10
5
(K10a
56
)
A knot diagram
1
Linearized knot diagam
7 9 8 3 10 1 2 4 5 6
Solving Sequence
4,9
8 3 5 10 6 2 7 1
c
8
c
3
c
4
c
9
c
5
c
2
c
7
c
1
c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
15
− 3u
13
− u
12
+ 6u
11
+ 2u
10
− 6u
9
− 4u
8
+ 5u
7
+ 3u
6
− 3u
5
− 3u
4
+ 3u
3
+ u
2
− u − 1i
I
u
2
= hu + 1i
* 2 irreducible components of dim
C
= 0, with total 16 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
=
hu
15
−3u
13
−u
12
+6u
11
+2u
10
−6u
9
−4u
8
+5u
7
+3u
6
−3u
5
−3u
4
+3u
3
+u
2
−u−1i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
8
=
1
−u
2
a
3
=
u
−u
3
+ u
a
5
=
−u
3
u
5
− u
3
+ u
a
10
=
u
8
− u
6
+ u
4
+ 1
−u
10
+ 2u
8
− 3u
6
+ 2u
4
− u
2
a
6
=
u
13
− 2u
11
+ 3u
9
− 2u
7
+ 2u
5
− 2u
3
+ u
−u
12
+ 2u
10
+ u
9
− 4u
8
− u
7
+ 3u
6
+ u
5
− 3u
4
+ u
3
+ u
2
− 1
a
2
=
u
3
−u
3
+ u
a
7
=
u
8
− u
6
+ u
4
+ 1
−u
8
+ 2u
6
− 2u
4
a
1
=
−u
13
+ 2u
11
− 3u
9
+ 2u
7
− 2u
5
+ 2u
3
− u
u
13
− 3u
11
+ 5u
9
− 4u
7
+ 2u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −4u
14
+12u
12
+4u
11
−20u
10
−8u
9
+16u
8
+12u
7
−8u
6
−8u
5
+8u
4
+4u
3
−8u
2
−4u+10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
9
, c
10
u
15
+ 2u
14
+ ··· + u + 1
c
2
u
15
− 3u
14
+ ··· + 21u − 5
c
3
, c
8
u
15
− 3u
13
+ ··· − u + 1
c
4
u
15
+ 6u
14
+ ··· + 3u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
9
, c
10
y
15
− 22y
14
+ ··· + 3y −1
c
2
y
15
− 3y
14
+ ··· + 241y −25
c
3
, c
8
y
15
− 6y
14
+ ··· + 3y −1
c
4
y
15
+ 6y
14
+ ··· − 17y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.938536 + 0.379610I
−1.44795 − 1.44538I 0.924314 + 0.710077I
u = 0.938536 − 0.379610I
−1.44795 + 1.44538I 0.924314 − 0.710077I
u = −0.496009 + 0.834142I
17.7129 − 1.8405I 12.03822 + 0.10978I
u = −0.496009 − 0.834142I
17.7129 + 1.8405I 12.03822 − 0.10978I
u = −1.004360 + 0.506467I
−0.48193 + 4.24481I 5.44692 − 7.82705I
u = −1.004360 − 0.506467I
−0.48193 − 4.24481I 5.44692 + 7.82705I
u = 0.483842 + 0.722916I
6.64012 + 1.24233I 12.05713 − 0.59928I
u = 0.483842 − 0.722916I
6.64012 − 1.24233I 12.05713 + 0.59928I
u = 1.16849
11.7390 6.35620
u = 1.053770 + 0.600336I
4.96865 − 6.29824I 9.18075 + 5.76248I
u = 1.053770 − 0.600336I
4.96865 + 6.29824I 9.18075 − 5.76248I
u = −1.090290 + 0.650224I
15.9255 + 7.3739I 9.68126 − 4.56542I
u = −1.090290 − 0.650224I
15.9255 − 7.3739I 9.68126 + 4.56542I
u = −0.469738 + 0.412319I
0.983732 − 0.215278I 10.49328 + 1.71815I
u = −0.469738 − 0.412319I
0.983732 + 0.215278I 10.49328 − 1.71815I
5
II. I
u
2
= hu + 1i
(i) Arc colorings
a
4
=
0
−1
a
9
=
1
0
a
8
=
1
−1
a
3
=
−1
0
a
5
=
1
−1
a
10
=
2
−1
a
6
=
−1
0
a
2
=
−1
0
a
7
=
2
−1
a
1
=
1
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
6
, c
7
, c
8
c
9
, c
10
u − 1
c
2
u
c
4
u + 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
7
c
8
, c
9
, c
10
y −1
c
2
y
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
1.64493 6.00000
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
9
, c
10
(u − 1)(u
15
+ 2u
14
+ ··· + u + 1)
c
2
u(u
15
− 3u
14
+ ··· + 21u − 5)
c
3
, c
8
(u − 1)(u
15
− 3u
13
+ ··· − u + 1)
c
4
(u + 1)(u
15
+ 6u
14
+ ··· + 3u + 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
9
, c
10
(y −1)(y
15
− 22y
14
+ ··· + 3y −1)
c
2
y(y
15
− 3y
14
+ ··· + 241y −25)
c
3
, c
8
(y −1)(y
15
− 6y
14
+ ··· + 3y −1)
c
4
(y −1)(y
15
+ 6y
14
+ ··· − 17y −1)
11
