11n
58
(K11n
58
)
1
Arc Sequences
4 1 9 2 9 10 11 3 1 7 6
Solving Sequence
1,4
2 3
5,9
6 10 8 11 7
c
1
c
2
c
4
c
5
c
9
c
8
c
11
c
7
c
3
, 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
22
+ 6u
21
+ ··· 6u 1, 8u
21
41u
20
+ ··· + 4a + 7, 10u
21
57u
20
+ ··· + 4b + 18i
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
= 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
3
=
0
1
a
5
=
1
0
a
9
=
0
b
a
6
=
1
b
2
a
10
=
b
b
a
8
=
0
b
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 0.02714 + 4.76366I
u = 1.00000
a = 0
b = 0.339110 + 0.822375I
1.31583 + 1.53058I 0.02714 4.76366I
u = 1.00000
a = 0
b = 0.455697 1.200152I
4.22763 + 4.40083I 4.43089 2.80751I
u = 1.00000
a = 0
b = 0.455697 + 1.200152I
4.22763 4.40083I 4.43089 + 2.80751I
u = 1.00000
a = 0
b = 0.766826
0.756147 2.80750
3
II. I
u
2
= hu
22
+ 6u
21
+ · · · 6u 1, 8u
21
41u
20
+ · · · + 4a + 7, 10u
21
57u
20
+ · · · + 4b + 18i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
5
=
u
u
3
+ u
a
9
=
2u
21
+
41
4
u
20
+ ···
13
2
u
7
4
5
2
u
21
+
57
4
u
20
+ ···
83
4
u
9
2
a
6
=
u
4
u
2
2u + 1
0.0625000u
21
0.312500u
20
+ ··· + 0.312500u + 0.0625000
a
10
=
1
2
u
21
4u
20
+ ··· +
57
4
u +
11
4
5
2
u
21
+
57
4
u
20
+ ···
83
4
u
9
2
a
8
=
u
21
+
17
4
u
20
+ ···
1
2
u
3
4
7
4
u
21
+
43
4
u
20
+ ···
33
2
u
7
2
a
11
=
0.187500u
21
0.937500u
20
+ ··· + 0.937500u + 1.18750
5
8
u
21
13
4
u
20
+ ··· +
29
8
u +
3
4
a
7
=
0.687500u
21
3.56250u
20
+ ··· + 1.93750u + 0.812500
3
4
u
21
+
31
8
u
20
+ ···
13
4
u
7
8
a
7
=
0.687500u
21
3.56250u
20
+ ··· + 1.93750u + 0.812500
3
4
u
21
+
31
8
u
20
+ ···
13
4
u
7
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 = 1.17332 1.06805I
a = 0.687089 + 1.085971I
b = 0.20188 1.87354I
16.2359 9.9783I 6.35264 + 4.88027I
u = 1.17332 + 1.06805I
a = 0.687089 1.085971I
b = 0.20188 + 1.87354I
16.2359 + 9.9783I 6.35264 4.88027I
u = 1.10072 1.06687I
a = 0.712322 1.067561I
b = 0.21701 + 1.82297I
9.80847 6.35147I 3.22096 + 4.88727I
u = 1.10072 + 1.06687I
a = 0.712322 + 1.067561I
b = 0.21701 1.82297I
9.80847 + 6.35147I 3.22096 4.88727I
u = 1.03181 1.11638I
a = 0.747624 + 1.069700I
b = 0.19072 1.76612I
10.04627 1.61926I 3.79117 0.60262I
u = 1.03181 + 1.11638I
a = 0.747624 1.069700I
b = 0.19072 + 1.76612I
10.04627 + 1.61926I 3.79117 + 0.60262I
u = 1.01491 1.20726I
a = 0.771753 1.096590I
b = 0.12854 + 1.74696I
16.8152 + 1.7067I 7.03198 0.67482I
u = 1.01491 + 1.20726I
a = 0.771753 + 1.096590I
b = 0.12854 1.74696I
16.8152 1.7067I 7.03198 + 0.67482I
u = 0.586491 0.548563I
a = 0.937796 + 0.745999I
b = 0.751111 1.182444I
5.30289 5.58097I 6.98899 + 5.83204I
u = 0.586491 + 0.548563I
a = 0.937796 0.745999I
b = 0.751111 + 1.182444I
5.30289 + 5.58097I 6.98899 5.83204I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.413660 0.450768I
a = 1.114712 0.742364I
b = 0.691977 + 0.865216I
0.10442 2.33425I 2.92732 + 5.10863I
u = 0.413660 + 0.450768I
a = 1.114712 + 0.742364I
b = 0.691977 0.865216I
0.10442 + 2.33425I 2.92732 5.10863I
u = 0.363445
a = 1.54532
b = 0.882309
1.97038 6.11975
u = 0.020785 0.558119I
a = 1.09400 + 1.07284I
b = 0.220616 0.726513I
1.10436 + 0.93215I 5.59687 3.71705I
u = 0.020785 + 0.558119I
a = 1.09400 1.07284I
b = 0.220616 + 0.726513I
1.10436 0.93215I 5.59687 + 3.71705I
u = 0.004750 0.900116I
a = 1.00304 1.00248I
b = 0.052435 + 0.949797I
7.28227 + 3.02618I 8.05288 2.57798I
u = 0.004750 + 0.900116I
a = 1.00304 + 1.00248I
b = 0.052435 0.949797I
7.28227 3.02618I 8.05288 + 2.57798I
u = 0.924539 0.471917I
a = 0.699573 0.650325I
b = 0.278529 + 0.415657I
3.07940 + 2.15283I 3.96233 2.53077I
u = 0.924539 + 0.471917I
a = 0.699573 + 0.650325I
b = 0.278529 0.415657I
3.07940 2.15283I 3.96233 + 2.53077I
u = 0.978070 0.165118I
a = 0.398312 + 0.417939I
b = 0.180827 0.224807I
1.75639 + 0.64723I 5.17438 + 1.08919I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.978070 + 0.165118I
a = 0.398312 0.417939I
b = 0.180827 + 0.224807I
1.75639 0.64723I 5.17438 1.08919I
u = 1.25110
a = 0.672163
b = 0.373570
1.80329 6.37871
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
5
(u
22
+ 6u
21
+ ··· 6u 1)
c
2
(u + 1)
5
(u
22
+ 2u
21
+ ··· + 2u + 1)
c
3
, c
8
u
5
(u
22
+ u
21
+ ··· + 64u + 32)
c
4
(u + 1)
5
(u
22
+ 6u
21
+ ··· 6u 1)
c
5
(u
5
+ u
4
+ ··· + u + 1)(u
22
+ 2u
21
+ ··· + 2996u 1960)
c
6
, c
7
(u
5
u
4
+ ··· + u + 1)(u
22
+ 2u
21
+ ··· 4u
2
1)
c
9
(u
5
+ u
4
+ ··· + u + 1)(u
22
+ 2u
21
+ ··· + 2u + 1)
c
10
(u
5
+ u
4
+ ··· + u 1)(u
22
+ 2u
21
+ ··· 4u
2
1)
c
11
(u
5
3u
4
+ ··· u + 1)(u
22
+ 6u
21
+ ··· 64u 17)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
5
(y
22
2y
21
+ ··· 2y + 1)
c
2
(y 1)
5
(y
22
+ 42y
21
+ ··· 62y + 1)
c
3
, c
8
y
5
(y
22
33y
21
+ ··· 7680y + 1024)
c
5
(y
5
+ 3y
4
+ 4y
3
+ y
2
y 1)
(y
22
+ 66y
21
+ ··· + 61827024y + 3841600)
c
6
, c
7
, c
10
(y
5
5y
4
+ ··· y 1)(y
22
22y
21
+ ··· + 8y + 1)
c
9
(y
5
+ 3y
4
+ ··· y 1)(y
22
+ 30y
21
+ ··· + 8y + 1)
c
11
(y
5
y
4
+ ··· + 3y 1)(y
22
14y
21
+ ··· 2056y + 289)
8