10
24
(K10a
71
)
1
Arc Sequences
7 5 9 3 2 10 1 4 6 8
Solving Sequence
4,8
9 3 5 2 6 10 1 7
c
8
c
3
c
4
c
2
c
5
c
9
c
10
c
7
c
1
, c
6
Representation Ideals
I = I
u
1
I
u
1
= hu
27
u
26
+ ··· u
2
1i
There are 1 irreducible components with 27 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
27
u
26
+ · · · u
2
1i
(i) Arc colorings
a
4
=
1
0
a
8
=
0
u
a
9
=
u
u
a
3
=
u
2
+ 1
u
2
a
5
=
u
4
+ u
2
+ 1
u
4
a
2
=
u
6
+ u
4
+ 2u
2
+ 1
u
6
+ u
2
a
6
=
u
8
+ u
6
+ 3u
4
+ 2u
2
+ 1
u
8
+ 2u
4
a
10
=
u
15
2u
13
6u
11
8u
9
10u
7
8u
5
4u
3
u
15
u
13
4u
11
3u
9
4u
7
2u
5
+ u
a
1
=
u
15
2u
13
6u
11
8u
9
10u
7
8u
5
4u
3
u
17
3u
15
7u
13
12u
11
13u
9
12u
7
6u
5
+ u
a
7
=
u
22
3u
20
+ ··· + 2u
2
+ 1
u
22
2u
20
+ ··· + 4u
4
+ u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
25
4u
24
+ 12u
23
12u
22
+ 44u
21
40u
20
+ 84u
19
76u
18
+
156u
17
124u
16
+ 196u
15
152u
14
+ 216u
13
136u
12
+ 160u
11
88u
10
+ 88u
9
24u
8
+
8u
7
+ 16u
6
16u
5
+ 16u
4
16u
3
4u
2
+ 4u 6
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.849312 0.907405I
11.72203 + 3.15301I 1.82291 2.60032I
u = 0.849312 + 0.907405I
11.72203 3.15301I 1.82291 + 2.60032I
u = 0.834094 0.813675I
2.75404 0.96140I 5.27084 + 1.18503I
u = 0.834094 + 0.813675I
2.75404 + 0.96140I 5.27084 1.18503I
u = 0.788550 0.963250I
2.29246 + 7.02686I 6.18454 6.08794I
u = 0.788550 + 0.963250I
2.29246 7.02686I 6.18454 + 6.08794I
u = 0.611045 0.149463I
2.54425 3.27708I 0.72206 + 2.87566I
u = 0.611045 + 0.149463I
2.54425 + 3.27708I 0.72206 2.87566I
u = 0.334942 0.978682I
0.01754 + 6.65682I 6.80212 7.22011I
u = 0.334942 + 0.978682I
0.01754 6.65682I 6.80212 + 7.22011I
u = 0.213473 0.634883I
0.352229 + 0.953640I 6.23281 7.10310I
u = 0.213473 + 0.634883I
0.352229 0.953640I 6.23281 + 7.10310I
u = 0.195439 0.958891I
0.823094 0.986974I 8.82659 0.25321I
u = 0.195439 + 0.958891I
0.823094 + 0.986974I 8.82659 + 0.25321I
u = 0.276294 0.962998I
4.29886 2.79673I 12.25981 + 4.61920I
u = 0.276294 + 0.962998I
4.29886 + 2.79673I 12.25981 4.61920I
u = 0.542850
1.51171 6.25830
u = 0.544854 0.629227I
4.34194 2.01066I 0.08108 + 3.90758I
u = 0.544854 + 0.629227I
4.34194 + 2.01066I 0.08108 3.90758I
u = 0.770594 0.919853I
4.70022 3.05015I 2.91169 + 1.99178I
u = 0.770594 + 0.919853I
4.70022 + 3.05015I 2.91169 1.99178I
u = 0.790499 0.862813I
4.87925 2.83072I 2.20196 + 3.74350I
u = 0.790499 + 0.862813I
4.87925 + 2.83072I 2.20196 3.74350I
u = 0.805943 0.978114I
7.22305 10.97745I 1.68833 + 7.27184I
u = 0.805943 + 0.978114I
7.22305 + 10.97745I 1.68833 7.27184I
u = 0.867246 0.813948I
7.73615 + 4.75862I 0.67410 2.41055I
u = 0.867246 + 0.813948I
7.73615 4.75862I 0.67410 + 2.41055I
3
II. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
7
, c
10
(u
27
+ u
26
+ ··· + 2u 1)
c
2
, c
4
, c
5
(u
27
+ 7u
26
+ ··· 2u 1)
c
3
, c
8
(u
27
+ u
26
+ ··· + u
2
+ 1)
c
6
, c
9
(u
27
+ u
26
+ ··· + 4u + 1)
4
III. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
7
, c
10
(y
27
+ 23y
26
+ ··· 2y 1)
c
2
, c
4
, c
5
(y
27
+ 27y
26
+ ··· + 14y 1)
c
3
, c
8
(y
27
+ 7y
26
+ ··· 2y 1)
c
6
, c
9
(y
27
13y
26
+ ··· 2y 1)
5