11n
3
(K11n
3
)
A knot diagram
1
Linearized knot diagam
5 1 8 2 3 9 11 3 1 7 10
Solving Sequence
7,10
11
3,8
4 1 2 9 6 5
c
10
c
7
c
3
c
11
c
2
c
9
c
6
c
5
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−5u
26
+ 11u
25
+ ··· + 2b − 5, u
25
− 2u
24
+ ··· + 2a + 5u, u
27
− 3u
26
+ ··· + 3u − 1i
I
u
2
= hu
2
a + b − a, −u
2
a + a
2
− au + u + 1, u
3
+ u
2
− 1i
* 2 irreducible components of dim
C
= 0, with total 33 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−5u
26
+11u
25
+· · ·+2b−5, u
25
−2u
24
+· · ·+2a+5u, u
27
−3u
26
+· · ·+3u−1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
3
=
−
1
2
u
25
+ u
24
+ ··· +
15
2
u
2
−
5
2
u
5
2
u
26
−
11
2
u
25
+ ··· − 5u +
5
2
a
8
=
u
−u
3
+ u
a
4
=
−u
26
+
5
2
u
25
+ ··· +
3
2
u − 2
1
2
u
26
−
3
2
u
25
+ ··· + 5u
2
+
1
2
a
1
=
−u
2
+ 1
−u
2
a
2
=
−u
25
+
3
2
u
24
+ ··· −
3
2
u −
1
2
2u
26
− 5u
25
+ ··· − 4u + 2
a
9
=
u
4
− u
2
+ 1
u
4
a
6
=
−u
9
+ 2u
7
− 3u
5
+ 2u
3
− u
−u
9
+ u
7
− u
5
+ u
a
5
=
1
2
u
25
− u
24
+ ··· +
3
2
u − 1
−
1
2
u
26
+
3
2
u
25
+ ··· + 3u −
1
2
a
5
=
1
2
u
25
− u
24
+ ··· +
3
2
u − 1
−
1
2
u
26
+
3
2
u
25
+ ··· + 3u −
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes =
23
2
u
26
− 24u
25
+ ··· −
69
2
u +
25
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
27
+ 4u
26
+ ··· + 4u + 1
c
2
u
27
+ 6u
26
+ ··· + 4u − 1
c
3
, c
8
u
27
− u
26
+ ··· − 32u − 64
c
5
u
27
− 4u
26
+ ··· + 6988u + 1153
c
6
u
27
+ 3u
26
+ ··· − u − 1
c
7
, c
10
u
27
− 3u
26
+ ··· + 3u − 1
c
9
, c
11
u
27
− 11u
26
+ ··· − 9u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
27
+ 6y
26
+ ··· + 4y −1
c
2
y
27
+ 34y
26
+ ··· + 136y −1
c
3
, c
8
y
27
+ 35y
26
+ ··· + 1024y −4096
c
5
y
27
+ 62y
26
+ ··· − 7660244y −1329409
c
6
y
27
− 47y
26
+ ··· − 9y −1
c
7
, c
10
y
27
− 11y
26
+ ··· − 9y −1
c
9
, c
11
y
27
+ 13y
26
+ ··· + 127y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.456584 + 0.907647I
a = −1.48477 − 0.61163I
b = −1.25986 + 0.88956I
7.22245 + 1.02048I 3.69178 − 1.94630I
u = 0.456584 − 0.907647I
a = −1.48477 + 0.61163I
b = −1.25986 − 0.88956I
7.22245 − 1.02048I 3.69178 + 1.94630I
u = 0.964922 + 0.396644I
a = −0.252553 + 0.564190I
b = 1.21469 + 1.07115I
2.07980 + 1.34949I 7.85827 − 1.99966I
u = 0.964922 − 0.396644I
a = −0.252553 − 0.564190I
b = 1.21469 − 1.07115I
2.07980 − 1.34949I 7.85827 + 1.99966I
u = 0.524863 + 0.914721I
a = 1.55069 + 0.62247I
b = 1.39313 − 0.99095I
6.79111 − 5.64536I 3.12437 + 2.66728I
u = 0.524863 − 0.914721I
a = 1.55069 − 0.62247I
b = 1.39313 + 0.99095I
6.79111 + 5.64536I 3.12437 − 2.66728I
u = −0.974186 + 0.462885I
a = 0.529071 − 0.939257I
b = 0.169302 − 0.547843I
1.69325 − 4.38642I 5.48340 + 6.16823I
u = −0.974186 − 0.462885I
a = 0.529071 + 0.939257I
b = 0.169302 + 0.547843I
1.69325 + 4.38642I 5.48340 − 6.16823I
u = −0.838777 + 0.356107I
a = −1.00459 + 1.02558I
b = −0.373891 + 0.589972I
0.942185 + 1.033980I 3.86006 + 2.18156I
u = −0.838777 − 0.356107I
a = −1.00459 − 1.02558I
b = −0.373891 − 0.589972I
0.942185 − 1.033980I 3.86006 − 2.18156I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.685911 + 0.573005I
a = 1.282770 − 0.152453I
b = 0.13458 − 1.76175I
−1.54191 − 1.16661I 0.74423 + 1.49202I
u = 0.685911 − 0.573005I
a = 1.282770 + 0.152453I
b = 0.13458 + 1.76175I
−1.54191 + 1.16661I 0.74423 − 1.49202I
u = −0.821117 + 0.748475I
a = −0.420089 − 0.037335I
b = −0.229956 + 0.061310I
−3.44772 − 1.77523I −1.86458 + 4.75426I
u = −0.821117 − 0.748475I
a = −0.420089 + 0.037335I
b = −0.229956 − 0.061310I
−3.44772 + 1.77523I −1.86458 − 4.75426I
u = 0.960749 + 0.570989I
a = 0.404574 − 1.204920I
b = −1.74421 − 1.76576I
−0.68885 + 5.76088I 3.34925 − 6.52520I
u = 0.960749 − 0.570989I
a = 0.404574 + 1.204920I
b = −1.74421 + 1.76576I
−0.68885 − 5.76088I 3.34925 + 6.52520I
u = 0.866757
a = −0.0957077
b = 0.622584
1.43125 6.84050
u = −0.922854 + 0.737156I
a = 0.035124 − 0.374638I
b = −0.045924 − 0.210874I
−3.14099 − 3.86941I 0.026459 + 0.626604I
u = −0.922854 − 0.737156I
a = 0.035124 + 0.374638I
b = −0.045924 + 0.210874I
−3.14099 + 3.86941I 0.026459 − 0.626604I
u = −1.242910 + 0.029552I
a = 0.02740 − 1.50043I
b = 0.006371 − 0.831742I
13.47880 − 3.52700I 8.35802 + 2.34346I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.242910 − 0.029552I
a = 0.02740 + 1.50043I
b = 0.006371 + 0.831742I
13.47880 + 3.52700I 8.35802 − 2.34346I
u = 1.135150 + 0.655354I
a = 0.445958 + 1.202700I
b = 2.34089 + 0.91299I
9.30444 + 4.72653I 5.93859 − 2.37138I
u = 1.135150 − 0.655354I
a = 0.445958 − 1.202700I
b = 2.34089 − 0.91299I
9.30444 − 4.72653I 5.93859 + 2.37138I
u = 1.118230 + 0.692002I
a = −0.51886 − 1.31753I
b = −2.49328 − 0.90508I
8.6132 + 11.5634I 4.87720 − 6.78953I
u = 1.118230 − 0.692002I
a = −0.51886 + 1.31753I
b = −2.49328 + 0.90508I
8.6132 − 11.5634I 4.87720 + 6.78953I
u = 0.020052 + 0.347458I
a = −1.54687 − 0.86649I
b = −0.423137 + 0.416168I
−0.075638 + 1.377020I −0.36727 − 4.75192I
u = 0.020052 − 0.347458I
a = −1.54687 + 0.86649I
b = −0.423137 − 0.416168I
−0.075638 − 1.377020I −0.36727 + 4.75192I
7
II. I
u
2
= hu
2
a + b − a, −u
2
a + a
2
− au + u + 1, u
3
+ u
2
− 1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
3
=
a
−u
2
a + a
a
8
=
u
u
2
+ u − 1
a
4
=
a
−u
2
a + a
a
1
=
−u
2
+ 1
−u
2
a
2
=
u
2
a + au
0
a
9
=
u
u
2
+ u − 1
a
6
=
u
2
− 1
u
2
a
5
=
a − u − 1
−u
2
a + a
a
5
=
a − u − 1
−u
2
a + a
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
a + 6au − u
2
+ 2u + 1
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
(u
2
+ u + 1)
3
c
3
, c
8
u
6
c
4
(u
2
− u + 1)
3
c
6
, c
9
(u
3
+ u
2
+ 2u + 1)
2
c
7
(u
3
− u
2
+ 1)
2
c
10
(u
3
+ u
2
− 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
2
, c
4
c
5
(y
2
+ y + 1)
3
c
3
, c
8
y
6
c
6
, c
9
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
c
7
, c
10
(y
3
− y
2
+ 2y −1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = −0.818128 + 0.292480I
b = −1.024480 − 0.839835I
−3.02413 − 4.85801I 0.94625 + 7.60556I
u = −0.877439 + 0.744862I
a = 0.155769 − 0.854759I
b = 1.239560 − 0.467306I
−3.02413 − 0.79824I 2.23639 − 1.26697I
u = −0.877439 − 0.744862I
a = −0.818128 − 0.292480I
b = −1.024480 + 0.839835I
−3.02413 + 4.85801I 0.94625 − 7.60556I
u = −0.877439 − 0.744862I
a = 0.155769 + 0.854759I
b = 1.239560 + 0.467306I
−3.02413 + 0.79824I 2.23639 + 1.26697I
u = 0.754878
a = 0.662359 + 1.147240I
b = 0.284920 + 0.493496I
1.11345 − 2.02988I 5.31735 + 5.84990I
u = 0.754878
a = 0.662359 − 1.147240I
b = 0.284920 − 0.493496I
1.11345 + 2.02988I 5.31735 − 5.84990I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u + 1)
3
)(u
27
+ 4u
26
+ ··· + 4u + 1)
c
2
((u
2
+ u + 1)
3
)(u
27
+ 6u
26
+ ··· + 4u − 1)
c
3
, c
8
u
6
(u
27
− u
26
+ ··· − 32u − 64)
c
4
((u
2
− u + 1)
3
)(u
27
+ 4u
26
+ ··· + 4u + 1)
c
5
((u
2
+ u + 1)
3
)(u
27
− 4u
26
+ ··· + 6988u + 1153)
c
6
((u
3
+ u
2
+ 2u + 1)
2
)(u
27
+ 3u
26
+ ··· − u − 1)
c
7
((u
3
− u
2
+ 1)
2
)(u
27
− 3u
26
+ ··· + 3u − 1)
c
9
((u
3
+ u
2
+ 2u + 1)
2
)(u
27
− 11u
26
+ ··· − 9u − 1)
c
10
((u
3
+ u
2
− 1)
2
)(u
27
− 3u
26
+ ··· + 3u − 1)
c
11
((u
3
− u
2
+ 2u − 1)
2
)(u
27
− 11u
26
+ ··· − 9u − 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y
2
+ y + 1)
3
)(y
27
+ 6y
26
+ ··· + 4y −1)
c
2
((y
2
+ y + 1)
3
)(y
27
+ 34y
26
+ ··· + 136y −1)
c
3
, c
8
y
6
(y
27
+ 35y
26
+ ··· + 1024y −4096)
c
5
((y
2
+ y + 1)
3
)(y
27
+ 62y
26
+ ··· − 7660244y −1329409)
c
6
((y
3
+ 3y
2
+ 2y −1)
2
)(y
27
− 47y
26
+ ··· − 9y −1)
c
7
, c
10
((y
3
− y
2
+ 2y −1)
2
)(y
27
− 11y
26
+ ··· − 9y −1)
c
9
, c
11
((y
3
+ 3y
2
+ 2y −1)
2
)(y
27
+ 13y
26
+ ··· + 127y −1)
13
