11n
100
(K11n
100
)
1
Arc Sequences
6 1 7 10 2 4 3 1 11 4 9
Solving Sequence
2,5
6
1,10
4 7 11 3 8 9
c
5
c
1
c
4
c
6
c
10
c
3
c
7
c
9
c
2
, c
8
, c
11
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hu
2
u + 1, a 1, b u + 1i
I
u
2
= hb
6
+ 2b
5
+ 7b
4
+ 4b
3
+ 8b
2
2b + 5, b
5
7b
3
b
2
17b + 11u + 3,
16b
5
33b
4
90b
3
49b
2
52b + 55a + 15i
I
u
3
= hu
4
+ u
3
+ 2u
2
+ 2u + 1, u
3
+ a + 2u + 1, u
3
+ u
2
+ b + 3u + 2i
I
u
4
= hu
22
+ u
21
+ ··· 4u + 1,
27401760606u
21
+ 32283303162u
20
+ ··· + 121906198036b + 36114460969,
306481557535u
21
+ 364610096714u
20
+ ··· + 243812396072a 76785066689i
There are 4 irreducible components with 34 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, a 1, b u + 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
u
u 1
a
10
=
1
u 1
a
4
=
u
u 1
a
7
=
1
u + 1
a
11
=
u
u 2
a
3
=
0
u
a
8
=
1
2u
a
9
=
0
u
a
9
=
0
u
(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 = 1.00000
b = 0.500000 0.866025I
2.02988I 3.46410I
u = 0.500000 + 0.866025I
a = 1.00000
b = 0.500000 + 0.866025I
2.02988I 3.46410I
3
II. I
u
2
= hb
6
+ 2b
5
+ 7b
4
+ 4b
3
+ 8b
2
2b + 5, b
5
7b
3
b
2
17b + 11u +
3, 16b
5
33b
4
+ · · · + 55a + 15i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
0.0909091b
5
+ 0.636364b
3
+ ··· + 1.54545b 0.272727
a
6
=
0.0909091b
5
+ 0.636364b
3
+ ··· + 1.54545b 0.272727
0.0909091b
5
+ 0.636364b
3
+ ··· + 1.54545b 0.272727
a
1
=
0
1
a
10
=
16
55
b
5
+
3
5
b
4
+ ··· +
52
55
b
3
11
b
a
4
=
0.0363636b
5
0.200000b
4
+ ··· + 0.381818b + 0.909091
2
55
b
5
1
5
b
4
+ ··· +
21
55
b
1
11
a
7
=
1
55
b
5
2
5
b
4
+ ··· +
17
55
b
5
11
6
55
b
5
2
5
b
4
+ ··· +
102
55
b
8
11
a
11
=
12
55
b
5
+
1
5
b
4
+ ···
16
55
b
16
11
2
55
b
5
1
5
b
4
+ ··· +
21
55
b
1
11
a
3
=
1
1
a
8
=
4
55
b
5
2
5
b
4
+ ···
68
55
b
2
11
1
55
b
5
2
5
b
4
+ ··· +
17
55
b
5
11
a
9
=
4
55
b
5
2
5
b
4
+ ···
68
55
b
2
11
0.0909091b
5
+ 0.636364b
3
+ ··· + 1.54545b 0.272727
a
9
=
4
55
b
5
2
5
b
4
+ ···
68
55
b
2
11
0.0909091b
5
+ 0.636364b
3
+ ··· + 1.54545b 0.272727
(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 = 1.00000I
a = 0.877439 0.744862I
b = 0.77736 1.96950I
3.02413 + 2.82812I 3.50976 2.97945I
u = 1.00000I
a = 0.877439 + 0.744862I
b = 0.77736 + 1.96950I
3.02413 2.82812I 3.50976 + 2.97945I
u = 1.00000I
a = 0.754878
b = 0.56984 1.32472I
1.11345 3.01951
u = 1.00000I
a = 0.754878
b = 0.56984 + 1.32472I
1.11345 3.01951
u = 1.00000I
a = 0.877439 0.744862I
b = 0.347200 0.644782I
3.02413 + 2.82812I 3.50976 2.97945I
u = 1.00000I
a = 0.877439 + 0.744862I
b = 0.347200 + 0.644782I
3.02413 2.82812I 3.50976 + 2.97945I
5
III. I
u
3
= hu
4
+ u
3
+ 2u
2
+ 2u + 1, u
3
+ a + 2u + 1, u
3
+ u
2
+ b + 3u + 2i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
10
=
u
3
2u 1
u
3
u
2
3u 2
a
4
=
u
2
+ 1
u
2
a
7
=
u
3
u
3
+ u
a
11
=
u
2
+ 1
u
3
2u 1
a
3
=
u
3
u
2
2u
u
3
2u
2
2u 1
a
8
=
u
3
2u 1
1
a
9
=
0
u
a
9
=
0
u
(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 = 0.621744 0.440597I
a = 0.121744 + 1.306622I
b = 0.448952 + 1.199342I
2.02988I 3.46410I
u = 0.621744 + 0.440597I
a = 0.121744 1.306622I
b = 0.448952 1.199342I
2.02988I 3.46410I
u = 0.121744 1.306622I
a = 0.621744 + 0.440597I
b = 0.05105 + 2.06537I
2.02988I 3.46410I
u = 0.121744 + 1.306622I
a = 0.621744 0.440597I
b = 0.05105 2.06537I
2.02988I 3.46410I
7
IV.
I
u
4
= hu
22
+u
21
+· · ·4u+1, 2.74×10
10
u
21
+3.23×10
10
u
20
+· · ·+1.22×10
11
b+
3.61 × 10
10
, 3.06 × 10
11
u
21
+ 3.65 × 10
11
u
20
+ · · · + 2.44 × 10
11
a 7.68 × 10
10
i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
10
=
1.25704u
21
1.49545u
20
+ ··· 21.5598u + 0.314935
0.224777u
21
0.264821u
20
+ ··· 4.42658u 0.296248
a
4
=
0.520062u
21
0.941512u
20
+ ··· 6.20121u 4.83139
0.0606928u
21
+ 0.0275143u
20
+ ··· + 1.35703u 1.46367
a
7
=
1.80972u
21
1.88500u
20
+ ··· 30.6024u + 5.10082
0.421450u
21
0.474311u
20
+ ··· 5.91163u + 0.520062
a
11
=
0.329536u
21
0.900218u
20
+ ··· 1.13762u 7.60006
0.0752832u
21
0.00470735u
20
+ ··· + 2.13807u 1.80972
a
3
=
u
4
+ u
2
+ 1
u
4
a
8
=
1.46826u
21
1.49856u
20
+ ··· 25.1993u + 4.50547
0.426044u
21
0.448508u
20
+ ··· 5.90074u + 0.527894
a
9
=
1.85194u
21
+ 1.93505u
20
+ ··· + 31.9010u 5.07839
0.388271u
21
+ 0.410694u
20
+ ··· + 5.69074u 0.580755
a
9
=
1.85194u
21
+ 1.93505u
20
+ ··· + 31.9010u 5.07839
0.388271u
21
+ 0.410694u
20
+ ··· + 5.69074u 0.580755
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.153311 0.765089I
a = 0.824436 + 0.428112I
b = 1.09341 + 1.04813I
9.48603 4.13683I 2.53393 + 2.55439I
u = 1.153311 + 0.765089I
a = 0.824436 0.428112I
b = 1.09341 1.04813I
9.48603 + 4.13683I 2.53393 2.55439I
u = 1.07708 1.03335I
a = 0.536672 0.615239I
b = 0.98592 2.18536I
13.19150 + 3.89903I 4.72901 2.42961I
u = 1.07708 + 1.03335I
a = 0.536672 + 0.615239I
b = 0.98592 + 2.18536I
13.19150 3.89903I 4.72901 + 2.42961I
u = 0.90302 1.18611I
a = 0.302390 + 0.821075I
b = 0.31975 + 2.72877I
8.10703 + 11.57362I 0.88963 6.62056I
u = 0.90302 + 1.18611I
a = 0.302390 0.821075I
b = 0.31975 2.72877I
8.10703 11.57362I 0.88963 + 6.62056I
u = 0.595604 0.824126I
a = 0.833215 0.205043I
b = 0.368315 + 0.238347I
0.94737 + 2.13228I 3.49508 3.26961I
u = 0.595604 + 0.824126I
a = 0.833215 + 0.205043I
b = 0.368315 0.238347I
0.94737 2.13228I 3.49508 + 3.26961I
u = 0.217043 0.710982I
a = 0.364202 + 0.538543I
b = 0.340220 + 0.348326I
0.380875 + 1.140107I 4.34193 6.22750I
u = 0.217043 + 0.710982I
a = 0.364202 0.538543I
b = 0.340220 0.348326I
0.380875 1.140107I 4.34193 + 6.22750I
9
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.029711 1.049821I
a = 0.806902 0.721985I
b = 0.219039 1.369133I
4.65093 + 2.79195I 9.45575 3.06805I
u = 0.029711 + 1.049821I
a = 0.806902 + 0.721985I
b = 0.219039 + 1.369133I
4.65093 2.79195I 9.45575 + 3.06805I
u = 0.106877 0.224236I
a = 1.90450 + 4.32043I
b = 0.711930 + 0.911129I
1.42494 2.89189I 2.45935 + 2.97630I
u = 0.106877 + 0.224236I
a = 1.90450 4.32043I
b = 0.711930 0.911129I
1.42494 + 2.89189I 2.45935 2.97630I
u = 0.678005 0.384923I
a = 0.747656 + 1.056654I
b = 1.04546 + 1.57059I
3.57528 1.09357I 5.97662 + 1.94696I
u = 0.678005 + 0.384923I
a = 0.747656 1.056654I
b = 1.04546 1.57059I
3.57528 + 1.09357I 5.97662 1.94696I
u = 0.778255 0.824454I
a = 0.283395 1.014924I
b = 0.68028 2.27208I
0.39802 7.14623I 0.40139 + 7.68801I
u = 0.778255 + 0.824454I
a = 0.283395 + 1.014924I
b = 0.68028 + 2.27208I
0.39802 + 7.14623I 0.40139 7.68801I
u = 0.89009 1.10133I
a = 0.704841 + 0.246159I
b = 0.497167 + 0.394085I
6.44298 5.66281I 0.84387 + 2.45088I
u = 0.89009 + 1.10133I
a = 0.704841 0.246159I
b = 0.497167 0.394085I
6.44298 + 5.66281I 0.84387 2.45088I
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.022545 0.827721I
a = 0.316750 0.661900I
b = 0.032238 0.626280I
7.32664 1.36370I 0.05052 + 1.94758I
u = 1.022545 + 0.827721I
a = 0.316750 + 0.661900I
b = 0.032238 + 0.626280I
7.32664 + 1.36370I 0.05052 1.94758I
10
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
5
(u
2
+ 1)
3
(u
2
u + 1)(u
4
+ u
3
+ ··· + 2u + 1)(u
22
+ u
21
+ ··· 4u + 1)
c
2
(u + 1)
6
(u
2
+ u + 1)(u
4
+ 3u
3
+ 2u
2
+ 1)(u
22
+ 3u
21
+ ··· + 24u + 1)
c
3
, c
6
, c
7
(u
2
+ 1)
3
(u
2
u + 1)(u
4
+ u
3
+ ··· + 2u + 1)(u
22
+ u
21
+ ··· 10u + 1)
c
4
, c
10
(u
2
+ u + 1)
3
(u
6
+ u
4
+ 2u
2
+ 1)(u
22
2u
21
+ ··· u + 2)
c
8
, c
9
(u
2
+ u + 1)
3
(1 + 2u + u
2
+ u
3
)
2
(u
22
+ 8u
21
+ ··· + 19u + 4)
c
11
(u
2
+ u + 1)
3
(1 + 2u u
2
+ u
3
)
2
(u
22
+ 8u
21
+ ··· + 19u + 4)
11
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y + 1)
6
(y
2
+ y + 1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
22
+ 3y
21
+ ··· + 24y + 1)
c
2
(y 1)
6
(y
2
+ y + 1)(y
4
5y
3
+ 6y
2
+ 4y + 1)
(y
22
+ 39y
21
+ ··· + 112y + 1)
c
3
, c
6
, c
7
(y + 1)
6
(y
2
+ y + 1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
22
+ 31y
21
+ ··· + 56y + 1)
c
4
, c
10
(y
2
+ y + 1)
3
(1 + 2y + y
2
+ y
3
)
2
(y
22
+ 8y
21
+ ··· + 19y + 4)
c
8
, c
9
, c
11
(y
2
+ y + 1)
3
(1 + 2y + 3y
2
+ y
3
)
2
(y
22
+ 12y
21
+ ··· + 623y + 16)
12