11n
83
(K11n
83
)
1
Arc Sequences
6 1 9 7 2 10 1 11 6 4 8
Solving Sequence
1,6
2
3,10
7 8 5 4 9 11
c
1
c
2
c
6
c
7
c
5
c
4
c
9
c
11
c
3
, c
8
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hb
4
+ 2b
3
+ 7b
2
+ 6b + 3, 2b
3
+ 3b
2
+ 10b + 5a + 2, 2b
3
+ 3b
2
+ 10b + 5u + 7i
I
u
2
= hu
2
+ u + 1, b + u, a + u + 1i
I
u
3
= hu
30
+ 8u
29
+ ··· + 59u + 9,
1.65693 × 10
21
u
29
1.25426 × 10
22
u
28
+ ··· + 7.78277 × 10
21
b 2.65124 × 10
22
,
1.16811 × 10
22
u
29
8.79857 × 10
22
u
28
+ ··· + 2.33483 × 10
22
a 1.51866 × 10
23
i
There are 3 irreducible components with 36 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
4
+ 2b
3
+ 7b
2
+ 6b + 3, 2b
3
+ 3b
2
+ 10b + 5a + 2, 2b
3
+ 3b
2
+ 10b + 5u + 7i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
2
5
b
3
3
5
b
2
2b
7
5
a
2
=
1
2
5
b
3
3
5
b
2
2b
2
5
a
3
=
2
5
b
3
+
3
5
b
2
+ 2b +
7
5
2
5
b
3
3
5
b
2
2b
2
5
a
10
=
2
5
b
3
3
5
b
2
2b
2
5
b
a
7
=
2
5
b
3
+
3
5
b
2
+ 2b +
2
5
2
5
b
3
3
5
b
2
3b
7
5
a
8
=
4
5
b
3
+
6
5
b
2
+ 5b +
9
5
2
5
b
3
3
5
b
2
3b
7
5
a
5
=
2
5
b
3
+
3
5
b
2
+ 2b +
7
5
2
5
b
3
3
5
b
2
2b
2
5
a
4
=
4
5
b
3
+
6
5
b
2
+ 5b +
19
5
b
3
2b
2
5b 4
a
9
=
2
5
b
3
3
5
b
2
2b
2
5
2
5
b
3
+
3
5
b
2
+ 3b +
7
5
a
11
=
1
5
b
3
4
5
b
2
2b
16
5
2
a
11
=
1
5
b
3
4
5
b
2
2b
16
5
2
(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 0.548188I
6.57974 2.02988I 6.00000 + 3.46410I
u = 0.500000 0.866025I
a = 0.500000 0.866025I
b = 0.500000 + 0.548188I
6.57974 + 2.02988I 6.00000 3.46410I
u = 0.500000 0.866025I
a = 0.500000 0.866025I
b = 0.50000 2.28024I
6.57974 + 2.02988I 6.00000 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 0.50000 + 2.28024I
6.57974 2.02988I 6.00000 + 3.46410I
3
II. I
u
2
= hu
2
+ u + 1, b + u, a + u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u + 1
a
3
=
u
u + 1
a
10
=
u 1
u
a
7
=
u 1
0
a
8
=
u 1
0
a
5
=
u
u + 1
a
4
=
u + 1
u + 1
a
9
=
u 1
0
a
11
=
1
0
a
11
=
1
0
(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 = 0.500000 + 0.866025I
1.64493 + 2.02988I 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 0.866025I
b = 0.500000 0.866025I
1.64493 2.02988I 3.46410I
5
III. I
u
3
=
hu
30
+8u
29
+· · ·+59u+9, 1.66×10
21
u
29
1.25×10
22
u
28
+· · ·+7.78×10
21
b
2.65× 10
22
, 1.17× 10
22
u
29
8.80 × 10
22
u
28
+· · · +2.33 × 10
22
a 1.52 × 10
23
i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
10
=
0.500296u
29
+ 3.76840u
28
+ ··· + 34.8145u + 6.50436
0.212897u
29
+ 1.61159u
28
+ ··· + 17.2043u + 3.40655
a
7
=
0.195327u
29
1.50953u
28
+ ··· 27.9406u 7.82719
0.200124u
29
1.50933u
28
+ ··· 14.4262u 3.01543
a
8
=
0.00479750u
29
0.000201005u
28
+ ··· 13.5144u 4.81176
0.200124u
29
1.50933u
28
+ ··· 14.4262u 3.01543
a
5
=
u
u
3
+ u
a
4
=
0.582123u
29
+ 4.29047u
28
+ ··· + 43.1171u + 10.9976
0.243597u
29
+ 1.73550u
28
+ ··· + 12.1493u + 1.59905
a
9
=
0.500296u
29
+ 3.76840u
28
+ ··· + 34.8145u + 6.50436
0.275979u
29
+ 2.11887u
28
+ ··· + 26.5057u + 5.51226
a
11
=
0.378505u
29
2.81515u
28
+ ··· 27.1883u 5.12752
0.163272u
29
+ 1.28132u
28
+ ··· + 17.0522u + 4.24326
a
11
=
0.378505u
29
2.81515u
28
+ ··· 27.1883u 5.12752
0.163272u
29
+ 1.28132u
28
+ ··· + 17.0522u + 4.24326
(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 = 1.194547 0.646359I
a = 0.806434 0.092426I
b = 1.66643 + 0.93121I
4.62105 + 5.90679I 4.52831 6.63187I
u = 1.194547 + 0.646359I
a = 0.806434 + 0.092426I
b = 1.66643 0.93121I
4.62105 5.90679I 4.52831 + 6.63187I
u = 0.956886 0.832908I
a = 0.124030 + 0.650953I
b = 1.108775 + 0.402600I
4.02348 + 1.34734I 3.26214 0.58804I
u = 0.956886 + 0.832908I
a = 0.124030 0.650953I
b = 1.108775 0.402600I
4.02348 1.34734I 3.26214 + 0.58804I
u = 0.827863 0.788133I
a = 0.610296 0.163041I
b = 0.907811 0.765680I
0.80409 + 2.85458I 2.92837 5.79821I
u = 0.827863 + 0.788133I
a = 0.610296 + 0.163041I
b = 0.907811 + 0.765680I
0.80409 2.85458I 2.92837 + 5.79821I
u = 0.473408 0.105538I
a = 2.08968 0.90137I
b = 0.08348 1.42196I
7.52373 + 0.32119I 8.17779 + 0.83002I
u = 0.473408 + 0.105538I
a = 2.08968 + 0.90137I
b = 0.08348 + 1.42196I
7.52373 0.32119I 8.17779 0.83002I
u = 0.42849 1.69223I
a = 0.623225 + 0.533526I
b = 1.23958 + 1.77796I
2.92088 + 12.01220I 0.99975 7.16123I
u = 0.42849 + 1.69223I
a = 0.623225 0.533526I
b = 1.23958 1.77796I
2.92088 12.01220I 0.99975 + 7.16123I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.289006 0.960575I
a = 0.341844 0.310737I
b = 0.004028 + 1.301356I
4.98680 + 2.32242I 0.98983 4.15024I
u = 0.289006 + 0.960575I
a = 0.341844 + 0.310737I
b = 0.004028 1.301356I
4.98680 2.32242I 0.98983 + 4.15024I
u = 0.288296 0.396727I
a = 1.58170 + 1.42670I
b = 0.0250531 + 0.0445070I
1.85260 + 1.11432I 2.87293 + 2.42323I
u = 0.288296 + 0.396727I
a = 1.58170 1.42670I
b = 0.0250531 0.0445070I
1.85260 1.11432I 2.87293 2.42323I
u = 0.22083 1.70141I
a = 0.625937 0.523734I
b = 0.49633 1.34756I
7.86584 + 6.86749I 2.91158 4.96982I
u = 0.22083 + 1.70141I
a = 0.625937 + 0.523734I
b = 0.49633 + 1.34756I
7.86584 6.86749I 2.91158 + 4.96982I
u = 0.215656 1.206983I
a = 0.182869 1.070299I
b = 0.07834 1.70158I
3.45780 + 2.70205I 2.31582 3.42763I
u = 0.215656 + 1.206983I
a = 0.182869 + 1.070299I
b = 0.07834 + 1.70158I
3.45780 2.70205I 2.31582 + 3.42763I
u = 0.17358 1.65624I
a = 0.598313 + 0.464993I
b = 0.030527 + 0.633530I
4.52151 + 5.10629I 1.02367 3.27936I
u = 0.17358 + 1.65624I
a = 0.598313 0.464993I
b = 0.030527 0.633530I
4.52151 5.10629I 1.02367 + 3.27936I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.01716 1.58025I
a = 0.639515 + 0.509136I
b = 0.029559 + 0.549776I
4.75687 + 1.39165I 1.30584 1.43090I
u = 0.01716 + 1.58025I
a = 0.639515 0.509136I
b = 0.029559 0.549776I
4.75687 1.39165I 1.30584 + 1.43090I
u = 0.09265 1.59384I
a = 0.666239 0.534839I
b = 0.455313 1.306323I
8.24005 0.28940I 3.67278 + 0.05033I
u = 0.09265 + 1.59384I
a = 0.666239 + 0.534839I
b = 0.455313 + 1.306323I
8.24005 + 0.28940I 3.67278 0.05033I
u = 0.131917 0.513534I
a = 1.123523 + 0.076835I
b = 0.192615 0.679919I
1.028652 + 0.891272I 5.65753 3.58094I
u = 0.131917 + 0.513534I
a = 1.123523 0.076835I
b = 0.192615 + 0.679919I
1.028652 0.891272I 5.65753 + 3.58094I
u = 0.29191 1.42887I
a = 0.744398 + 0.614810I
b = 1.13746 + 1.62619I
3.80529 5.57168I 0.42098 + 2.74433I
u = 0.29191 + 1.42887I
a = 0.744398 0.614810I
b = 1.13746 1.62619I
3.80529 + 5.57168I 0.42098 2.74433I
u = 0.534931 0.013556I
a = 1.31577 1.24844I
b = 0.939218 + 0.571832I
0.93475 2.30280I 1.24616 + 3.74570I
u = 0.534931 + 0.013556I
a = 1.31577 + 1.24844I
b = 0.939218 0.571832I
0.93475 + 2.30280I 1.24616 3.74570I
9
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)
3
(u
30
+ 8u
29
+ ··· + 59u + 9)
c
2
(u
2
+ u + 1)
3
(u
30
+ 32u
29
+ ··· + 47u + 81)
c
3
(u
2
+ u + 1)(u
4
2u
3
+ ··· 6u + 9)(u
30
8u
28
+ ··· + 3557u + 451)
c
4
(u
2
+ u + 1)(u
4
+ 2u
3
+ ··· + 6u + 9)(u
30
+ 4u
29
+ ··· 295u + 1601)
c
5
(u
2
u + 1)
3
(u
30
+ 8u
29
+ ··· + 59u + 9)
c
6
(u 1)
4
(u + 1)
2
(u
30
+ 3u
29
+ ··· 4u + 3)
c
7
, c
8
, c
11
u
2
(u
2
+ 2)
2
(u
30
+ u
29
+ ··· + 16u + 4)
c
9
(u 1)
2
(u + 1)
4
(u
30
+ 3u
29
+ ··· 4u + 3)
c
10
(u
2
u + 1)(u
2
+ u + 1)
2
(u
30
+ 2u
29
+ ··· u + 3)
10
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
2
+ y + 1)
3
(y
30
+ 32y
29
+ ··· + 47y + 81)
c
2
(y
2
+ y + 1)
3
(y
30
64y
29
+ ··· 221233y + 6561)
c
3
(y
2
+ y + 1)(y
4
2y
3
5y
2
18y + 81)
(y
30
16y
29
+ ··· 5277497y + 203401)
c
4
(y
2
+ y + 1)(y
4
2y
3
5y
2
18y + 81)
(y
30
+ 24y
29
+ ··· + 28807823y + 2563201)
c
6
, c
9
(y 1)
6
(y
30
9y
29
+ ··· 58y + 9)
c
7
, c
8
, c
11
y
2
(y + 2)
4
(y
30
+ 25y
29
+ ··· 32y + 16)
c
10
(y
2
+ y + 1)
3
(y
30
+ 8y
29
+ ··· + 59y + 9)
11