10
162
(K10n
40
)
A knot diagram
1
Linearized knot diagam
5 7 9 8 10 2 1 4 2 7
Solving Sequence
1,5 2,8
4 7 3 6 10 9
c
1
c
4
c
7
c
2
c
6
c
10
c
9
c
3
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hb − u, −u
9
+ u
8
+ 3u
7
− 2u
6
− 8u
5
+ 3u
4
+ 8u
3
+ a − 5u + 2,
u
10
− u
9
− 3u
8
+ 3u
7
+ 7u
6
− 5u
5
− 6u
4
+ 4u
3
+ 3u
2
− 3u + 1i
I
u
2
= h−30u
11
+ 6u
10
+ 76u
9
− 92u
8
+ 34u
7
+ 209u
6
− 204u
5
− 228u
4
+ 66u
3
+ 529u
2
+ 95b + 28u − 416,
− 336u
11
+ 50u
10
+ ··· + 1045a − 3305,
u
12
− u
11
− 2u
10
+ 5u
9
− 4u
8
− 5u
7
+ 11u
6
+ u
5
− 6u
4
− 16u
3
+ 16u
2
+ 10u − 11i
I
u
3
= hb + u, u
2
+ a − 1, u
5
− u
4
− u
3
+ u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 27 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
= hb − u, −u
9
+ u
8
+ · · · + a + 2, u
10
− u
9
+ · · · − 3u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
−u
2
a
8
=
u
9
− u
8
− 3u
7
+ 2u
6
+ 8u
5
− 3u
4
− 8u
3
+ 5u − 2
u
a
4
=
u
9
− u
8
− 2u
7
+ 3u
6
+ 3u
5
− 3u
4
+ u
3
+ 3u
2
− 2u
−u
8
+ u
7
+ 2u
6
− 2u
5
− 4u
4
+ 2u
3
+ u
2
a
7
=
u
9
− u
8
− 3u
7
+ 2u
6
+ 8u
5
− 3u
4
− 8u
3
+ 4u − 2
u
a
3
=
−2u
9
+ u
8
+ 5u
7
− u
6
− 12u
5
− 2u
4
+ 7u
3
+ 2u
2
− 4u + 3
u
9
− u
8
− 2u
7
+ 2u
6
+ 4u
5
− u
4
− u
3
− u
2
a
6
=
u
9
− 4u
7
+ 10u
5
+ u
4
− 9u
3
− u
2
+ 4u − 2
−u
8
+ u
7
+ 3u
6
− 4u
5
− 5u
4
+ 4u
3
+ 3u
2
− 2u + 1
a
10
=
−u
7
+ u
6
+ 2u
5
− 2u
4
− 4u
3
+ u
2
+ u
u
2
a
9
=
−u
9
+ u
8
+ u
7
− u
6
− 2u
5
− u
4
− 3u
3
+ 2u
2
+ u
u
9
− u
8
− 3u
7
+ 4u
6
+ 5u
5
− 5u
4
− 3u
3
+ 4u
2
− u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −5u
9
+ 4u
8
+ 16u
7
−11u
6
−39u
5
+ 15u
4
+ 38u
3
−11u
2
−21u + 14
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
, c
10
u
10
− u
9
− 3u
8
+ 3u
7
+ 7u
6
− 5u
5
− 6u
4
+ 4u
3
+ 3u
2
− 3u + 1
c
2
, c
5
, c
6
u
10
+ 7u
8
− u
7
+ 20u
6
− 6u
5
+ 25u
4
− 8u
3
+ 10u
2
− 2u + 1
c
3
, c
4
, c
8
u
10
+ 5u
9
+ ··· + 18u + 4
c
9
u
10
− 9u
9
+ ··· − 20u + 8
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
10
y
10
− 7y
9
+ ··· − 3y + 1
c
2
, c
5
, c
6
y
10
+ 14y
9
+ ··· + 16y + 1
c
3
, c
4
, c
8
y
10
+ 9y
9
+ ··· + 68y + 16
c
9
y
10
− 5y
9
+ ··· + 496y + 64
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.834890 + 0.288236I
a = −1.16719 − 0.85231I
b = 0.834890 + 0.288236I
−1.336140 − 0.440636I 5.86082 − 0.80149I
u = 0.834890 − 0.288236I
a = −1.16719 + 0.85231I
b = 0.834890 − 0.288236I
−1.336140 + 0.440636I 5.86082 + 0.80149I
u = −0.989389 + 0.553558I
a = −0.604538 + 1.276350I
b = −0.989389 + 0.553558I
7.86026 − 2.34852I 3.25800 + 2.98056I
u = −0.989389 − 0.553558I
a = −0.604538 − 1.276350I
b = −0.989389 − 0.553558I
7.86026 + 2.34852I 3.25800 − 2.98056I
u = −1.093020 + 0.614392I
a = 0.571463 + 0.630872I
b = −1.093020 + 0.614392I
−3.41629 − 5.60135I 2.31471 + 5.03009I
u = −1.093020 − 0.614392I
a = 0.571463 − 0.630872I
b = −1.093020 − 0.614392I
−3.41629 + 5.60135I 2.31471 − 5.03009I
u = 0.329249 + 0.368284I
a = 0.479615 + 1.097570I
b = 0.329249 + 0.368284I
0.201388 + 1.011140I 3.39938 − 6.83831I
u = 0.329249 − 0.368284I
a = 0.479615 − 1.097570I
b = 0.329249 − 0.368284I
0.201388 − 1.011140I 3.39938 + 6.83831I
u = 1.41827 + 0.76674I
a = 0.220652 + 0.935375I
b = 1.41827 + 0.76674I
2.44804 + 10.69340I 5.16708 − 5.74333I
u = 1.41827 − 0.76674I
a = 0.220652 − 0.935375I
b = 1.41827 − 0.76674I
2.44804 − 10.69340I 5.16708 + 5.74333I
5
II. I
u
2
= h−30u
11
+ 6u
10
+ · · · + 95b − 416, −336u
11
+ 50u
10
+ · · · + 1045a −
3305, u
12
− u
11
+ · · · + 10u − 11i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
−u
2
a
8
=
0.321531u
11
− 0.0478469u
10
+ ··· + 0.449761u + 3.16268
0.315789u
11
− 0.0631579u
10
+ ··· − 0.294737u + 4.37895
a
4
=
0.107177u
11
+ 0.0612440u
10
+ ··· − 1.03254u − 0.359809
0.284211u
11
+ 0.126316u
10
+ ··· − u + 1.53684
a
7
=
0.00574163u
11
+ 0.0153110u
10
+ ··· + 0.744498u − 1.21627
0.315789u
11
− 0.0631579u
10
+ ··· − 0.294737u + 4.37895
a
3
=
0.123445u
11
− 0.0181818u
10
+ ··· + 0.0277512u + 2.61340
−0.0210526u
11
− 0.147368u
10
+ ··· + 1.27368u + 0.936842
a
6
=
−0.478469u
11
− 0.00574163u
10
+ ··· + 1.18660u − 5.82679
4
5
u
11
−
1
19
u
10
+ ··· −
2
95
u +
944
95
a
10
=
−0.144498u
11
− 0.129187u
10
+ ··· + 1.24593u − 1.67656
0.284211u
11
+ 0.273684u
10
+ ··· − 2.42105u + 2.07368
a
9
=
−0.123445u
11
+ 0.0181818u
10
+ ··· − 0.0277512u − 2.61340
0.0315789u
11
+ 0.126316u
10
+ ··· − 0.968421u + 0.221053
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
8
19
u
11
−
36
95
u
10
−
88
95
u
9
+
32
19
u
8
−
128
95
u
7
−
44
19
u
6
+
68
19
u
5
+
8
5
u
4
−
316
95
u
3
−
148
19
u
2
+
524
95
u +
706
95
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
, c
10
u
12
− u
11
+ ··· + 10u − 11
c
2
, c
5
, c
6
u
12
+ u
11
+ ··· − 26u − 1
c
3
, c
4
, c
8
(u
3
− u
2
+ 2u − 1)
4
c
9
(u
2
+ u − 1)
6
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
10
y
12
− 5y
11
+ ··· − 452y + 121
c
2
, c
5
, c
6
y
12
+ 7y
11
+ ··· − 680y + 1
c
3
, c
4
, c
8
(y
3
+ 3y
2
+ 2y −1)
4
c
9
(y
2
− 3y + 1)
6
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.968966 + 0.268874I
a = −0.141468 − 1.309750I
b = 0.45076 − 1.47409I
−0.92371 + 2.82812I 5.50976 − 2.97945I
u = 0.968966 − 0.268874I
a = −0.141468 + 1.309750I
b = 0.45076 + 1.47409I
−0.92371 − 2.82812I 5.50976 + 2.97945I
u = −0.610709 + 0.902723I
a = −0.292966 − 0.433049I
b = −0.610709 − 0.902723I
−5.06130 −6 − 1.019511 + 0.10I
u = −0.610709 − 0.902723I
a = −0.292966 + 0.433049I
b = −0.610709 + 0.902723I
−5.06130 −6 − 1.019511 + 0.10I
u = −0.816782
a = −0.697665
b = 1.28332
2.83439 −1.01950
u = 1.008300 + 0.692219I
a = −0.459918 − 0.980637I
b = −1.55059 − 0.23187I
6.97197 + 2.82812I 5.50976 − 2.97945I
u = 1.008300 − 0.692219I
a = −0.459918 + 0.980637I
b = −1.55059 + 0.23187I
6.97197 − 2.82812I 5.50976 + 2.97945I
u = 1.28332
a = 0.444035
b = −0.816782
2.83439 −1.01950
u = 0.45076 + 1.47409I
a = 0.851722 + 0.114540I
b = 0.968966 − 0.268874I
−0.92371 − 2.82812I 5.50976 + 2.97945I
u = 0.45076 − 1.47409I
a = 0.851722 − 0.114540I
b = 0.968966 + 0.268874I
−0.92371 + 2.82812I 5.50976 − 2.97945I
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.55059 + 0.23187I
a = −0.012373 − 0.844848I
b = 1.008300 − 0.692219I
6.97197 − 2.82812I 5.50976 + 2.97945I
u = −1.55059 − 0.23187I
a = −0.012373 + 0.844848I
b = 1.008300 + 0.692219I
6.97197 + 2.82812I 5.50976 − 2.97945I
10
III. I
u
3
= hb + u, u
2
+ a − 1, u
5
− u
4
− u
3
+ u
2
+ 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
−u
2
a
8
=
−u
2
+ 1
−u
a
4
=
u
4
− u
3
− u
2
+ u − 1
u
4
− u
2
+ u
a
7
=
−u
2
+ u + 1
−u
a
3
=
u
4
− u
3
− u
2
+ 2u
−u
2
+ 1
a
6
=
u
4
− u
3
− 2u
2
+ 2u + 1
0
a
10
=
u
3
− u
2
− u + 1
u
2
a
9
=
u
3
− u
−u
4
+ 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
4
− u
3
+ 6u
2
+ 5
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
5
− u
4
− u
3
+ u
2
+ 1
c
2
, c
5
u
5
+ u
3
− u
2
− u + 1
c
3
, c
4
u
5
+ 3u
3
+ 2u + 1
c
6
u
5
+ u
3
+ u
2
− u − 1
c
8
u
5
+ 3u
3
+ 2u − 1
c
9
u
5
− 2u
4
+ u
3
− 2u
2
+ 2u + 1
c
10
u
5
+ u
4
− u
3
− u
2
− 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
10
y
5
− 3y
4
+ 3y
3
+ y
2
− 2y −1
c
2
, c
5
, c
6
y
5
+ 2y
4
− y
3
− 3y
2
+ 3y −1
c
3
, c
4
, c
8
y
5
+ 6y
4
+ 13y
3
+ 12y
2
+ 4y −1
c
9
y
5
− 2y
4
− 3y
3
+ 4y
2
+ 8y −1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.15950
a = −0.344435
b = 1.15950
3.66375 11.0100
u = −0.144591 + 0.695997I
a = 1.46351 + 0.20127I
b = 0.144591 − 0.695997I
−2.68365 + 1.36579I 1.66321 − 1.28728I
u = −0.144591 − 0.695997I
a = 1.46351 − 0.20127I
b = 0.144591 + 0.695997I
−2.68365 − 1.36579I 1.66321 + 1.28728I
u = 1.224340 + 0.455764I
a = −0.291288 − 1.116020I
b = −1.224340 − 0.455764I
9.07644 + 2.10101I 10.83155 − 1.02320I
u = 1.224340 − 0.455764I
a = −0.291288 + 1.116020I
b = −1.224340 + 0.455764I
9.07644 − 2.10101I 10.83155 + 1.02320I
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
7
(u
5
− u
4
− u
3
+ u
2
+ 1)
· (u
10
− u
9
− 3u
8
+ 3u
7
+ 7u
6
− 5u
5
− 6u
4
+ 4u
3
+ 3u
2
− 3u + 1)
· (u
12
− u
11
+ ··· + 10u − 11)
c
2
, c
5
(u
5
+ u
3
− u
2
− u + 1)
· (u
10
+ 7u
8
− u
7
+ 20u
6
− 6u
5
+ 25u
4
− 8u
3
+ 10u
2
− 2u + 1)
· (u
12
+ u
11
+ ··· − 26u − 1)
c
3
, c
4
((u
3
− u
2
+ 2u − 1)
4
)(u
5
+ 3u
3
+ 2u + 1)(u
10
+ 5u
9
+ ··· + 18u + 4)
c
6
(u
5
+ u
3
+ u
2
− u − 1)
· (u
10
+ 7u
8
− u
7
+ 20u
6
− 6u
5
+ 25u
4
− 8u
3
+ 10u
2
− 2u + 1)
· (u
12
+ u
11
+ ··· − 26u − 1)
c
8
((u
3
− u
2
+ 2u − 1)
4
)(u
5
+ 3u
3
+ 2u − 1)(u
10
+ 5u
9
+ ··· + 18u + 4)
c
9
((u
2
+ u − 1)
6
)(u
5
− 2u
4
+ ··· + 2u + 1)(u
10
− 9u
9
+ ··· − 20u + 8)
c
10
(u
5
+ u
4
− u
3
− u
2
− 1)
· (u
10
− u
9
− 3u
8
+ 3u
7
+ 7u
6
− 5u
5
− 6u
4
+ 4u
3
+ 3u
2
− 3u + 1)
· (u
12
− u
11
+ ··· + 10u − 11)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
10
(y
5
− 3y
4
+ 3y
3
+ y
2
− 2y −1)(y
10
− 7y
9
+ ··· − 3y + 1)
· (y
12
− 5y
11
+ ··· − 452y + 121)
c
2
, c
5
, c
6
(y
5
+ 2y
4
− y
3
− 3y
2
+ 3y −1)(y
10
+ 14y
9
+ ··· + 16y + 1)
· (y
12
+ 7y
11
+ ··· − 680y + 1)
c
3
, c
4
, c
8
(y
3
+ 3y
2
+ 2y −1)
4
(y
5
+ 6y
4
+ 13y
3
+ 12y
2
+ 4y −1)
· (y
10
+ 9y
9
+ ··· + 68y + 16)
c
9
(y
2
− 3y + 1)
6
(y
5
− 2y
4
− 3y
3
+ 4y
2
+ 8y −1)
· (y
10
− 5y
9
+ ··· + 496y + 64)
16
