11n
1
(K11n
1
)
A knot diagram
1
Linearized knot diagam
5 1 9 2 3 9 11 4 1 7 10
Solving Sequence
1,5
2 3
6,9
10 4 8 11 7
c
1
c
2
c
5
c
9
c
4
c
8
c
11
c
7
c
3
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−2u
18
+ 7u
17
+ ··· + 4b + 5, 5u
18
− 18u
17
+ ··· + 4a − 5, u
19
− 4u
18
+ ··· − 12u
2
+ 1i
I
u
2
= h−au + b, a
3
+ a
2
u + a
2
+ 2au − 1, u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 25 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−2u
18
+7u
17
+· · ·+4b+5, 5u
18
−18u
17
+· · ·+4a−5, u
19
−4u
18
+· · ·−12u
2
+1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
−u
2
a
3
=
u
2
+ 1
−u
2
a
6
=
−u
5
− 2u
3
− u
u
5
+ u
3
+ u
a
9
=
−
5
4
u
18
+
9
2
u
17
+ ··· +
11
2
u +
5
4
1
2
u
18
−
7
4
u
17
+ ··· −
5
4
u −
5
4
a
10
=
−0.750000u
18
+ 2.75000u
17
+ ··· − 19.2500u
2
+ 4.25000u
1
2
u
18
−
7
4
u
17
+ ··· −
5
4
u −
5
4
a
4
=
−u
u
3
+ u
a
8
=
−
3
4
u
18
+
5
2
u
17
+ ··· +
9
2
u +
3
4
1
2
u
18
−
9
4
u
17
+ ··· +
1
4
u −
3
4
a
11
=
−
1
4
u
17
+
3
4
u
16
+ ··· +
3
4
u +
7
4
1
4
u
18
− u
17
+ ··· − u −
1
4
a
7
=
1
4
u
18
−
3
4
u
17
+ ··· −
7
4
u − 1
−
1
4
u
18
+ u
17
+ ··· + 2u +
1
4
a
7
=
1
4
u
18
−
3
4
u
17
+ ··· −
7
4
u − 1
−
1
4
u
18
+ u
17
+ ··· + 2u +
1
4
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 3u
18
−
23
2
u
17
+ 43u
16
− 99u
15
+ 208u
14
− 350u
13
+
1049
2
u
12
− 708u
11
+
1657
2
u
10
−
926u
9
+ 884u
8
− 803u
7
+
1299
2
u
6
− 448u
5
+
615
2
u
4
− 136u
3
+
133
2
u
2
− 17u + 4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
19
+ 4u
18
+ ··· + 12u
2
− 1
c
2
u
19
+ 14u
18
+ ··· + 24u − 1
c
3
, c
8
u
19
+ u
18
+ ··· + 160u − 64
c
5
u
19
− 4u
18
+ ··· + u − 2
c
6
u
19
+ 3u
18
+ ··· + 2759u − 937
c
7
, c
10
u
19
− 3u
18
+ ··· + u − 1
c
9
, c
11
u
19
− 3u
18
+ ··· + 11u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
19
+ 14y
18
+ ··· + 24y −1
c
2
y
19
− 14y
18
+ ··· + 612y −1
c
3
, c
8
y
19
+ 35y
18
+ ··· − 15360y −4096
c
5
y
19
− 42y
18
+ ··· + 13y −4
c
6
y
19
+ 89y
18
+ ··· + 15394803y −877969
c
7
, c
10
y
19
− 3y
18
+ ··· + 11y −1
c
9
, c
11
y
19
+ 29y
18
+ ··· + 11y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.578849 + 0.831148I
a = −0.236375 − 0.103692I
b = −0.223009 + 0.136441I
0.57171 − 2.29308I 0.43534 + 5.97155I
u = −0.578849 − 0.831148I
a = −0.236375 + 0.103692I
b = −0.223009 − 0.136441I
0.57171 + 2.29308I 0.43534 − 5.97155I
u = −0.379573 + 1.066790I
a = −0.384017 + 0.248046I
b = 0.118852 + 0.503818I
−1.34954 − 2.72131I 3.10172 + 4.42849I
u = −0.379573 − 1.066790I
a = −0.384017 − 0.248046I
b = 0.118852 − 0.503818I
−1.34954 + 2.72131I 3.10172 − 4.42849I
u = −0.066477 + 0.849480I
a = 0.390573 − 0.852872I
b = −0.698534 − 0.388480I
0.275217 + 0.309939I 6.84413 − 1.19842I
u = −0.066477 − 0.849480I
a = 0.390573 + 0.852872I
b = −0.698534 + 0.388480I
0.275217 − 0.309939I 6.84413 + 1.19842I
u = 1.158440 + 0.036055I
a = −0.02963 − 1.59757I
b = −0.02328 + 1.85176I
−13.35980 − 3.50957I 4.64823 + 2.14006I
u = 1.158440 − 0.036055I
a = −0.02963 + 1.59757I
b = −0.02328 − 1.85176I
−13.35980 + 3.50957I 4.64823 − 2.14006I
u = 0.239802 + 1.225750I
a = 1.218930 − 0.275212I
b = −0.62964 − 1.42811I
−5.41009 + 5.31951I 2.17850 − 4.32462I
u = 0.239802 − 1.225750I
a = 1.218930 + 0.275212I
b = −0.62964 + 1.42811I
−5.41009 − 5.31951I 2.17850 + 4.32462I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.105421 + 1.309260I
a = −0.964760 + 0.228937I
b = 0.401445 + 1.238990I
−6.74492 − 0.62050I 0.784704 + 1.156660I
u = 0.105421 − 1.309260I
a = −0.964760 − 0.228937I
b = 0.401445 − 1.238990I
−6.74492 + 0.62050I 0.784704 − 1.156660I
u = 0.493476 + 0.148245I
a = −0.12816 − 2.14991I
b = −0.255468 + 1.079930I
−2.12600 − 2.49879I 4.77209 + 3.99040I
u = 0.493476 − 0.148245I
a = −0.12816 + 2.14991I
b = −0.255468 − 1.079930I
−2.12600 + 2.49879I 4.77209 − 3.99040I
u = 0.58478 + 1.39635I
a = 1.225590 + 0.379250I
b = −0.18714 − 1.93313I
−17.6117 + 9.7005I 2.62109 − 4.88323I
u = 0.58478 − 1.39635I
a = 1.225590 − 0.379250I
b = −0.18714 + 1.93313I
−17.6117 − 9.7005I 2.62109 + 4.88323I
u = 0.54417 + 1.43391I
a = −1.179380 − 0.320996I
b = 0.18150 + 1.86580I
−17.9983 + 2.5634I 2.10495 − 0.56524I
u = 0.54417 − 1.43391I
a = −1.179380 + 0.320996I
b = 0.18150 − 1.86580I
−17.9983 − 2.5634I 2.10495 + 0.56524I
u = −0.202383
a = −1.82553
b = −0.369456
0.846922 12.0190
6
II. I
u
2
= h−au + b, a
3
+ a
2
u + a
2
+ 2au − 1, u
2
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u + 1
a
3
=
−u
u + 1
a
6
=
−1
0
a
9
=
a
au
a
10
=
au + a
au
a
4
=
−u
u + 1
a
8
=
a
au
a
11
=
−a
2
+ 1
−a
2
u − a
2
a
7
=
a
2
u − 1
−a
2
u − a
2
a
7
=
a
2
u − 1
−a
2
u − a
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −5a
2
u − 6a
2
− 5au − a + u + 14
7
(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
8
(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
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = −1.239560 + 0.467306I
b = 0.215080 − 1.307140I
−3.02413 + 0.79824I 2.74410 − 0.29766I
u = −0.500000 + 0.866025I
a = 1.024480 − 0.839835I
b = 0.215080 + 1.307140I
−3.02413 − 4.85801I 4.03424 + 5.28153I
u = −0.500000 + 0.866025I
a = −0.284920 − 0.493496I
b = 0.569840
1.11345 − 2.02988I 12.72167 + 1.07831I
u = −0.500000 − 0.866025I
a = 1.024480 + 0.839835I
b = 0.215080 − 1.307140I
−3.02413 − 0.79824I 2.74410 + 0.29766I
u = −0.500000 − 0.866025I
a = −1.239560 − 0.467306I
b = 0.215080 + 1.307140I
−3.02413 + 4.85801I 4.03424 − 5.28153I
u = −0.500000 − 0.866025I
a = −0.284920 + 0.493496I
b = 0.569840
1.11345 + 2.02988I 12.72167 − 1.07831I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u + 1)
3
)(u
19
+ 4u
18
+ ··· + 12u
2
− 1)
c
2
((u
2
+ u + 1)
3
)(u
19
+ 14u
18
+ ··· + 24u − 1)
c
3
, c
8
u
6
(u
19
+ u
18
+ ··· + 160u − 64)
c
4
((u
2
− u + 1)
3
)(u
19
+ 4u
18
+ ··· + 12u
2
− 1)
c
5
((u
2
+ u + 1)
3
)(u
19
− 4u
18
+ ··· + u − 2)
c
6
((u
3
+ u
2
+ 2u + 1)
2
)(u
19
+ 3u
18
+ ··· + 2759u − 937)
c
7
((u
3
− u
2
+ 1)
2
)(u
19
− 3u
18
+ ··· + u − 1)
c
9
((u
3
+ u
2
+ 2u + 1)
2
)(u
19
− 3u
18
+ ··· + 11u − 1)
c
10
((u
3
+ u
2
− 1)
2
)(u
19
− 3u
18
+ ··· + u − 1)
c
11
((u
3
− u
2
+ 2u − 1)
2
)(u
19
− 3u
18
+ ··· + 11u − 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y
2
+ y + 1)
3
)(y
19
+ 14y
18
+ ··· + 24y −1)
c
2
((y
2
+ y + 1)
3
)(y
19
− 14y
18
+ ··· + 612y −1)
c
3
, c
8
y
6
(y
19
+ 35y
18
+ ··· − 15360y −4096)
c
5
((y
2
+ y + 1)
3
)(y
19
− 42y
18
+ ··· + 13y −4)
c
6
((y
3
+ 3y
2
+ 2y −1)
2
)(y
19
+ 89y
18
+ ··· + 1.53948 × 10
7
y −877969)
c
7
, c
10
((y
3
− y
2
+ 2y −1)
2
)(y
19
− 3y
18
+ ··· + 11y −1)
c
9
, c
11
((y
3
+ 3y
2
+ 2y −1)
2
)(y
19
+ 29y
18
+ ··· + 11y −1)
12
