11a
236
(K11a
236
)
1
Arc Sequences
7 1 11 10 9 8 2 3 4 5 6
Solving Sequence
2,7
8 1 3 9 6 5 11 4 10
c
7
c
1
c
2
c
8
c
6
c
5
c
11
c
3
c
10
c
4
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu 1i
I
u
2
= hu
48
+ 2u
47
+ ··· + 4u + 1i
There are 2 irreducible components with 49 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 1i
(i) Arc colorings
a
2
=
1
0
a
7
=
0
1
a
8
=
1
1
a
1
=
1
1
a
3
=
0
1
a
9
=
1
0
a
6
=
1
0
a
5
=
1
0
a
11
=
0
1
a
4
=
0
1
a
10
=
1
1
a
10
=
1
1
(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
4.93480 18.0000
3
II. I
u
2
= hu
48
+ 2u
47
+ · · · + 4u + 1i
(i) Arc colorings
a
2
=
1
0
a
7
=
0
u
a
8
=
u
u
a
1
=
1
u
2
a
3
=
u
2
+ 1
u
4
a
9
=
u
7
2u
5
+ 2u
3
2u
u
9
+ u
7
u
5
+ u
a
6
=
u
3
u
3
+ u
a
5
=
u
19
4u
17
+ 10u
15
18u
13
+ 23u
11
24u
9
+ 18u
7
10u
5
+ 5u
3
u
21
+ 3u
19
7u
17
+ 10u
15
10u
13
+ 7u
11
u
9
2u
7
+ 3u
5
3u
3
+ u
a
11
=
u
8
+ u
6
u
4
+ 1
u
8
2u
6
+ 2u
4
2u
2
a
4
=
u
20
+ 3u
18
7u
16
+ 10u
14
10u
12
+ 7u
10
u
8
2u
6
+ 3u
4
3u
2
+ 1
u
20
4u
18
+ 10u
16
18u
14
+ 23u
12
24u
10
+ 18u
8
10u
6
+ 5u
4
a
10
=
2u
47
+ 3u
46
+ ··· + 4u + 2
u
47
3u
46
+ ··· 6u 2
a
10
=
2u
47
+ 3u
46
+ ··· + 4u + 2
u
47
3u
46
+ ··· 6u 2
(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 = 1.09915
11.4368 21.8603
u = 1.070418 0.070047I
2.46715 3.58742I 13.8838 + 4.2943I
u = 1.070418 + 0.070047I
2.46715 + 3.58742I 13.8838 4.2943I
u = 1.033896 0.692265I
2.41639 13.71870I 13.3452 + 9.2783I
u = 1.033896 + 0.692265I
2.41639 + 13.71870I 13.3452 9.2783I
u = 1.009837 0.589266I
4.16473 + 0.96747I 15.6034 + 0.1358I
u = 1.009837 + 0.589266I
4.16473 0.96747I 15.6034 0.1358I
u = 1.006857 0.688557I
0.39402 5.62399I 11.59767 + 4.05733I
u = 1.006857 + 0.688557I
0.39402 + 5.62399I 11.59767 4.05733I
u = 0.996479 0.655041I
0.98972 5.75638I 13.3076 + 6.8650I
u = 0.996479 + 0.655041I
0.98972 + 5.75638I 13.3076 6.8650I
u = 0.882555 0.461026I
3.01031 4.88758I 15.6251 + 6.5404I
u = 0.882555 + 0.461026I
3.01031 + 4.88758I 15.6251 6.5404I
u = 0.863763 0.744913I
6.99140 2.82021I 3.95134 + 3.08292I
u = 0.863763 + 0.744913I
6.99140 + 2.82021I 3.95134 3.08292I
u = 0.665767 0.762214I
0.632478 + 0.108144I 9.55124 + 0.86883I
u = 0.665767 + 0.762214I
0.632478 0.108144I 9.55124 0.86883I
u = 0.644420 0.678291I
0.048446 + 0.560613I 11.68780 1.95261I
u = 0.644420 + 0.678291I
0.048446 0.560613I 11.68780 + 1.95261I
u = 0.624433 0.797000I
1.18842 + 8.10290I 11.33189 4.69039I
u = 0.624433 + 0.797000I
1.18842 8.10290I 11.33189 + 4.69039I
u = 0.421695
0.568709 17.6426
u = 0.376949 0.637932I
2.56451 5.60912I 12.01724 + 5.48028I
u = 0.376949 + 0.637932I
2.56451 + 5.60912I 12.01724 5.48028I
u = 0.209593 0.518446I
1.32155 + 1.52053I 9.58208 0.33085I
u = 0.209593 + 0.518446I
1.32155 1.52053I 9.58208 + 0.33085I
u = 0.339798 0.564531I
1.97900 + 1.93404I 6.47322 4.04899I
u = 0.339798 + 0.564531I
1.97900 1.93404I 6.47322 + 4.04899I
u = 0.555627 0.727194I
5.98604 1.18604I 15.5397 + 0.4606I
u = 0.555627 + 0.727194I
5.98604 + 1.18604I 15.5397 0.4606I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.639372 0.784790I
3.51281 4.08944I 6.57921 + 3.06594I
u = 0.639372 + 0.784790I
3.51281 + 4.08944I 6.57921 3.06594I
u = 0.785703 0.584322I
1.37338 + 2.15146I 8.25248 5.45590I
u = 0.785703 + 0.584322I
1.37338 2.15146I 8.25248 + 5.45590I
u = 0.841196 0.750320I
3.16432 1.24428I 8.04097 + 0.56162I
u = 0.841196 + 0.750320I
3.16432 + 1.24428I 8.04097 0.56162I
u = 0.885079 0.741658I
3.03130 + 6.89085I 8.49212 6.46442I
u = 0.885079 + 0.741658I
3.03130 6.89085I 8.49212 + 6.46442I
u = 0.921125
4.91974 18.6276
u = 0.977993 0.605871I
0.66805 + 2.46888I 10.26548 1.31520I
u = 0.977993 + 0.605871I
0.66805 2.46888I 10.26548 + 1.31520I
u = 1.02158
4.93269 18.1533
u = 1.024261 0.692231I
2.35847 + 9.67537I 8.68198 7.82216I
u = 1.024261 + 0.692231I
2.35847 9.67537I 8.68198 + 7.82216I
u = 1.031616 0.650416I
7.35665 + 6.46067I 17.5687 5.3712I
u = 1.031616 + 0.650416I
7.35665 6.46067I 17.5687 + 5.3712I
u = 1.093389 0.071584I
7.29017 + 7.41299I 18.4799 5.5389I
u = 1.093389 + 0.071584I
7.29017 7.41299I 18.4799 + 5.5389I
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
7
(u 1)(u
48
+ 2u
47
+ ··· + 4u + 1)
c
2
, c
6
(u + 1)(u
48
+ 16u
47
+ ··· + 8u + 1)
c
3
, c
5
(u)(u
48
+ 3u
47
+ ··· 20u
2
+ 1)
c
4
, c
9
, c
10
(u 1)(u
48
+ 2u
47
+ ··· + 4u + 1)
c
8
, c
11
(u + 1)(u
48
14u
46
+ ··· 4u + 1)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
7
(y 1)(y
48
16y
47
+ ··· 8y + 1)
c
2
, c
6
(y 1)(y
48
+ 32y
47
+ ··· 8y + 1)
c
3
, c
5
(y)(y
48
+ 27y
47
+ ··· 40y + 1)
c
4
(y 1)(y
48
40y
47
+ ··· 8y + 1)
c
8
, c
11
(y 1)(y
48
28y
47
+ ··· + 72y + 1)
c
9
(y 1)(y
48
40y
47
+ ··· 8y + 1)
c
10
(y 1)(y
48
40y
47
+ ··· 8y + 1)
8