10
134
(K10n
6
)
A knot diagram
1
Linearized knot diagam
8 1 6 3 9 4 10 2 6 3
Solving Sequence
3,6
4
7,10
1 2 9 5 8
c
3
c
6
c
10
c
2
c
9
c
5
c
8
c
1
, c
4
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= h7u
13
+ 19u
12
+ ··· + 4b − 9, 3u
13
+ 9u
12
+ ··· + 2a − 1,
u
14
+ 4u
13
− 2u
12
− 21u
11
+ 2u
10
+ 53u
9
− 13u
8
− 77u
7
+ 38u
6
+ 57u
5
− 37u
4
− 9u
3
+ 12u
2
+ u − 1i
I
u
2
= hb
3
+ b
2
+ 2b + 1, a, u − 1i
* 2 irreducible components of dim
C
= 0, with total 17 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
= h7u
13
+19u
12
+· · ·+4b−9, 3u
13
+9u
12
+· · ·+2a−1, u
14
+4u
13
+· · ·+u−1i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
4
=
1
u
2
a
7
=
−u
−u
3
+ u
a
10
=
−
3
2
u
13
−
9
2
u
12
+ ··· − 8u +
1
2
−
7
4
u
13
−
19
4
u
12
+ ··· − 5u +
9
4
a
1
=
1
4
u
13
+
1
4
u
12
+ ··· − 3u −
7
4
−
7
4
u
13
−
19
4
u
12
+ ··· − 5u +
9
4
a
2
=
−
1
4
u
13
−
3
4
u
12
+ ··· −
3
2
u +
9
4
1
4
u
13
+
3
4
u
12
+ ··· −
1
2
u −
1
4
a
9
=
−
3
2
u
13
−
9
2
u
12
+ ··· − 8u +
1
2
3
4
u
13
+
3
4
u
12
+ ··· − 2u +
3
4
a
5
=
−u
2
+ 1
u
2
a
8
=
−1
−
1
4
u
13
−
3
4
u
12
+ ··· +
3
2
u +
1
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
13
−
17
2
u
12
+ 4u
11
+
93
2
u
10
−
11
2
u
9
−
241
2
u
8
+ 38u
7
+ 172u
6
−
215
2
u
5
− 113u
4
+ 102u
3
−
1
2
u
2
−
51
2
u −
13
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
14
+ 2u
13
+ ··· − 4u − 1
c
2
, c
10
u
14
+ 6u
13
+ ··· + 8u + 1
c
3
, c
6
u
14
− 4u
13
+ ··· − u − 1
c
4
u
14
+ 20u
13
+ ··· + 25u + 1
c
5
, c
9
u
14
+ u
13
+ ··· + 20u + 8
c
7
u
14
− 2u
13
+ ··· − 2u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
14
− 6y
13
+ ··· − 8y + 1
c
2
, c
10
y
14
+ 6y
13
+ ··· − 8y + 1
c
3
, c
6
y
14
− 20y
13
+ ··· − 25y + 1
c
4
y
14
− 48y
13
+ ··· − 153y + 1
c
5
, c
9
y
14
− 21y
13
+ ··· − 144y + 64
c
7
y
14
− 30y
13
+ ··· − 8y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.879051 + 0.720119I
a = 0.739858 − 0.863536I
b = 0.731209 + 1.048470I
−1.98336 − 4.24963I −13.14655 + 5.18533I
u = 0.879051 − 0.720119I
a = 0.739858 + 0.863536I
b = 0.731209 − 1.048470I
−1.98336 + 4.24963I −13.14655 − 5.18533I
u = 1.305050 + 0.250183I
a = 0.247411 − 0.791940I
b = 0.744850 − 0.696808I
−3.10381 + 1.41191I −13.8732 − 3.8151I
u = 1.305050 − 0.250183I
a = 0.247411 + 0.791940I
b = 0.744850 + 0.696808I
−3.10381 − 1.41191I −13.8732 + 3.8151I
u = 0.517778 + 0.426572I
a = −1.035790 + 0.663451I
b = 0.134884 − 0.480979I
−0.660151 − 0.090610I −10.51478 + 0.23122I
u = 0.517778 − 0.426572I
a = −1.035790 − 0.663451I
b = 0.134884 + 0.480979I
−0.660151 + 0.090610I −10.51478 − 0.23122I
u = −0.412302 + 0.084821I
a = −0.201506 + 1.398290I
b = 0.267015 + 1.222640I
2.37413 − 2.69540I −3.68064 + 2.88879I
u = −0.412302 − 0.084821I
a = −0.201506 − 1.398290I
b = 0.267015 − 1.222640I
2.37413 + 2.69540I −3.68064 − 2.88879I
u = 0.303096
a = −1.73095
b = 0.243596
−0.780136 −12.5300
u = −1.72923 + 0.15134I
a = −1.078920 + 0.093411I
b = −0.629782 + 0.920041I
−9.20540 + 2.45847I −11.50081 − 0.42962I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.72923 − 0.15134I
a = −1.078920 − 0.093411I
b = −0.629782 − 0.920041I
−9.20540 − 2.45847I −11.50081 + 0.42962I
u = −1.77359 + 0.25173I
a = 1.114820 − 0.148082I
b = 0.82970 − 1.55473I
−11.19680 + 8.39292I −13.3988 − 4.5885I
u = −1.77359 − 0.25173I
a = 1.114820 + 0.148082I
b = 0.82970 + 1.55473I
−11.19680 − 8.39292I −13.3988 + 4.5885I
u = −1.87660
a = 1.15919
b = 1.60066
−15.8216 −16.2410
6
II. I
u
2
= hb
3
+ b
2
+ 2b + 1, a, u − 1i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
1
a
4
=
1
1
a
7
=
−1
0
a
10
=
0
b
a
1
=
−b
b
a
2
=
b
2
+ 1
−b
2
a
9
=
0
b
a
5
=
0
1
a
8
=
−1
−b
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −b
2
− 3b − 15
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
3
− u
2
+ 1
c
2
u
3
+ u
2
+ 2u + 1
c
3
(u − 1)
3
c
4
, c
6
(u + 1)
3
c
5
, c
9
u
3
c
7
, c
10
u
3
− u
2
+ 2u − 1
c
8
u
3
+ u
2
− 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
3
− y
2
+ 2y −1
c
2
, c
7
, c
10
y
3
+ 3y
2
+ 2y −1
c
3
, c
4
, c
6
(y −1)
3
c
5
, c
9
y
3
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0
b = −0.215080 + 1.307140I
1.37919 + 2.82812I −12.69240 − 3.35914I
u = 1.00000
a = 0
b = −0.215080 − 1.307140I
1.37919 − 2.82812I −12.69240 + 3.35914I
u = 1.00000
a = 0
b = −0.569840
−2.75839 −13.6150
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
3
− u
2
+ 1)(u
14
+ 2u
13
+ ··· − 4u − 1)
c
2
(u
3
+ u
2
+ 2u + 1)(u
14
+ 6u
13
+ ··· + 8u + 1)
c
3
((u − 1)
3
)(u
14
− 4u
13
+ ··· − u − 1)
c
4
((u + 1)
3
)(u
14
+ 20u
13
+ ··· + 25u + 1)
c
5
, c
9
u
3
(u
14
+ u
13
+ ··· + 20u + 8)
c
6
((u + 1)
3
)(u
14
− 4u
13
+ ··· − u − 1)
c
7
(u
3
− u
2
+ 2u − 1)(u
14
− 2u
13
+ ··· − 2u − 1)
c
8
(u
3
+ u
2
− 1)(u
14
+ 2u
13
+ ··· − 4u − 1)
c
10
(u
3
− u
2
+ 2u − 1)(u
14
+ 6u
13
+ ··· + 8u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
8
(y
3
− y
2
+ 2y −1)(y
14
− 6y
13
+ ··· − 8y + 1)
c
2
, c
10
(y
3
+ 3y
2
+ 2y −1)(y
14
+ 6y
13
+ ··· − 8y + 1)
c
3
, c
6
((y −1)
3
)(y
14
− 20y
13
+ ··· − 25y + 1)
c
4
((y −1)
3
)(y
14
− 48y
13
+ ··· − 153y + 1)
c
5
, c
9
y
3
(y
14
− 21y
13
+ ··· − 144y + 64)
c
7
(y
3
+ 3y
2
+ 2y −1)(y
14
− 30y
13
+ ··· − 8y + 1)
12
