11n
99
(K11n
99
)
A knot diagram
1
Linearized knot diagam
7 1 8 7 10 2 4 11 1 5 9
Solving Sequence
8,11
9
1,4
3 2 7 5 6 10
c
8
c
11
c
3
c
2
c
7
c
4
c
6
c
10
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−1338u
11
− 2403u
10
+ ··· + 24722b + 1642, −2813u
11
− 3694u
10
+ ··· + 49444a − 23561,
u
12
− 2u
11
− 3u
10
+ 7u
9
+ 3u
8
− 10u
7
+ 12u
6
− 17u
5
− 14u
4
+ 34u
3
+ 2u
2
− 7u − 4i
I
u
2
= hu
5
− u
4
+ u
2
a − u
3
+ 2u
2
+ b − a − 1, −u
5
+ 2u
3
a + 2u
4
− 2u
2
a + a
2
− au − 2u
2
+ 2a + 2u − 1,
u
6
− u
5
− u
4
+ 2u
3
− u + 1i
I
u
3
= h−au + b − a + 1, a
2
+ 2au − 4a − 6u + 10, u
2
− u − 1i
I
u
4
= hb − 1, 2a + 1, u + 1i
* 4 irreducible components of dim
C
= 0, with total 29 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−1338u
11
− 2403u
10
+ · · · + 24722b + 1642, −2813u
11
− 3694u
10
+
· · · + 49444a − 23561, u
12
− 2u
11
+ · · · − 7u − 4i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
u
a
9
=
1
−u
2
a
1
=
u
−u
3
+ u
a
4
=
0.0568926u
11
+ 0.0747108u
10
+ ··· + 2.87250u + 0.476519
0.0541218u
11
+ 0.0972009u
10
+ ··· + 0.203098u − 0.0664186
a
3
=
0.00277081u
11
− 0.0224901u
10
+ ··· + 2.66940u + 0.542937
0.0541218u
11
+ 0.0972009u
10
+ ··· + 0.203098u − 0.0664186
a
2
=
0.0205687u
11
+ 0.0812232u
10
+ ··· + 1.83784u + 0.263996
−0.273036u
11
+ 0.227126u
10
+ ··· − 1.67482u − 0.902597
a
7
=
−0.0335531u
11
− 0.0159777u
10
+ ··· − 1.36526u + 0.330414
0.0697355u
11
− 0.157269u
10
+ ··· + 1.27813u + 0.343419
a
5
=
0.0774614u
11
+ 0.155934u
10
+ ··· + 3.71034u + 0.740515
−0.218914u
11
+ 0.324327u
10
+ ··· − 2.47173u − 0.969015
a
6
=
0.182105u
11
− 0.175188u
10
+ ··· − 1.19481u + 0.409554
0.00392363u
11
− 0.236227u
10
+ ··· + 1.73259u + 0.429415
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
10
=
−u
2
+ 1
u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
43847
49444
u
11
+
112737
49444
u
10
+
32482
12361
u
9
−
383109
49444
u
8
−
28567
12361
u
7
+
985
94
u
6
−
299113
24722
u
5
+
967541
49444
u
4
+
605165
49444
u
3
−
1852919
49444
u
2
+
20717
49444
u +
92353
12361
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
6
, c
7
u
12
− u
11
+ ··· + u − 1
c
2
u
12
+ 15u
11
+ ··· − 7u + 1
c
5
, c
10
u
12
− 3u
11
+ ··· − 30u + 8
c
8
, c
9
, c
11
u
12
+ 2u
11
+ ··· + 7u − 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
6
, c
7
y
12
+ 15y
11
+ ··· − 7y + 1
c
2
y
12
− 37y
11
+ ··· − 75y + 1
c
5
, c
10
y
12
+ 3y
11
+ ··· − 180y + 64
c
8
, c
9
, c
11
y
12
− 10y
11
+ ··· − 65y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.10462
a = −0.677570
b = 0.286848
2.08000 2.69920
u = 0.811259
a = 0.740294
b = −1.25251
2.82934 −4.50160
u = 0.038537 + 1.279810I
a = 0.05191 − 1.51413I
b = 0.22221 − 1.69295I
−13.9790 − 4.8530I 0.50797 + 2.30086I
u = 0.038537 − 1.279810I
a = 0.05191 + 1.51413I
b = 0.22221 + 1.69295I
−13.9790 + 4.8530I 0.50797 − 2.30086I
u = 1.51234 + 0.05980I
a = 0.151707 + 0.418299I
b = −0.376488 + 0.828316I
6.26923 − 1.30619I 9.66269 + 5.18573I
u = 1.51234 − 0.05980I
a = 0.151707 − 0.418299I
b = −0.376488 − 0.828316I
6.26923 + 1.30619I 9.66269 − 5.18573I
u = 1.41659 + 0.65856I
a = −0.972994 + 0.888579I
b = 0.46036 + 1.59632I
−9.7228 + 11.6344I 3.05947 − 5.63312I
u = 1.41659 − 0.65856I
a = −0.972994 − 0.888579I
b = 0.46036 − 1.59632I
−9.7228 − 11.6344I 3.05947 + 5.63312I
u = −0.312665 + 0.284545I
a = −0.598564 + 1.005440I
b = −0.257303 + 0.306472I
0.367468 − 0.926038I 6.49064 + 7.55473I
u = −0.312665 − 0.284545I
a = −0.598564 − 1.005440I
b = −0.257303 − 0.306472I
0.367468 + 0.926038I 6.49064 − 7.55473I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.50812 + 0.67127I
a = 0.711581 + 0.645229I
b = −0.06595 + 1.61394I
−9.24108 − 2.07346I 2.05543 + 1.04459I
u = −1.50812 − 0.67127I
a = 0.711581 − 0.645229I
b = −0.06595 − 1.61394I
−9.24108 + 2.07346I 2.05543 − 1.04459I
6
II. I
u
2
= hu
5
− u
4
+ u
2
a − u
3
+ 2u
2
+ b − a − 1, −u
5
+ 2u
4
+ · · · + 2a −
1, u
6
− u
5
− u
4
+ 2u
3
− u + 1i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
u
a
9
=
1
−u
2
a
1
=
u
−u
3
+ u
a
4
=
a
−u
5
+ u
4
− u
2
a + u
3
− 2u
2
+ a + 1
a
3
=
u
5
− u
4
+ u
2
a − u
3
+ 2u
2
− 1
−u
5
+ u
4
− u
2
a + u
3
− 2u
2
+ a + 1
a
2
=
u
5
a − u
4
a + u
5
− 2u
3
a − u
4
+ 2u
2
a − 2u
3
+ au + 2u
2
− a + 2u − 1
2u
5
a − 3u
3
a + u
4
− 2u
3
+ 2au − 2u
2
+ 2u + 1
a
7
=
u
5
a − u
4
a + u
5
− 2u
3
a − u
4
+ 2u
2
a + au − a + 1
u
5
a − u
5
− 2u
3
a + u
4
+ 2u
3
+ au − 2u
2
− u + 1
a
5
=
u
4
− u
2
+ 1
−2u
5
+ u
4
+ 4u
3
− 2u
2
− 2u + 2
a
6
=
−2u
4
+ 2u
2
− u − 2
4u
5
− 2u
4
− 6u
3
+ 4u
2
+ 3u − 4
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
10
=
−u
2
+ 1
u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
4
+ 4u
2
− 4u + 2
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
6
, c
7
u
12
+ 3u
11
+ ··· + 12u + 9
c
2
u
12
+ 11u
11
+ ··· + 432u + 81
c
5
, c
8
, c
9
c
10
, c
11
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
6
, c
7
y
12
+ 11y
11
+ ··· + 432y + 81
c
2
y
12
− 21y
11
+ ··· − 8100y + 6561
c
5
, c
8
, c
9
c
10
, c
11
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.002190 + 0.295542I
a = 1.63266 − 0.89783I
b = −0.250689 + 0.621966I
−1.39926 − 0.92430I 7.71672 + 0.79423I
u = −1.002190 + 0.295542I
a = −1.31279 − 1.72080I
b = −0.007520 − 1.191130I
−1.39926 − 0.92430I 7.71672 + 0.79423I
u = −1.002190 − 0.295542I
a = 1.63266 + 0.89783I
b = −0.250689 − 0.621966I
−1.39926 + 0.92430I 7.71672 − 0.79423I
u = −1.002190 − 0.295542I
a = −1.31279 + 1.72080I
b = −0.007520 + 1.191130I
−1.39926 + 0.92430I 7.71672 − 0.79423I
u = 0.428243 + 0.664531I
a = 0.0287467 + 0.1266650I
b = 0.793458 − 0.920250I
−5.18047 − 0.92430I 0.283283 + 0.794226I
u = 0.428243 + 0.664531I
a = −1.13932 + 1.53189I
b = 0.12359 + 1.51263I
−5.18047 − 0.92430I 0.283283 + 0.794226I
u = 0.428243 − 0.664531I
a = 0.0287467 − 0.1266650I
b = 0.793458 + 0.920250I
−5.18047 + 0.92430I 0.283283 − 0.794226I
u = 0.428243 − 0.664531I
a = −1.13932 − 1.53189I
b = 0.12359 − 1.51263I
−5.18047 + 0.92430I 0.283283 − 0.794226I
u = 1.073950 + 0.558752I
a = −0.598264 + 0.445195I
b = 1.172060 + 0.407463I
−3.28987 + 5.69302I 4.00000 − 5.51057I
u = 1.073950 + 0.558752I
a = 0.888970 − 1.003950I
b = −0.33089 − 1.60761I
−3.28987 + 5.69302I 4.00000 − 5.51057I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.073950 − 0.558752I
a = −0.598264 − 0.445195I
b = 1.172060 − 0.407463I
−3.28987 − 5.69302I 4.00000 + 5.51057I
u = 1.073950 − 0.558752I
a = 0.888970 + 1.003950I
b = −0.33089 + 1.60761I
−3.28987 − 5.69302I 4.00000 + 5.51057I
11
III. I
u
3
= h−au + b − a + 1, a
2
+ 2au − 4a − 6u + 10, u
2
− u − 1i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
u
a
9
=
1
−u − 1
a
1
=
u
−u − 1
a
4
=
a
au + a − 1
a
3
=
−au + 1
au + a − 1
a
2
=
−au + u + 1
au + a − u − 2
a
7
=
a + 2u − 3
−1
a
5
=
au + a − 1
0
a
6
=
−2au − a + u
3au + 2a − u − 1
a
10
=
−u
u
a
10
=
−u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
6
, c
7
(u
2
+ 1)
2
c
2
(u + 1)
4
c
5
, c
10
u
4
+ 3u
2
+ 1
c
8
, c
9
(u
2
− u − 1)
2
c
11
(u
2
+ u − 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
6
, c
7
(y + 1)
4
c
2
(y −1)
4
c
5
, c
10
(y
2
+ 3y + 1)
2
c
8
, c
9
, c
11
(y
2
− 3y + 1)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 2.61803 + 2.61803I
b = 1.000000I
−2.30291 4.00000
u = −0.618034
a = 2.61803 − 2.61803I
b = − 1.000000I
−2.30291 4.00000
u = 1.61803
a = 0.381966 + 0.381966I
b = 1.000000I
5.59278 4.00000
u = 1.61803
a = 0.381966 − 0.381966I
b = − 1.000000I
5.59278 4.00000
15
IV. I
u
4
= hb − 1, 2a + 1, u + 1i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
−1
a
9
=
1
−1
a
1
=
−1
0
a
4
=
−0.5
1
a
3
=
−1.5
1
a
2
=
−0.5
1
a
7
=
0.5
1
a
5
=
0
2
a
6
=
0
2
a
10
=
0
−1
a
10
=
0
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 14.25
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
8
, c
9
u + 1
c
2
, c
6
, c
7
c
11
u − 1
c
5
, c
10
u
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
9
, c
11
y −1
c
5
, c
10
y
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −0.500000
b = 1.00000
3.28987 14.2500
19
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
(u + 1)(u
2
+ 1)
2
(u
12
− u
11
+ ··· + u − 1)(u
12
+ 3u
11
+ ··· + 12u + 9)
c
2
(u − 1)(u + 1)
4
(u
12
+ 11u
11
+ ··· + 432u + 81)
· (u
12
+ 15u
11
+ ··· − 7u + 1)
c
5
, c
10
u(u
4
+ 3u
2
+ 1)(u
6
+ u
5
+ ··· + u + 1)
2
(u
12
− 3u
11
+ ··· − 30u + 8)
c
6
, c
7
(u − 1)(u
2
+ 1)
2
(u
12
− u
11
+ ··· + u − 1)(u
12
+ 3u
11
+ ··· + 12u + 9)
c
8
, c
9
(u + 1)(u
2
− u − 1)
2
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)
2
· (u
12
+ 2u
11
+ ··· + 7u − 4)
c
11
(u − 1)(u
2
+ u − 1)
2
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)
2
· (u
12
+ 2u
11
+ ··· + 7u − 4)
20
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
6
, c
7
(y −1)(y + 1)
4
(y
12
+ 11y
11
+ ··· + 432y + 81)
· (y
12
+ 15y
11
+ ··· − 7y + 1)
c
2
((y −1)
5
)(y
12
− 37y
11
+ ··· − 75y + 1)
· (y
12
− 21y
11
+ ··· − 8100y + 6561)
c
5
, c
10
y(y
2
+ 3y + 1)
2
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
· (y
12
+ 3y
11
+ ··· − 180y + 64)
c
8
, c
9
, c
11
(y −1)(y
2
− 3y + 1)
2
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
· (y
12
− 10y
11
+ ··· − 65y + 16)
21
