11n
25
(K11n
25
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 8 10 3 11 1 6 9
Solving Sequence
1,10
9 11
3,8
4 2 5 7 6
c
9
c
11
c
8
c
3
c
1
c
4
c
7
c
6
c
2
, c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−78512077u
27
− 145372804u
26
+ ··· + 67221149b − 176175,
− 62468877u
27
− 129722654u
26
+ ··· + 67221149a + 343854265, u
28
+ 2u
27
+ ··· − 5u + 1i
I
u
2
= hu
4
+ u
3
− u
2
+ b − u, u
4
+ u
3
− 2u
2
+ a − u + 1, u
5
+ u
4
− 2u
3
− u
2
+ u − 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−7.85 × 10
7
u
27
− 1.45 × 10
8
u
26
+ · · · + 6.72 × 10
7
b − 1.76 × 10
5
, −6.25 ×
10
7
u
27
−1.30×10
8
u
26
+· · · +6.72×10
7
a+3.44×10
8
, u
28
+2u
27
+· · · −5u +1i
(i) Arc colorings
a
1
=
0
u
a
10
=
1
0
a
9
=
1
−u
2
a
11
=
u
−u
3
+ u
a
3
=
0.929304u
27
+ 1.92979u
26
+ ··· + 11.1441u − 5.11527
1.16797u
27
+ 2.16261u
26
+ ··· + 2.86935u + 0.00262083
a
8
=
−u
2
+ 1
u
4
− 2u
2
a
4
=
1.01767u
27
+ 2.01755u
26
+ ··· + 12.4640u − 4.97118
0.885451u
27
+ 1.93727u
26
+ ··· + 3.16598u − 0.0228994
a
2
=
0.929304u
27
+ 1.92979u
26
+ ··· + 11.1441u − 5.11527
1.40015u
27
+ 2.23327u
26
+ ··· + 3.44274u + 0.0738023
a
5
=
0.823260u
27
+ 0.824473u
26
+ ··· + 0.360292u + 0.711827
−2.02865u
27
− 1.42566u
26
+ ··· + 4.66019u − 0.824392
a
7
=
−1.43559u
27
− 1.43578u
26
+ ··· + 3.66414u − 1.71740
1.00626u
27
+ 1.00303u
26
+ ··· − 0.736587u + 0.00303364
a
6
=
−2.44185u
27
− 2.43882u
26
+ ··· + 4.40073u − 1.72043
1.00626u
27
+ 1.00303u
26
+ ··· − 0.736587u + 0.00303364
a
6
=
−2.44185u
27
− 2.43882u
26
+ ··· + 4.40073u − 1.72043
1.00626u
27
+ 1.00303u
26
+ ··· − 0.736587u + 0.00303364
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
337541215
67221149
u
27
−
737621299
67221149
u
26
+ ··· −
124989334
67221149
u −
310642967
67221149
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
28
− 6u
27
+ ··· + 5u − 1
c
2
u
28
+ 6u
27
+ ··· + 17u + 1
c
3
, c
7
u
28
+ 3u
27
+ ··· + 128u + 32
c
5
u
28
− 6u
27
+ ··· − 3079u − 1609
c
6
, c
10
u
28
− 2u
27
+ ··· + u − 1
c
8
, c
9
, c
11
u
28
− 2u
27
+ ··· + 5u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
28
− 6y
27
+ ··· − 17y + 1
c
2
y
28
+ 38y
27
+ ··· − 17y + 1
c
3
, c
7
y
28
− 33y
27
+ ··· − 14848y + 1024
c
5
y
28
+ 22y
27
+ ··· + 19658749y + 2588881
c
6
, c
10
y
28
+ 6y
27
+ ··· + 5y + 1
c
8
, c
9
, c
11
y
28
− 22y
27
+ ··· + 5y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.078918 + 0.962099I
a = −2.18951 + 0.08530I
b = −2.13157 + 0.38039I
7.91461 − 7.14026I −1.25171 + 4.70902I
u = 0.078918 − 0.962099I
a = −2.18951 − 0.08530I
b = −2.13157 − 0.38039I
7.91461 + 7.14026I −1.25171 − 4.70902I
u = 1.062800 + 0.195424I
a = −0.155091 − 0.173758I
b = 0.944249 + 0.890794I
−2.13241 − 0.82619I −5.35184 − 1.04773I
u = 1.062800 − 0.195424I
a = −0.155091 + 0.173758I
b = 0.944249 − 0.890794I
−2.13241 + 0.82619I −5.35184 + 1.04773I
u = −0.064858 + 0.917024I
a = 2.10546 − 0.51061I
b = 1.81015 − 0.32496I
8.51104 + 0.29713I −0.154356 − 0.088934I
u = −0.064858 − 0.917024I
a = 2.10546 + 0.51061I
b = 1.81015 + 0.32496I
8.51104 − 0.29713I −0.154356 + 0.088934I
u = 1.08514
a = −1.05851
b = −3.52629
−3.64067 25.0750
u = −1.206550 + 0.074740I
a = 1.56629 − 0.83850I
b = 0.209734 + 0.919216I
−5.80044 + 1.71298I −12.51650 − 3.41779I
u = −1.206550 − 0.074740I
a = 1.56629 + 0.83850I
b = 0.209734 − 0.919216I
−5.80044 − 1.71298I −12.51650 + 3.41779I
u = −1.184170 + 0.243247I
a = −0.333050 − 1.009550I
b = 0.117217 + 0.500518I
−2.65368 + 4.42550I −6.50791 − 7.50568I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.184170 − 0.243247I
a = −0.333050 + 1.009550I
b = 0.117217 − 0.500518I
−2.65368 − 4.42550I −6.50791 + 7.50568I
u = 0.429683 + 0.631440I
a = −0.286291 − 0.034580I
b = 0.231084 + 0.498413I
−0.69638 − 1.96456I −1.12748 + 4.70329I
u = 0.429683 − 0.631440I
a = −0.286291 + 0.034580I
b = 0.231084 − 0.498413I
−0.69638 + 1.96456I −1.12748 − 4.70329I
u = 1.29906
a = −0.104420
b = 0.740536
−2.76755 −1.63490
u = −1.238710 + 0.461764I
a = −1.08774 + 1.05934I
b = −1.71542 − 0.63051I
4.88950 + 4.61956I −3.15122 − 3.64430I
u = −1.238710 − 0.461764I
a = −1.08774 − 1.05934I
b = −1.71542 + 0.63051I
4.88950 − 4.61956I −3.15122 + 3.64430I
u = 1.236490 + 0.512229I
a = 0.561715 + 1.232250I
b = 2.01617 − 0.03791I
4.35345 + 1.91548I −3.75065 − 1.71492I
u = 1.236490 − 0.512229I
a = 0.561715 − 1.232250I
b = 2.01617 + 0.03791I
4.35345 − 1.91548I −3.75065 + 1.71492I
u = 1.338030 + 0.423130I
a = −0.503580 − 1.252560I
b = −1.76982 − 0.03806I
4.11604 − 5.09421I −3.90084 + 3.07789I
u = 1.338030 − 0.423130I
a = −0.503580 + 1.252560I
b = −1.76982 + 0.03806I
4.11604 + 5.09421I −3.90084 − 3.07789I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.35468 + 0.45041I
a = 0.88989 − 1.31460I
b = 2.05008 + 0.73045I
3.42542 + 12.18300I −4.89577 − 7.01706I
u = −1.35468 − 0.45041I
a = 0.88989 + 1.31460I
b = 2.05008 − 0.73045I
3.42542 − 12.18300I −4.89577 + 7.01706I
u = −0.018893 + 0.524896I
a = −0.530824 − 0.538417I
b = −0.574926 + 0.352337I
0.76347 − 1.52310I 1.12747 + 4.03193I
u = −0.018893 − 0.524896I
a = −0.530824 + 0.538417I
b = −0.574926 − 0.352337I
0.76347 + 1.52310I 1.12747 − 4.03193I
u = −1.46446 + 0.20354I
a = 0.333881 − 0.215100I
b = −0.390847 + 0.079026I
−6.89024 + 4.97150I −3.93501 − 6.51666I
u = −1.46446 − 0.20354I
a = 0.333881 + 0.215100I
b = −0.390847 − 0.079026I
−6.89024 − 4.97150I −3.93501 + 6.51666I
u = 0.194298 + 0.209673I
a = −3.28968 + 1.85114I
b = 0.596766 + 1.022050I
−1.90419 − 0.70187I −5.30439 − 2.49815I
u = 0.194298 − 0.209673I
a = −3.28968 − 1.85114I
b = 0.596766 − 1.022050I
−1.90419 + 0.70187I −5.30439 + 2.49815I
7
II.
I
u
2
= hu
4
+u
3
−u
2
+b −u, u
4
+u
3
−2u
2
+a −u + 1, u
5
+u
4
−2u
3
−u
2
+u −1i
(i) Arc colorings
a
1
=
0
u
a
10
=
1
0
a
9
=
1
−u
2
a
11
=
u
−u
3
+ u
a
3
=
−u
4
− u
3
+ 2u
2
+ u − 1
−u
4
− u
3
+ u
2
+ u
a
8
=
−u
2
+ 1
u
4
− 2u
2
a
4
=
−u
4
− u
3
+ 2u
2
+ u − 1
−u
4
− u
3
+ u
2
+ u
a
2
=
−u
4
− u
3
+ 2u
2
+ u − 1
−u
4
− u
3
+ u
2
+ 2u
a
5
=
0
−u
a
7
=
−u
2
+ 1
u
4
− 2u
2
a
6
=
−u
4
+ u
2
+ 1
u
4
− 2u
2
a
6
=
−u
4
+ u
2
+ 1
u
4
− 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
4
− 5u
3
+ 2u
2
+ 8u − 10
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
5
c
2
, c
4
(u + 1)
5
c
3
, c
7
u
5
c
5
u
5
− 3u
4
+ 4u
3
− u
2
− u + 1
c
6
u
5
− u
4
+ 2u
3
− u
2
+ u − 1
c
8
, c
9
u
5
+ u
4
− 2u
3
− u
2
+ u − 1
c
10
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
c
11
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
5
c
3
, c
7
y
5
c
5
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1
c
6
, c
10
y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1
c
8
, c
9
, c
11
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.21774
a = −0.821196
b = −1.30408
−4.04602 −10.7190
u = 0.309916 + 0.549911I
a = −0.77780 + 1.38013I
b = 0.428550 + 1.039280I
−1.97403 − 1.53058I −6.52924 + 5.40154I
u = 0.309916 − 0.549911I
a = −0.77780 − 1.38013I
b = 0.428550 − 1.039280I
−1.97403 + 1.53058I −6.52924 − 5.40154I
u = −1.41878 + 0.21917I
a = 0.688402 + 0.106340I
b = −0.276511 + 0.728237I
−7.51750 + 4.40083I −11.11126 − 1.16747I
u = −1.41878 − 0.21917I
a = 0.688402 − 0.106340I
b = −0.276511 − 0.728237I
−7.51750 − 4.40083I −11.11126 + 1.16747I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
28
− 6u
27
+ ··· + 5u − 1)
c
2
((u + 1)
5
)(u
28
+ 6u
27
+ ··· + 17u + 1)
c
3
, c
7
u
5
(u
28
+ 3u
27
+ ··· + 128u + 32)
c
4
((u + 1)
5
)(u
28
− 6u
27
+ ··· + 5u − 1)
c
5
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)(u
28
− 6u
27
+ ··· − 3079u − 1609)
c
6
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)(u
28
− 2u
27
+ ··· + u − 1)
c
8
, c
9
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)(u
28
− 2u
27
+ ··· + 5u + 1)
c
10
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)(u
28
− 2u
27
+ ··· + u − 1)
c
11
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)(u
28
− 2u
27
+ ··· + 5u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
5
)(y
28
− 6y
27
+ ··· − 17y + 1)
c
2
((y −1)
5
)(y
28
+ 38y
27
+ ··· − 17y + 1)
c
3
, c
7
y
5
(y
28
− 33y
27
+ ··· − 14848y + 1024)
c
5
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
· (y
28
+ 22y
27
+ ··· + 19658749y + 2588881)
c
6
, c
10
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)(y
28
+ 6y
27
+ ··· + 5y + 1)
c
8
, c
9
, c
11
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)(y
28
− 22y
27
+ ··· + 5y + 1)
13
