11n
60
(K11n
60
)
1
Arc Sequences
4 1 8 2 9 10 11 4 1 7 6
Solving Sequence
1,4
2
5,9
6 10 8 3 11 7
c
1
c
4
c
5
c
9
c
8
c
3
c
11
c
7
c
2
, c
6
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
5
b
4
+ 2b
3
b
2
+ b 1, a, u 1i
I
u
2
= hu
20
+ 6u
19
+ ··· 7u 1, 5u
19
+ 21u
18
+ ··· + 4b 8, 8u
19
+ 41u
18
+ ··· + 4a 19i
There are 2 irreducible components with 25 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
5
b
4
+ 2b
3
b
2
+ b 1, a, u 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
1
0
a
9
=
0
b
a
6
=
1
b
2
a
10
=
b
b
a
8
=
0
b
a
3
=
0
1
a
11
=
b
2
+ 1
b
4
a
7
=
b
4
b
2
1
b
4
a
7
=
b
4
b
2
1
b
4
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 0
b = 0.339110 0.822375I
1.31583 1.53058I 6.94263 + 4.09764I
u = 1.00000
a = 0
b = 0.339110 + 0.822375I
1.31583 + 1.53058I 6.94263 4.09764I
u = 1.00000
a = 0
b = 0.455697 1.200152I
4.22763 + 4.40083I 2.94226 4.18967I
u = 1.00000
a = 0
b = 0.455697 + 1.200152I
4.22763 4.40083I 2.94226 + 4.18967I
u = 1.00000
a = 0
b = 0.766826
0.756147 2.23022
3
II. I
u
2
=
hu
20
+6u
19
+· · ·7u1, 5u
19
+21u
18
+· · ·+4b8, 8u
19
+41u
18
+· · ·+4a19i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
u
u
3
+ u
a
9
=
2u
19
41
4
u
18
+ ··· +
97
4
u +
19
4
5
4
u
19
21
4
u
18
+ ··· + 7u + 2
a
6
=
1
0.0625000u
19
0.312500u
18
+ ··· + 2.37500u + 0.0625000
a
10
=
3
4
u
19
5u
18
+ ··· +
69
4
u +
11
4
5
4
u
19
21
4
u
18
+ ··· + 7u + 2
a
8
=
2u
19
41
4
u
18
+ ··· +
97
4
u +
19
4
7
4
u
19
+ 7u
18
+ ···
13
4
u +
1
4
a
3
=
u
2
+ 1
u
2
a
11
=
1
16
u
19
+
5
16
u
18
+ ···
19
8
u +
15
16
1
8
u
19
+
3
4
u
18
+ ···
11
8
u
1
4
a
7
=
1
16
u
19
+
7
16
u
18
+ ··· + u
3
16
1
4
u
19
11
8
u
18
+ ··· +
25
8
u +
3
8
a
7
=
1
16
u
19
+
7
16
u
18
+ ··· + u
3
16
1
4
u
19
11
8
u
18
+ ··· +
25
8
u +
3
8
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.87718 0.09620I
a = 1.161329 0.053142I
b = 1.58626 0.63668I
12.36329 0.65965I 5.32167 1.11773I
u = 1.87718 + 0.09620I
a = 1.161329 + 0.053142I
b = 1.58626 + 0.63668I
12.36329 + 0.65965I 5.32167 + 1.11773I
u = 1.86982 0.19925I
a = 1.163223 + 0.109868I
b = 1.47750 + 1.30770I
14.4810 5.2607I 7.68359 + 3.29127I
u = 1.86982 + 0.19925I
a = 1.163223 0.109868I
b = 1.47750 1.30770I
14.4810 + 5.2607I 7.68359 3.29127I
u = 1.83140 0.26446I
a = 1.148327 0.148781I
b = 1.17277 1.68864I
9.22640 9.48716I 3.68120 + 4.58970I
u = 1.83140 + 0.26446I
a = 1.148327 + 0.148781I
b = 1.17277 + 1.68864I
9.22640 + 9.48716I 3.68120 4.58970I
u = 1.71247
a = 1.06232
b = 0.561952
4.28328 1.87429
u = 0.391766 0.184540I
a = 0.42625 + 1.37075I
b = 0.557944 + 1.112972I
5.26329 + 4.10687I 5.09324 3.12286I
u = 0.391766 + 0.184540I
a = 0.42625 1.37075I
b = 0.557944 1.112972I
5.26329 4.10687I 5.09324 + 3.12286I
u = 0.209222 0.168474I
a = 0.70160 1.78397I
b = 0.427622 0.631782I
0.003151 + 1.284383I 0.00392 4.85690I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.209222 + 0.168474I
a = 0.70160 + 1.78397I
b = 0.427622 + 0.631782I
0.003151 1.284383I 0.00392 + 4.85690I
u = 0.189223 0.491141I
a = 1.18556 + 0.95484I
b = 0.683238 0.333693I
3.00168 0.46375I 0.380412 0.924408I
u = 0.189223 + 0.491141I
a = 1.18556 0.95484I
b = 0.683238 + 0.333693I
3.00168 + 0.46375I 0.380412 + 0.924408I
u = 0.804346
a = 0.662496
b = 0.133287
1.18420 8.02953
u = 1.006047 0.733534I
a = 0.671331 0.898752I
b = 1.12411 + 0.96042I
0.65473 + 4.99045I 3.53301 4.53534I
u = 1.006047 + 0.733534I
a = 0.671331 + 0.898752I
b = 1.12411 0.96042I
0.65473 4.99045I 3.53301 + 4.53534I
u = 1.117649 0.548619I
a = 0.542775 + 0.831412I
b = 1.136316 0.278580I
3.69282 + 1.48185I 9.31815 1.78036I
u = 1.117649 + 0.548619I
a = 0.542775 0.831412I
b = 1.136316 + 0.278580I
3.69282 1.48185I 9.31815 + 1.78036I
u = 1.320530 0.374618I
a = 0.335130 0.855556I
b = 1.131460 0.566768I
0.14678 1.84866I 3.98019 + 2.83860I
u = 1.320530 + 0.374618I
a = 0.335130 + 0.855556I
b = 1.131460 + 0.566768I
0.14678 + 1.84866I 3.98019 2.83860I
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
5
(u
20
+ 6u
19
+ ··· 7u 1)
c
2
(u + 1)
5
(u
20
+ 30u
19
+ ··· + 13u + 1)
c
3
, c
8
u
5
(u
20
+ u
19
+ ··· 32u + 32)
c
4
(u + 1)
5
(u
20
+ 6u
19
+ ··· 7u 1)
c
5
(u
5
+ u
4
+ ··· + u + 1)(u
20
+ 2u
19
+ ··· 3u 1)
c
6
, c
7
(u
5
u
4
+ ··· + u + 1)(u
20
+ 2u
19
+ ··· + 3u 1)
c
9
(u
5
+ u
4
+ ··· + u + 1)(u
20
+ 6u
19
+ ··· 81u 9)
c
10
(u
5
+ u
4
+ ··· + u 1)(u
20
+ 2u
19
+ ··· + 3u 1)
c
11
(u
5
3u
4
+ ··· u + 1)(u
20
+ 6u
19
+ ··· + 35u + 5)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
5
(y
20
30y
19
+ ··· 13y + 1)
c
2
(y 1)
5
(y
20
74y
19
+ ··· + 355y + 1)
c
3
, c
8
y
5
(y
20
33y
19
+ ··· + 3584y + 1024)
c
5
(y
5
+ 3y
4
+ ··· y 1)(y
20
42y
19
+ ··· 11y + 1)
c
6
, c
7
, c
10
(y
5
5y
4
+ ··· y 1)(y
20
18y
19
+ ··· 11y + 1)
c
9
(y
5
+ 3y
4
+ ··· y 1)(y
20
6y
19
+ ··· 5031y + 81)
c
11
(y
5
y
4
+ ··· + 3y 1)(y
20
+ 6y
19
+ ··· 335y + 25)
8