11n
91
(K11n
91
)
1
Arc Sequences
6 1 9 10 2 3 11 4 3 7 10
Solving Sequence
7,10
11
1,3
2 6 9 4 5 8
c
10
c
11
c
2
c
6
c
9
c
3
c
4
c
8
c
1
, c
5
, c
7
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
2
u + 1, b, a + u 1i
I
u
2
= hb
2
+ 2, a
2
a + 1, a + u + 1i
I
u
3
= hu
21
+ 2u
20
+ ··· + 19u 3, 9.38772 × 10
25
u
20
+ 2.00275 × 10
26
u
19
+ ··· + 4.50163 × 10
26
b + 1.25177 × 10
27
,
1.04018 × 10
26
u
20
+ 9.99735 × 10
25
u
19
+ ··· + 6.75244 × 10
26
a + 4.35050 × 10
27
i
There are 3 irreducible components with 27 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, a + u 1i
(i) Arc colorings
a
7
=
0
u
a
10
=
u + 1
0
a
11
=
u + 1
u
a
1
=
0
u
a
3
=
1
0
a
2
=
1
u + 1
a
6
=
u
u
a
9
=
u + 1
0
a
4
=
1
0
a
5
=
1
0
a
8
=
u + 1
0
a
8
=
u + 1
0
(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
1.64493 + 2.02988I 12.00000 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 0.866025I
b = 0
1.64493 2.02988I 12.00000 + 3.46410I
3
II. I
u
2
= hb
2
+ 2, a
2
a + 1, a + u + 1i
(i) Arc colorings
a
7
=
0
a 1
a
10
=
a
b
a
11
=
a
b + a 1
a
1
=
0
a 1
a
3
=
1
0
a
2
=
1
a
a
6
=
a 1
a 1
a
9
=
b + a
b
a
4
=
ba 1
2
a
5
=
1
0
a
8
=
a
b
a
8
=
a
b
(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 = 1.41421I
3.28987 2.02988I 6.00000 + 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 1.41421I
3.28987 + 2.02988I 6.00000 3.46410I
u = 0.500000 0.866025I
a = 0.500000 0.866025I
b = 1.41421I
3.28987 2.02988I 6.00000 + 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 1.41421I
3.28987 + 2.02988I 6.00000 3.46410I
5
III. I
u
3
=
hu
21
+2u
20
+· · ·+19u 3, 9.39×10
25
u
20
+2.00×10
26
u
19
+· · ·+4.50×10
26
b+
1.25 × 10
27
, 1.04 × 10
26
u
20
+ 1.00 × 10
26
u
19
+ · · · + 6.75 × 10
26
a + 4.35 × 10
27
i
(i) Arc colorings
a
7
=
0
u
a
10
=
0.154045u
20
0.148055u
19
+ ··· + 12.2786u 6.44286
0.208540u
20
0.444893u
19
+ ··· + 10.5886u 2.78070
a
11
=
0.154045u
20
0.148055u
19
+ ··· + 12.2786u 6.44286
0.299642u
20
0.662645u
19
+ ··· + 14.0914u 3.26081
a
1
=
1.05775u
20
+ 2.44311u
19
+ ··· 40.3046u + 7.26209
0.131677u
20
+ 0.323791u
19
+ ··· 2.61018u 0.457296
a
3
=
1
0
a
2
=
1.03604u
20
+ 2.39568u
19
+ ··· 38.0066u + 6.46059
0.109961u
20
+ 0.276363u
19
+ ··· 0.312219u 1.25880
a
6
=
u
u
a
9
=
0.0544952u
20
+ 0.296838u
19
+ ··· + 1.68999u 3.66216
0.208540u
20
0.444893u
19
+ ··· + 10.5886u 2.78070
a
4
=
1.08694u
20
2.47351u
19
+ ··· + 41.2336u 6.56038
0.180057u
20
0.463889u
19
+ ··· + 5.07513u + 1.02995
a
5
=
0.480036u
20
0.878057u
19
+ ··· + 18.3126u 8.67965
0.327604u
20
0.704871u
19
+ ··· + 12.8352u 3.17326
a
8
=
0.801232u
20
+ 1.72335u
19
+ ··· 37.0150u + 11.7370
0.431556u
20
+ 0.981633u
19
+ ··· 16.6492u + 3.43831
a
8
=
0.801232u
20
+ 1.72335u
19
+ ··· 37.0150u + 11.7370
0.431556u
20
+ 0.981633u
19
+ ··· 16.6492u + 3.43831
(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 = 2.31424 0.38605I
a = 0.460907 + 0.520650I
b = 0.18378 + 1.78761I
16.6802 9.9035I 2.37944 + 4.78800I
u = 2.31424 + 0.38605I
a = 0.460907 0.520650I
b = 0.18378 1.78761I
16.6802 + 9.9035I 2.37944 4.78800I
u = 2.11224 0.57307I
a = 0.251873 + 0.691388I
b = 0.01070 + 1.85472I
17.8929 + 1.4476I 1.24285 0.69389I
u = 2.11224 + 0.57307I
a = 0.251873 0.691388I
b = 0.01070 1.85472I
17.8929 1.4476I 1.24285 + 0.69389I
u = 1.121828 0.041171I
a = 0.626443 + 0.590425I
b = 0.271568 + 1.062539I
2.57223 + 2.30104I 5.12347 3.68698I
u = 1.121828 + 0.041171I
a = 0.626443 0.590425I
b = 0.271568 1.062539I
2.57223 2.30104I 5.12347 + 3.68698I
u = 0.528725 0.172484I
a = 0.99577 1.21455I
b = 0.713808 0.270061I
2.21076 + 2.09468I 3.00496 3.91489I
u = 0.528725 + 0.172484I
a = 0.99577 + 1.21455I
b = 0.713808 + 0.270061I
2.21076 2.09468I 3.00496 + 3.91489I
u = 0.409958 1.144135I
a = 0.091963 0.565647I
b = 0.033686 0.472962I
0.91652 2.35539I 2.06528 + 5.30676I
u = 0.409958 + 1.144135I
a = 0.091963 + 0.565647I
b = 0.033686 + 0.472962I
0.91652 + 2.35539I 2.06528 5.30676I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.241575 0.166383I
a = 2.28432 3.50132I
b = 0.21844 1.42669I
5.12357 0.62763I 1.407681 0.031302I
u = 0.241575 + 0.166383I
a = 2.28432 + 3.50132I
b = 0.21844 + 1.42669I
5.12357 + 0.62763I 1.407681 + 0.031302I
u = 0.331843 0.436309I
a = 1.06175 + 1.81838I
b = 0.264392 0.442489I
1.11524 + 1.37460I 4.04933 + 2.02582I
u = 0.331843 + 0.436309I
a = 1.06175 1.81838I
b = 0.264392 + 0.442489I
1.11524 1.37460I 4.04933 2.02582I
u = 0.374304
a = 1.37846
b = 0.417301
0.754310 13.3555
u = 0.957992 0.868352I
a = 0.371789 + 0.232830I
b = 0.130919 + 1.391487I
5.17740 + 1.74265I 0.53526 2.03708I
u = 0.957992 + 0.868352I
a = 0.371789 0.232830I
b = 0.130919 1.391487I
5.17740 1.74265I 0.53526 + 2.03708I
u = 1.48329 0.10999I
a = 0.606364 + 0.613627I
b = 0.567067 + 1.129179I
6.41753 6.55364I 2.05255 + 5.68240I
u = 1.48329 + 0.10999I
a = 0.606364 0.613627I
b = 0.567067 1.129179I
6.41753 + 6.55364I 2.05255 5.68240I
u = 2.28513 0.54730I
a = 0.334967 + 0.551115I
b = 0.08814 + 1.77927I
12.95414 + 3.94853I 4.46142 1.91994I
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 2.28513 + 0.54730I
a = 0.334967 0.551115I
b = 0.08814 1.77927I
12.95414 3.94853I 4.46142 + 1.91994I
8
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
u + 1)
2
(u
2
+ u + 1)(u
21
+ 2u
20
+ ··· + 7u + 3)
c
2
(u
2
+ u + 1)
3
(u
21
+ 14u
20
+ ··· + 49u 9)
c
3
, c
8
, c
9
u
2
(u
2
+ 2)
2
(u
21
+ u
20
+ ··· + 8u 4)
c
4
u
2
(u
2
+ 2)
2
(u
21
+ u
20
+ ··· 64u + 548)
c
5
(u
2
u + 1)(u
2
+ u + 1)
2
(u
21
+ 2u
20
+ ··· + 7u + 3)
c
6
(u
2
u + 1)
2
(u
2
+ u + 1)(u
21
+ 2u
20
+ ··· + 19u 3)
c
7
(u 1)
4
(u + 1)
2
(u
21
+ 3u
20
+ ··· 2u + 3)
c
10
(u 1)
2
(u + 1)
4
(u
21
+ 3u
20
+ ··· 2u + 3)
c
11
(u 1)
6
(u
21
+ 3u
20
+ ··· + 22u + 9)
9
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
2
+ y + 1)
3
(y
21
+ 14y
20
+ ··· + 49y 9)
c
2
(y
2
+ y + 1)
3
(y
21
10y
20
+ ··· + 7117y 81)
c
3
, c
8
, c
9
y
2
(y + 2)
4
(y
21
+ 31y
20
+ ··· 160y 16)
c
4
y
2
(y + 2)
4
(y
21
+ 91y
20
+ ··· 4121248y 300304)
c
6
(y
2
+ y + 1)
3
(y
21
34y
20
+ ··· + 193y 9)
c
7
, c
10
(y 1)
6
(y
21
3y
20
+ ··· + 22y 9)
c
11
(y 1)
6
(y
21
+ 37y
20
+ ··· + 2158y 81)
10