11n
26
(K11n
26
)
1
Arc Sequences
4 1 6 2 10 4 5 11 6 8 9
Solving Sequence
5,10 2,6
4 7 8 1 3 9 11
c
5
c
4
c
6
c
7
c
1
c
3
c
9
c
11
c
2
, c
8
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hb
5
2b
4
b
3
+ 2b
2
1, u 1, b
3
2b
2
b + a + 1i
I
u
2
= hu
3
+ u
2
1, a, u
2
+ b u 1i
I
u
3
= hu
28
+ 7u
27
+ ··· + 5u + 1, 201191203u
27
1108083459u
26
+ ··· + 533902792a 71402890,
410351181u
27
2261934046u
26
+ ··· + 1067805584b + 861074247i
There are 3 irreducible components with 36 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
2b
4
b
3
+ 2b
2
1, u 1, b
3
2b
2
b + a + 1i
(i) Arc colorings
a
5
=
0
1
a
10
=
b
3
+ 2b
2
+ b 1
b
a
2
=
1
0
a
6
=
b
4
2b
3
b
2
+ b + 1
b
4
2b
3
b
2
+ b + 1
a
4
=
1
1
a
7
=
b
4
2b
3
b
2
+ b + 1
b
4
2b
3
b
2
+ b + 1
a
8
=
b
4
2b
3
b
2
+ b + 1
0
a
1
=
0
1
a
3
=
1
1
a
9
=
b
2
b 1
b
3
b
2
b
a
11
=
b
4
2b
3
b
2
+ 2b + 1
b
a
11
=
b
4
2b
3
b
2
+ 2b + 1
b
(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 = 0.339110 + 0.822375I
b = 0.767660 0.216900I
1.31583 + 1.53058I 5.47076 5.40154I
u = 1.00000
a = 0.339110 0.822375I
b = 0.767660 + 0.216900I
1.31583 1.53058I 5.47076 + 5.40154I
u = 1.00000
a = 0.455697 1.200152I
b = 0.732208 0.471915I
4.22763 + 4.40083I 0.88874 1.16747I
u = 1.00000
a = 0.455697 + 1.200152I
b = 0.732208 + 0.471915I
4.22763 4.40083I 0.88874 + 1.16747I
u = 1.00000
a = 0.766826
b = 2.07090
0.756147 1.28099
3
II. I
u
2
= hu
3
+ u
2
1, a, u
2
+ b u 1i
(i) Arc colorings
a
5
=
0
u
a
10
=
0
u
2
+ u + 1
a
2
=
1
0
a
6
=
0
u
a
4
=
u
u
a
7
=
u
2
1
u
2
+ u 1
a
8
=
u
2
1
u
2
a
1
=
u
2
+ 1
u
2
a
3
=
u
u
2
+ u 1
a
9
=
0
u
2
+ u + 1
a
11
=
u
2
+ 1
u + 1
a
11
=
u
2
+ 1
u + 1
(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 = 0.877439 0.744862I
a = 0
b = 0.337641 + 0.562280I
4.66906 2.82812I 4.21508 + 1.30714I
u = 0.877439 + 0.744862I
a = 0
b = 0.337641 0.562280I
4.66906 + 2.82812I 4.21508 1.30714I
u = 0.754878
a = 0
b = 2.32472
0.531480 4.56984
5
III.
I
u
3
= hu
28
+ 7u
27
+ · · · + 5u + 1, 2.01 × 10
8
u
27
1.11 × 10
9
u
26
+ · · · + 5.34 ×
10
8
a7.14×10
7
, 4.10×10
8
u
27
2.26×10
9
u
26
+· · ·+1.07×10
9
b+8.61×10
8
i
(i) Arc colorings
a
5
=
0
u
a
10
=
0.376831u
27
+ 2.07544u
26
+ ··· + 3.81852u + 0.133738
0.384294u
27
+ 2.11830u
26
+ ··· 1.74330u 0.806396
a
2
=
1
0
a
6
=
1.25986u
27
7.66743u
26
+ ··· 2.89643u 0.200716
0.691498u
27
4.38940u
26
+ ··· 3.69213u 0.804996
a
4
=
u
u
a
7
=
1.50549u
27
9.60132u
26
+ ··· 5.83056u 0.901213
0.937133u
27
6.32329u
26
+ ··· 6.62625u 1.50549
a
8
=
1.50549u
27
9.60132u
26
+ ··· 5.83056u 0.901213
0.700497u
27
4.65784u
26
+ ··· 3.44608u 0.568361
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
u
2
+ 1
u
4
a
9
=
1.98352u
27
+ 13.0710u
26
+ ··· + 21.1333u + 4.92136
0.0464332u
27
0.509563u
26
+ ··· 0.750264u + 0.0699424
a
11
=
0.0496820u
27
0.221917u
26
+ ··· + 9.10463u + 2.37366
0.339712u
27
2.56601u
26
+ ··· 3.10113u 0.544675
a
11
=
0.0496820u
27
0.221917u
26
+ ··· + 9.10463u + 2.37366
0.339712u
27
2.56601u
26
+ ··· 3.10113u 0.544675
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.21166 0.95824I
a = 0.992665 0.493458I
b = 2.40331 + 0.30645I
15.7181 11.7289I 1.85525 + 5.52053I
u = 1.21166 + 0.95824I
a = 0.992665 + 0.493458I
b = 2.40331 0.30645I
15.7181 + 11.7289I 1.85525 5.52053I
u = 1.08042 0.99381I
a = 0.826721 + 0.565549I
b = 2.27480 0.57676I
8.71794 6.87707I 0.35894 + 4.81213I
u = 1.08042 + 0.99381I
a = 0.826721 0.565549I
b = 2.27480 + 0.57676I
8.71794 + 6.87707I 0.35894 4.81213I
u = 1.03272 1.05137I
a = 0.661202 + 0.788679I
b = 0.635607 + 0.580507I
11.20706 3.83748I 1.59779 + 2.22620I
u = 1.03272 + 1.05137I
a = 0.661202 0.788679I
b = 0.635607 0.580507I
11.20706 + 3.83748I 1.59779 2.22620I
u = 0.95494 1.07229I
a = 0.706388 0.731921I
b = 1.85864 + 0.72430I
9.13873 0.66915I 1.241168 0.226691I
u = 0.95494 + 1.07229I
a = 0.706388 + 0.731921I
b = 1.85864 0.72430I
9.13873 + 0.66915I 1.241168 + 0.226691I
u = 0.894453 0.824309I
a = 0.381171 0.361966I
b = 0.335788 0.244659I
3.90340 3.06304I 6.04954 + 3.66902I
u = 0.894453 + 0.824309I
a = 0.381171 + 0.361966I
b = 0.335788 + 0.244659I
3.90340 + 3.06304I 6.04954 3.66902I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.84120 1.23310I
a = 0.705663 + 0.965443I
b = 1.43671 0.43273I
16.9931 + 3.8383I 3.05153 1.22806I
u = 0.84120 + 1.23310I
a = 0.705663 0.965443I
b = 1.43671 + 0.43273I
16.9931 3.8383I 3.05153 + 1.22806I
u = 0.744635 0.318856I
a = 1.04254 + 1.00447I
b = 0.743836 + 0.653886I
4.82268 5.10002I 4.96882 + 7.61668I
u = 0.744635 + 0.318856I
a = 1.04254 1.00447I
b = 0.743836 0.653886I
4.82268 + 5.10002I 4.96882 7.61668I
u = 0.414733 0.266521I
a = 1.76186 0.85665I
b = 0.816071 0.434174I
0.04395 1.99045I 0.01306 + 4.61620I
u = 0.414733 + 0.266521I
a = 1.76186 + 0.85665I
b = 0.816071 + 0.434174I
0.04395 + 1.99045I 0.01306 4.61620I
u = 0.136281 0.404052I
a = 0.05147 2.09880I
b = 0.515428 + 0.118133I
2.62794 0.46347I 2.20728 0.53901I
u = 0.136281 + 0.404052I
a = 0.05147 + 2.09880I
b = 0.515428 0.118133I
2.62794 + 0.46347I 2.20728 + 0.53901I
u = 0.220238 0.502777I
a = 1.54836 0.65502I
b = 1.107366 + 0.858066I
1.30841 + 1.56433I 2.39227 4.63205I
u = 0.220238 + 0.502777I
a = 1.54836 + 0.65502I
b = 1.107366 0.858066I
1.30841 1.56433I 2.39227 + 4.63205I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.423937 0.981765I
a = 1.43972 + 0.31762I
b = 1.60951 0.53420I
8.83250 + 3.91759I 2.90293 3.10234I
u = 0.423937 + 0.981765I
a = 1.43972 0.31762I
b = 1.60951 + 0.53420I
8.83250 3.91759I 2.90293 + 3.10234I
u = 0.668462
a = 0.455572
b = 0.336502
1.01341 10.2409
u = 0.917213
a = 0.380776
b = 4.19808
0.303143 47.2617
u = 1.051397 0.149533I
a = 0.176815 + 0.574305I
b = 1.305116 0.429518I
1.169037 + 0.736496I 2.56323 + 2.93619I
u = 1.051397 + 0.149533I
a = 0.176815 0.574305I
b = 1.305116 + 0.429518I
1.169037 0.736496I 2.56323 2.93619I
u = 1.322623 0.396919I
a = 0.367492 0.981366I
b = 1.283548 0.368371I
5.47968 + 1.56446I 1.30115 0.62804I
u = 1.322623 + 0.396919I
a = 0.367492 + 0.981366I
b = 1.283548 + 0.368371I
5.47968 1.56446I 1.30115 + 0.62804I
9
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
5
(u
3
+ u
2
1)(u
28
+ 7u
27
+ ··· + 5u + 1)
c
2
(u + 1)
5
(u
3
+ u
2
+ 2u + 1)(u
28
+ 5u
27
+ ··· 3u + 1)
c
3
, c
6
u
5
(u
3
u
2
+ 2u 1)(u
28
+ 2u
27
+ ··· 24u
2
32)
c
4
(u + 1)
5
(u
3
+ u
2
1)(u
28
+ 7u
27
+ ··· + 5u + 1)
c
5
u
3
(u
5
+ u
4
+ ··· + u + 1)(u
28
+ 2u
27
+ ··· + 20u + 8)
c
7
(u
3
3u
2
+ 2u + 1)(u
5
3u
4
+ ··· u + 1)(u
28
+ 3u
27
+ ··· u 1)
c
8
(u + 1)
3
(u
5
u
4
+ ··· + u + 1)(u
28
+ 5u
27
+ ··· 8u 1)
c
9
u
3
(u
5
u
4
+ ··· + u 1)(u
28
+ 2u
27
+ ··· + 20u + 8)
c
10
(u + 1)
3
(u
5
+ u
4
+ ··· + u 1)(u
28
+ 5u
27
+ ··· 8u 1)
c
11
(u 1)
3
(u
5
+ u
4
+ ··· + u 1)(u
28
+ 5u
27
+ ··· 8u 1)
10
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
5
(y
3
y
2
+ 2y 1)(y
28
5y
27
+ ··· + 3y + 1)
c
2
(y 1)
5
(y
3
+ 3y
2
+ 2y 1)(y
28
+ 43y
27
+ ··· + 3y + 1)
c
3
, c
6
y
5
(y
3
+ 3y
2
+ 2y 1)(y
28
+ 36y
27
+ ··· + 1536y + 1024)
c
5
, c
9
y
3
(y
5
+ 3y
4
+ ··· y 1)(y
28
+ 24y
27
+ ··· 848y + 64)
c
7
(y
3
5y
2
+ 10y 1)(y
5
y
4
+ 8y
3
3y
2
+ 3y 1)
(y
28
37y
27
+ ··· 35y + 1)
c
8
, c
10
, c
11
(y 1)
3
(y
5
5y
4
+ ··· y 1)(y
28
31y
27
+ ··· 128y + 1)
11