11n
97
(K11n
97
)
A knot diagram
1
Linearized knot diagam
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
Ideals for irreducible components
2
of X
par
I
u
1
= h−30727u
11
− 34887u
10
+ ··· + 3039144b − 2595171,
− 4836461u
11
+ 12982939u
10
+ ··· + 36469728a + 72471903,
u
12
− 2u
11
+ 11u
10
− 18u
9
+ 46u
8
− 52u
7
+ 89u
6
− 74u
5
+ 120u
4
− 38u
3
+ 52u
2
+ 9i
I
u
2
= hb, −u
3
− 2u
2
+ 2a − 3u − 1, u
4
+ u
3
+ u
2
+ 1i
I
u
3
= h−au + b + u + 1, a
2
+ 2au − a − u − 2, u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 20 representations.
1
The image of knot diagram is generated by the software “Draw programme” developed by An-
drew Bartholomew(http://www.layer8.co.uk/maths/draw/index.htm#Running-draw), where we modi-
fied some parts for our purpose(https://github.com/CATsTAILs/LinksPainter).
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
1
I.
I
u
1
= h−3.07 × 10
4
u
11
− 3.49 × 10
4
u
10
+ · · · + 3.04 × 10
6
b − 2.60 × 10
6
, −4.84 ×
10
6
u
11
+1.30×10
7
u
10
+· · ·+3.65×10
7
a+7.25×10
7
, u
12
−2u
11
+· · ·+52u
2
+9i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
−u
2
a
1
=
−u
u
3
+ u
a
10
=
0.132616u
11
− 0.355992u
10
+ ··· + 7.05457u − 1.98718
0.0101104u
11
+ 0.0114792u
10
+ ··· − 0.601162u + 0.853915
a
4
=
−0.103120u
11
+ 0.218828u
10
+ ··· − 3.36335u + 0.0167258
0.0178412u
11
− 0.0449113u
10
+ ··· + 1.11635u + 0.000284291
a
7
=
−0.0000315878u
11
+ 0.0179044u
10
+ ··· − 0.854914u + 2.11635
0.00740208u
11
− 0.0289272u
10
+ ··· + 0.112721u − 0.541350
a
3
=
−u
3
u
5
+ u
3
+ u
a
8
=
0.0217851u
11
− 0.0147232u
10
+ ··· − 0.838473u + 2.20500
0.0577531u
11
− 0.0685147u
10
+ ··· + 0.179624u − 0.829029
a
9
=
0.149942u
11
− 0.378276u
10
+ ··· + 8.21373u − 2.31454
−0.0310324u
11
+ 0.110398u
10
+ ··· − 1.60439u + 1.29258
a
11
=
0.167308u
11
− 0.446132u
10
+ ··· + 7.46196u − 1.32551
−0.00135137u
11
− 0.00746559u
10
+ ··· − 1.11597u + 0.589375
a
11
=
0.167308u
11
− 0.446132u
10
+ ··· + 7.46196u − 1.32551
−0.00135137u
11
− 0.00746559u
10
+ ··· − 1.11597u + 0.589375
(ii) Obstruction class = −1
(iii) Cusp Shapes =
4442973
8104384
u
11
−
9147283
8104384
u
10
+ ··· +
28543939
8104384
u +
25892857
8104384
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
12
− 2u
11
+ ··· + 52u
2
+ 9
c
2
u
12
+ 18u
11
+ ··· + 936u + 81
c
3
, c
6
, c
7
u
12
− 2u
11
+ ··· + 12u + 9
c
4
, c
10
u
12
− 8u
11
+ ··· − 48u + 64
c
8
, c
9
, c
11
u
12
+ 7u
11
+ ··· − 3u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
12
+ 18y
11
+ ··· + 936y + 81
c
2
y
12
− 42y
11
+ ··· − 88128y + 6561
c
3
, c
6
, c
7
y
12
+ 2y
11
+ ··· + 648y + 81
c
4
, c
10
y
12
− 30y
11
+ ··· + 13056y + 4096
c
8
, c
9
, c
11
y
12
− y
11
+ ··· + 559y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.217703 + 0.714491I
a = −0.323268 − 0.564378I
b = −0.299646 + 0.378751I
−0.382669 + 1.142140I −4.20479 − 6.27644I
u = 0.217703 − 0.714491I
a = −0.323268 + 0.564378I
b = −0.299646 − 0.378751I
−0.382669 − 1.142140I −4.20479 + 6.27644I
u = −0.640918 + 1.176710I
a = −0.126668 − 0.646625I
b = 0.682857 − 1.234360I
9.18153 − 2.19341I 4.66853 + 1.23820I
u = −0.640918 − 1.176710I
a = −0.126668 + 0.646625I
b = 0.682857 + 1.234360I
9.18153 + 2.19341I 4.66853 − 1.23820I
u = 1.15244 + 0.97674I
a = −0.320357 + 0.241963I
b = 1.73050 + 0.90375I
−1.62774 + 2.71130I 0.00178 − 2.31651I
u = 1.15244 − 0.97674I
a = −0.320357 − 0.241963I
b = 1.73050 − 0.90375I
−1.62774 − 2.71130I 0.00178 + 2.31651I
u = −0.148425 + 0.443858I
a = 0.38903 + 3.01143I
b = 0.403960 − 0.536532I
2.12411 − 0.85388I 7.33787 − 1.04083I
u = −0.148425 − 0.443858I
a = 0.38903 − 3.01143I
b = 0.403960 + 0.536532I
2.12411 + 0.85388I 7.33787 + 1.04083I
u = 0.62854 + 1.75953I
a = −1.237110 − 0.324756I
b = −2.03160 + 1.52165I
−9.75530 + 10.18300I 1.33053 − 4.09142I
u = 0.62854 − 1.75953I
a = −1.237110 + 0.324756I
b = −2.03160 − 1.52165I
−9.75530 − 10.18300I 1.33053 + 4.09142I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.20933 + 2.25945I
a = 1.035040 − 0.220318I
b = 3.51393 − 0.31919I
−12.69940 − 0.47600I −0.258917 + 0.098219I
u = −0.20933 − 2.25945I
a = 1.035040 + 0.220318I
b = 3.51393 + 0.31919I
−12.69940 + 0.47600I −0.258917 − 0.098219I
6
II. I
u
2
= hb, −u
3
− 2u
2
+ 2a − 3u − 1, u
4
+ u
3
+ u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
−u
2
a
1
=
−u
u
3
+ u
a
10
=
1
2
u
3
+ u
2
+
3
2
u +
1
2
0
a
4
=
1
0
a
7
=
u
2
+ 1
−u
2
a
3
=
−u
3
u
3
+ u
2
+ 1
a
8
=
u
−u
3
− u
a
9
=
1
2
u
3
+ u
2
+
5
2
u +
1
2
−u
3
− u
a
11
=
1
2
u
3
+ u
2
+
3
2
u +
1
2
0
a
11
=
1
2
u
3
+ u
2
+
3
2
u +
1
2
0
(ii) Obstruction class = 1
(iii) Cusp Shapes =
1
4
u
3
−
9
2
u
2
−
9
4
u −
5
4
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
4
− u
3
+ u
2
+ 1
c
2
, c
6
, c
7
u
4
+ u
3
+ 3u
2
+ 2u + 1
c
3
u
4
− u
3
+ 3u
2
− 2u + 1
c
4
, c
10
u
4
c
5
u
4
+ u
3
+ u
2
+ 1
c
8
, c
9
(u + 1)
4
c
11
(u − 1)
4
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
4
+ y
3
+ 3y
2
+ 2y + 1
c
2
, c
3
, c
6
c
7
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
c
4
, c
10
y
4
c
8
, c
9
, c
11
(y −1)
4
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.351808 + 0.720342I
a = 0.38053 + 1.53420I
b = 0
1.43393 + 1.41510I −0.38954 − 3.92814I
u = 0.351808 − 0.720342I
a = 0.38053 − 1.53420I
b = 0
1.43393 − 1.41510I −0.38954 + 3.92814I
u = −0.851808 + 0.911292I
a = −0.130534 + 0.427872I
b = 0
8.43568 − 3.16396I 1.51454 + 5.24252I
u = −0.851808 − 0.911292I
a = −0.130534 − 0.427872I
b = 0
8.43568 + 3.16396I 1.51454 − 5.24252I
10
III. I
u
3
= h−au + b + u + 1, a
2
+ 2au − a − u − 2, u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
1
a
1
=
−u
0
a
10
=
a
au − u − 1
a
4
=
−a − 2u + 2
−a − u + 2
a
7
=
−au + 2u + 3
−au + 2u + 2
a
3
=
u
u
a
8
=
−au + 2u + 2
−au + 2u + 1
a
9
=
au + a − u − 1
au − u − 1
a
11
=
−2au + 2u + 3
−au + 2u + 1
a
11
=
−2au + 2u + 3
−au + 2u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
6
, c
7
(u
2
+ 1)
2
c
2
(u + 1)
4
c
4
, c
10
u
4
+ 3u
2
+ 1
c
8
, c
9
(u
2
− u − 1)
2
c
11
(u
2
+ u − 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
6
, c
7
(y + 1)
4
c
2
(y −1)
4
c
4
, c
10
(y
2
+ 3y + 1)
2
c
8
, c
9
, c
11
(y
2
− 3y + 1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = −0.618034 − 1.000000I
b = − 1.61803I
8.88264 4.00000
u = 1.000000I
a = 1.61803 − 1.00000I
b = 0.618034I
0.986960 4.00000
u = − 1.000000I
a = −0.618034 + 1.000000I
b = 1.61803I
8.88264 4.00000
u = − 1.000000I
a = 1.61803 + 1.00000I
b = − 0.618034I
0.986960 4.00000
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
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
+ 3u
2
+ 2u + 1)(u
12
+ 18u
11
+ ··· + 936u + 81)
c
3
((u
2
+ 1)
2
)(u
4
− u
3
+ 3u
2
− 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
+ 3u
2
+ 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)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
((y + 1)
4
)(y
4
+ y
3
+ 3y
2
+ 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)
16
