11n
97
(K11n
97
)
1
Arc Sequences
6 1 7 10 2 4 3 11 1 4 9
Solving Sequence
2,5
6
1,10
4 7 3 8 9 11
c
5
c
1
c
4
c
6
c
3
c
7
c
9
c
11
c
2
, c
8
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hb
4
+ 4b
3
+ 9b
2
+ 10b + 5, b
2
+ 2b + a + 3, b
3
+ 3b
2
+ 5b + u + 3i
I
u
2
= hu
4
u
3
+ u
2
+ 1, a, u
3
+ 2u
2
+ 2b u 1i
I
u
3
= hu
12
+ 2u
11
+ 11u
10
+ 18u
9
+ 46u
8
+ 52u
7
+ 89u
6
+ 74u
5
+ 120u
4
+ 38u
3
+ 52u
2
+ 9,
865057u
11
1822295u
10
+ ··· + 9117432a + 5481057,
1103339u
11
+ 2037325u
10
+ ··· + 12156576b 14509383i
There are 3 irreducible components with 20 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
+ 4b
3
+ 9b
2
+ 10b + 5, b
2
+ 2b + a + 3, b
3
+ 3b
2
+ 5b + u + 3i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
b
3
3b
2
5b 3
a
6
=
b
3
3b
2
5b 3
b
3
3b
2
5b 3
a
1
=
0
1
a
10
=
b
2
2b 3
b
a
4
=
2b
3
6b
2
11b 7
2b
3
6b
2
11b 8
a
7
=
b
3
2b
2
3b + 1
2b
3
5b
2
8b 2
a
3
=
1
1
a
8
=
b
2
+ 2b + 4
b
3
2b
2
3b + 1
a
9
=
b
2
2b 3
b
2
b 3
a
11
=
b
2
+ 2b + 4
b
3
b
2
b + 3
a
11
=
b
2
+ 2b + 4
b
3
b
2
b + 3
(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 = 1.00000I
a = 0.618034
b = 1.00000 1.61803I
0.986960 4.00000
u = 1.00000I
a = 0.618034
b = 1.00000 + 1.61803I
0.986960 4.00000
u = 1.00000I
a = 1.61803
b = 1.000000 0.618034I
8.88264 4.00000
u = 1.00000I
a = 1.61803
b = 1.000000 + 0.618034I
8.88264 4.00000
3
II. I
u
2
= hu
4
u
3
+ u
2
+ 1, a, u
3
+ 2u
2
+ 2b u 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
10
=
0
1
2
u
3
u
2
+
1
2
u +
1
2
a
4
=
0
u
a
7
=
u
u
3
+ u
a
3
=
u
3
u
3
u
2
1
a
8
=
u
2
1
u
2
a
9
=
u
2
1
1
2
u
3
2u
2
+
1
2
u +
1
2
a
11
=
0
1
2
u
3
u
2
+
1
2
u +
1
2
a
11
=
0
1
2
u
3
u
2
+
1
2
u +
1
2
(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 = 0.351808 0.720342I
a = 0
b = 0.971274 0.813859I
1.43393 + 1.41510I 0.38954 3.92814I
u = 0.351808 + 0.720342I
a = 0
b = 0.971274 + 0.813859I
1.43393 1.41510I 0.38954 + 3.92814I
u = 0.851808 0.911292I
a = 0
b = 0.278726 + 0.483420I
8.43568 3.16396I 1.51454 + 5.24252I
u = 0.851808 + 0.911292I
a = 0
b = 0.278726 0.483420I
8.43568 + 3.16396I 1.51454 5.24252I
5
III.
I
u
3
= hu
12
+ 2u
11
+ · · · +52u
2
+ 9, 8.65 × 10
5
u
11
1.82 × 10
6
u
10
+ · · · + 9.12 ×
10
6
a +5.48 ×10
6
, 1.10 × 10
6
u
11
+2.04× 10
6
u
10
+· · ·+ 1.22 ×10
7
b 1.45 ×10
7
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
=
0.0948795u
11
+ 0.199869u
10
+ ··· + 2.05218u 0.601162
0.0907607u
11
0.167590u
10
+ ··· 1.98718u + 1.19354
a
4
=
0.0000315878u
11
+ 0.0179044u
10
+ ··· + 0.854914u + 1.11635
0.0125878u
11
+ 0.00989687u
10
+ ··· + 0.0167258u 0.928080
a
7
=
0.0601501u
11
0.112898u
10
+ ··· 0.617007u + 0.112721
0.0178412u
11
+ 0.0449113u
10
+ ··· + 2.11635u 0.000284291
a
3
=
u
4
+ u
2
+ 1
u
4
a
8
=
0.0921143u
11
0.126476u
10
+ ··· 2.27470u + 0.179624
0.0288469u
11
+ 0.137264u
10
+ ··· + 2.20500u + 0.196066
a
9
=
0.143620u
11
+ 0.256207u
10
+ ··· + 2.04124u 1.60439
0.0783932u
11
0.166673u
10
+ ··· 2.31454u + 1.34947
a
11
=
0.0654861u
11
+ 0.129621u
10
+ ··· + 0.771316u 1.11597
0.111516u
11
0.189888u
10
+ ··· 1.32551u + 1.50577
a
11
=
0.0654861u
11
+ 0.129621u
10
+ ··· + 0.771316u 1.11597
0.111516u
11
0.189888u
10
+ ··· 1.32551u + 1.50577
(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 = 1.15244 0.97674I
a = 1.260670 0.284266I
b = 0.605527 + 0.034060I
1.62774 + 2.71130I 0.00178 2.31651I
u = 1.15244 + 0.97674I
a = 1.260670 + 0.284266I
b = 0.605527 0.034060I
1.62774 2.71130I 0.00178 + 2.31651I
u = 0.62854 1.75953I
a = 0.401159 + 1.297932I
b = 0.20615 + 2.38084I
9.75530 + 10.18301I 1.33053 4.09142I
u = 0.62854 + 1.75953I
a = 0.401159 1.297932I
b = 0.20615 2.38084I
9.75530 10.18301I 1.33053 + 4.09142I
u = 0.217703 0.714491I
a = 0.368136 + 0.531554I
b = 0.332867 + 0.353839I
0.382669 + 1.142139I 4.20479 6.27644I
u = 0.217703 + 0.714491I
a = 0.368136 0.531554I
b = 0.332867 0.353839I
0.382669 1.142139I 4.20479 + 6.27644I
u = 0.148425 0.443858I
a = 1.36095 0.45501I
b = 1.39439 + 0.27430I
2.12411 0.85388I 7.33787 1.04083I
u = 0.148425 + 0.443858I
a = 1.36095 + 0.45501I
b = 1.39439 0.27430I
2.12411 + 0.85388I 7.33787 + 1.04083I
u = 0.20933 2.25945I
a = 0.28293 1.52901I
b = 0.28113 2.38474I
12.69940 0.47600I 0.258917 + 0.098219I
u = 0.20933 + 2.25945I
a = 0.28293 + 1.52901I
b = 0.28113 + 2.38474I
12.69940 + 0.47600I 0.258917 0.098219I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.640918 1.176705I
a = 1.052756 0.006906I
b = 0.842070 0.265383I
9.18153 2.19341I 4.66853 + 1.23820I
u = 0.640918 + 1.176705I
a = 1.052756 + 0.006906I
b = 0.842070 + 0.265383I
9.18153 + 2.19341I 4.66853 1.23820I
8
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ 1)
2
(u
4
u
3
+ u
2
+ 1)(u
12
+ 2u
11
+ ··· + 52u
2
+ 9)
c
2
(u + 1)
4
(u
4
+ u
3
+ ··· + 2u + 1)(u
12
+ 18u
11
+ ··· + 936u + 81)
c
3
(u
2
+ 1)
2
(u
4
u
3
+ ··· 2u + 1)(u
12
+ 2u
11
+ ··· 12u + 9)
c
4
, c
10
u
4
(u
4
+ 3u
2
+ 1)(u
12
+ 8u
11
+ ··· + 48u + 64)
c
5
(u
2
+ 1)
2
(u
4
+ u
3
+ u
2
+ 1)(u
12
+ 2u
11
+ ··· + 52u
2
+ 9)
c
6
, c
7
(u
2
+ 1)
2
(u
4
+ u
3
+ ··· + 2u + 1)(u
12
+ 2u
11
+ ··· 12u + 9)
c
8
, c
9
(u + 1)
4
(u
2
u 1)
2
(u
12
+ 7u
11
+ ··· 3u + 4)
c
11
(u 1)
4
(u
2
+ u 1)
2
(u
12
+ 7u
11
+ ··· 3u + 4)
9
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y + 1)
4
(y
4
+ y
3
+ ··· + 2y + 1)(y
12
+ 18y
11
+ ··· + 936y + 81)
c
2
(y 1)
4
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
12
42y
11
+ ··· 88128y + 6561)
c
3
, c
6
, c
7
(y + 1)
4
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
12
+ 2y
11
+ ··· + 648y + 81)
c
4
, c
10
y
4
(y
2
+ 3y + 1)
2
(y
12
30y
11
+ ··· + 13056y + 4096)
c
8
, c
9
, c
11
(y 1)
4
(y
2
3y + 1)
2
(y
12
y
11
+ ··· + 559y + 16)
10