11n
27
(K11n
27
)
1
Arc Sequences
4 1 7 2 8 10 4 11 1 6 9
Solving Sequence
9,11 1,4
2 5 10 8 7 3 6
c
11
c
1
c
4
c
9
c
8
c
7
c
3
c
6
c
2
, c
5
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hb
2
+ b 1, u + 1, a 1i
I
u
2
= hu
5
4u
3
5u
2
2u 1, 3u
4
+ 4u
3
+ 10u
2
+ 5a 4, 2u
4
u
3
10u
2
+ 5b 5u + 1i
I
u
3
= hu
16
u
15
+ ··· + 71u 41,
6.72277 × 10
19
u
15
+ 8.40759 × 10
18
u
14
+ ··· + 4.95570 × 10
21
b 9.94823 × 10
20
,
2.71636 × 10
21
u
15
1.06484 × 10
21
u
14
+ ··· + 1.01592 × 10
23
a + 2.70442 × 10
22
i
There are 3 irreducible components with 23 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
2
+ b 1, u + 1, a 1i
(i) Arc colorings
a
9
=
0
1
a
11
=
1
b
a
1
=
1
b + 1
a
4
=
1
0
a
2
=
b
b + 1
a
5
=
0
b + 2
a
10
=
1
b
a
8
=
1
b 1
a
7
=
b 2
b 1
a
3
=
2b 2
b 2
a
6
=
b 2
b 1
a
6
=
b 2
b 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
a = 1.00000
b = 1.61803
2.63189 21.0000
u = 1.00000
a = 1.00000
b = 0.618034
10.5276 21.0000
3
II. I
u
2
=
hu
5
4u
3
5u
2
2u1, 3u
4
+4u
3
+10u
2
+5a4, 2u
4
u
3
10u
2
+5b5u+1i
(i) Arc colorings
a
9
=
0
u
a
11
=
3
5
u
4
4
5
u
3
2u
2
+
4
5
2
5
u
4
+
1
5
u
3
+ 2u
2
+ u
1
5
a
1
=
3
5
u
4
4
5
u
3
2u
2
+
4
5
1
a
4
=
1
0
a
2
=
3
5
u
4
4
5
u
3
2u
2
+
9
5
1
a
5
=
3
5
u
4
+
4
5
u
3
+ 2u
2
4
5
1
a
10
=
1
5
u
4
+
3
5
u
3
u +
2
5
4
5
u
4
+
2
5
u
3
+ 3u
2
+ 3u +
3
5
a
8
=
1
5
u
4
3
5
u
3
+ u
2
5
0
a
7
=
1
5
u
4
3
5
u
3
+ u
2
5
0
a
3
=
1
0
a
6
=
1
5
u
4
2
5
u
3
+ 2u
2
+ u
3
5
1
a
6
=
1
5
u
4
2
5
u
3
+ 2u
2
+ u
3
5
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 = 1.142272 0.509071I
a = 0.964913 + 0.621896I
b = 0.732208 + 0.471915I
7.51750 + 4.40083I 14.3774 5.8297I
u = 1.142272 + 0.509071I
a = 0.964913 0.621896I
b = 0.732208 0.471915I
7.51750 4.40083I 14.3774 + 5.8297I
u = 0.118634 0.489365I
a = 1.206354 0.340852I
b = 0.767660 0.216900I
1.97403 1.53058I 10.50099 + 3.45976I
u = 0.118634 + 0.489365I
a = 1.206354 + 0.340852I
b = 0.767660 + 0.216900I
1.97403 + 1.53058I 10.50099 3.45976I
u = 2.52181
a = 0.482881
b = 2.07090
4.04602 8.24327
5
III. I
u
3
=
hu
16
u
15
+· · ·+71u41, 6.72×10
19
u
15
+8.41×10
18
u
14
+· · ·+4.96×10
21
b
9.95 × 10
20
, 2.72 × 10
21
u
15
1.06 × 10
21
u
14
+ · · · + 1.02 × 10
23
a + 2.70 × 10
22
i
(i) Arc colorings
a
9
=
0
u
a
11
=
0.0267380u
15
+ 0.0104816u
14
+ ··· 1.36193u 0.266204
0.0135658u
15
0.00169655u
14
+ ··· 0.705658u + 0.200743
a
1
=
0.0267380u
15
+ 0.0104816u
14
+ ··· 1.36193u 0.266204
0.000819945u
15
+ 0.00450546u
14
+ ··· 0.647712u 0.465768
a
4
=
1
0
a
2
=
0.0259180u
15
+ 0.00597614u
14
+ ··· 0.714218u + 0.199564
0.000819945u
15
+ 0.00450546u
14
+ ··· 0.647712u 0.465768
a
5
=
0.0215167u
15
0.00210266u
14
+ ··· 0.182334u + 0.235316
0.0200970u
15
+ 0.00152051u
14
+ ··· 0.214731u + 0.743230
a
10
=
0.0181385u
15
+ 0.0114287u
14
+ ··· 2.65417u + 1.48574
0.00200178u
15
0.00569618u
14
+ ··· + 0.920660u + 0.0582048
a
8
=
0.0181385u
15
0.0114287u
14
+ ··· + 2.65417u 1.48574
0.0153044u
15
+ 0.00614683u
14
+ ··· 0.276252u + 1.15405
a
7
=
0.00283417u
15
0.00528184u
14
+ ··· + 2.37792u 0.331684
0.0153044u
15
+ 0.00614683u
14
+ ··· 0.276252u + 1.15405
a
3
=
0.0109636u
15
+ 0.00522172u
14
+ ··· 1.53465u + 1.24351
0.0155996u
15
0.0115747u
14
+ ··· + 0.139188u 0.457919
a
6
=
0.0174875u
15
0.0291539u
14
+ ··· + 1.83034u + 0.340220
0.000819945u
15
0.00450546u
14
+ ··· + 0.647712u + 0.465768
a
6
=
0.0174875u
15
0.0291539u
14
+ ··· + 1.83034u + 0.340220
0.000819945u
15
0.00450546u
14
+ ··· + 0.647712u + 0.465768
(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 = 2.75413 0.83327I
a = 0.513894 0.307699I
b = 2.94107 0.66400I
15.4222 + 9.3337I 13.42948 3.49093I
u = 2.75413 + 0.83327I
a = 0.513894 + 0.307699I
b = 2.94107 + 0.66400I
15.4222 9.3337I 13.42948 + 3.49093I
u = 1.10854
a = 0.656633
b = 0.869046
10.0599 3.26618
u = 1.00968 1.74590I
a = 0.091891 + 0.435115I
b = 0.21748 + 1.39785I
3.23204 + 0.76102I 14.1078 + 3.1845I
u = 1.00968 + 1.74590I
a = 0.091891 0.435115I
b = 0.21748 1.39785I
3.23204 0.76102I 14.1078 3.1845I
u = 0.764692 0.192667I
a = 1.77958 0.70366I
b = 0.658398 + 0.409536I
6.69050 + 3.49798I 9.87558 1.25665I
u = 0.764692 + 0.192667I
a = 1.77958 + 0.70366I
b = 0.658398 0.409536I
6.69050 3.49798I 9.87558 + 1.25665I
u = 0.022638 0.459801I
a = 0.938004 + 0.768501I
b = 0.256531 + 0.411391I
0.428790 + 1.166927I 5.36023 5.57896I
u = 0.022638 + 0.459801I
a = 0.938004 0.768501I
b = 0.256531 0.411391I
0.428790 1.166927I 5.36023 + 5.57896I
u = 0.530023
a = 0.159422
b = 0.678379
1.09573 8.61157
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.871184
a = 1.28231
b = 1.13180
2.15355 1.76419
u = 1.018812 0.314256I
a = 0.290313 + 1.176814I
b = 1.030660 0.784875I
16.5506 3.5813I 12.12116 + 2.15994I
u = 1.018812 + 0.314256I
a = 0.290313 1.176814I
b = 1.030660 + 0.784875I
16.5506 + 3.5813I 12.12116 2.15994I
u = 2.19773 0.03597I
a = 0.787135 0.050402I
b = 2.31778 0.82044I
12.66301 2.31460I 13.80105 + 1.17558I
u = 2.19773 + 0.03597I
a = 0.787135 + 0.050402I
b = 2.31778 + 0.82044I
12.66301 + 2.31460I 13.80105 1.17558I
u = 3.28596
a = 0.520308
b = 3.50482
19.0073 12.9675
8
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
5
(u
2
+ u 1)(u
16
+ 7u
15
+ ··· 3u + 1)
c
2
(u + 1)
5
(u
2
+ 3u + 1)(u
16
+ 29u
15
+ ··· + 17u + 1)
c
3
u
5
(u
2
+ u 1)(u
16
+ 2u
15
+ ··· + 72u
2
32)
c
4
(u + 1)
5
(u
2
u 1)(u
16
+ 7u
15
+ ··· 3u + 1)
c
5
(u
2
3u + 1)(u
5
3u
4
+ ··· u + 1)(u
16
+ 3u
15
+ ··· u 1)
c
6
u
2
(u
5
u
4
+ ··· + u 1)(u
16
+ 2u
15
+ ··· + 20u 4)
c
7
u
5
(u
2
u 1)(u
16
+ 2u
15
+ ··· + 72u
2
32)
c
8
, c
9
(u 1)
2
(u
5
+ u
4
+ ··· + u 1)(u
16
+ 4u
15
+ ··· + 10u + 1)
c
10
u
2
(u
5
+ u
4
+ ··· + u + 1)(u
16
+ 2u
15
+ ··· + 20u 4)
c
11
(u + 1)
2
(u
5
u
4
+ ··· + u + 1)(u
16
+ 4u
15
+ ··· + 10u + 1)
9
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
5
(y
2
3y + 1)(y
16
29y
15
+ ··· 17y + 1)
c
2
(y 1)
5
(y
2
7y + 1)(y
16
77y
15
+ ··· 1761y + 1)
c
3
, c
7
y
5
(y
2
3y + 1)(y
16
36y
15
+ ··· 4608y + 1024)
c
5
(y
2
7y + 1)(y
5
y
4
+ ··· + 3y 1)(y
16
37y
15
+ ··· 11y + 1)
c
6
, c
10
y
2
(y
5
+ 3y
4
+ ··· y 1)(y
16
+ 18y
15
+ ··· 168y + 16)
c
8
, c
9
, c
11
(y 1)
2
(y
5
5y
4
+ ··· y 1)(y
16
20y
15
+ ··· 146y + 1)
10