.2467u 1.54863
0.350831u
21
+ 0.132673u
20
+ ··· + 5.94112u 5.01057
a
9
=
0.0521698u
21
+ 0.0483176u
20
+ ··· + 5.30555u + 3.46194
0.350831u
21
+ 0.132673u
20
+ ··· + 5.94112u 5.01057
a
2
=
0.167853u
21
0.0489427u
20
+ ··· 5.40869u 3.86909
0.206228u
21
+ 0.0277419u
20
+ ··· + 2.68638u 2.72478
a
5
=
0
u
a
3
=
0.167853u
21
0.0489427u
20
+ ··· 5.40869u 3.86909
0.0231829u
21
0.00720060u
20
+ ··· + 1.17570u 0.773627
a
7
=
0.403001u
21
+ 0.180991u
20
+ ··· + 11.2467u 1.54863
0.165508u
21
+ 0.0684145u
20
+ ··· + 2.31411u 3.01248
a
4
=
0.362388u
21
0.0108766u
20
+ ··· 4.78694u + 7.03606
0.153546u
21
0.0550249u
20
+ ··· 3.33238u + 0.203122
a
6
=
0.497841u
21
0.128328u
20
+ ··· 2.30037u + 9.23083
0.313935u
21
0.00250308u
20
+ ··· 3.43028u + 4.04486
a
10
=
0.411032u
21
0.0490938u
20
+ ··· 9.20236u + 7.50308
0.130882u
21
0.0712371u
20
+ ··· 3.21454u 0.583105
(ii) Obstruction class = 1
(iii) Cusp Shapes = 1.71558u
21
0.254831u
20
+ ··· 21.9461u + 14.1775
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.57096 0.94066I
a = 0.258512 + 0.561773I
b = 1.38990 + 0.37870I
0.37121 + 4.08988I 7.66142 3.87499I
u = 1.57096 + 0.94066I
a = 0.258512 0.561773I
b = 1.38990 0.37870I
0.37121 4.08988I 7.66142 + 3.87499I
u = 1.083737 0.669508I
a = 0.393332 + 0.920545I
b = 0.137382 + 0.980052I
3.27405 + 6.32540I 5.56731 5.28995I
u = 1.083737 + 0.669508I
a = 0.393332 0.920545I
b = 0.137382 0.980052I
3.27405 6.32540I 5.56731 + 5.28995I
u = 0.860192
a = 1.29256
b = 0.260308
2.22827 0.635126
u = 0.751068
a = 2.13675
b = 1.22419
6.34803 16.4998
u = 0.625608 0.424523I
a = 1.25797 0.94803I
b = 1.195652 0.411381I
1.13790 + 4.89828I 6.90240 4.82636I
u = 0.625608 + 0.424523I
a = 1.25797 + 0.94803I
b = 1.195652 + 0.411381I
1.13790 4.89828I 6.90240 + 4.82636I
u = 0.515831
a = 0.182336
b = 0.736463
1.10354 8.74826
u = 0.447977 1.208301I
a = 0.344000 0.992002I
b = 1.207457 0.170395I
4.60553 + 3.49423I 13.3144 6.3296I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.447977 + 1.208301I
a = 0.344000 + 0.992002I
b = 1.207457 + 0.170395I
4.60553 3.49423I 13.3144 + 6.3296I
u = 0.094354 0.983533I
a = 0.329848 0.728206I
b = 0.096382 0.403421I
0.85664 1.35693I 6.38441 + 4.83589I
u = 0.094354 + 0.983533I
a = 0.329848 + 0.728206I
b = 0.096382 + 0.403421I
0.85664 + 1.35693I 6.38441 4.83589I
u = 0.313746 0.099352I
a = 1.64681 + 2.55270I
b = 1.154467 + 0.562023I
0.148418 0.912400I 7.06168 + 2.22739I
u = 0.313746 + 0.099352I
a = 1.64681 2.55270I
b = 1.154467 0.562023I
0.148418 + 0.912400I 7.06168 2.22739I
u = 0.72608 2.68987I
a = 0.026741 0.331798I
b = 1.080884 0.106938I
3.24923 0.58535I 11.5610 9.1342I
u = 0.72608 + 2.68987I
a = 0.026741 + 0.331798I
b = 1.080884 + 0.106938I
3.24923 + 0.58535I 11.5610 + 9.1342I
u = 1.06827
a = 0.772350
b = 1.66272
10.1504 0.707927
u = 1.217790 0.169386I
a = 0.441293 + 0.779766I
b = 0.123407 + 0.853958I
4.43145 + 0.35468I 3.17978 + 0.18562I
u = 1.217790 + 0.169386I
a = 0.441293 0.779766I
b = 0.123407 0.853958I
4.43145 0.35468I 3.17978 0.18562I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.40572 1.40325I
a = 0.186270 + 0.696267I
b = 1.38743 + 0.45171I
1.50863 11.44273I 9.41507 + 7.02258I
u = 1.40572 + 1.40325I
a = 0.186270 0.696267I
b = 1.38743 0.45171I
1.50863 + 11.44273I 9.41507 7.02258I
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
u 1)(u
22
+ 2u
21
+ ··· 5u + 1)
c
2
, c
5
u
2
(u
22
+ 3u
21
+ ··· + 28u + 4)
c
3
(u
2
u 1)(u
22
+ 2u
21
+ ··· + u + 1)
c
4
(u 1)
2
(u
22
+ 3u
21
+ ··· + 12u + 1)
c
6
(u 1)
2
(u
22
+ 9u
21
+ ··· + 120u + 1)
c
7
(u + 1)
2
(u
22
+ 3u
21
+ ··· + 12u + 1)
c
8
, c
9
(u
2
+ u 1)(u
22
+ 2u
21
+ ··· 5u + 1)
c
10
(u
2
+ u 1)(u
22
+ 2u
21
+ ··· + u + 1)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
8
, c
9
(y
2
3y + 1)(y
22
18y
21
+ ··· 9y + 1)
c
2
, c
5
y
2
(y
22
15y
21
+ ··· 264y + 16)
c
3
, c
10
(y
2
3y + 1)(y
22
6y
21
+ ··· 9y + 1)
c
4
, c
7
(y 1)
2
(y
22
9y
21
+ ··· 120y + 1)
c
6
(y 1)
2
(y
22
+ 11y
21
+ ··· 12776y + 1)
8