11n
82
(K11n
82
)
A knot diagram
1
Linearized knot diagam
6 1 9 7 2 10 11 1 6 4 8
Solving Sequence
1,6
2
3,9
10 7 5 4 8 11
c
1
c
2
c
9
c
6
c
5
c
4
c
8
c
11
c
3
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−30181087379u
14
− 63482369835u
13
+ ··· + 286209328348b − 329512457510,
222459805170u
14
+ 486064128543u
13
+ ··· + 286209328348a + 3182822169919,
u
15
+ 2u
14
+ ··· + 10u − 1i
I
u
2
= hb
2
− 2, a − u − 1, u
2
+ u + 1i
I
u
3
= hb, a + u + 1, u
2
+ u + 1i
* 3 irreducible components of dim
C
= 0, with total 21 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−3.02 × 10
10
u
14
− 6.35 × 10
10
u
13
+ · · · + 2.86 × 10
11
b − 3.30 ×
10
11
, 2.22 × 10
11
u
14
+ 4.86 × 10
11
u
13
+ · · · + 2.86 × 10
11
a + 3.18 ×
10
12
, u
15
+ 2u
14
+ · · · + 10u − 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
−u
2
a
3
=
u
2
+ 1
−u
2
a
9
=
−0.777263u
14
− 1.69828u
13
+ ··· − 20.7871u − 11.1206
0.105451u
14
+ 0.221804u
13
+ ··· + 3.04137u + 1.15130
a
10
=
−0.777263u
14
− 1.69828u
13
+ ··· − 20.7871u − 11.1206
0.0983887u
14
+ 0.190365u
13
+ ··· + 2.38106u + 1.29506
a
7
=
1.21619u
14
+ 2.44009u
13
+ ··· + 22.7698u + 12.4895
−0.0983778u
14
− 0.210858u
13
+ ··· − 1.81717u − 1.45285
a
5
=
−u
u
3
+ u
a
4
=
−1.42791u
14
− 2.95841u
13
+ ··· − 31.0375u − 14.0034
0.156497u
14
+ 0.304479u
13
+ ··· + 2.72370u + 1.69949
a
8
=
−0.671811u
14
− 1.47648u
13
+ ··· − 17.7458u − 9.96931
0.105451u
14
+ 0.221804u
13
+ ··· + 3.04137u + 1.15130
a
11
=
1.29506u
14
+ 2.68850u
13
+ ··· + 27.7863u + 15.3316
−0.150085u
14
− 0.291905u
13
+ ··· − 3.03487u − 1.79788
a
11
=
1.29506u
14
+ 2.68850u
13
+ ··· + 27.7863u + 15.3316
−0.150085u
14
− 0.291905u
13
+ ··· − 3.03487u − 1.79788
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
127706987601
286209328348
u
14
−
268342154629
286209328348
u
13
+ ··· −
2023089158227
286209328348
u −
132460189755
71552332087
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
15
− 2u
14
+ ··· + 10u + 1
c
2
u
15
+ 26u
14
+ ··· + 130u − 1
c
3
u
15
− 27u
13
+ ··· − 294u + 181
c
4
u
15
+ 2u
14
+ ··· − 26u − 29
c
6
, c
9
u
15
− 3u
14
+ ··· + 11u + 7
c
7
, c
8
, c
11
u
15
+ u
14
+ ··· − 12u + 4
c
10
u
15
+ 2u
14
+ ··· + 2u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
15
+ 26y
14
+ ··· + 130y −1
c
2
y
15
− 70y
14
+ ··· + 19426y −1
c
3
y
15
− 54y
14
+ ··· + 330786y −32761
c
4
y
15
+ 18y
14
+ ··· − 5646y −841
c
6
, c
9
y
15
+ y
14
+ ··· − 61y −49
c
7
, c
8
, c
11
y
15
− 25y
14
+ ··· + 112y −16
c
10
y
15
+ 2y
14
+ ··· + 10y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.803098 + 0.748670I
a = 0.646315 + 0.226464I
b = −0.574139 − 0.064590I
0.89161 − 2.83187I −3.17067 + 6.54660I
u = −0.803098 − 0.748670I
a = 0.646315 − 0.226464I
b = −0.574139 + 0.064590I
0.89161 + 2.83187I −3.17067 − 6.54660I
u = 0.289457 + 0.470166I
a = −1.237560 − 0.324915I
b = 0.565034 + 0.450274I
−1.172960 − 0.777601I −6.04145 + 2.77158I
u = 0.289457 − 0.470166I
a = −1.237560 + 0.324915I
b = 0.565034 − 0.450274I
−1.172960 + 0.777601I −6.04145 − 2.77158I
u = −0.231441 + 0.401782I
a = −1.51486 − 1.67785I
b = −0.280610 + 0.385572I
1.80414 − 1.09347I 2.23827 − 2.22165I
u = −0.231441 − 0.401782I
a = −1.51486 + 1.67785I
b = −0.280610 − 0.385572I
1.80414 + 1.09347I 2.23827 + 2.22165I
u = 0.31482 + 1.53771I
a = 0.672510 + 0.519039I
b = −1.36446 − 0.54656I
−7.20728 − 4.71372I −5.60542 + 4.01319I
u = 0.31482 − 1.53771I
a = 0.672510 − 0.519039I
b = −1.36446 + 0.54656I
−7.20728 + 4.71372I −5.60542 − 4.01319I
u = −0.70342 + 1.47150I
a = −0.520487 + 0.299718I
b = 1.49730 − 0.27028I
−6.05090 − 1.57623I −5.49718 + 1.52700I
u = −0.70342 − 1.47150I
a = −0.520487 − 0.299718I
b = 1.49730 + 0.27028I
−6.05090 + 1.57623I −5.49718 − 1.52700I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.0846656
a = −13.3666
b = 1.46557
−3.38897 −2.51440
u = 0.44610 + 2.10573I
a = 0.766270 − 0.413924I
b = −1.95777 + 0.08885I
−19.5760 + 1.0165I −5.21772 + 0.08752I
u = 0.44610 − 2.10573I
a = 0.766270 + 0.413924I
b = −1.95777 − 0.08885I
−19.5760 − 1.0165I −5.21772 − 0.08752I
u = −0.35476 + 2.19036I
a = −0.628878 − 0.529570I
b = 1.88186 + 0.21032I
−18.8095 − 8.6900I −4.44865 + 3.93161I
u = −0.35476 − 2.19036I
a = −0.628878 + 0.529570I
b = 1.88186 − 0.21032I
−18.8095 + 8.6900I −4.44865 − 3.93161I
6
II. I
u
2
= hb
2
− 2, a − u − 1, u
2
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u + 1
a
3
=
−u
u + 1
a
9
=
u + 1
b
a
10
=
u + 1
b + u
a
7
=
−u − 1
−b
a
5
=
−u
u + 1
a
4
=
b − u + 1
−bu − u + 1
a
8
=
b + u + 1
b
a
11
=
−bu − b − 1
−2
a
11
=
−bu − b − 1
−2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
10
(u
2
+ u + 1)
2
c
3
u
4
− 2u
3
+ 5u
2
+ 2u + 1
c
4
u
4
+ 2u
3
+ 5u
2
− 2u + 1
c
5
(u
2
− u + 1)
2
c
6
(u − 1)
4
c
7
, c
8
, c
11
(u
2
− 2)
2
c
9
(u + 1)
4
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
10
(y
2
+ y + 1)
2
c
3
, c
4
y
4
+ 6y
3
+ 35y
2
+ 6y + 1
c
6
, c
9
(y −1)
4
c
7
, c
8
, c
11
(y −2)
4
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 1.41421
−3.28987 − 2.02988I −2.00000 + 3.46410I
u = −0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = −1.41421
−3.28987 − 2.02988I −2.00000 + 3.46410I
u = −0.500000 − 0.866025I
a = 0.500000 − 0.866025I
b = 1.41421
−3.28987 + 2.02988I −2.00000 − 3.46410I
u = −0.500000 − 0.866025I
a = 0.500000 − 0.866025I
b = −1.41421
−3.28987 + 2.02988I −2.00000 − 3.46410I
10
III. I
u
3
= hb, a + u + 1, u
2
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u + 1
a
3
=
−u
u + 1
a
9
=
−u − 1
0
a
10
=
−u − 1
−u
a
7
=
−u − 1
0
a
5
=
−u
u + 1
a
4
=
−u + 1
u + 1
a
8
=
−u − 1
0
a
11
=
1
0
a
11
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u + 2
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
u
2
+ u + 1
c
5
, c
10
u
2
− u + 1
c
6
(u + 1)
2
c
7
, c
8
, c
11
u
2
c
9
(u − 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
10
y
2
+ y + 1
c
6
, c
9
(y −1)
2
c
7
, c
8
, c
11
y
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = −0.500000 − 0.866025I
b = 0
1.64493 − 2.02988I 0. + 3.46410I
u = −0.500000 − 0.866025I
a = −0.500000 + 0.866025I
b = 0
1.64493 + 2.02988I 0. − 3.46410I
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u + 1)
3
)(u
15
− 2u
14
+ ··· + 10u + 1)
c
2
((u
2
+ u + 1)
3
)(u
15
+ 26u
14
+ ··· + 130u − 1)
c
3
(u
2
+ u + 1)(u
4
− 2u
3
+ ··· + 2u + 1)(u
15
− 27u
13
+ ··· − 294u + 181)
c
4
(u
2
+ u + 1)(u
4
+ 2u
3
+ ··· − 2u + 1)(u
15
+ 2u
14
+ ··· − 26u − 29)
c
5
((u
2
− u + 1)
3
)(u
15
− 2u
14
+ ··· + 10u + 1)
c
6
((u − 1)
4
)(u + 1)
2
(u
15
− 3u
14
+ ··· + 11u + 7)
c
7
, c
8
, c
11
u
2
(u
2
− 2)
2
(u
15
+ u
14
+ ··· − 12u + 4)
c
9
((u − 1)
2
)(u + 1)
4
(u
15
− 3u
14
+ ··· + 11u + 7)
c
10
(u
2
− u + 1)(u
2
+ u + 1)
2
(u
15
+ 2u
14
+ ··· + 2u − 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
((y
2
+ y + 1)
3
)(y
15
+ 26y
14
+ ··· + 130y −1)
c
2
((y
2
+ y + 1)
3
)(y
15
− 70y
14
+ ··· + 19426y −1)
c
3
(y
2
+ y + 1)(y
4
+ 6y
3
+ 35y
2
+ 6y + 1)
· (y
15
− 54y
14
+ ··· + 330786y −32761)
c
4
(y
2
+ y + 1)(y
4
+ 6y
3
+ ··· + 6y + 1)(y
15
+ 18y
14
+ ··· − 5646y −841)
c
6
, c
9
((y −1)
6
)(y
15
+ y
14
+ ··· − 61y −49)
c
7
, c
8
, c
11
y
2
(y −2)
4
(y
15
− 25y
14
+ ··· + 112y −16)
c
10
((y
2
+ y + 1)
3
)(y
15
+ 2y
14
+ ··· + 10y −1)
16
