9
25
(K9a
4
)
1
Arc Sequences
5 9 6 2 7 4 1 3 8
Solving Sequence
3,6
4
7,9
2 5 1 8
c
3
c
6
c
2
c
5
c
1
c
8
c
4
, c
7
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
2
+ 3, u 1, 2b a 1i
I
u
2
= hu
25
+ 3u
24
+ ··· 4u 1, u
24
2u
23
+ ··· + a + 5u, u
24
2u
23
+ ··· + 2b + 1i
There are 2 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
= ha
2
+ 3, u 1, 2b a 1i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
1
a
4
=
1
1
a
7
=
1
0
a
9
=
a
1
2
a +
1
2
a
2
=
1
2
a
1
2
1
2
a
1
2
a
5
=
1
1
a
1
=
1
2
a
1
2
1
2
a
1
2
a
8
=
1
2
a
1
2
1
2
a +
1
2
a
8
=
1
2
a
1
2
1
2
a +
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2a 9
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.73205I
b = 0.500000 0.866025I
1.64493 2.02988I 9.00000 + 3.46410I
u = 1.00000
a = 1.73205I
b = 0.500000 + 0.866025I
1.64493 + 2.02988I 9.00000 3.46410I
3
II. I
u
2
=
hu
25
+3u
24
+· · ·4u1, u
24
2u
23
+· · ·+a+5u, u
24
2u
23
+· · ·+2b+1i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
4
=
1
u
2
a
7
=
u
u
3
+ u
a
9
=
u
24
+ 2u
23
+ ··· 5u
2
5u
1
2
u
24
+ u
23
+ ···
7
2
u
1
2
a
2
=
7
2
u
24
+ 9u
23
+ ···
33
2
u
9
2
3
2
u
24
+ 4u
23
+ ···
13
2
u
5
2
a
5
=
u
3
u
5
u
3
+ u
a
1
=
3
2
u
24
+ 3u
23
+ ···
15
2
u
3
2
5
2
u
24
+ 6u
23
+ ···
21
2
u
7
2
a
8
=
1
2
u
24
+ u
23
+ ···
3
2
u +
1
2
1
2
u
24
+ u
23
+ ···
7
2
u
1
2
a
8
=
1
2
u
24
+ u
23
+ ···
3
2
u +
1
2
1
2
u
24
+ u
23
+ ···
7
2
u
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
24
+ 7u
23
5u
22
42u
21
25u
20
+ 108u
19
+ 144u
18
128u
17
357u
16
+ 2u
15
+ 526u
14
+ 286u
13
498u
12
538u
11
+ 238u
10
+ 584u
9
+ 35u
8
389u
7
165u
6
+ 164u
5
+ 119u
4
22u
3
38u
2
6u 5
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.139241 0.687767I
a = 1.14519 + 1.80727I
b = 0.753308 + 1.027548I
4.33274 11.39031I 3.28983 + 7.76664I
u = 1.139241 + 0.687767I
a = 1.14519 1.80727I
b = 0.753308 1.027548I
4.33274 + 11.39031I 3.28983 7.76664I
u = 1.089150 0.711472I
a = 0.490999 + 0.203095I
b = 0.865451 0.706038I
5.32382 5.36637I 1.53322 + 3.05337I
u = 1.089150 + 0.711472I
a = 0.490999 0.203095I
b = 0.865451 + 0.706038I
5.32382 + 5.36637I 1.53322 3.05337I
u = 1.012756 0.537221I
a = 0.77689 2.25052I
b = 0.204213 1.096694I
1.91594 5.41987I 7.35697 + 6.54919I
u = 1.012756 + 0.537221I
a = 0.77689 + 2.25052I
b = 0.204213 + 1.096694I
1.91594 + 5.41987I 7.35697 6.54919I
u = 0.840318 0.621070I
a = 0.114344 0.489930I
b = 0.723797 0.117969I
2.10182 2.44039I 0.16599 + 3.61173I
u = 0.840318 + 0.621070I
a = 0.114344 + 0.489930I
b = 0.723797 + 0.117969I
2.10182 + 2.44039I 0.16599 3.61173I
u = 0.706780 0.369020I
a = 0.42079 + 1.91115I
b = 0.427994 + 1.010942I
0.62342 + 1.39976I 3.04278 0.06062I
u = 0.706780 + 0.369020I
a = 0.42079 1.91115I
b = 0.427994 1.010942I
0.62342 1.39976I 3.04278 + 0.06062I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.563663 0.911236I
a = 0.646213 + 0.436873I
b = 0.842489 + 0.787076I
6.92874 0.59688I 0.46758 + 1.80507I
u = 0.563663 + 0.911236I
a = 0.646213 0.436873I
b = 0.842489 0.787076I
6.92874 + 0.59688I 0.46758 1.80507I
u = 0.479273 0.936834I
a = 0.544317 0.502084I
b = 0.776571 0.974090I
6.34798 + 5.44271I 0.49829 3.51350I
u = 0.479273 + 0.936834I
a = 0.544317 + 0.502084I
b = 0.776571 + 0.974090I
6.34798 5.44271I 0.49829 + 3.51350I
u = 0.144497 0.357570I
a = 1.21724 + 0.74670I
b = 0.316251 + 0.806276I
0.33578 + 1.50728I 2.97928 4.31266I
u = 0.144497 + 0.357570I
a = 1.21724 0.74670I
b = 0.316251 0.806276I
0.33578 1.50728I 2.97928 + 4.31266I
u = 0.781818 0.585895I
a = 0.235890 + 0.629868I
b = 0.734813 0.804167I
1.52493 0.43356I 3.08804 0.04506I
u = 0.781818 + 0.585895I
a = 0.235890 0.629868I
b = 0.734813 + 0.804167I
1.52493 + 0.43356I 3.08804 + 0.04506I
u = 0.819709
a = 0.530934
b = 0.251925
1.19408 8.44382
u = 0.903290 0.591334I
a = 1.72740 + 1.15219I
b = 0.719637 + 0.929655I
1.14086 + 5.11531I 4.18255 5.48464I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.903290 + 0.591334I
a = 1.72740 1.15219I
b = 0.719637 0.929655I
1.14086 5.11531I 4.18255 + 5.48464I
u = 1.073952 0.294320I
a = 1.15104 1.96262I
b = 0.071208 0.875733I
3.46537 + 1.05922I 11.39395 0.37058I
u = 1.073952 + 0.294320I
a = 1.15104 + 1.96262I
b = 0.071208 + 0.875733I
3.46537 1.05922I 11.39395 + 0.37058I
u = 1.306764 0.052319I
a = 0.27343 + 1.51011I
b = 0.691717 + 0.872891I
0.20167 2.66172I 2.71477 + 3.57661I
u = 1.306764 + 0.052319I
a = 0.27343 1.51011I
b = 0.691717 0.872891I
0.20167 + 2.66172I 2.71477 3.57661I
7
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
u
2
(u
25
+ u
24
+ ··· + 4u 4)
c
2
(u
2
u + 1)(u
25
+ 2u
24
+ ··· + 3u + 1)
c
3
(u 1)
2
(u
25
+ 3u
24
+ ··· 4u 1)
c
5
(u 1)
2
(u
25
+ 11u
24
+ ··· 2u + 1)
c
6
(u + 1)
2
(u
25
+ 3u
24
+ ··· 4u 1)
c
7
(u
2
u + 1)(u
25
+ 8u
24
+ ··· + 11u 1)
c
8
(u
2
+ u + 1)(u
25
+ 2u
24
+ ··· + 3u + 1)
c
9
(u
2
+ u + 1)(u
25
+ 8u
24
+ ··· + 11u 1)
8
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
y
2
(y
25
+ 15y
24
+ ··· 88y 16)
c
2
, c
8
(y
2
+ y + 1)(y
25
+ 8y
24
+ ··· + 11y 1)
c
3
, c
6
(y 1)
2
(y
25
11y
24
+ ··· 2y 1)
c
5
(y 1)
2
(y
25
+ 9y
24
+ ··· 2y 1)
c
7
, c
9
(y
2
+ y + 1)(y
25
+ 20y
24
+ ··· + 251y 1)
9