10
46
(K10a
81
)
1
Arc Sequences
8 5 6 9 3 10 1 2 4 7
Solving Sequence
2,5
3
6,9
4 8 1 7 10
c
2
c
5
c
4
c
8
c
1
c
7
c
10
c
3
, c
6
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
2
b 1, a, u 1i
I
u
2
= hu
17
+ 3u
16
+ ··· 2u 1, u
16
9u
14
+ ··· + 2b + 1, 2u
16
3u
15
+ ··· + a + 3i
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, a, u 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
1
a
3
=
1
1
a
6
=
1
0
a
9
=
0
b
a
4
=
0
1
a
8
=
b
b
a
1
=
b
b 1
a
7
=
1
b 1
a
10
=
0
b
(ii) Obstruction class = 1
(iii) Cusp Shapes = 17
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 0
b = 0.618034
2.63189 17.0000
u = 1.00000
a = 0
b = 1.61803
10.5276 17.0000
3
II.
I
u
2
= hu
17
+ 3 u
16
+· · ·−2u1, u
16
9u
14
+· · ·+2b+1, 2u
16
3u
15
+· · ·+a+3i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
3
=
1
u
2
a
6
=
u
u
3
+ u
a
9
=
2u
16
+ 3u
15
+ ··· + 3u 3
1
2
u
16
+
9
2
u
14
+ ··· +
3
2
u
1
2
a
4
=
u
2
+ 1
u
4
+ 2u
2
a
8
=
3
2
u
16
+ 3u
15
+ ··· +
9
2
u
7
2
1
2
u
16
+
9
2
u
14
+ ··· +
3
2
u
1
2
a
1
=
1
2
u
16
u
15
+ ··· +
7
2
u +
1
2
1
2
u
16
u
15
+ ··· +
3
2
u +
1
2
a
7
=
u
6
+ 3u
4
2u
3
2u
2
+ 4u 1
1
2
u
16
u
15
+ ··· +
1
2
u +
1
2
a
10
=
u
16
u
15
+ ··· + 7u 1
3
2
u
16
2u
15
+ ··· +
7
2
u +
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
16
u
15
12u
14
+ 9u
13
+ 50u
12
46u
11
88u
10
+ 143u
9
+
40u
8
228u
7
+ 82u
6
+ 140u
5
126u
4
+ 2u
3
+ 50u
2
3u 10
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.61959 0.31356I
a = 0.207503 + 1.033203I
b = 1.77104 0.09789I
17.4178 7.7170I 15.2806 + 3.2820I
u = 1.61959 + 0.31356I
a = 0.207503 1.033203I
b = 1.77104 + 0.09789I
17.4178 + 7.7170I 15.2806 3.2820I
u = 1.55782 0.20538I
a = 0.148124 0.974069I
b = 1.160735 + 0.369892I
11.54314 5.69036I 14.9028 + 4.0871I
u = 1.55782 + 0.20538I
a = 0.148124 + 0.974069I
b = 1.160735 0.369892I
11.54314 + 5.69036I 14.9028 4.0871I
u = 1.50356 0.06755I
a = 0.053037 + 0.917950I
b = 0.389835 0.662254I
6.67400 2.15086I 12.06720 + 3.08735I
u = 1.50356 + 0.06755I
a = 0.053037 0.917950I
b = 0.389835 + 0.662254I
6.67400 + 2.15086I 12.06720 3.08735I
u = 0.532039
a = 1.16083
b = 1.64837
9.71406 6.33025
u = 0.235031
a = 2.00493
b = 0.726749
1.27609 7.02090
u = 0.353541 0.303071I
a = 0.65186 1.30452I
b = 0.245709 + 0.306515I
0.413031 + 0.944940I 7.13539 7.21571I
u = 0.353541 + 0.303071I
a = 0.65186 + 1.30452I
b = 0.245709 0.306515I
0.413031 0.944940I 7.13539 + 7.21571I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.560836 0.704658I
a = 0.850949 + 0.924525I
b = 1.061546 0.132627I
4.50346 + 2.40856I 13.38977 3.98608I
u = 0.560836 + 0.704658I
a = 0.850949 0.924525I
b = 1.061546 + 0.132627I
4.50346 2.40856I 13.38977 + 3.98608I
u = 0.626661 0.929444I
a = 0.949200 0.879220I
b = 1.74789 + 0.03164I
14.6712 + 3.0771I 13.60428 2.54829I
u = 0.626661 + 0.929444I
a = 0.949200 + 0.879220I
b = 1.74789 0.03164I
14.6712 3.0771I 13.60428 + 2.54829I
u = 1.09525
a = 0.426868
b = 0.288922
2.06625 1.60996
u = 1.38948
a = 0.818409
b = 1.10417
6.53818 13.8716
u = 1.56221
a = 0.960254
b = 1.75801
16.9433 14.4074
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
10
(u
2
u 1)(u
17
+ 2u
16
+ ··· u 1)
c
2
, c
3
(u 1)
2
(u
17
+ 3u
16
+ ··· 2u 1)
c
4
, c
9
u
2
(u
17
+ u
16
+ ··· + 8u + 4)
c
5
(u + 1)
2
(u
17
+ 3u
16
+ ··· 2u 1)
c
6
, c
7
, c
8
(u
2
+ u 1)(u
17
+ 2u
16
+ ··· u 1)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
, c
7
c
8
, c
10
(y
2
3y + 1)(y
17
24y
16
+ ··· + 15y 1)
c
2
, c
3
, c
5
(y 1)
2
(y
17
19y
16
+ ··· + 26y 1)
c
4
, c
9
y
2
(y
17
+ 15y
16
+ ··· + 72y 16)
8