10
127
(K10n
16
)
A knot diagram
1
Linearized knot diagam
4 8 5 2 8 10 1 2 6 7
Solving Sequence
1,4
2 5
3,8
7 10 6 9
c
1
c
4
c
3
c
7
c
10
c
6
c
9
c
2
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−3u
15
− 8u
14
+ ··· + 2b − 5, −3u
15
− 6u
14
+ ··· + 2a − 7, u
16
+ 3u
15
+ ··· + 3u + 1i
I
u
2
= hb − a, a
2
− a − 1, u − 1i
* 2 irreducible components of dim
C
= 0, with total 18 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−3u
15
−8u
14
+· · ·+2b−5, −3u
15
−6u
14
+· · ·+2a−7, u
16
+3u
15
+· · ·+3u+1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
3
=
u
3
u
5
− u
3
+ u
a
8
=
3
2
u
15
+ 3u
14
+ ··· + u +
7
2
3
2
u
15
+ 4u
14
+ ··· + 3u +
5
2
a
7
=
3u
15
+ 7u
14
+ ··· + 4u + 6
3
2
u
15
+ 4u
14
+ ··· + 3u +
5
2
a
10
=
u
15
+ 2u
14
+ ··· + 4u + 1
1
2
u
15
+ u
14
+ ··· + 2u +
1
2
a
6
=
−
1
2
u
15
− u
14
+ ··· − 3u +
1
2
−
1
2
u
15
− u
14
+ ··· − u −
1
2
a
9
=
−
3
2
u
15
− 4u
14
+ ··· − u −
9
2
−
1
2
u
15
− u
14
+ ··· +
11
2
u
2
−
3
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
15
− 3u
14
+ u
13
+ 12u
12
+ 10u
11
− 19u
10
− 29u
9
+ 10u
8
+
44u
7
+ 4u
6
− 40u
5
− 18u
4
+ 26u
3
+ 19u
2
− 7u − 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
16
− 3u
15
+ ··· − 3u + 1
c
2
, c
8
u
16
− u
15
+ ··· + 4u + 4
c
3
u
16
+ 5u
15
+ ··· + 15u + 1
c
5
u
16
− 2u
15
+ ··· + 2u + 1
c
6
, c
7
, c
9
c
10
u
16
+ 2u
15
+ ··· − 6u
2
+ 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
16
− 5y
15
+ ··· − 15y + 1
c
2
, c
8
y
16
− 15y
15
+ ··· − 152y + 16
c
3
y
16
+ 15y
15
+ ··· − 75y + 1
c
5
y
16
+ 18y
15
+ ··· − 12y + 1
c
6
, c
7
, c
9
c
10
y
16
− 18y
15
+ ··· − 12y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.817221 + 0.650517I
a = 0.69329 + 1.38874I
b = −1.47026 − 0.07876I
−6.61455 − 2.48847I −10.73866 + 2.85289I
u = 0.817221 − 0.650517I
a = 0.69329 − 1.38874I
b = −1.47026 + 0.07876I
−6.61455 + 2.48847I −10.73866 − 2.85289I
u = 1.09835
a = −0.682687
b = −0.347472
−2.11624 −0.212820
u = −0.616496 + 0.976582I
a = −0.323356 − 0.180239I
b = 1.45750 + 0.22598I
−0.88412 − 2.45544I −7.41928 + 0.95551I
u = −0.616496 − 0.976582I
a = −0.323356 + 0.180239I
b = 1.45750 − 0.22598I
−0.88412 + 2.45544I −7.41928 − 0.95551I
u = −0.839144 + 0.905830I
a = 0.354184 + 0.747930I
b = −0.427794 − 0.712268I
5.17546 + 0.91530I −4.32887 + 0.19716I
u = −0.839144 − 0.905830I
a = 0.354184 − 0.747930I
b = −0.427794 + 0.712268I
5.17546 − 0.91530I −4.32887 − 0.19716I
u = −0.997540 + 0.847971I
a = −0.383254 − 1.181700I
b = −0.593993 + 0.677497I
4.68170 + 5.57131I −5.69073 − 5.47773I
u = −0.997540 − 0.847971I
a = −0.383254 + 1.181700I
b = −0.593993 − 0.677497I
4.68170 − 5.57131I −5.69073 + 5.47773I
u = −0.688577
a = −2.20439
b = −1.64693
−9.85589 −4.30720
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.35209
a = 1.39091
b = 1.48463
−8.27471 −10.1750
u = 0.549818 + 0.327281I
a = −0.426191 − 1.322820I
b = 0.349186 + 0.338218I
−0.629599 − 1.102380I −6.95123 + 6.20216I
u = 0.549818 − 0.327281I
a = −0.426191 + 1.322820I
b = 0.349186 − 0.338218I
−0.629599 + 1.102380I −6.95123 − 6.20216I
u = −1.127720 + 0.779615I
a = 0.38145 + 1.56857I
b = 1.56155 − 0.22278I
−2.44912 + 8.89682I −9.23385 − 5.21727I
u = −1.127720 − 0.779615I
a = 0.38145 − 1.56857I
b = 1.56155 + 0.22278I
−2.44912 − 8.89682I −9.23385 + 5.21727I
u = −0.334148
a = 1.90392
b = 0.757410
−1.34177 −6.57950
6
II. I
u
2
= hb − a, a
2
− a − 1, u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
−1
0
a
3
=
1
1
a
8
=
a
a
a
7
=
2a
a
a
10
=
−2a − 1
−a − 1
a
6
=
−a − 2
−a − 1
a
9
=
a
a
(ii) Obstruction class = 1
(iii) Cusp Shapes = −17
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u − 1)
2
c
2
, c
8
u
2
c
4
(u + 1)
2
c
5
, c
6
, c
7
u
2
+ u − 1
c
9
, c
10
u
2
− u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
(y − 1)
2
c
2
, c
8
y
2
c
5
, c
6
, c
7
c
9
, c
10
y
2
− 3y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −0.618034
b = −0.618034
−2.63189 −17.0000
u = 1.00000
a = 1.61803
b = 1.61803
−10.5276 −17.0000
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
2
)(u
16
− 3u
15
+ ··· − 3u + 1)
c
2
, c
8
u
2
(u
16
− u
15
+ ··· + 4u + 4)
c
3
((u − 1)
2
)(u
16
+ 5u
15
+ ··· + 15u + 1)
c
4
((u + 1)
2
)(u
16
− 3u
15
+ ··· − 3u + 1)
c
5
(u
2
+ u − 1)(u
16
− 2u
15
+ ··· + 2u + 1)
c
6
, c
7
(u
2
+ u − 1)(u
16
+ 2u
15
+ ··· − 6u
2
+ 1)
c
9
, c
10
(u
2
− u − 1)(u
16
+ 2u
15
+ ··· − 6u
2
+ 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y − 1)
2
)(y
16
− 5y
15
+ ··· − 15y + 1)
c
2
, c
8
y
2
(y
16
− 15y
15
+ ··· − 152y + 16)
c
3
((y − 1)
2
)(y
16
+ 15y
15
+ ··· − 75y + 1)
c
5
(y
2
− 3y + 1)(y
16
+ 18y
15
+ ··· − 12y + 1)
c
6
, c
7
, c
9
c
10
(y
2
− 3y + 1)(y
16
− 18y
15
+ ··· − 12y + 1)
12
