11n
27
(K11n
27
)
A knot diagram
1
Linearized knot diagam
4 1 7 2 8 10 4 11 1 6 9
Solving Sequence
8,11
9
1,4
2 10 7 3 6 5
c
8
c
11
c
1
c
9
c
7
c
3
c
6
c
5
c
2
, c
4
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h2085u
15
− 3383u
14
+ ··· + 4922b + 953, −953u
15
+ 5897u
14
+ ··· + 4922a + 34271,
u
16
− 4u
15
+ ··· − 10u + 1i
I
u
2
= hb, −u
2
+ a − u + 1, u
5
+ u
4
− 2u
3
− u
2
+ u − 1i
I
u
3
= hb − a − 1, a
2
+ a − 1, u − 1i
* 3 irreducible components of dim
C
= 0, with total 23 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
= h2085u
15
− 3383u
14
+ · · · + 4922b + 953, −953u
15
+ 5897u
14
+ · · · +
4922a + 34271, u
16
− 4u
15
+ · · · − 10u + 1i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
u
a
9
=
1
u
2
a
1
=
−u
−u
3
+ u
a
4
=
0.193620u
15
− 1.19809u
14
+ ··· − 13.7284u − 6.96282
−0.423608u
15
+ 0.687322u
14
+ ··· − 4.02662u − 0.193620
a
2
=
−0.108899u
15
− 0.139374u
14
+ ··· − 12.8663u − 4.79846
−0.576392u
15
+ 1.31268u
14
+ ··· − 4.97338u + 0.193620
a
10
=
−u
2
+ 1
−u
4
+ 2u
2
a
7
=
0.878505u
15
− 2.34579u
14
+ ··· − 4.05607u − 3.47664
1.00264u
15
− 2.16640u
14
+ ··· + 3.58818u − 0.728769
a
3
=
0.0534336u
15
− 0.366315u
14
+ ··· − 11.4854u − 4.89740
0.340309u
15
− 0.439456u
14
+ ··· − 2.29277u − 0.129825
a
6
=
0.136936u
15
− 0.626981u
14
+ ··· − 5.58228u − 3.01463
−0.576392u
15
+ 1.31268u
14
+ ··· − 4.97338u + 0.193620
a
5
=
−0.439456u
15
+ 0.685697u
14
+ ··· − 10.5557u − 2.82101
−0.576392u
15
+ 1.31268u
14
+ ··· − 4.97338u + 0.193620
a
5
=
−0.439456u
15
+ 0.685697u
14
+ ··· − 10.5557u − 2.82101
−0.576392u
15
+ 1.31268u
14
+ ··· − 4.97338u + 0.193620
(ii) Obstruction class = −1
(iii) Cusp Shapes =
2411
2461
u
15
−
4233
2461
u
14
+ ··· +
28620
2461
u −
24444
2461
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
16
− 7u
15
+ ··· + 3u + 1
c
2
u
16
+ 29u
15
+ ··· + 17u + 1
c
3
, c
7
u
16
+ 2u
15
+ ··· + 72u
2
− 32
c
5
u
16
− 3u
15
+ ··· + u − 1
c
6
, c
10
u
16
− 2u
15
+ ··· − 20u − 4
c
8
, c
9
, c
11
u
16
− 4u
15
+ ··· − 10u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
16
− 29y
15
+ ··· − 17y + 1
c
2
y
16
− 77y
15
+ ··· − 1761y + 1
c
3
, c
7
y
16
− 36y
15
+ ··· − 4608y + 1024
c
5
y
16
− 37y
15
+ ··· − 11y + 1
c
6
, c
10
y
16
+ 18y
15
+ ··· − 168y + 16
c
8
, c
9
, c
11
y
16
− 20y
15
+ ··· − 146y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.852448 + 0.278896I
a = 0.25417 − 1.55665I
b = 0.434441 + 0.614956I
−3.23204 + 0.76102I −14.1078 + 3.1845I
u = −0.852448 − 0.278896I
a = 0.25417 + 1.55665I
b = 0.434441 − 0.614956I
−3.23204 − 0.76102I −14.1078 − 3.1845I
u = −1.11713
a = 1.01314
b = 0.393183
−2.15355 −1.76420
u = 0.727902
a = 1.19391
b = 1.74112
−10.0599 −3.26620
u = −0.665595 + 1.107720I
a = 0.109829 − 0.996426I
b = −2.57424 + 0.30502I
−16.5506 + 3.5813I −12.12116 − 2.15994I
u = −0.665595 − 1.107720I
a = 0.109829 + 0.996426I
b = −2.57424 − 0.30502I
−16.5506 − 3.5813I −12.12116 + 2.15994I
u = −0.374592 + 0.413898I
a = 0.238036 − 0.835224I
b = −0.289006 + 0.411875I
−0.428790 + 1.166930I −5.36023 − 5.57896I
u = −0.374592 − 0.413898I
a = 0.238036 + 0.835224I
b = −0.289006 − 0.411875I
−0.428790 − 1.166930I −5.36023 + 5.57896I
u = 1.49641 + 0.19521I
a = −0.467729 − 0.212663I
b = −0.140592 − 0.934027I
−6.69050 − 3.49798I −9.87558 + 1.25665I
u = 1.49641 − 0.19521I
a = −0.467729 + 0.212663I
b = −0.140592 + 0.934027I
−6.69050 + 3.49798I −9.87558 − 1.25665I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.70971
a = 2.04995
b = 2.70628
−19.0073 −12.9680
u = 1.67172 + 0.41923I
a = −1.56150 + 0.78878I
b = −2.44091 − 0.95623I
15.4222 − 9.3337I −13.42948 + 3.49093I
u = 1.67172 − 0.41923I
a = −1.56150 − 0.78878I
b = −2.44091 + 0.95623I
15.4222 + 9.3337I −13.42948 − 3.49093I
u = 1.73172 + 0.08246I
a = 1.31289 − 0.53629I
b = 1.88369 − 1.19369I
−12.66300 − 2.31460I −13.80105 + 1.17558I
u = 1.73172 − 0.08246I
a = 1.31289 + 0.53629I
b = 1.88369 + 1.19369I
−12.66300 + 2.31460I −13.80105 − 1.17558I
u = 0.0844975
a = −8.02839
b = −0.587345
−1.09573 −8.61160
6
II. I
u
2
= hb, −u
2
+ a − u + 1, u
5
+ u
4
− 2u
3
− u
2
+ u − 1i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
u
a
9
=
1
u
2
a
1
=
−u
−u
3
+ u
a
4
=
u
2
+ u − 1
0
a
2
=
u
2
− 1
−u
3
+ u
a
10
=
−u
2
+ 1
−u
4
+ 2u
2
a
7
=
1
0
a
3
=
u
2
+ u − 1
0
a
6
=
−u
3
+ 2u
u
3
− u
a
5
=
u
u
3
− u
a
5
=
u
u
3
− u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
4
− 3u
3
− 2u
2
+ 8u − 14
7
(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
8
(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
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.21774
a = 1.70062
b = 0
−4.04602 −8.24330
u = 0.309916 + 0.549911I
a = −0.896438 + 0.890762I
b = 0
−1.97403 − 1.53058I −10.50099 + 3.45976I
u = 0.309916 − 0.549911I
a = −0.896438 − 0.890762I
b = 0
−1.97403 + 1.53058I −10.50099 − 3.45976I
u = −1.41878 + 0.21917I
a = −0.453870 − 0.402731I
b = 0
−7.51750 + 4.40083I −14.3774 − 5.8297I
u = −1.41878 − 0.21917I
a = −0.453870 + 0.402731I
b = 0
−7.51750 − 4.40083I −14.3774 + 5.8297I
10
III. I
u
3
= hb − a − 1, a
2
+ a − 1, u − 1i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
1
a
9
=
1
1
a
1
=
−1
0
a
4
=
a
a + 1
a
2
=
−2
−a − 2
a
10
=
0
1
a
7
=
0
−a − 2
a
3
=
a
−a − 2
a
6
=
0
−a − 2
a
5
=
−a − 2
−a − 2
a
5
=
−a − 2
−a − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −21
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
u
2
+ u − 1
c
2
u
2
+ 3u + 1
c
4
, c
7
u
2
− u − 1
c
5
u
2
− 3u + 1
c
6
, c
10
u
2
c
8
, c
9
(u − 1)
2
c
11
(u + 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
y
2
− 3y + 1
c
2
, c
5
y
2
− 7y + 1
c
6
, c
10
y
2
c
8
, c
9
, c
11
(y −1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0.618034
b = 1.61803
−10.5276 −21.0000
u = 1.00000
a = −1.61803
b = −0.618034
−2.63189 −21.0000
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
2
+ u − 1)(u
16
− 7u
15
+ ··· + 3u + 1)
c
2
((u + 1)
5
)(u
2
+ 3u + 1)(u
16
+ 29u
15
+ ··· + 17u + 1)
c
3
u
5
(u
2
+ u − 1)(u
16
+ 2u
15
+ ··· + 72u
2
− 32)
c
4
((u + 1)
5
)(u
2
− u − 1)(u
16
− 7u
15
+ ··· + 3u + 1)
c
5
(u
2
− 3u + 1)(u
5
− 3u
4
+ ··· − u + 1)(u
16
− 3u
15
+ ··· + u − 1)
c
6
u
2
(u
5
− u
4
+ ··· + u − 1)(u
16
− 2u
15
+ ··· − 20u − 4)
c
7
u
5
(u
2
− u − 1)(u
16
+ 2u
15
+ ··· + 72u
2
− 32)
c
8
, c
9
((u − 1)
2
)(u
5
+ u
4
+ ··· + u − 1)(u
16
− 4u
15
+ ··· − 10u + 1)
c
10
u
2
(u
5
+ u
4
+ ··· + u + 1)(u
16
− 2u
15
+ ··· − 20u − 4)
c
11
((u + 1)
2
)(u
5
− u
4
+ ··· + u + 1)(u
16
− 4u
15
+ ··· − 10u + 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
5
)(y
2
− 3y + 1)(y
16
− 29y
15
+ ··· − 17y + 1)
c
2
((y −1)
5
)(y
2
− 7y + 1)(y
16
− 77y
15
+ ··· − 1761y + 1)
c
3
, c
7
y
5
(y
2
− 3y + 1)(y
16
− 36y
15
+ ··· − 4608y + 1024)
c
5
(y
2
− 7y + 1)(y
5
− y
4
+ ··· + 3y −1)(y
16
− 37y
15
+ ··· − 11y + 1)
c
6
, c
10
y
2
(y
5
+ 3y
4
+ ··· − y −1)(y
16
+ 18y
15
+ ··· − 168y + 16)
c
8
, c
9
, c
11
((y −1)
2
)(y
5
− 5y
4
+ ··· − y −1)(y
16
− 20y
15
+ ··· − 146y + 1)
16
