11n
145
(K11n
145
)
A knot diagram
1
Linearized knot diagam
7 1 6 11 1 11 2 5 4 6 9
Solving Sequence
9,11 1,7
2 3 6 5 4 8 10
c
11
c
1
c
2
c
6
c
5
c
4
c
8
c
10
c
3
, c
7
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
11
+ u
10
+ 2u
9
+ u
8
+ 4u
7
+ 2u
6
+ 3u
5
− u
4
+ u
3
+ 2u
2
+ b + 2u + 1,
3u
11
+ 5u
10
+ 6u
9
+ 3u
8
+ 12u
7
+ 11u
6
+ 6u
5
− 6u
4
+ 5u
3
+ 8u
2
+ a + 8u + 2,
u
12
+ 2u
11
+ 3u
10
+ 2u
9
+ 5u
8
+ 5u
7
+ 5u
6
− u
5
+ 2u
4
+ 2u
3
+ 5u
2
+ 2u + 1i
I
u
2
= hu
6
− 2u
5
+ 2u
4
+ b − u + 1, −2u
7
+ 6u
6
− 9u
5
+ 5u
4
− 2u
3
+ 4u
2
+ a − 7u + 2,
u
8
− 3u
7
+ 5u
6
− 4u
5
+ 3u
4
− 3u
3
+ 4u
2
− 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 20 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
11
+u
10
+· · ·+b +1, 3u
11
+5u
10
+· · ·+a +2, u
12
+2u
11
+· · ·+2u +1i
(i) Arc colorings
a
9
=
0
u
a
11
=
1
0
a
1
=
1
−u
2
a
7
=
−3u
11
− 5u
10
+ ··· − 8u − 2
−u
11
− u
10
− 2u
9
− u
8
− 4u
7
− 2u
6
− 3u
5
+ u
4
− u
3
− 2u
2
− 2u − 1
a
2
=
−2u
11
− 3u
10
+ ··· − 2u − 1
−u
11
− u
9
− 3u
7
+ u
6
− u
5
+ u
4
− u
3
− 2u
2
a
3
=
−2u
11
− 2u
10
− 3u
9
− 8u
7
− 3u
6
− 3u
5
+ 6u
4
− 7u
3
− 3u
2
− 2u
u
11
+ u
10
+ 2u
9
+ u
8
+ 4u
7
+ 2u
6
+ 3u
5
− u
4
+ u
3
+ 2u
2
+ 2u + 1
a
6
=
−2u
11
− 4u
10
+ ··· − 6u − 1
−u
11
− u
10
− 2u
9
− u
8
− 4u
7
− 2u
6
− 3u
5
+ u
4
− u
3
− 2u
2
− 2u − 1
a
5
=
−u
11
− 3u
10
− 4u
9
− 2u
8
− 5u
7
− 8u
6
− 6u
5
+ 4u
4
− u
3
− 5u
2
− 6u − 2
−2u
10
− 2u
9
− 2u
8
− u
7
− 7u
6
− 3u
5
+ 2u
3
− 4u
2
− 3u − 2
a
4
=
−u
11
− u
10
− 2u
9
− 4u
7
− u
6
− 3u
5
+ 4u
4
− 3u
3
− u
2
− 3u
−2u
10
− 2u
9
− 2u
8
− u
7
− 7u
6
− 3u
5
+ 2u
3
− 4u
2
− 3u − 2
a
8
=
u
11
+ 2u
10
+ 2u
9
+ 2u
8
+ 3u
7
+ 6u
6
+ u
5
− u
3
+ 6u
2
+ u + 2
u
11
+ u
10
+ 2u
9
+ u
8
+ 3u
7
+ 3u
6
+ 2u
5
− u
4
+ 4u
2
+ 2u + 1
a
10
=
−u
11
− u
10
− u
9
− u
8
− 3u
7
− 2u
6
− u
3
− 2u
2
− u
u
11
+ 2u
10
+ u
9
+ 2u
8
+ 3u
7
+ 5u
6
− u
5
+ u
4
+ 4u
2
+ 2u + 1
a
10
=
−u
11
− u
10
− u
9
− u
8
− 3u
7
− 2u
6
− u
3
− 2u
2
− u
u
11
+ 2u
10
+ u
9
+ 2u
8
+ 3u
7
+ 5u
6
− u
5
+ u
4
+ 4u
2
+ 2u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −3u
11
− 7u
10
− 8u
9
− 7u
8
− 14u
7
− 21u
6
− 8u
5
− u
3
− 16u
2
− 4u − 9
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
12
− 11u
11
+ ··· − 48u + 16
c
2
u
12
+ 9u
11
+ ··· + 896u + 256
c
3
u
12
− 4u
11
+ ··· − 4u + 1
c
4
u
12
− u
11
+ ··· + 4u + 10
c
5
, c
8
u
12
+ 2u
11
+ ··· − 16u
2
+ 1
c
6
, c
9
, c
10
u
12
− 10u
10
+ ··· − u + 1
c
11
u
12
+ 2u
11
+ ··· + 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
12
− 9y
11
+ ··· − 896y + 256
c
2
y
12
+ 59y
11
+ ··· + 253952y + 65536
c
3
y
12
− 30y
11
+ ··· − 16y + 1
c
4
y
12
− 25y
11
+ ··· − 896y + 100
c
5
, c
8
y
12
+ 26y
11
+ ··· − 32y + 1
c
6
, c
9
, c
10
y
12
− 20y
11
+ ··· + 3y + 1
c
11
y
12
+ 2y
11
+ ··· + 6y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.800801 + 0.482482I
a = 0.183380 + 0.498565I
b = 0.698671 + 0.235425I
1.43652 + 0.62326I 4.44877 + 0.65496I
u = 0.800801 − 0.482482I
a = 0.183380 − 0.498565I
b = 0.698671 − 0.235425I
1.43652 − 0.62326I 4.44877 − 0.65496I
u = −0.372107 + 0.751953I
a = −0.81363 + 1.82447I
b = 0.243009 + 0.355918I
−9.07675 − 1.52290I −4.89097 + 7.64925I
u = −0.372107 − 0.751953I
a = −0.81363 − 1.82447I
b = 0.243009 − 0.355918I
−9.07675 + 1.52290I −4.89097 − 7.64925I
u = 0.690074 + 1.109750I
a = −0.458254 − 0.957045I
b = −0.715998 − 0.535824I
−0.63619 + 5.03255I −5.39872 − 6.82429I
u = 0.690074 − 1.109750I
a = −0.458254 + 0.957045I
b = −0.715998 + 0.535824I
−0.63619 − 5.03255I −5.39872 + 6.82429I
u = −0.981759 + 0.915783I
a = 1.31213 − 0.66172I
b = 2.49508 + 1.02200I
11.48080 + 2.02735I −0.446845 + 0.080322I
u = −0.981759 − 0.915783I
a = 1.31213 + 0.66172I
b = 2.49508 − 1.02200I
11.48080 − 2.02735I −0.446845 − 0.080322I
u = −0.924436 + 1.006840I
a = −0.21708 + 2.14841I
b = −2.45017 + 1.30392I
11.1731 − 9.0121I −0.90831 + 4.12550I
u = −0.924436 − 1.006840I
a = −0.21708 − 2.14841I
b = −2.45017 − 1.30392I
11.1731 + 9.0121I −0.90831 − 4.12550I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.212573 + 0.487267I
a = 0.49344 − 1.36791I
b = −0.270602 − 0.384607I
−1.218070 − 0.691278I −5.30392 + 1.72582I
u = −0.212573 − 0.487267I
a = 0.49344 + 1.36791I
b = −0.270602 + 0.384607I
−1.218070 + 0.691278I −5.30392 − 1.72582I
6
II.
I
u
2
= hu
6
−2u
5
+2u
4
+b−u+1, −2u
7
+6u
6
+· · ·+a+2, u
8
−3u
7
+· · ·−2u+1i
(i) Arc colorings
a
9
=
0
u
a
11
=
1
0
a
1
=
1
−u
2
a
7
=
2u
7
− 6u
6
+ 9u
5
− 5u
4
+ 2u
3
− 4u
2
+ 7u − 2
−u
6
+ 2u
5
− 2u
4
+ u − 1
a
2
=
−4u
7
+ 11u
6
− 16u
5
+ 9u
4
− 5u
3
+ 8u
2
− 12u + 4
−u
7
+ 3u
6
− 5u
5
+ 4u
4
− 2u
3
+ u
2
− 3u + 2
a
3
=
−4u
7
+ 10u
6
− 14u
5
+ 6u
4
− 4u
3
+ 7u
2
− 11u + 1
u
6
− 2u
5
+ 2u
4
− u + 1
a
6
=
2u
7
− 5u
6
+ 7u
5
− 3u
4
+ 2u
3
− 4u
2
+ 6u − 1
−u
6
+ 2u
5
− 2u
4
+ u − 1
a
5
=
2u
7
− 6u
6
+ 9u
5
− 6u
4
+ 3u
3
− 5u
2
+ 7u − 3
u
7
− 3u
6
+ 5u
5
− 4u
4
+ 2u
3
− 2u
2
+ 3u − 2
a
4
=
u
7
− 3u
6
+ 4u
5
− 2u
4
+ u
3
− 3u
2
+ 4u − 1
u
7
− 3u
6
+ 5u
5
− 4u
4
+ 2u
3
− 2u
2
+ 3u − 2
a
8
=
−3u
7
+ 9u
6
− 14u
5
+ 9u
4
− 5u
3
+ 7u
2
− 11u + 3
−u
7
+ 4u
6
− 7u
5
+ 6u
4
− 3u
3
+ 4u
2
− 5u + 3
a
10
=
−u
7
+ 2u
6
− 2u
5
− u
4
+ u
3
− u − 1
−u
7
+ 3u
6
− 5u
5
+ 4u
4
− 3u
3
+ 3u
2
− 3u + 1
a
10
=
−u
7
+ 2u
6
− 2u
5
− u
4
+ u
3
− u − 1
−u
7
+ 3u
6
− 5u
5
+ 4u
4
− 3u
3
+ 3u
2
− 3u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
7
+ 5u
5
− 11u
4
+ 2u
3
− 4u
2
+ 7u − 9
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
8
+ u
7
− 4u
6
− 2u
5
+ 7u
4
+ u
3
− 5u
2
+ 2
c
2
u
8
+ 9u
7
+ 34u
6
+ 72u
5
+ 97u
4
+ 87u
3
+ 53u
2
+ 20u + 4
c
3
u
8
− 9u
7
+ 31u
6
− 51u
5
+ 42u
4
− 20u
3
+ 9u
2
− 2u + 1
c
4
u
8
+ u
6
+ 2u
5
+ 3u
4
+ 3u
3
− u
2
+ 2
c
5
, c
8
u
8
− u
7
− u
6
− u
5
+ 2u
4
+ 2u
3
+ 3u
2
+ 2u + 1
c
6
, c
9
u
8
− u
7
+ 2u
6
− u
5
− u
4
+ 3u
3
− u
2
− u + 1
c
7
u
8
− u
7
− 4u
6
+ 2u
5
+ 7u
4
− u
3
− 5u
2
+ 2
c
10
u
8
+ u
7
+ 2u
6
+ u
5
− u
4
− 3u
3
− u
2
+ u + 1
c
11
u
8
− 3u
7
+ 5u
6
− 4u
5
+ 3u
4
− 3u
3
+ 4u
2
− 2u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
8
− 9y
7
+ 34y
6
− 72y
5
+ 97y
4
− 87y
3
+ 53y
2
− 20y + 4
c
2
y
8
− 13y
7
+ 54y
6
− 48y
5
+ 133y
4
+ 105y
3
+ 105y
2
+ 24y + 16
c
3
y
8
− 19y
7
+ 127y
6
− 339y
5
+ 248y
4
+ 214y
3
+ 85y
2
+ 14y + 1
c
4
y
8
+ 2y
7
+ 7y
6
− y
4
− 11y
3
+ 13y
2
− 4y + 4
c
5
, c
8
y
8
− 3y
7
+ 3y
6
+ 5y
5
+ 8y
4
+ 10y
3
+ 5y
2
+ 2y + 1
c
6
, c
9
, c
10
y
8
+ 3y
7
− y
5
+ 3y
4
− 5y
3
+ 5y
2
− 3y + 1
c
11
y
8
+ y
7
+ 7y
6
+ 4y
5
+ 15y
4
+ 9y
3
+ 10y
2
+ 4y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.601219 + 0.700245I
a = −1.25463 − 1.13763I
b = 0.04009 − 1.51942I
−5.61184 − 2.16662I −1.31720 + 3.92427I
u = −0.601219 − 0.700245I
a = −1.25463 + 1.13763I
b = 0.04009 + 1.51942I
−5.61184 + 2.16662I −1.31720 − 3.92427I
u = 0.975658 + 0.743632I
a = 0.205573 + 0.088262I
b = 0.804982 + 0.154967I
1.42321 + 1.81732I 3.74465 − 3.23500I
u = 0.975658 − 0.743632I
a = 0.205573 − 0.088262I
b = 0.804982 − 0.154967I
1.42321 − 1.81732I 3.74465 + 3.23500I
u = 0.235731 + 0.563343I
a = 0.73791 + 2.94527I
b = −0.641984 + 0.737766I
−9.14900 + 0.88713I −6.29124 + 4.01225I
u = 0.235731 − 0.563343I
a = 0.73791 − 2.94527I
b = −0.641984 − 0.737766I
−9.14900 − 0.88713I −6.29124 − 4.01225I
u = 0.88983 + 1.14020I
a = −0.188850 − 0.848475I
b = −0.703087 − 0.423228I
0.17815 + 5.07460I 4.36379 − 8.11889I
u = 0.88983 − 1.14020I
a = −0.188850 + 0.848475I
b = −0.703087 + 0.423228I
0.17815 − 5.07460I 4.36379 + 8.11889I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
8
+ u
7
+ ··· − 5u
2
+ 2)(u
12
− 11u
11
+ ··· − 48u + 16)
c
2
(u
8
+ 9u
7
+ 34u
6
+ 72u
5
+ 97u
4
+ 87u
3
+ 53u
2
+ 20u + 4)
· (u
12
+ 9u
11
+ ··· + 896u + 256)
c
3
(u
8
− 9u
7
+ 31u
6
− 51u
5
+ 42u
4
− 20u
3
+ 9u
2
− 2u + 1)
· (u
12
− 4u
11
+ ··· − 4u + 1)
c
4
(u
8
+ u
6
+ 2u
5
+ 3u
4
+ 3u
3
− u
2
+ 2)(u
12
− u
11
+ ··· + 4u + 10)
c
5
, c
8
(u
8
− u
7
− u
6
− u
5
+ 2u
4
+ 2u
3
+ 3u
2
+ 2u + 1)
· (u
12
+ 2u
11
+ ··· − 16u
2
+ 1)
c
6
, c
9
(u
8
− u
7
+ ··· − u + 1)(u
12
− 10u
10
+ ··· − u + 1)
c
7
(u
8
− u
7
+ ··· − 5u
2
+ 2)(u
12
− 11u
11
+ ··· − 48u + 16)
c
10
(u
8
+ u
7
+ ··· + u + 1)(u
12
− 10u
10
+ ··· − u + 1)
c
11
(u
8
− 3u
7
+ 5u
6
− 4u
5
+ 3u
4
− 3u
3
+ 4u
2
− 2u + 1)
· (u
12
+ 2u
11
+ ··· + 2u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
(y
8
− 9y
7
+ 34y
6
− 72y
5
+ 97y
4
− 87y
3
+ 53y
2
− 20y + 4)
· (y
12
− 9y
11
+ ··· − 896y + 256)
c
2
(y
8
− 13y
7
+ 54y
6
− 48y
5
+ 133y
4
+ 105y
3
+ 105y
2
+ 24y + 16)
· (y
12
+ 59y
11
+ ··· + 253952y + 65536)
c
3
(y
8
− 19y
7
+ 127y
6
− 339y
5
+ 248y
4
+ 214y
3
+ 85y
2
+ 14y + 1)
· (y
12
− 30y
11
+ ··· − 16y + 1)
c
4
(y
8
+ 2y
7
+ 7y
6
− y
4
− 11y
3
+ 13y
2
− 4y + 4)
· (y
12
− 25y
11
+ ··· − 896y + 100)
c
5
, c
8
(y
8
− 3y
7
+ 3y
6
+ 5y
5
+ 8y
4
+ 10y
3
+ 5y
2
+ 2y + 1)
· (y
12
+ 26y
11
+ ··· − 32y + 1)
c
6
, c
9
, c
10
(y
8
+ 3y
7
+ ··· − 3y + 1)(y
12
− 20y
11
+ ··· + 3y + 1)
c
11
(y
8
+ y
7
+ 7y
6
+ 4y
5
+ 15y
4
+ 9y
3
+ 10y
2
+ 4y + 1)
· (y
12
+ 2y
11
+ ··· + 6y + 1)
12
