11n
89
(K11n
89
)
1
Arc Sequences
6 1 9 10 2 3 11 3 4 7 10
Solving Sequence
7,11 3,8
9 4 6 10 5 1 2
c
7
c
8
c
3
c
6
c
10
c
4
c
11
c
2
c
1
, c
5
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
2
u + 1, b + u, a + u 1i
I
u
2
= hb
4
+ 2b
3
b
2
2b + 7, 2b
3
3b
2
2b + 11u + 5, 2b
3
3b
2
2b + 11a 6i
I
u
3
= hu
36
+ 2u
35
+ ··· + 6u + 13, 4.71208 × 10
61
u
35
+ 8.17244 × 10
60
u
34
+ ··· + 5.00069 × 10
63
b + 8.68163 × 10
63
,
8.35233 × 10
62
u
35
1.25676 × 10
63
u
34
+ ··· + 2.50034 × 10
63
a 4.45980 × 10
64
i
There are 3 irreducible components with 42 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
2
u + 1, b + u, a + u 1i
(i) Arc colorings
a
7
=
0
u
a
11
=
u + 1
u
a
3
=
1
0
a
8
=
u + 1
0
a
9
=
u + 1
0
a
4
=
1
0
a
6
=
u
u
a
10
=
u + 1
0
a
5
=
1
0
a
1
=
0
u
a
2
=
1
u + 1
a
2
=
1
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 = 0.500000 0.866025I
a = 0.500000 + 0.866025I
b = 0.500000 + 0.866025I
1.64493 + 2.02988I 12.00000 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 0.866025I
b = 0.500000 0.866025I
1.64493 2.02988I 12.00000 + 3.46410I
3
II.
I
u
2
= hb
4
+2b
3
b
2
2b+7, 2b
3
3b
2
2b+11 u+5, 2b
3
3b
2
2b+11a6i
(i) Arc colorings
a
7
=
0
0.181818b
3
+ 0.272727b
2
+ 0.181818b 0.454545
a
11
=
0.181818b
3
+ 0.272727b
2
+ 0.181818b + 0.545455
b
a
3
=
1
0
a
8
=
0.181818b
3
0.272727b
2
0.181818b 0.545455
0.181818b
3
+ 0.272727b
2
0.818182b 0.454545
a
9
=
b 1
0.181818b
3
+ 0.272727b
2
0.818182b 0.454545
a
4
=
0.0909091b
3
0.363636b
2
0.909091b 0.727273
2
a
6
=
0.181818b
3
+ 0.272727b
2
+ 0.181818b 0.454545
0.181818b
3
+ 0.272727b
2
+ 0.181818b 0.454545
a
10
=
0.181818b
3
+ 0.272727b
2
+ 0.181818b + 0.545455
0.181818b
3
0.272727b
2
+ 0.818182b + 0.454545
a
5
=
1
0
a
1
=
0
0.181818b
3
+ 0.272727b
2
+ 0.181818b 0.454545
a
2
=
1
0.181818b
3
+ 0.272727b
2
+ 0.181818b + 0.545455
a
2
=
1
0.181818b
3
+ 0.272727b
2
+ 0.181818b + 0.545455
(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 = 0.500000 0.866025I
a = 0.500000 0.866025I
b = 1.91421 0.86603I
6.57974 2.02988I 14.0000 + 3.4641I
u = 0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 1.91421 + 0.86603I
6.57974 + 2.02988I 14.0000 3.4641I
u = 0.500000 0.866025I
a = 0.500000 0.866025I
b = 0.914214 0.866025I
6.57974 2.02988I 14.0000 + 3.4641I
u = 0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 0.914214 + 0.866025I
6.57974 + 2.02988I 14.0000 3.4641I
5
III. I
u
3
=
hu
36
+2u
35
+· · ·+6u +13, 4.71×10
61
u
35
+8.17×10
60
u
34
+· · ·+5.00×10
63
b+
8.68× 10
63
, 8.35×10
62
u
35
1.26 ×10
63
u
34
+· · · +2.50 ×10
63
a 4.46× 10
64
i
(i) Arc colorings
a
7
=
0
u
a
11
=
0.334047u
35
+ 0.502634u
34
+ ··· 31.1990u + 17.8368
0.00942286u
35
0.00163426u
34
+ ··· + 2.73146u 1.73609
a
3
=
1
0
a
8
=
0.210635u
35
+ 0.379907u
34
+ ··· 22.1626u + 7.02975
0.0586116u
35
+ 0.0710441u
34
+ ··· 5.17845u + 4.15687
a
9
=
0.269247u
35
+ 0.450951u
34
+ ··· 27.3410u + 11.1866
0.0586116u
35
+ 0.0710441u
34
+ ··· 5.17845u + 4.15687
a
4
=
0.133545u
35
+ 0.257668u
34
+ ··· 15.8063u + 3.53274
0.0822309u
35
+ 0.114882u
34
+ ··· 8.72202u + 4.50191
a
6
=
u
u
a
10
=
0.334047u
35
+ 0.502634u
34
+ ··· 31.1990u + 17.8368
0.0211232u
35
+ 0.0448917u
34
+ ··· 0.618385u + 0.414906
a
5
=
0.405598u
35
+ 0.678718u
34
+ ··· 42.8853u + 16.2252
0.183051u
35
+ 0.304780u
34
+ ··· 17.7817u + 6.94404
a
1
=
0.534157u
35
+ 0.885263u
34
+ ··· 48.7047u + 20.9867
0.0711566u
35
0.123075u
34
+ ··· + 7.94583u 2.89311
a
2
=
0.468111u
35
+ 0.774365u
34
+ ··· 42.0493u + 18.3569
0.137202u
35
0.233973u
34
+ ··· + 14.6012u 5.52287
a
2
=
0.468111u
35
+ 0.774365u
34
+ ··· 42.0493u + 18.3569
0.137202u
35
0.233973u
34
+ ··· + 14.6012u 5.52287
(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 = 1.72488 0.64270I
a = 0.447593 + 0.609037I
b = 1.79130 + 1.37903I
1.51226 12.78077I 8.52716 + 7.80857I
u = 1.72488 + 0.64270I
a = 0.447593 0.609037I
b = 1.79130 1.37903I
1.51226 + 12.78077I 8.52716 7.80857I
u = 1.60135 0.28882I
a = 0.455623 0.552067I
b = 0.685533 + 0.078345I
3.35690 6.29314I 6.15059 + 3.94061I
u = 1.60135 + 0.28882I
a = 0.455623 + 0.552067I
b = 0.685533 0.078345I
3.35690 + 6.29314I 6.15059 3.94061I
u = 1.48227 0.34724I
a = 0.456843 0.645607I
b = 1.266798 0.563919I
3.22625 3.00094I 6.75961 + 2.89336I
u = 1.48227 + 0.34724I
a = 0.456843 + 0.645607I
b = 1.266798 + 0.563919I
3.22625 + 3.00094I 6.75961 2.89336I
u = 1.407201 0.026096I
a = 0.581550 + 0.648030I
b = 0.481253 0.080718I
3.88454 2.88072I 5.75517 + 2.19113I
u = 1.407201 + 0.026096I
a = 0.581550 0.648030I
b = 0.481253 + 0.080718I
3.88454 + 2.88072I 5.75517 2.19113I
u = 1.091498 0.385226I
a = 0.555170 + 0.983462I
b = 1.40509 + 1.14928I
3.43518 3.60339I 6.07450 + 2.50762I
u = 1.091498 + 0.385226I
a = 0.555170 0.983462I
b = 1.40509 1.14928I
3.43518 + 3.60339I 6.07450 2.50762I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.992449 0.407103I
a = 1.170396 0.425257I
b = 1.61805 0.12059I
3.84173 3.51209I 9.42797 + 4.38206I
u = 0.992449 + 0.407103I
a = 1.170396 + 0.425257I
b = 1.61805 + 0.12059I
3.84173 + 3.51209I 9.42797 4.38206I
u = 0.420405 1.157563I
a = 0.071031 0.532455I
b = 0.410669 1.206823I
0.87457 2.36441I 2.54968 + 4.98912I
u = 0.420405 + 1.157563I
a = 0.071031 + 0.532455I
b = 0.410669 + 1.206823I
0.87457 + 2.36441I 2.54968 4.98912I
u = 0.335905 0.009537I
a = 0.68674 2.05565I
b = 0.692193 + 0.136688I
2.03068 + 1.81473I 2.83557 4.64572I
u = 0.335905 + 0.009537I
a = 0.68674 + 2.05565I
b = 0.692193 0.136688I
2.03068 1.81473I 2.83557 + 4.64572I
u = 0.020013 0.439599I
a = 0.36979 2.34279I
b = 1.401881 0.072692I
6.55786 + 0.26724I 12.41884 + 1.59842I
u = 0.020013 + 0.439599I
a = 0.36979 + 2.34279I
b = 1.401881 + 0.072692I
6.55786 0.26724I 12.41884 1.59842I
u = 0.18257 1.53684I
a = 0.282687 + 0.585246I
b = 0.46964 + 1.88597I
4.64327 + 4.73682I 11.38513 6.59168I
u = 0.18257 + 1.53684I
a = 0.282687 0.585246I
b = 0.46964 1.88597I
4.64327 4.73682I 11.38513 + 6.59168I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.273798 0.439077I
a = 0.90645 + 2.01694I
b = 0.0399825 + 0.0159679I
1.14397 + 1.32004I 4.86119 + 2.11551I
u = 0.273798 + 0.439077I
a = 0.90645 2.01694I
b = 0.0399825 0.0159679I
1.14397 1.32004I 4.86119 2.11551I
u = 0.363292
a = 1.30125
b = 0.187974
0.718633 13.9022
u = 0.448670
a = 2.62621
b = 1.49912
6.40417 14.3787
u = 0.55262 1.43337I
a = 0.470375 + 0.337027I
b = 0.21399 + 1.65792I
4.33907 + 0.62358I 10.41420 + 0.63892I
u = 0.55262 + 1.43337I
a = 0.470375 0.337027I
b = 0.21399 1.65792I
4.33907 0.62358I 10.41420 0.63892I
u = 0.730843 0.488737I
a = 0.349601 0.724796I
b = 1.296770 0.039394I
5.02009 + 2.85591I 6.91764 5.56949I
u = 0.730843 + 0.488737I
a = 0.349601 + 0.724796I
b = 1.296770 + 0.039394I
5.02009 2.85591I 6.91764 + 5.56949I
u = 1.40736 0.25612I
a = 0.543345 0.510783I
b = 0.610573 + 0.189166I
0.05037 + 1.45720I 9.46667 0.65893I
u = 1.40736 + 0.25612I
a = 0.543345 + 0.510783I
b = 0.610573 0.189166I
0.05037 1.45720I 9.46667 + 0.65893I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.42072 0.09612I
a = 0.508652 0.711232I
b = 1.239880 0.450718I
7.38383 1.35991I 2.81294 + 0.73046I
u = 1.42072 + 0.09612I
a = 0.508652 + 0.711232I
b = 1.239880 + 0.450718I
7.38383 + 1.35991I 2.81294 0.73046I
u = 1.42836 0.79532I
a = 0.356300 + 0.714879I
b = 1.57977 + 1.46639I
1.38910 + 7.43800I 11.20734 4.77385I
u = 1.42836 + 0.79532I
a = 0.356300 0.714879I
b = 1.57977 1.46639I
1.38910 7.43800I 11.20734 + 4.77385I
u = 1.67372 0.34301I
a = 0.497930 0.609628I
b = 1.34994 0.54291I
6.59180 + 8.00171I 4.29537 5.98015I
u = 1.67372 + 0.34301I
a = 0.497930 + 0.609628I
b = 1.34994 + 0.54291I
6.59180 8.00171I 4.29537 + 5.98015I
10
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
u + 1)
2
(u
2
+ u + 1)(u
36
+ 2u
35
+ ··· + 6u + 1)
c
2
(u
2
+ u + 1)
3
(u
36
+ 20u
35
+ ··· 6u + 1)
c
3
, c
4
, c
8
c
9
u
2
(u
2
2)
2
(u
36
+ u
35
+ ··· 12u 4)
c
5
(u
2
u + 1)(u
2
+ u + 1)
2
(u
36
+ 2u
35
+ ··· + 6u + 1)
c
6
(u
2
u + 1)
2
(u
2
+ u + 1)(u
36
+ 2u
35
+ ··· + 6u + 13)
c
7
(u 1)
4
(u + 1)
2
(u
36
+ 3u
35
+ ··· 13u 7)
c
10
(u 1)
2
(u + 1)
4
(u
36
+ 3u
35
+ ··· 13u 7)
c
11
(u 1)
6
(u
36
+ 13u
35
+ ··· + 687u + 49)
11
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
2
+ y + 1)
3
(y
36
+ 20y
35
+ ··· 6y + 1)
c
2
(y
2
+ y + 1)
3
(y
36
4y
35
+ ··· 142y + 1)
c
3
, c
4
, c
8
c
9
y
2
(y 2)
4
(y
36
31y
35
+ ··· + 80y + 16)
c
6
(y
2
+ y + 1)
3
(y
36
28y
35
+ ··· 3962y + 169)
c
7
, c
10
(y 1)
6
(y
36
13y
35
+ ··· 687y + 49)
c
11
(y 1)
6
(y
36
+ 27y
35
+ ··· 44983y + 2401)
12