11n
6
(K11n
6
)
A knot diagram
1
Linearized knot diagam
5 1 7 2 3 10 4 11 7 1 9
Solving Sequence
1,5
2 3
6,10
7 4 8 9 11
c
1
c
2
c
5
c
6
c
4
c
7
c
9
c
11
c
3
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−4445u
18
+ 25166u
17
+ ··· + 157228b − 130422,
102053u
18
− 516382u
17
+ ··· + 157228a + 1635396, u
19
− 5u
18
+ ··· + 18u − 1i
I
u
2
= h3a
2
u + a
2
− 4au + 7b + a + u − 9, a
3
+ a
2
u − a
2
+ 3au + 2a − 5u − 5, u
2
+ u + 1i
I
u
3
= hb − 1, u
4
− u
3
+ 2u
2
+ a − u + 1, u
5
− u
4
+ 2u
3
− u
2
+ u − 1i
* 3 irreducible components of dim
C
= 0, with total 30 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−4445u
18
+2.52×10
4
u
17
+· · ·+1.57×10
5
b−1.30×10
5
, 1.02×10
5
u
18
−
5.16 × 10
5
u
17
+ · · · + 1.57 × 10
5
a + 1.64 × 10
6
, u
19
− 5u
18
+ · · · + 18u − 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
10
=
−0.649077u
18
+ 3.28429u
17
+ ··· + 38.5711u − 10.4014
0.0282710u
18
− 0.160061u
17
+ ··· − 2.36210u + 0.829509
a
7
=
0.638684u
18
− 2.98006u
17
+ ··· − 23.7743u + 5.48846
−0.213359u
18
+ 1.10318u
17
+ ··· + 6.00785u − 0.638684
a
4
=
−u
u
3
+ u
a
8
=
0.586798u
18
− 2.76808u
17
+ ··· − 23.8489u + 5.47099
−0.195678u
18
+ 0.900151u
17
+ ··· + 5.28026u − 0.573772
a
9
=
−0.987191u
18
+ 4.82270u
17
+ ··· + 47.6110u − 13.0026
0.180432u
18
− 0.834985u
17
+ ··· − 5.84697u + 1.16762
a
11
=
−0.620805u
18
+ 3.12423u
17
+ ··· + 36.2090u − 9.57192
0.0282710u
18
− 0.160061u
17
+ ··· − 2.36210u + 0.829509
a
11
=
−0.620805u
18
+ 3.12423u
17
+ ··· + 36.2090u − 9.57192
0.0282710u
18
− 0.160061u
17
+ ··· − 2.36210u + 0.829509
(ii) Obstruction class = −1
(iii) Cusp Shapes =
48467
78614
u
18
−
220275
78614
u
17
+ ··· −
2697747
78614
u +
420630
39307
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
19
+ 5u
18
+ ··· + 18u + 1
c
2
u
19
+ 15u
18
+ ··· + 208u − 1
c
3
, c
7
u
19
+ 2u
18
+ ··· + 96u − 64
c
5
u
19
− 5u
18
+ ··· + 854u + 49
c
6
, c
9
u
19
+ 3u
18
+ ··· − 88u
2
− 32
c
8
, c
11
u
19
+ 8u
18
+ ··· − 15u − 1
c
10
u
19
+ 2u
18
+ ··· + 69u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
19
+ 15y
18
+ ··· + 208y −1
c
2
y
19
− 17y
18
+ ··· + 45036y −1
c
3
, c
7
y
19
− 40y
18
+ ··· + 17408y −4096
c
5
y
19
− 49y
18
+ ··· + 501760y −2401
c
6
, c
9
y
19
+ 39y
18
+ ··· − 5632y −1024
c
8
, c
11
y
19
+ 2y
18
+ ··· + 69y −1
c
10
y
19
+ 54y
18
+ ··· − 2699y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.459827 + 0.896977I
a = 2.21570 + 0.93164I
b = 0.888580 + 0.149996I
1.34528 − 1.87445I 29.9213 + 13.6703I
u = −0.459827 − 0.896977I
a = 2.21570 − 0.93164I
b = 0.888580 − 0.149996I
1.34528 + 1.87445I 29.9213 − 13.6703I
u = −0.351109 + 0.745933I
a = 0.875169 − 0.043573I
b = 0.0093474 + 0.0139592I
−0.22305 − 1.43330I −1.61645 + 4.92513I
u = −0.351109 − 0.745933I
a = 0.875169 + 0.043573I
b = 0.0093474 − 0.0139592I
−0.22305 + 1.43330I −1.61645 − 4.92513I
u = 0.389305 + 1.111150I
a = −0.862837 − 0.197630I
b = 0.668732 + 0.907469I
−4.23087 + 5.58158I −3.54918 − 7.60584I
u = 0.389305 − 1.111150I
a = −0.862837 + 0.197630I
b = 0.668732 − 0.907469I
−4.23087 − 5.58158I −3.54918 + 7.60584I
u = −0.265172 + 1.190510I
a = 0.759357 + 0.604707I
b = −0.460697 + 0.797639I
−1.40496 − 0.89543I −0.827466 + 0.267848I
u = −0.265172 − 1.190510I
a = 0.759357 − 0.604707I
b = −0.460697 − 0.797639I
−1.40496 + 0.89543I −0.827466 − 0.267848I
u = 1.236570 + 0.125353I
a = −0.08471 + 3.83451I
b = 0.07758 − 3.21706I
−14.6816 − 4.7277I 1.60029 + 1.79597I
u = 1.236570 − 0.125353I
a = −0.08471 − 3.83451I
b = 0.07758 + 3.21706I
−14.6816 + 4.7277I 1.60029 − 1.79597I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.578208 + 0.363922I
a = 1.06872 + 1.66313I
b = −0.039673 − 0.833778I
−2.02190 − 1.85032I 0.40467 + 3.82422I
u = 0.578208 − 0.363922I
a = 1.06872 − 1.66313I
b = −0.039673 + 0.833778I
−2.02190 + 1.85032I 0.40467 − 3.82422I
u = 0.12128 + 1.49762I
a = 1.41899 + 0.08782I
b = −2.23717 − 0.89116I
−8.30737 + 0.41800I −1.92088 − 0.17258I
u = 0.12128 − 1.49762I
a = 1.41899 − 0.08782I
b = −2.23717 + 0.89116I
−8.30737 − 0.41800I −1.92088 + 0.17258I
u = 0.66518 + 1.37702I
a = −2.46188 − 1.46194I
b = 0.83728 + 3.06795I
−18.5579 + 11.4070I 0.02962 − 4.80086I
u = 0.66518 − 1.37702I
a = −2.46188 + 1.46194I
b = 0.83728 − 3.06795I
−18.5579 − 11.4070I 0.02962 + 4.80086I
u = 0.55124 + 1.54379I
a = 2.12819 + 1.66552I
b = −1.08511 − 3.58320I
19.5193 + 1.7269I −0.878629 − 0.706920I
u = 0.55124 − 1.54379I
a = 2.12819 − 1.66552I
b = −1.08511 + 3.58320I
19.5193 − 1.7269I −0.878629 + 0.706920I
u = 0.0686432
a = −8.11341
b = 0.682273
1.19847 8.67340
6
II.
I
u
2
= h3a
2
u+a
2
−4au+7b+a+u−9, a
3
+a
2
u−a
2
+ 3 au+2a−5u−5, 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
10
=
a
−
3
7
a
2
u +
4
7
au + ··· −
1
7
a +
9
7
a
7
=
−
1
7
a
2
u −
1
7
au + ··· +
2
7
a +
3
7
−
4
7
a
2
u +
3
7
au + ··· +
1
7
a +
5
7
a
4
=
−u
u + 1
a
8
=
−
1
7
a
2
u −
1
7
au + ··· +
2
7
a +
3
7
−
4
7
a
2
u +
3
7
au + ··· +
1
7
a +
5
7
a
9
=
−
3
7
a
2
u −
3
7
au + ··· −
1
7
a +
16
7
−
1
7
a
2
u −
1
7
au + ··· +
2
7
a +
3
7
a
11
=
−
3
7
a
2
u +
4
7
au + ··· +
6
7
a +
9
7
−
3
7
a
2
u +
4
7
au + ··· −
1
7
a +
9
7
a
11
=
−
3
7
a
2
u +
4
7
au + ··· +
6
7
a +
9
7
−
3
7
a
2
u +
4
7
au + ··· −
1
7
a +
9
7
(ii) Obstruction class = 1
(iii) Cusp Shapes =
17
7
a
2
u +
29
7
a
2
−
4
7
au −
6
7
a +
99
7
u +
12
7
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
7
u
6
c
4
(u
2
− u + 1)
3
c
6
, c
10
(u
3
+ u
2
+ 2u + 1)
2
c
8
(u
3
− u
2
+ 1)
2
c
9
(u
3
− u
2
+ 2u − 1)
2
c
11
(u
3
+ u
2
− 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
7
y
6
c
6
, c
9
, c
10
(y
3
+ 3y
2
+ 2y −1)
2
c
8
, c
11
(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 = 1.46996 − 0.49350I
b = 0.569840
1.11345 − 2.02988I 2.22484 + 11.58609I
u = −0.500000 + 0.866025I
a = −1.11700 + 1.21217I
b = 0.215080 − 1.307140I
−3.02413 − 4.85801I 0.92725 + 3.71146I
u = −0.500000 + 0.866025I
a = 1.14704 − 1.58470I
b = 0.215080 + 1.307140I
−3.02413 + 0.79824I −2.65209 − 0.57512I
u = −0.500000 − 0.866025I
a = 1.46996 + 0.49350I
b = 0.569840
1.11345 + 2.02988I 2.22484 − 11.58609I
u = −0.500000 − 0.866025I
a = −1.11700 − 1.21217I
b = 0.215080 + 1.307140I
−3.02413 + 4.85801I 0.92725 − 3.71146I
u = −0.500000 − 0.866025I
a = 1.14704 + 1.58470I
b = 0.215080 − 1.307140I
−3.02413 − 0.79824I −2.65209 + 0.57512I
10
III. I
u
3
= hb − 1, u
4
− u
3
+ 2u
2
+ a − u + 1, u
5
− u
4
+ 2u
3
− u
2
+ u − 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
4
− u
2
− 1
u
4
− u
3
+ u
2
+ 1
a
10
=
−u
4
+ u
3
− 2u
2
+ u − 1
1
a
7
=
−u
4
− u
2
− 1
u
4
− u
3
+ u
2
+ 1
a
4
=
−u
u
3
+ u
a
8
=
−1
0
a
9
=
−u
4
+ u
3
− 2u
2
+ u − 1
1
a
11
=
−u
4
+ u
3
− 2u
2
+ u
1
a
11
=
−u
4
+ u
3
− 2u
2
+ u
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
4
+ 5u
3
− 4u
2
+ 3
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
5
− u
4
+ 2u
3
− u
2
+ u − 1
c
2
u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1
c
3
u
5
+ u
4
− 2u
3
− u
2
+ u − 1
c
4
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
c
5
, c
7
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
c
6
, c
9
u
5
c
8
, c
10
(u + 1)
5
c
11
(u − 1)
5
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1
c
2
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1
c
3
, c
5
, c
7
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
c
6
, c
9
y
5
c
8
, c
10
, c
11
(y −1)
5
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.339110 + 0.822375I
a = 0.428550 + 1.039280I
b = 1.00000
1.31583 − 1.53058I 8.47842 − 1.00973I
u = −0.339110 − 0.822375I
a = 0.428550 − 1.039280I
b = 1.00000
1.31583 + 1.53058I 8.47842 + 1.00973I
u = 0.766826
a = −1.30408
b = 1.00000
−0.756147 1.86520
u = 0.455697 + 1.200150I
a = −0.276511 + 0.728237I
b = 1.00000
−4.22763 + 4.40083I −2.41100 − 1.19010I
u = 0.455697 − 1.200150I
a = −0.276511 − 0.728237I
b = 1.00000
−4.22763 − 4.40083I −2.41100 + 1.19010I
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u + 1)
3
)(u
5
− u
4
+ ··· + u − 1)(u
19
+ 5u
18
+ ··· + 18u + 1)
c
2
(u
2
+ u + 1)
3
(u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)
· (u
19
+ 15u
18
+ ··· + 208u − 1)
c
3
u
6
(u
5
+ u
4
+ ··· + u − 1)(u
19
+ 2u
18
+ ··· + 96u − 64)
c
4
((u
2
− u + 1)
3
)(u
5
+ u
4
+ ··· + u + 1)(u
19
+ 5u
18
+ ··· + 18u + 1)
c
5
((u
2
+ u + 1)
3
)(u
5
− u
4
+ ··· + u + 1)(u
19
− 5u
18
+ ··· + 854u + 49)
c
6
u
5
(u
3
+ u
2
+ 2u + 1)
2
(u
19
+ 3u
18
+ ··· − 88u
2
− 32)
c
7
u
6
(u
5
− u
4
+ ··· + u + 1)(u
19
+ 2u
18
+ ··· + 96u − 64)
c
8
((u + 1)
5
)(u
3
− u
2
+ 1)
2
(u
19
+ 8u
18
+ ··· − 15u − 1)
c
9
u
5
(u
3
− u
2
+ 2u − 1)
2
(u
19
+ 3u
18
+ ··· − 88u
2
− 32)
c
10
((u + 1)
5
)(u
3
+ u
2
+ 2u + 1)
2
(u
19
+ 2u
18
+ ··· + 69u − 1)
c
11
((u − 1)
5
)(u
3
+ u
2
− 1)
2
(u
19
+ 8u
18
+ ··· − 15u − 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
2
+ y + 1)
3
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
· (y
19
+ 15y
18
+ ··· + 208y −1)
c
2
(y
2
+ y + 1)
3
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
· (y
19
− 17y
18
+ ··· + 45036y −1)
c
3
, c
7
y
6
(y
5
− 5y
4
+ ··· − y −1)(y
19
− 40y
18
+ ··· + 17408y −4096)
c
5
(y
2
+ y + 1)
3
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
· (y
19
− 49y
18
+ ··· + 501760y −2401)
c
6
, c
9
y
5
(y
3
+ 3y
2
+ 2y −1)
2
(y
19
+ 39y
18
+ ··· − 5632y −1024)
c
8
, c
11
((y −1)
5
)(y
3
− y
2
+ 2y −1)
2
(y
19
+ 2y
18
+ ··· + 69y −1)
c
10
((y −1)
5
)(y
3
+ 3y
2
+ 2y −1)
2
(y
19
+ 54y
18
+ ··· − 2699y −1)
16
