10
144
(K10n
28
)
A knot diagram
1
Linearized knot diagam
5 8 7 6 2 10 3 4 6 7
Solving Sequence
3,7
4 8
2,10
6 5 1 9
c
3
c
7
c
2
c
6
c
5
c
1
c
9
c
4
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
8
− 2u
7
+ 5u
6
− 6u
5
+ 7u
4
− 5u
3
+ 3u
2
+ b − 2u + 1,
u
9
− u
8
+ 5u
7
− 4u
6
+ 8u
5
− 5u
4
+ 3u
3
− 2u
2
+ 2a − 2u,
u
10
− 3u
9
+ 9u
8
− 16u
7
+ 24u
6
− 27u
5
+ 23u
4
− 16u
3
+ 8u
2
− 4u + 2i
I
u
2
= hu
4
a + u
3
a − u
4
+ 2u
2
a − u
3
+ au − 2u
2
+ b − a − u, −u
5
− u
4
+ u
2
a − 4u
3
+ a
2
− 3u
2
+ a − 4u − 2,
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
I
u
3
= hb + 2u − 1, 2a + u, u
2
+ 2i
I
v
1
= ha, b + 1, v + 1i
* 4 irreducible components of dim
C
= 0, with total 25 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
8
− 2u
7
+ · · · + b + 1, u
9
− u
8
+ · · · + 2a − 2u, u
10
− 3u
9
+ · · · − 4u + 2i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
−u
2
a
8
=
u
u
a
2
=
u
2
+ 1
u
2
a
10
=
−
1
2
u
9
+
1
2
u
8
+ ··· + u
2
+ u
−u
8
+ 2u
7
− 5u
6
+ 6u
5
− 7u
4
+ 5u
3
− 3u
2
+ 2u − 1
a
6
=
−
1
2
u
9
+
3
2
u
8
+ ··· − 2u + 2
u
8
− 2u
7
+ 5u
6
− 7u
5
+ 7u
4
− 6u
3
+ 2u
2
− u + 1
a
5
=
1
2
u
9
−
3
2
u
8
+ ··· + 2u − 1
u
7
− 2u
6
+ 4u
5
− 5u
4
+ 4u
3
− 3u
2
+ u − 1
a
1
=
1
2
u
9
−
1
2
u
8
+ ··· − u
2
− u
u
9
− 2u
8
+ 6u
7
− 8u
6
+ 11u
5
− 9u
4
+ 6u
3
− 3u
2
+ u − 1
a
9
=
u
3
+ 2u
−u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
9
+ 6u
8
− 18u
7
+ 26u
6
− 38u
5
+ 28u
4
− 18u
3
+ 4u
2
+ 2u
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
9
, c
10
u
10
+ u
9
− u
8
− 2u
7
+ 3u
6
+ 4u
5
− 4u
3
+ u + 1
c
2
, c
3
, c
7
u
10
+ 3u
9
+ 9u
8
+ 16u
7
+ 24u
6
+ 27u
5
+ 23u
4
+ 16u
3
+ 8u
2
+ 4u + 2
c
4
u
10
+ 3u
9
+ 11u
8
+ 18u
7
+ 33u
6
+ 32u
5
+ 34u
4
+ 18u
3
+ 8u
2
+ u + 1
c
8
u
10
− 3u
9
+ 3u
8
− 8u
6
+ 17u
5
+ 17u
4
− 58u
3
+ 48u
2
− 16u + 10
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
9
, c
10
y
10
− 3y
9
+ 11y
8
− 18y
7
+ 33y
6
− 32y
5
+ 34y
4
− 18y
3
+ 8y
2
− y + 1
c
2
, c
3
, c
7
y
10
+ 9y
9
+ ··· + 16y + 4
c
4
y
10
+ 13y
9
+ ··· + 15y + 1
c
8
y
10
− 3y
9
+ ··· + 704y + 100
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.880108 + 0.189454I
a = 0.91534 − 1.10455I
b = −0.474443 − 0.824770I
3.61170 + 6.23908I −1.40880 − 5.42921I
u = 0.880108 − 0.189454I
a = 0.91534 + 1.10455I
b = −0.474443 + 0.824770I
3.61170 − 6.23908I −1.40880 + 5.42921I
u = 0.453532 + 1.055340I
a = −0.931418 + 0.352143I
b = −0.363378 + 0.743264I
0.94791 − 1.45588I −3.02190 + 1.71983I
u = 0.453532 − 1.055340I
a = −0.931418 − 0.352143I
b = −0.363378 − 0.743264I
0.94791 + 1.45588I −3.02190 − 1.71983I
u = −0.246909 + 0.578012I
a = −0.485195 + 0.815685I
b = −0.178372 + 0.508008I
−0.143235 − 1.179710I −1.77268 + 5.86187I
u = −0.246909 − 0.578012I
a = −0.485195 − 0.815685I
b = −0.178372 − 0.508008I
−0.143235 + 1.179710I −1.77268 − 5.86187I
u = 0.38382 + 1.39954I
a = 0.279302 + 0.892816I
b = 0.00363 + 3.07096I
−1.41581 + 10.79660I −5.84814 − 6.97307I
u = 0.38382 − 1.39954I
a = 0.279302 − 0.892816I
b = 0.00363 − 3.07096I
−1.41581 − 10.79660I −5.84814 + 6.97307I
u = 0.02945 + 1.49900I
a = 0.221969 − 0.511453I
b = 0.51256 − 2.49603I
−7.11290 − 1.33139I −5.94848 + 5.33149I
u = 0.02945 − 1.49900I
a = 0.221969 + 0.511453I
b = 0.51256 + 2.49603I
−7.11290 + 1.33139I −5.94848 − 5.33149I
5
II. I
u
2
= hu
4
a − u
4
+ · · · + b − a, −u
5
− u
4
+ u
2
a − 4u
3
+ a
2
− 3u
2
+ a − 4u −
2, u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
−u
2
a
8
=
u
u
a
2
=
u
2
+ 1
u
2
a
10
=
a
−u
4
a − u
3
a + u
4
− 2u
2
a + u
3
− au + 2u
2
+ a + u
a
6
=
−u
3
a + u
4
+ u
3
− au + 2u
2
+ u + 1
u
5
a + u
4
a + u
3
a + u
4
+ u
2
a + u
3
+ u
2
+ u
a
5
=
u
5
+ u
4
+ 3u
3
+ 2u
2
+ 2u + 1
u
5
a + u
4
a + u
5
+ 2u
3
a + u
4
+ u
2
a + 2u
3
+ u
2
+ u
a
1
=
−a
u
4
a + u
3
a − u
4
+ u
2
a − u
3
+ au − 2u
2
− a − u
a
9
=
u
3
+ 2u
−u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
4
+ 4u
3
+ 8u
2
+ 4u − 2
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
9
, c
10
u
12
+ u
11
− 2u
10
− 4u
9
+ u
8
+ 5u
7
− u
6
− 7u
5
− u
4
+ 9u
3
+ 6u
2
− 2u − 3
c
2
, c
3
, c
7
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)
2
c
4
u
12
+ 5u
11
+ ··· + 40u + 9
c
8
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
9
, c
10
y
12
− 5y
11
+ ··· − 40y + 9
c
2
, c
3
, c
7
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1)
2
c
4
y
12
+ 3y
11
+ ··· − 196y + 81
c
8
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.873214
a = −0.881252 + 1.009130I
b = 0.186123 + 0.436575I
4.37022 0.269500
u = −0.873214
a = −0.881252 − 1.009130I
b = 0.186123 − 0.436575I
4.37022 0.269500
u = 0.138835 + 1.234450I
a = 0.185128 − 1.031140I
b = 0.02999 − 3.18010I
−6.25011 + 1.97241I −7.42428 − 3.68478I
u = 0.138835 + 1.234450I
a = 0.319451 + 0.688377I
b = −1.23755 + 0.99495I
−6.25011 + 1.97241I −7.42428 − 3.68478I
u = 0.138835 − 1.234450I
a = 0.185128 + 1.031140I
b = 0.02999 + 3.18010I
−6.25011 − 1.97241I −7.42428 + 3.68478I
u = 0.138835 − 1.234450I
a = 0.319451 − 0.688377I
b = −1.23755 − 0.99495I
−6.25011 − 1.97241I −7.42428 + 3.68478I
u = −0.408802 + 1.276380I
a = −0.340041 + 0.871835I
b = −0.11686 + 2.25474I
0.40571 − 4.59213I −3.41886 + 3.20482I
u = −0.408802 + 1.276380I
a = 0.802059 + 0.171737I
b = 0.869443 + 0.391246I
0.40571 − 4.59213I −3.41886 + 3.20482I
u = −0.408802 − 1.276380I
a = −0.340041 − 0.871835I
b = −0.11686 − 2.25474I
0.40571 + 4.59213I −3.41886 − 3.20482I
u = −0.408802 − 1.276380I
a = 0.802059 − 0.171737I
b = 0.869443 − 0.391246I
0.40571 + 4.59213I −3.41886 − 3.20482I
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.413150
a = 1.61251
b = 1.08931
−2.55102 1.41680
u = 0.413150
a = −2.78320
b = 0.448389
−2.55102 1.41680
10
III. I
u
3
= hb + 2u − 1, 2a + u, u
2
+ 2i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
2
a
8
=
u
u
a
2
=
−1
−2
a
10
=
−
1
2
u
−2u + 1
a
6
=
−
1
2
u
−u + 1
a
5
=
−
1
2
u + 1
−u + 3
a
1
=
−
1
2
u
−u + 1
a
9
=
0
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
(u + 1)
2
c
2
, c
3
, c
7
c
8
u
2
+ 2
c
4
, c
5
, c
9
c
10
(u − 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
9
, c
10
(y −1)
2
c
2
, c
3
, c
7
c
8
(y + 2)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.414210I
a = − 0.707107I
b = 1.00000 − 2.82843I
−8.22467 −12.0000
u = − 1.414210I
a = 0.707107I
b = 1.00000 + 2.82843I
−8.22467 −12.0000
14
IV. I
v
1
= ha, b + 1, v + 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
−1
0
a
4
=
1
0
a
8
=
−1
0
a
2
=
1
0
a
10
=
0
−1
a
6
=
−1
1
a
5
=
0
1
a
1
=
1
−1
a
9
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
u − 1
c
2
, c
3
, c
7
c
8
u
c
5
, c
9
, c
10
u + 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
9
, c
10
y −1
c
2
, c
3
, c
7
c
8
y
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = −1.00000
−3.28987 −12.0000
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
(u − 1)(u + 1)
2
(u
10
+ u
9
− u
8
− 2u
7
+ 3u
6
+ 4u
5
− 4u
3
+ u + 1)
· (u
12
+ u
11
− 2u
10
− 4u
9
+ u
8
+ 5u
7
− u
6
− 7u
5
− u
4
+ 9u
3
+ 6u
2
− 2u − 3)
c
2
, c
3
, c
7
u(u
2
+ 2)(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)
2
· (u
10
+ 3u
9
+ 9u
8
+ 16u
7
+ 24u
6
+ 27u
5
+ 23u
4
+ 16u
3
+ 8u
2
+ 4u + 2)
c
4
(u − 1)
3
· (u
10
+ 3u
9
+ 11u
8
+ 18u
7
+ 33u
6
+ 32u
5
+ 34u
4
+ 18u
3
+ 8u
2
+ u + 1)
· (u
12
+ 5u
11
+ ··· + 40u + 9)
c
5
, c
9
, c
10
(u − 1)
2
(u + 1)(u
10
+ u
9
− u
8
− 2u
7
+ 3u
6
+ 4u
5
− 4u
3
+ u + 1)
· (u
12
+ u
11
− 2u
10
− 4u
9
+ u
8
+ 5u
7
− u
6
− 7u
5
− u
4
+ 9u
3
+ 6u
2
− 2u − 3)
c
8
u(u
2
+ 2)(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)
2
· (u
10
− 3u
9
+ 3u
8
− 8u
6
+ 17u
5
+ 17u
4
− 58u
3
+ 48u
2
− 16u + 10)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
9
, c
10
(y −1)
3
· (y
10
− 3y
9
+ 11y
8
− 18y
7
+ 33y
6
− 32y
5
+ 34y
4
− 18y
3
+ 8y
2
− y + 1)
· (y
12
− 5y
11
+ ··· − 40y + 9)
c
2
, c
3
, c
7
y(y + 2)
2
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1)
2
· (y
10
+ 9y
9
+ ··· + 16y + 4)
c
4
((y −1)
3
)(y
10
+ 13y
9
+ ··· + 15y + 1)(y
12
+ 3y
11
+ ··· − 196y + 81)
c
8
y(y + 2)
2
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
2
· (y
10
− 3y
9
+ ··· + 704y + 100)
20
