11n
5
(K11n
5
)
1
Arc Sequences
5 1 7 2 3 10 4 11 7 8 10
Solving Sequence
8,10
11
1,4
7 3 2 6 5 9
c
10
c
11
c
7
c
3
c
2
c
6
c
5
c
9
c
1
, c
4
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
5
+ u
4
2u
3
u
2
+ u 1, u
4
+ u
3
2u
2
+ a u + 1, u
4
+ u
3
2u
2
+ b u + 1i
I
u
2
= hu
40
2u
39
+ ··· + u 1, 9.16658 × 10
39
u
39
1.59405 × 10
40
u
38
+ ··· + 9.67095 × 10
39
b + 2.15406 × 10
39
,
1.11354 × 10
40
u
39
1.93364 × 10
40
u
38
+ ··· + 9.67095 × 10
39
a + 5.08844 × 10
39
i
There are 2 irreducible components with 45 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
5
+u
4
2u
3
u
2
+u1, u
4
+u
3
2u
2
+au+1, u
4
+u
3
2u
2
+bu+1i
(i) Arc colorings
a
8
=
0
u
a
10
=
u
4
u
3
+ 2u
2
+ u 1
u
4
u
3
+ 2u
2
+ u 1
a
11
=
u
4
u
3
+ 2u
2
+ u 1
u
4
u
3
+ 2u
2
1
a
1
=
0
u
a
4
=
1
0
a
7
=
u
u
a
3
=
u
2
+ 1
u
2
a
2
=
u
2
+ 1
u
4
2u
2
a
6
=
u
u
a
5
=
u
3
+ 2u
u
3
+ u
a
9
=
u
4
u
3
+ 2u
2
+ u 1
u
4
u
3
+ 2u
2
+ u 1
a
9
=
u
4
u
3
+ 2u
2
+ u 1
u
4
u
3
+ 2u
2
+ u 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 = 1.41878 0.21917I
a = 0.688402 0.106340I
b = 0.688402 0.106340I
7.51750 4.40083I 14.4110 + 1.1901I
u = 1.41878 + 0.21917I
a = 0.688402 + 0.106340I
b = 0.688402 + 0.106340I
7.51750 + 4.40083I 14.4110 1.1901I
u = 0.309916 0.549911I
a = 0.77780 1.38013I
b = 0.77780 1.38013I
1.97403 + 1.53058I 3.52158 + 1.00973I
u = 0.309916 + 0.549911I
a = 0.77780 + 1.38013I
b = 0.77780 + 1.38013I
1.97403 1.53058I 3.52158 1.00973I
u = 1.21774
a = 0.821196
b = 0.821196
4.04602 10.1348
3
II.
I
u
2
= hu
40
2u
39
+· · ·+u1, 9.17×10
39
u
39
1.59×10
40
u
38
+· · ·+9.67×10
39
b+
2.15 × 10
39
, 1.11 × 10
40
u
39
1.93 × 10
40
u
38
+ · · · + 9.67 × 10
39
a + 5.09 × 10
39
i
(i) Arc colorings
a
8
=
0
u
a
10
=
1.15143u
39
+ 1.99944u
38
+ ··· + 0.957478u 0.526157
0.947847u
39
+ 1.64829u
38
+ ··· + 2.80548u 0.222735
a
11
=
1.15143u
39
+ 1.99944u
38
+ ··· + 0.957478u 0.526157
1.15143u
39
+ 1.99944u
38
+ ··· + 1.95748u 0.526157
a
1
=
0.0802655u
39
+ 0.0403044u
38
+ ··· + 0.427077u + 0.157661
0.380473u
39
+ 0.568782u
38
+ ··· 0.322046u 0.290241
a
4
=
1
0
a
7
=
u
u
a
3
=
u
2
+ 1
u
2
a
2
=
0.0172308u
39
0.0529909u
38
+ ··· + 0.373414u + 1.27235
0.0200871u
39
0.146528u
38
+ ··· + 0.668344u + 0.578068
a
6
=
0.381156u
39
+ 0.660665u
38
+ ··· 0.709162u 0.568128
0.461421u
39
+ 0.700969u
38
+ ··· 0.282085u 0.410467
a
5
=
0.735406u
39
+ 1.13358u
38
+ ··· 0.842898u 0.666205
0.435198u
39
+ 0.605100u
38
+ ··· 0.0937756u 0.218303
a
9
=
1.12586u
39
+ 1.98702u
38
+ ··· + 0.809910u 0.582171
0.922279u
39
+ 1.63587u
38
+ ··· + 2.65792u 0.278749
a
9
=
1.12586u
39
+ 1.98702u
38
+ ··· + 0.809910u 0.582171
0.922279u
39
+ 1.63587u
38
+ ··· + 2.65792u 0.278749
(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 = 1.51868 0.20012I
a = 0.504310 + 0.008724I
b = 0.874745 + 0.095369I
7.01606 4.59273I 0.02801 + 6.01256I
u = 1.51868 + 0.20012I
a = 0.504310 0.008724I
b = 0.874745 0.095369I
7.01606 + 4.59273I 0.02801 6.01256I
u = 1.093021 0.888850I
a = 0.998834 + 0.071384I
b = 1.54898 + 1.09445I
4.47319 8.54344I 1.84932 + 4.74667I
u = 1.093021 + 0.888850I
a = 0.998834 0.071384I
b = 1.54898 1.09445I
4.47319 + 8.54344I 1.84932 4.74667I
u = 1.054049 0.808417I
a = 0.930522 0.073496I
b = 1.68413 + 0.73753I
5.28266 4.71026I 0.90242 + 5.00895I
u = 1.054049 + 0.808417I
a = 0.930522 + 0.073496I
b = 1.68413 0.73753I
5.28266 + 4.71026I 0.90242 5.00895I
u = 0.799783 0.916888I
a = 0.319317 + 0.987947I
b = 1.032672 0.198651I
6.09690 1.71389I 0.373315 + 0.701321I
u = 0.799783 + 0.916888I
a = 0.319317 0.987947I
b = 1.032672 + 0.198651I
6.09690 + 1.71389I 0.373315 0.701321I
u = 0.797251 1.038723I
a = 0.104291 + 0.925897I
b = 0.886790 + 0.124926I
5.41673 + 1.52162I 0.135230 0.682803I
u = 0.797251 + 1.038723I
a = 0.104291 0.925897I
b = 0.886790 0.124926I
5.41673 1.52162I 0.135230 + 0.682803I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.752901 0.442189I
a = 1.36315 + 0.44618I
b = 0.401158 1.069343I
1.98806 6.13047I 7.04352 + 9.47962I
u = 0.752901 + 0.442189I
a = 1.36315 0.44618I
b = 0.401158 + 1.069343I
1.98806 + 6.13047I 7.04352 9.47962I
u = 0.711514 0.271124I
a = 1.70079 + 0.13033I
b = 0.303552 0.853386I
3.56782 0.26795I 11.60981 + 1.69985I
u = 0.711514 + 0.271124I
a = 1.70079 0.13033I
b = 0.303552 + 0.853386I
3.56782 + 0.26795I 11.60981 1.69985I
u = 0.577742
a = 2.08389
b = 0.810576
2.39024 2.93566
u = 0.392808 0.498293I
a = 0.399459 1.090883I
b = 0.38285 1.84809I
1.02172 + 2.83866I 5.26921 + 0.34131I
u = 0.392808 + 0.498293I
a = 0.399459 + 1.090883I
b = 0.38285 + 1.84809I
1.02172 2.83866I 5.26921 0.34131I
u = 0.103618 0.421336I
a = 0.44769 2.36070I
b = 0.21261 3.21486I
1.92691 1.82552I 15.7454 + 26.9436I
u = 0.103618 + 0.421336I
a = 0.44769 + 2.36070I
b = 0.21261 + 3.21486I
1.92691 + 1.82552I 15.7454 26.9436I
u = 0.422420 0.531130I
a = 0.436609 + 0.016413I
b = 0.066812 0.708931I
0.196422 + 1.363410I 2.16894 4.86353I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.422420 + 0.531130I
a = 0.436609 0.016413I
b = 0.066812 + 0.708931I
0.196422 1.363410I 2.16894 + 4.86353I
u = 0.460095 0.445935I
a = 0.278491 0.419058I
b = 0.357134 0.973895I
0.154276 + 1.379412I 2.57505 4.68653I
u = 0.460095 + 0.445935I
a = 0.278491 + 0.419058I
b = 0.357134 + 0.973895I
0.154276 1.379412I 2.57505 + 4.68653I
u = 0.643082 1.012103I
a = 0.193054 + 0.732565I
b = 0.621491 0.083394I
0.145121 + 0.763133I 4.51767 0.23788I
u = 0.643082 + 1.012103I
a = 0.193054 0.732565I
b = 0.621491 + 0.083394I
0.145121 0.763133I 4.51767 + 0.23788I
u = 0.655458 0.476008I
a = 1.128361 + 0.264501I
b = 0.408849 0.969317I
0.42554 + 1.82346I 2.95478 4.65601I
u = 0.655458 + 0.476008I
a = 1.128361 0.264501I
b = 0.408849 + 0.969317I
0.42554 1.82346I 2.95478 + 4.65601I
u = 0.753278 0.081360I
a = 2.00566 + 0.05030I
b = 0.133758 0.305112I
4.40562 + 3.00647I 12.14909 4.76014I
u = 0.753278 + 0.081360I
a = 2.00566 0.05030I
b = 0.133758 + 0.305112I
4.40562 3.00647I 12.14909 + 4.76014I
u = 0.810417 0.872688I
a = 0.412435 + 1.014587I
b = 1.080361 0.335145I
4.57008 + 7.07850I 2.02910 5.97329I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.810417 + 0.872688I
a = 0.412435 1.014587I
b = 1.080361 + 0.335145I
4.57008 7.07850I 2.02910 + 5.97329I
u = 0.816149 1.085625I
a = 0.024199 + 0.913417I
b = 0.839085 + 0.253020I
3.33510 6.89323I 3.18635 + 5.25790I
u = 0.816149 + 1.085625I
a = 0.024199 0.913417I
b = 0.839085 0.253020I
3.33510 + 6.89323I 3.18635 5.25790I
u = 1.028319 0.763633I
a = 0.883281 0.139874I
b = 1.71233 + 0.55004I
3.86288 0.92225I 2.44177 0.36369I
u = 1.028319 + 0.763633I
a = 0.883281 + 0.139874I
b = 1.71233 0.55004I
3.86288 + 0.92225I 2.44177 + 0.36369I
u = 1.10053 0.91184I
a = 1.016520 + 0.117647I
b = 1.49494 + 1.20130I
2.4064 + 14.1156I 4.53533 8.71801I
u = 1.10053 + 0.91184I
a = 1.016520 0.117647I
b = 1.49494 1.20130I
2.4064 14.1156I 4.53533 + 8.71801I
u = 1.14394 0.85054I
a = 0.889142 + 0.076491I
b = 1.36393 + 0.91140I
1.33633 + 6.06144I 7.30135 4.78570I
u = 1.14394 + 0.85054I
a = 0.889142 0.076491I
b = 1.36393 0.91140I
1.33633 6.06144I 7.30135 + 4.78570I
u = 1.35764
a = 0.493788
b = 0.941286
3.32615 1.98009
8
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
5
u
4
+ ··· + u 1)(u
40
+ 2u
39
+ ··· + 5u + 1)
c
2
(u
5
+ 3u
4
+ ··· u 1)(u
40
+ 18u
39
+ ··· u + 1)
c
3
(u
5
+ u
4
+ ··· + u 1)(u
40
+ 2u
39
+ ··· u 1)
c
4
(u
5
+ u
4
+ ··· + u + 1)(u
40
+ 2u
39
+ ··· + 5u + 1)
c
5
(u
5
u
4
+ ··· + u + 1)(u
40
+ 2u
39
+ ··· 61u + 17)
c
6
, c
9
u
5
(u
40
+ 5u
39
+ ··· + 64u + 32)
c
7
(u
5
u
4
+ ··· + u + 1)(u
40
+ 2u
39
+ ··· u 1)
c
8
(u 1)
5
(u
40
+ 6u
39
+ ··· + 5u 1)
c
10
(u + 1)
5
(u
40
+ 6u
39
+ ··· + 5u 1)
c
11
(u + 1)
5
(u
40
+ 14u
39
+ ··· + 23u + 1)
9
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
5
+ 3y
4
+ ··· y 1)(y
40
+ 18y
39
+ ··· y + 1)
c
2
(y
5
y
4
+ ··· + 3y 1)(y
40
+ 10y
39
+ ··· 101y + 1)
c
3
, c
7
(y
5
5y
4
+ ··· y 1)(y
40
10y
39
+ ··· y + 1)
c
5
(y
5
5y
4
+ ··· y 1)(y
40
+ 2y
39
+ ··· + 223y + 289)
c
6
, c
9
y
5
(y
40
33y
39
+ ··· 8704y + 1024)
c
8
, c
10
(y 1)
5
(y
40
14y
39
+ ··· 23y + 1)
c
11
(y 1)
5
(y
40
+ 30y
39
+ ··· + 89y + 1)
10