11n
180
(K11n
180
)
1
Arc Sequences
6 8 1 9 10 1 4 2 5 4 8
Solving Sequence
4,7 1,8
3 6 2 9 11 10 5
c
7
c
3
c
6
c
1
c
8
c
11
c
10
c
5
c
2
, c
4
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
9
3u
8
+ u
7
+ 2u
6
+ u
5
2u
4
+ 2u
3
2u
2
1, u
7
2u
6
u
5
+ 2u
4
+ u
3
u
2
+ b + u,
u
8
+ 2u
7
+ u
6
2u
5
u
4
+ 2u
3
2u
2
+ a ui
I
u
2
= hu
16
+ 12u
15
+ ··· 360u
2
+ 32, 7u
15
66u
14
+ ··· + 4b 36, 41u
15
460u
14
+ ··· + 32a 944i
I
u
3
= hb
20
+ b
19
+ ··· + 42b 1,
391024534512b
19
+ 5790186695099u + ··· 6634100171759b 9201275740538,
4.94036 × 10
15
b
19
+ 1.56311 × 10
16
b
18
+ ··· + 5.38493 × 10
17
a + 2.30093 × 10
17
i
There are 3 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
9
3u
8
+ u
7
+ 2u
6
+ u
5
2u
4
+ 2u
3
2u
2
1, u
7
2u
6
u
5
+
2u
4
+ u
3
u
2
+ b + u, u
8
+ 2u
7
+ · · · + a ui
(i) Arc colorings
a
4
=
0
u
a
7
=
u
8
2u
7
u
6
+ 2u
5
+ u
4
2u
3
+ 2u
2
+ u
u
7
+ 2u
6
+ u
5
2u
4
u
3
+ u
2
u
a
1
=
1
0
a
8
=
u
8
2u
7
u
6
+ 2u
5
+ u
4
2u
3
+ 2u
2
+ u
u
8
2u
7
+ 3u
2
+ 1
a
3
=
u
u
a
6
=
u
8
3u
7
+ u
6
+ 3u
5
u
4
3u
3
+ 3u
2
u
7
+ 2u
6
+ u
5
2u
4
u
3
+ u
2
u
a
2
=
u
8
+ 3u
7
u
6
2u
5
u
4
+ 2u
3
2u
2
+ 3u
u 1
a
9
=
u
6
2u
5
u
4
+ u
3
+ 2u
2
+ 1
u
7
u
6
3u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
a
11
=
u
7
+ 2u
6
+ u
5
u
4
2u
3
2u + 1
u
8
+ 2u
7
+ u
6
u
5
2u
4
2u
2
a
10
=
u
7
+ 2u
6
+ u
5
u
4
2u
3
2u + 1
u
2
+ 1
a
5
=
u
8
+ 2u
7
+ u
6
u
5
3u
4
+ u
3
1
u
8
+ 2u
7
+ u
6
u
5
3u
4
+ 2u
3
u
2
1
a
5
=
u
8
+ 2u
7
+ u
6
u
5
3u
4
+ u
3
1
u
8
+ 2u
7
+ u
6
u
5
3u
4
+ 2u
3
u
2
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 = 0.901563 0.564856I
a = 0.432004 + 0.508675I
b = 0.202634 1.393780I
1.79239 1.81153I 9.80943 + 3.54546I
u = 0.901563 + 0.564856I
a = 0.432004 0.508675I
b = 0.202634 + 1.393780I
1.79239 + 1.81153I 9.80943 3.54546I
u = 0.233182 0.559961I
a = 1.39335 0.25566I
b = 0.246149 + 1.137441I
4.84522 + 1.19732I 2.35469 1.53706I
u = 0.233182 + 0.559961I
a = 1.39335 + 0.25566I
b = 0.246149 1.137441I
4.84522 1.19732I 2.35469 + 1.53706I
u = 0.291663 0.753926I
a = 0.866867 0.853127I
b = 0.403820 0.933492I
0.28529 + 4.77597I 8.22483 4.15931I
u = 0.291663 + 0.753926I
a = 0.866867 + 0.853127I
b = 0.403820 + 0.933492I
0.28529 4.77597I 8.22483 + 4.15931I
u = 1.27478
a = 1.49648
b = 0.524196
11.1241 11.2464
u = 1.54182
a = 0.881123
b = 0.736086
4.17989 9.50993
u = 1.86956
a = 0.573604
b = 0.932501
7.27021 22.4658
3
II. I
u
2
= hu
16
+ 12u
15
+ · · · 360u
2
+ 32, 7u
15
66u
14
+ · · · + 4b
36, 41u
15
460u
14
+ · · · + 32a 944i
(i) Arc colorings
a
4
=
0
u
a
7
=
41
32
u
15
+
115
8
u
14
+ ··· 17u +
59
2
7
4
u
15
+
33
2
u
14
+ ··· +
23
2
u + 9
a
1
=
1
0
a
8
=
41
32
u
15
+
115
8
u
14
+ ··· 17u +
59
2
u
15
+
51
4
u
14
+ ···
59
2
u + 41
a
3
=
u
u
a
6
=
3.03125u
15
+ 30.8750u
14
+ ··· 5.50000u + 38.5000
7
4
u
15
+
33
2
u
14
+ ··· +
23
2
u + 9
a
2
=
1
32
u
15
5
16
u
14
+ ··· +
45
8
u
2
+ u
1
16
u
15
5
8
u
14
+ ··· + u 1
a
9
=
3.18750u
15
+ 31.9375u
14
+ ··· 8.75000u + 39.5000
33
16
u
15
+
153
8
u
14
+ ··· +
15
2
u + 14
a
11
=
31
32
u
15
171
16
u
14
+ ··· + 31u 30
15
16
u
15
83
8
u
14
+ ··· + 31u 31
a
10
=
31
32
u
15
171
16
u
14
+ ··· + 31u 30
1
16
u
15
5
8
u
14
+ ··· +
41
4
u
2
1
a
5
=
5
4
u
15
49
4
u
14
+ ··· 7u 11
17
4
u
15
345
8
u
14
+ ··· + 26u 66
a
5
=
5
4
u
15
49
4
u
14
+ ··· 7u 11
17
4
u
15
345
8
u
14
+ ··· + 26u 66
(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 = 1.68493 0.47840I
a = 1.090815 + 0.055715I
b = 0.94605 1.28824I
7.9978 12.6965I 13.6924 + 6.6132I
u = 1.68493 + 0.47840I
a = 1.090815 0.055715I
b = 0.94605 + 1.28824I
7.9978 + 12.6965I 13.6924 6.6132I
u = 1.61124 0.53845I
a = 1.041829 + 0.051732I
b = 0.78590 + 1.21081I
2.13858 8.47484I 10.02725 + 6.22402I
u = 1.61124 + 0.53845I
a = 1.041829 0.051732I
b = 0.78590 1.21081I
2.13858 + 8.47484I 10.02725 6.22402I
u = 1.45999 0.54919I
a = 1.051977 0.234978I
b = 0.636967 1.022314I
3.42042 3.44951I 11.26032 + 2.20716I
u = 1.45999 + 0.54919I
a = 1.051977 + 0.234978I
b = 0.636967 + 1.022314I
3.42042 + 3.44951I 11.26032 2.20716I
u = 1.312263 0.342243I
a = 1.36980 + 0.44221I
b = 0.646721 + 0.632174I
11.54012 0.83768I 14.6260 + 5.7551I
u = 1.312263 + 0.342243I
a = 1.36980 0.44221I
b = 0.646721 0.632174I
11.54012 + 0.83768I 14.6260 5.7551I
u = 0.646210 0.969563I
a = 0.294430 0.642409I
b = 0.161958 0.834243I
0.44623 2.35749I 11.39443 + 1.24540I
u = 0.646210 + 0.969563I
a = 0.294430 + 0.642409I
b = 0.161958 + 0.834243I
0.44623 + 2.35749I 11.39443 1.24540I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.132048 1.204662I
a = 0.014404 + 0.499826I
b = 0.465657 + 0.769869I
2.97261 + 1.81783I 9.12623 4.02809I
u = 0.132048 + 1.204662I
a = 0.014404 0.499826I
b = 0.465657 0.769869I
2.97261 1.81783I 9.12623 + 4.02809I
u = 0.215621
a = 1.31222
b = 0.343951
0.531160 18.6233
u = 0.27007 1.38948I
a = 0.071515 0.368633I
b = 0.675325 0.738298I
1.44057 + 5.86096I 14.3345 7.4758I
u = 0.27007 + 1.38948I
a = 0.071515 + 0.368633I
b = 0.675325 + 0.738298I
1.44057 5.86096I 14.3345 + 7.4758I
u = 0.937606
a = 0.432873
b = 0.786405
5.72940 16.4544
6
III.
I
u
3
= hb
20
+b
19
+· · ·+42b1, 5.79×10
12
u3.91×10
11
b
19
+· · ·6.63×10
12
b
9.20 × 10
12
, 4.94 × 10
15
b
19
+ 1.56 × 10
16
b
18
+ · · · + 5.38 × 10
17
a + 2.30 × 10
17
i
(i) Arc colorings
a
4
=
0
0.0675323b
19
+ 0.0696533b
18
+ ··· + 1.14575b + 1.58912
a
7
=
0.00917441b
19
0.0290275b
18
+ ··· 2.97011b 0.427290
b
a
1
=
1
0
a
8
=
0.00917441b
19
0.0290275b
18
+ ··· 2.97011b 0.427290
0.0135555b
19
0.0193971b
18
+ ··· + 0.528869b 1.29052
a
3
=
0.0675323b
19
+ 0.0696533b
18
+ ··· + 1.14575b + 1.58912
0.0675323b
19
+ 0.0696533b
18
+ ··· + 1.14575b + 1.58912
a
6
=
0.00917441b
19
0.0290275b
18
+ ··· 1.97011b 0.427290
b
a
2
=
0.0382020b
19
0.0178862b
18
+ ··· 0.812615b + 1.00917
b
2
a
9
=
0.0262870b
19
+ 0.0280973b
18
+ ··· + 1.55357b + 0.715364
0.0260722b
19
0.0434551b
18
+ ··· 0.526714b 1.26511
a
11
=
0.0351719b
19
0.0678491b
18
+ ··· 1.05432b + 0.388155
0.0547355b
19
0.0646556b
18
+ ··· 1.07587b 1.60972
a
10
=
0.0351719b
19
0.0678491b
18
+ ··· 1.05432b + 0.388155
0.0616907b
19
0.0535712b
18
+ ··· 0.424564b 2.57556
a
5
=
0.0230172b
19
0.0589354b
18
+ ··· 0.239674b 0.910333
0.0903855b
19
+ 0.0941738b
18
+ ··· + 1.27062b + 3.14168
a
5
=
0.0230172b
19
0.0589354b
18
+ ··· 0.239674b 0.910333
0.0903855b
19
+ 0.0941738b
18
+ ··· + 1.27062b + 3.14168
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.61803
a = 1.124851 + 0.047143I
b = 1.40649 0.78888I
9.82040 + 4.40083I 14.7443 3.4986I
u = 1.61803
a = 1.124851 0.047143I
b = 1.40649 + 0.78888I
9.82040 4.40083I 14.7443 + 3.4986I
u = 1.61803
a = 0.790561
b = 1.26448
6.34892 11.4811
u = 1.61803
a = 0.842560 + 0.263250I
b = 0.632978 0.771506I
4.27694 1.53058I 10.51511 + 4.43065I
u = 1.61803
a = 0.842560 0.263250I
b = 0.632978 + 0.771506I
4.27694 + 1.53058I 10.51511 4.43065I
u = 0.618034
a = 1.147361 0.672981I
b = 0.410028 1.268906I
1.92472 4.40083I 14.7443 + 3.4986I
u = 0.618034
a = 1.147361 + 0.672981I
b = 0.410028 + 1.268906I
1.92472 + 4.40083I 14.7443 3.4986I
u = 0.618034
a = 0.21019 + 2.15583I
b = 0.338497 0.825199I
3.61874 1.53058I 10.51511 + 4.43065I
u = 0.618034
a = 0.21019 2.15583I
b = 0.338497 + 0.825199I
3.61874 + 1.53058I 10.51511 4.43065I
u = 1.61803
a = 0.497659
b = 0.0237347
6.34892 11.4811
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.618034
a = 0.677607 + 0.002824I
b = 0.128916 1.333455I
3.61874 + 1.53058I 10.51511 4.43065I
u = 0.618034
a = 0.677607 0.002824I
b = 0.128916 + 1.333455I
3.61874 1.53058I 10.51511 + 4.43065I
u = 0.618034
a = 1.00379 + 1.46626I
b = 0.23696 1.46626I
1.54676 11.4811
u = 0.618034
a = 1.00379 1.46626I
b = 0.23696 + 1.46626I
1.54676 11.4811
u = 1.61803
a = 0.950790 + 0.411274I
b = 0.669154 1.153009I
9.82040 + 4.40083I 14.7443 3.4986I
u = 1.61803
a = 0.950790 0.411274I
b = 0.669154 + 1.153009I
9.82040 4.40083I 14.7443 + 3.4986I
u = 0.618034
a = 0.04567 + 2.46906I
b = 0.691664 0.527171I
1.92472 + 4.40083I 14.7443 3.4986I
u = 0.618034
a = 0.04567 2.46906I
b = 0.691664 + 0.527171I
1.92472 4.40083I 14.7443 + 3.4986I
u = 1.61803
a = 0.972088 + 0.050869I
b = 1.181670 0.559125I
4.27694 1.53058I 10.51511 + 4.43065I
u = 1.61803
a = 0.972088 0.050869I
b = 1.181670 + 0.559125I
4.27694 + 1.53058I 10.51511 4.43065I
9
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
8
(u
9
+ 3u
7
+ ··· u 1)(u
16
+ 3u
14
+ ··· + 3u 1)
(u
20
+ u
19
+ ··· + 42u 1)
c
2
, c
6
(u
9
+ 3u
7
+ ··· u + 1)(u
16
+ 3u
14
+ ··· + 3u 1)
(u
20
+ u
19
+ ··· + 42u 1)
c
3
(u
2
u 1)
10
(u
9
+ 3u
8
+ u
7
2u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 1)
(u
16
+ 12u
15
+ ··· 360u
2
+ 32)
c
4
, c
5
(1 + u + u
2
2u
3
u
4
+ u
5
)
4
(u
9
5u
7
+ ··· 2u + 1)
(u
16
+ 5u
15
+ ··· + 10u 4)
c
7
, c
11
(u
9
u
8
+ ··· + 3u
2
+ 1)(u
16
+ u
15
+ ··· 4u 1)
(u
20
+ u
19
+ ··· + 40u 29)
c
9
(1 + u + u
2
2u
3
u
4
+ u
5
)
4
(u
9
5u
7
+ ··· 2u 1)
(u
16
+ 5u
15
+ ··· + 10u 4)
c
10
(u
5
3u
4
+ 4u
3
u
2
u + 1)
4
(u
9
u
7
+ 8u
6
4u
5
+ 5u
4
2u
3
+ 3u
2
2u 1)
(u
16
+ 15u
15
+ ··· + 1722u + 196)
10
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
2
, c
6
c
8
(y
9
+ 6y
8
+ 13y
7
+ 9y
6
8y
5
16y
4
5y
3
+ 5y
2
+ 3y 1)
(y
16
+ 6y
15
+ ··· 5y + 1)(y
20
+ 7y
19
+ ··· 1772y + 1)
c
3
(y
2
3y + 1)
10
(y
9
7y
8
+ 15y
7
10y
6
+ y
5
+ 2y
4
8y
2
4y 1)
(y
16
10y
15
+ ··· 23040y + 1024)
c
4
, c
5
, c
9
(y
5
5y
4
+ 8y
3
3y
2
y 1)
4
(y
9
10y
8
+ 41y
7
86y
6
+ 90y
5
29y
4
19y
3
+ 6y
2
+ 8y 1)
(y
16
15y
15
+ ··· 140y + 16)
c
7
, c
11
(y
9
3y
8
5y
7
+ 5y
6
+ 16y
5
+ 8y
4
9y
3
13y
2
6y 1)
(y
16
27y
15
+ ··· 36y + 1)(y
20
13y
19
+ ··· 12736y + 841)
c
10
(y
5
y
4
+ 8y
3
3y
2
+ 3y 1)
4
(y
9
2y
8
7y
7
60y
6
64y
5
53y
4
+ 6y
3
+ 9y
2
+ 10y 1)
(y
16
3y
15
+ ··· 394156y + 38416)
11