11n
28
(K11n
28
)
1
Arc Sequences
4 1 8 2 8 10 3 1 11 6 9
Solving Sequence
6,10
7 11 9
1,3
2 8 4 5
c
6
c
10
c
9
c
11
c
2
c
8
c
3
c
4
c
1
, c
5
, c
7
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
4
+ b
2
b + 1, u 1, b
3
b + a + 1i
I
u
2
= hu
14
+ u
13
+ ··· + 5u + 1,
271965455479u
13
11438964108u
12
+ ··· + 2446025522016a + 14628820700249,
308923491301u
13
413049036210u
12
+ ··· + 4892051044032b 3107040395155i
There are 2 irreducible components with 18 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
+ b
2
b + 1, u 1, b
3
b + a + 1i
(i) Arc colorings
a
6
=
0
1
a
10
=
b
3
+ b 1
b
a
7
=
b
3
b
2
b
0
a
11
=
b
3
+ b 1
b
3
+ 1
a
9
=
b
3
+ b
2
+ 2b
b
3
b
2
b
a
1
=
b
3
+ 2
1
a
3
=
1
0
a
2
=
b
3
+ 3
1
a
8
=
b
3
b
2
b
0
a
4
=
1
0
a
5
=
b
3
2
1
a
5
=
b
3
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 = 1.00000
a = 0.351808 0.720342I
b = 0.547424 1.120873I
1.85594 1.41510I 5.13523 + 6.85627I
u = 1.00000
a = 0.351808 + 0.720342I
b = 0.547424 + 1.120873I
1.85594 + 1.41510I 5.13523 6.85627I
u = 1.00000
a = 0.851808 0.911292I
b = 0.547424 0.585652I
5.14581 + 3.16396I 0.63523 2.29471I
u = 1.00000
a = 0.851808 + 0.911292I
b = 0.547424 + 0.585652I
5.14581 3.16396I 0.63523 + 2.29471I
3
II. I
u
2
=
hu
14
+u
13
+· · · +5u+1, 2.72×10
11
u
13
1.14×10
10
u
12
+· · · +2.45×10
12
a+
1.46 × 10
13
, 3.09× 10
11
u
13
4.13 × 10
11
u
12
+ · · · + 4.89 × 10
12
b 3.11 × 10
12
i
(i) Arc colorings
a
6
=
0
u
a
10
=
0.111187u
13
+ 0.00467655u
12
+ ··· 5.37669u 5.98065
0.0631481u
13
+ 0.0844327u
12
+ ··· + 5.48597u + 0.635120
a
7
=
0.837327u
13
+ 0.807911u
12
+ ··· + 35.8519u + 0.145921
0.0747090u
13
0.0508090u
12
+ ··· + 0.907321u + 0.616101
a
11
=
0.111187u
13
+ 0.00467655u
12
+ ··· 5.37669u 5.98065
0.0680857u
13
+ 0.0928447u
12
+ ··· + 5.06461u + 0.528610
a
9
=
0.501986u
13
+ 0.489429u
12
+ ··· + 22.6456u 1.10759
0.0584250u
13
0.0317344u
12
+ ··· + 1.59756u + 0.586684
a
1
=
0.155664u
13
0.198576u
12
+ ··· 7.49944u 3.10013
u
a
3
=
1
0
a
2
=
0.0429122u
13
+ 0.0234198u
12
+ ··· + 2.32181u + 0.844336
u
2
a
8
=
0.762618u
13
+ 0.757102u
12
+ ··· + 36.7592u + 0.762022
0.0747090u
13
0.0508090u
12
+ ··· + 0.907321u + 0.616101
a
4
=
0.653072u
13
0.640947u
12
+ ··· 21.7576u + 1.26505
0.0251734u
13
0.0694250u
12
+ ··· 2.74279u 0.684274
a
5
=
0.540734u
13
0.518543u
12
+ ··· 18.1346u + 1.76849
0.0121243u
13
0.0580186u
12
+ ··· 2.53041u 0.653072
a
5
=
0.540734u
13
0.518543u
12
+ ··· 18.1346u + 1.76849
0.0121243u
13
0.0580186u
12
+ ··· 2.53041u 0.653072
(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 = 0.841212 1.029315I
a = 0.025061 0.872571I
b = 1.268210 0.615843I
7.00688 0.55948I 2.27714 + 0.75874I
u = 0.841212 + 1.029315I
a = 0.025061 + 0.872571I
b = 1.268210 + 0.615843I
7.00688 + 0.55948I 2.27714 0.75874I
u = 0.337193 1.184656I
a = 0.920439 0.366491I
b = 1.54747 + 0.60627I
6.27413 + 5.41755I 1.11952 5.07443I
u = 0.337193 + 1.184656I
a = 0.920439 + 0.366491I
b = 1.54747 0.60627I
6.27413 5.41755I 1.11952 + 5.07443I
u = 0.196229 0.341059I
a = 1.93607 + 0.51409I
b = 0.550908 0.305501I
0.65784 + 1.53044I 1.45925 4.48215I
u = 0.196229 + 0.341059I
a = 1.93607 0.51409I
b = 0.550908 + 0.305501I
0.65784 1.53044I 1.45925 + 4.48215I
u = 0.17769 2.99241I
a = 0.294120 + 0.345863I
b = 2.42912 + 0.26319I
18.9281 + 1.0022I 1.174737 + 0.209171I
u = 0.17769 + 2.99241I
a = 0.294120 0.345863I
b = 2.42912 0.26319I
18.9281 1.0022I 1.174737 0.209171I
u = 0.038211 0.159690I
a = 5.38896 + 0.83379I
b = 0.190892 0.700537I
0.35154 + 1.84409I 0.79789 4.83996I
u = 0.038211 + 0.159690I
a = 5.38896 0.83379I
b = 0.190892 + 0.700537I
0.35154 1.84409I 0.79789 + 4.83996I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.07654 2.96001I
a = 0.329002 + 0.327325I
b = 2.94643 + 0.61854I
18.7301 + 8.0616I 0.87427 4.09385I
u = 0.07654 + 2.96001I
a = 0.329002 0.327325I
b = 2.94643 0.61854I
18.7301 8.0616I 0.87427 + 4.09385I
u = 1.013993 0.325583I
a = 0.035467 0.501132I
b = 0.621959 1.014427I
1.89748 + 0.70166I 5.60702 + 2.76477I
u = 1.013993 + 0.325583I
a = 0.035467 + 0.501132I
b = 0.621959 + 1.014427I
1.89748 0.70166I 5.60702 2.76477I
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
4
(u
14
+ 5u
13
+ ··· + 3u + 1)
c
2
(u + 1)
4
(u
14
+ u
13
+ ··· + 5u + 1)
c
3
, c
7
u
4
(u
14
+ u
13
+ ··· + 72u + 16)
c
4
(u + 1)
4
(u
14
+ 5u
13
+ ··· + 3u + 1)
c
5
(u
4
u
3
+ 3u
2
2u + 1)(u
14
+ 2u
13
+ ··· 540u + 200)
c
6
(u
4
u
3
+ u
2
+ 1)(u
14
+ 2u
13
+ ··· + u + 1)
c
8
, c
9
(u
4
u
3
+ 3u
2
2u + 1)(u
14
+ 2u
13
+ ··· + 7u + 1)
c
10
(u
4
+ u
3
+ u
2
+ 1)(u
14
+ 2u
13
+ ··· + u + 1)
c
11
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
14
+ 2u
13
+ ··· + 7u + 1)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
4
(y
14
+ y
13
+ ··· + 5y + 1)
c
2
(y 1)
4
(y
14
+ 37y
13
+ ··· + 73y + 1)
c
3
, c
7
y
4
(y
14
27y
13
+ ··· 832y + 256)
c
5
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
14
+ 82y
13
+ ··· + 531600y + 40000)
c
6
, c
10
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
14
+ 2y
13
+ ··· + 7y + 1)
c
8
, c
9
, c
11
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
14
+ 22y
13
+ ··· + 7y + 1)
8