11n
100
(K11n
100
)
A knot diagram
1
Linearized knot diagam
6 1 7 10 2 4 3 1 11 4 9
Solving Sequence
4,11
10
2,5
6 7 3 8 9 1
c
10
c
4
c
5
c
6
c
3
c
7
c
9
c
11
c
1
, c
2
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h3u
20
+ 3u
19
+ ··· + 4b + 4, −2u
21
− 4u
20
+ ··· + 4a − 8, u
22
+ 2u
21
+ ··· + u + 2i
I
u
2
= hu
5
+ u
3
+ b + u − 1, −u
5
− u
4
− u
3
− u
2
+ a − 2u − 1, u
6
+ u
4
+ 2u
2
+ 1i
I
u
3
= hb + a − u + 1, a
2
− au + 2a + 1, u
2
− u + 1i
I
u
4
= hb + u − 2, a − u, u
2
− u + 1i
* 4 irreducible components of dim
C
= 0, with total 34 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
= h3u
20
+3u
19
+· · ·+4b+4, −2u
21
−4u
20
+· · ·+4a−8, u
22
+2u
21
+· · ·+u+2i
(i) Arc colorings
a
4
=
0
u
a
11
=
1
0
a
10
=
1
−u
2
a
2
=
1
2
u
21
+ u
20
+ ··· +
5
4
u + 2
−
3
4
u
20
−
3
4
u
19
+ ··· +
3
2
u − 1
a
5
=
−u
u
3
+ u
a
6
=
1
4
u
21
+ u
19
+ ··· +
1
4
u − 1
−
3
4
u
21
− u
20
+ ··· +
1
2
u − 1
a
7
=
1
4
u
21
+ u
19
+ ··· +
1
4
u − 1
−
1
4
u
21
−
3
4
u
19
+ ··· +
3
2
u
2
+
1
2
u
a
3
=
1
4
u
18
+
3
4
u
16
+ ··· +
1
2
u +
1
2
−
1
4
u
18
−
3
4
u
16
+ ··· −
3
4
u
2
+ u
a
8
=
−u
6
− u
4
− 2u
2
− 1
u
6
+ u
2
a
9
=
u
2
+ 1
−u
2
a
1
=
u
4
+ u
2
+ 1
−u
4
a
1
=
u
4
+ u
2
+ 1
−u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −2u
21
− 4u
20
− 8u
19
− 12u
18
− 18u
17
− 30u
16
− 32u
15
− 46u
14
− 36u
13
− 54u
12
−
44u
11
− 64u
10
− 38u
9
− 42u
8
− 26u
7
− 46u
6
− 24u
5
− 18u
4
+ 10u
3
+ 6u
2
+ 8u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
22
− u
21
+ ··· + 4u + 1
c
2
u
22
+ 3u
21
+ ··· + 24u + 1
c
3
, c
6
, c
7
u
22
− u
21
+ ··· + 10u + 1
c
4
, c
10
u
22
− 2u
21
+ ··· − u + 2
c
8
, c
9
, c
11
u
22
− 8u
21
+ ··· − 19u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
22
+ 3y
21
+ ··· + 24y + 1
c
2
y
22
+ 39y
21
+ ··· + 112y + 1
c
3
, c
6
, c
7
y
22
+ 31y
21
+ ··· + 56y + 1
c
4
, c
10
y
22
+ 8y
21
+ ··· + 19y + 4
c
8
, c
9
, c
11
y
22
+ 12y
21
+ ··· + 623y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.100185 + 1.004210I
a = 0.599236 + 0.939946I
b = −0.636317 − 0.511584I
3.57528 − 1.09357I 5.97662 + 1.94696I
u = −0.100185 − 1.004210I
a = 0.599236 − 0.939946I
b = −0.636317 + 0.511584I
3.57528 + 1.09357I 5.97662 − 1.94696I
u = −0.871760 + 0.414642I
a = −0.620508 − 0.603007I
b = 0.833056 − 0.720277I
7.32664 + 1.36370I −0.05052 − 1.94758I
u = −0.871760 − 0.414642I
a = −0.620508 + 0.603007I
b = 0.833056 + 0.720277I
7.32664 − 1.36370I −0.05052 + 1.94758I
u = 0.898472 + 0.557159I
a = 1.76169 + 0.04754I
b = −1.32483 − 1.39331I
6.44298 + 5.66281I −0.84387 − 2.45088I
u = 0.898472 − 0.557159I
a = 1.76169 − 0.04754I
b = −1.32483 + 1.39331I
6.44298 − 5.66281I −0.84387 + 2.45088I
u = −0.665247 + 0.564550I
a = 1.49996 − 0.51683I
b = −0.16586 + 1.42901I
−0.94737 − 2.13228I −3.49508 + 3.26961I
u = −0.665247 − 0.564550I
a = 1.49996 + 0.51683I
b = −0.16586 − 1.42901I
−0.94737 + 2.13228I −3.49508 − 3.26961I
u = −0.733981 + 0.868553I
a = −0.907314 − 0.753290I
b = 0.983819 + 0.049852I
−4.65093 + 2.79195I −9.45575 − 3.06805I
u = −0.733981 − 0.868553I
a = −0.907314 + 0.753290I
b = 0.983819 − 0.049852I
−4.65093 − 2.79195I −9.45575 + 3.06805I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.765248 + 0.888811I
a = −0.797359 + 0.717329I
b = 1.52476 + 0.32014I
−1.42494 − 2.89189I 2.45935 + 2.97630I
u = 0.765248 − 0.888811I
a = −0.797359 − 0.717329I
b = 1.52476 − 0.32014I
−1.42494 + 2.89189I 2.45935 − 2.97630I
u = −0.616205 + 1.023520I
a = −0.42505 + 1.35805I
b = −1.31664 − 1.90123I
0.39802 + 7.14623I −0.40139 − 7.68801I
u = −0.616205 − 1.023520I
a = −0.42505 − 1.35805I
b = −1.31664 + 1.90123I
0.39802 − 7.14623I −0.40139 + 7.68801I
u = −0.057721 + 1.217230I
a = 0.180750 − 1.262320I
b = −0.541329 + 0.344642I
13.19150 + 3.89903I 4.72901 − 2.42961I
u = −0.057721 − 1.217230I
a = 0.180750 + 1.262320I
b = −0.541329 − 0.344642I
13.19150 − 3.89903I 4.72901 + 2.42961I
u = −0.623287 + 1.124510I
a = −0.138066 − 0.552037I
b = 1.74630 + 0.37098I
9.48603 + 4.13683I 2.53393 − 2.55439I
u = −0.623287 − 1.124510I
a = −0.138066 + 0.552037I
b = 1.74630 − 0.37098I
9.48603 − 4.13683I 2.53393 + 2.55439I
u = 0.700819 + 1.100120I
a = 0.07522 − 1.65631I
b = −2.18014 + 1.42172I
8.10703 − 11.57360I 0.88963 + 6.62056I
u = 0.700819 − 1.100120I
a = 0.07522 + 1.65631I
b = −2.18014 − 1.42172I
8.10703 + 11.57360I 0.88963 − 6.62056I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.303847 + 0.375828I
a = 0.521447 + 0.925816I
b = 0.577176 + 0.274527I
−0.380875 − 1.140110I −4.34193 + 6.22750I
u = 0.303847 − 0.375828I
a = 0.521447 − 0.925816I
b = 0.577176 − 0.274527I
−0.380875 + 1.140110I −4.34193 − 6.22750I
7
II.
I
u
2
= hu
5
+ u
3
+ b + u − 1, −u
5
− u
4
− u
3
− u
2
+ a − 2u − 1, u
6
+ u
4
+ 2u
2
+ 1i
(i) Arc colorings
a
4
=
0
u
a
11
=
1
0
a
10
=
1
−u
2
a
2
=
u
5
+ u
4
+ u
3
+ u
2
+ 2u + 1
−u
5
− u
3
− u + 1
a
5
=
−u
u
3
+ u
a
6
=
−u
4
− 1
u
5
+ u
3
− u
2
+ 2u
a
7
=
−u
4
− 1
u
5
+ u
4
+ u
3
+ 2u + 1
a
3
=
u
5
+ u
3
+ 2u
−u
5
+ u
4
− u
3
− u + 1
a
8
=
0
u
4
+ u
2
+ 1
a
9
=
u
2
+ 1
−u
2
a
1
=
u
4
+ u
2
+ 1
−u
4
a
1
=
u
4
+ u
2
+ 1
−u
4
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
4
+ 4u
2
+ 4
8
(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)
3
c
2
(u + 1)
6
c
4
, c
10
u
6
+ u
4
+ 2u
2
+ 1
c
8
, c
9
(u
3
+ u
2
+ 2u + 1)
2
c
11
(u
3
− u
2
+ 2u − 1)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
6
, c
7
(y + 1)
6
c
2
(y −1)
6
c
4
, c
10
(y
3
+ y
2
+ 2y + 1)
2
c
8
, c
9
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.744862 + 0.877439I
a = −1.43972 + 1.40722I
b = 2.30714 + 0.21508I
−3.02413 − 2.82812I −3.50976 + 2.97945I
u = 0.744862 − 0.877439I
a = −1.43972 − 1.40722I
b = 2.30714 − 0.21508I
−3.02413 + 2.82812I −3.50976 − 2.97945I
u = −0.744862 + 0.877439I
a = −0.315159 − 0.082503I
b = −0.307141 + 0.215080I
−3.02413 + 2.82812I −3.50976 − 2.97945I
u = −0.744862 − 0.877439I
a = −0.315159 + 0.082503I
b = −0.307141 − 0.215080I
−3.02413 − 2.82812I −3.50976 + 2.97945I
u = 0.754878I
a = 0.75488 + 1.32472I
b = 1.000000 − 0.569840I
1.11345 3.01950
u = − 0.754878I
a = 0.75488 − 1.32472I
b = 1.000000 + 0.569840I
1.11345 3.01950
11
III. I
u
3
= hb + a − u + 1, a
2
− au + 2a + 1, u
2
− u + 1i
(i) Arc colorings
a
4
=
0
u
a
11
=
1
0
a
10
=
1
−u + 1
a
2
=
a
−a + u − 1
a
5
=
−u
u − 1
a
6
=
−au + a − u + 1
−1
a
7
=
−au + a − u + 1
au + u − 1
a
3
=
a
au − 2a + u − 1
a
8
=
−u
u
a
9
=
u
−u + 1
a
1
=
0
u
a
1
=
0
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u − 2
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
6
, c
7
u
4
− u
3
+ 2u
2
− 2u + 1
c
2
u
4
+ 3u
3
+ 2u
2
+ 1
c
4
, c
10
(u
2
+ u + 1)
2
c
8
, c
9
, 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
5
c
6
, c
7
y
4
+ 3y
3
+ 2y
2
+ 1
c
2
y
4
− 5y
3
+ 6y
2
+ 4y + 1
c
4
, c
8
, c
9
c
10
, c
11
(y
2
+ y + 1)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = −0.378256 − 0.440597I
b = −0.121744 + 1.306620I
− 2.02988I 0. + 3.46410I
u = 0.500000 + 0.866025I
a = −1.12174 + 1.30662I
b = 0.621744 − 0.440597I
− 2.02988I 0. + 3.46410I
u = 0.500000 − 0.866025I
a = −0.378256 + 0.440597I
b = −0.121744 − 1.306620I
2.02988I 0. − 3.46410I
u = 0.500000 − 0.866025I
a = −1.12174 − 1.30662I
b = 0.621744 + 0.440597I
2.02988I 0. − 3.46410I
15
IV. I
u
4
= hb + u − 2, a − u, u
2
− u + 1i
(i) Arc colorings
a
4
=
0
u
a
11
=
1
0
a
10
=
1
−u + 1
a
2
=
u
−u + 2
a
5
=
−u
u − 1
a
6
=
−1
2u
a
7
=
−1
u + 1
a
3
=
u
−u + 1
a
8
=
−u
u
a
9
=
u
−u + 1
a
1
=
0
u
a
1
=
0
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u − 2
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
10
u
2
+ u + 1
c
8
, c
9
, c
11
u
2
− u + 1
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
y
2
+ y + 1
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 1.50000 − 0.86603I
− 2.02988I 0. + 3.46410I
u = 0.500000 − 0.866025I
a = 0.500000 − 0.866025I
b = 1.50000 + 0.86603I
2.02988I 0. − 3.46410I
19
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
((u
2
+ 1)
3
)(u
2
+ u + 1)(u
4
− u
3
+ ··· − 2u + 1)(u
22
− u
21
+ ··· + 4u + 1)
c
2
((u + 1)
6
)(u
2
+ u + 1)(u
4
+ 3u
3
+ 2u
2
+ 1)(u
22
+ 3u
21
+ ··· + 24u + 1)
c
3
, c
6
, c
7
((u
2
+ 1)
3
)(u
2
+ u + 1)(u
4
− u
3
+ ··· − 2u + 1)(u
22
− u
21
+ ··· + 10u + 1)
c
4
, c
10
((u
2
+ u + 1)
3
)(u
6
+ u
4
+ 2u
2
+ 1)(u
22
− 2u
21
+ ··· − u + 2)
c
8
, c
9
((u
2
− u + 1)
3
)(u
3
+ u
2
+ 2u + 1)
2
(u
22
− 8u
21
+ ··· − 19u + 4)
c
11
((u
2
− u + 1)
3
)(u
3
− u
2
+ 2u − 1)
2
(u
22
− 8u
21
+ ··· − 19u + 4)
20
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
((y + 1)
6
)(y
2
+ y + 1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
22
+ 3y
21
+ ··· + 24y + 1)
c
2
(y −1)
6
(y
2
+ y + 1)(y
4
− 5y
3
+ 6y
2
+ 4y + 1)
· (y
22
+ 39y
21
+ ··· + 112y + 1)
c
3
, c
6
, c
7
((y + 1)
6
)(y
2
+ y + 1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
22
+ 31y
21
+ ··· + 56y + 1)
c
4
, c
10
((y
2
+ y + 1)
3
)(y
3
+ y
2
+ 2y + 1)
2
(y
22
+ 8y
21
+ ··· + 19y + 4)
c
8
, c
9
, c
11
((y
2
+ y + 1)
3
)(y
3
+ 3y
2
+ 2y −1)
2
(y
22
+ 12y
21
+ ··· + 623y + 16)
21
