11n
56
(K11n
56
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 9 10 4 1 11 7 6
Solving Sequence
6,10
7 11
1,4
2 3 9 5 8
c
6
c
10
c
11
c
1
c
2
c
9
c
5
c
8
c
3
, c
4
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
22
− u
21
+ ··· − u
3
+ b, −u
22
+ u
21
+ ··· + a + 1, u
23
− 2u
22
+ ··· − 2u + 1i
I
u
2
= h−u
3
+ b + u + 1, −u
4
− u
3
+ u
2
+ a + u, u
6
+ u
5
− u
4
− 2u
3
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 29 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
22
−u
21
+· · ·−u
3
+b, −u
22
+u
21
+· · ·+a+1, u
23
−2u
22
+· · ·−2u+1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
7
=
1
u
2
a
11
=
−u
−u
3
+ u
a
1
=
−u
3
−u
3
+ u
a
4
=
u
22
− u
21
+ ··· + 2u − 1
−u
22
+ u
21
+ ··· − 5u
4
+ u
3
a
2
=
u
20
− u
19
+ ··· − u + 1
u
22
− u
21
+ ··· − u
3
+ u
2
a
3
=
u
22
− u
21
+ ··· − 4u
3
+ u
2
u
22
− u
21
+ ··· + 2u
2
− u
a
9
=
u
3
u
5
− u
3
+ u
a
5
=
u
8
− u
6
+ u
4
+ 1
u
10
− 2u
8
+ 3u
6
− 2u
4
+ u
2
a
8
=
−u
11
+ 2u
9
− 2u
7
+ u
3
−u
11
+ 3u
9
− 4u
7
+ 3u
5
− u
3
+ u
a
8
=
−u
11
+ 2u
9
− 2u
7
+ u
3
−u
11
+ 3u
9
− 4u
7
+ 3u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 8u
22
− 10u
21
− 40u
20
+ 65u
19
+ 85u
18
− 190u
17
− 61u
16
+ 307u
15
− 81u
14
− 264u
13
+
228u
12
+ 59u
11
−203u
10
+ 86u
9
+ 76u
8
−58u
7
−12u
6
−8u
5
+ 26u
4
−7u
3
+ 3u
2
+ 8u −10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
23
− 7u
22
+ ··· − 7u + 1
c
2
u
23
+ 35u
22
+ ··· + 11u + 1
c
3
, c
7
u
23
− u
22
+ ··· + 128u + 64
c
5
u
23
− 2u
22
+ ··· − 108u − 36
c
6
, c
10
u
23
+ 2u
22
+ ··· − 2u − 1
c
8
u
23
+ 24u
21
+ ··· + 2u + 1
c
9
u
23
+ 12u
22
+ ··· + 2u + 1
c
11
u
23
+ 6u
22
+ ··· − 18u − 7
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
23
− 35y
22
+ ··· + 11y −1
c
2
y
23
− 87y
22
+ ··· + 15y −1
c
3
, c
7
y
23
+ 39y
22
+ ··· + 28672y −4096
c
5
y
23
+ 12y
22
+ ··· + 10296y −1296
c
6
, c
10
y
23
− 12y
22
+ ··· + 2y −1
c
8
y
23
+ 48y
22
+ ··· + 2y −1
c
9
y
23
+ 24y
21
+ ··· + 6y −1
c
11
y
23
+ 12y
22
+ ··· + 282y −49
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.797336 + 0.702236I
a = −1.35368 + 0.49960I
b = −0.843379 + 0.494457I
−9.64155 − 2.65369I −2.71409 + 2.86915I
u = 0.797336 − 0.702236I
a = −1.35368 − 0.49960I
b = −0.843379 − 0.494457I
−9.64155 + 2.65369I −2.71409 − 2.86915I
u = 1.027390 + 0.366873I
a = 0.534979 + 0.341386I
b = −0.263822 + 0.275329I
−1.85876 − 1.44380I −2.27537 + 0.68239I
u = 1.027390 − 0.366873I
a = 0.534979 − 0.341386I
b = −0.263822 − 0.275329I
−1.85876 + 1.44380I −2.27537 − 0.68239I
u = 0.255023 + 0.855822I
a = −0.831897 − 0.002920I
b = 0.09815 − 2.48508I
−12.76380 + 5.09874I −3.17808 − 1.98307I
u = 0.255023 − 0.855822I
a = −0.831897 + 0.002920I
b = 0.09815 + 2.48508I
−12.76380 − 5.09874I −3.17808 + 1.98307I
u = −1.079080 + 0.536804I
a = −0.144531 + 0.926453I
b = 0.323737 + 0.843029I
−0.53628 + 5.30661I 1.77241 − 5.11876I
u = −1.079080 − 0.536804I
a = −0.144531 − 0.926453I
b = 0.323737 − 0.843029I
−0.53628 − 5.30661I 1.77241 + 5.11876I
u = −1.141520 + 0.416414I
a = 0.33767 − 2.48715I
b = −1.03478 − 2.11364I
−5.57676 + 2.33070I −7.43736 − 2.84176I
u = −1.141520 − 0.416414I
a = 0.33767 + 2.48715I
b = −1.03478 + 2.11364I
−5.57676 − 2.33070I −7.43736 + 2.84176I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.693757 + 0.359279I
a = 0.576016 + 0.850672I
b = −0.158914 − 0.203116I
−0.87588 − 1.51254I −2.24997 + 5.09221I
u = 0.693757 − 0.359279I
a = 0.576016 − 0.850672I
b = −0.158914 + 0.203116I
−0.87588 + 1.51254I −2.24997 − 5.09221I
u = 1.146720 + 0.479206I
a = −1.60133 − 1.31891I
b = 0.29675 − 1.85371I
−5.12386 − 5.67209I −6.64054 + 5.01271I
u = 1.146720 − 0.479206I
a = −1.60133 + 1.31891I
b = 0.29675 + 1.85371I
−5.12386 + 5.67209I −6.64054 − 5.01271I
u = −0.401701 + 0.617973I
a = 0.562564 + 0.078186I
b = 0.450330 − 0.386164I
1.43616 − 0.72615I 6.25783 + 0.91942I
u = −0.401701 − 0.617973I
a = 0.562564 − 0.078186I
b = 0.450330 + 0.386164I
1.43616 + 0.72615I 6.25783 − 0.91942I
u = −1.235200 + 0.278047I
a = −0.73735 + 2.60704I
b = 1.07712 + 1.98419I
−17.5522 − 1.4869I −8.02521 − 0.25180I
u = −1.235200 − 0.278047I
a = −0.73735 − 2.60704I
b = 1.07712 − 1.98419I
−17.5522 + 1.4869I −8.02521 + 0.25180I
u = −0.718932
a = −1.72386
b = −1.61562
−2.53646 −1.61890
u = 1.182790 + 0.567983I
a = 2.25195 + 2.10373I
b = 0.17909 + 3.14085I
−15.5409 − 10.3372I −6.00224 + 5.46879I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.182790 − 0.567983I
a = 2.25195 − 2.10373I
b = 0.17909 − 3.14085I
−15.5409 + 10.3372I −6.00224 − 5.46879I
u = 0.113951 + 0.644421I
a = −0.232464 − 0.843186I
b = −0.316473 + 1.333070I
−2.25261 + 1.36983I −3.69794 − 1.43293I
u = 0.113951 − 0.644421I
a = −0.232464 + 0.843186I
b = −0.316473 − 1.333070I
−2.25261 − 1.36983I −3.69794 + 1.43293I
7
II. I
u
2
= h−u
3
+ b + u + 1, −u
4
− u
3
+ u
2
+ a + u, u
6
+ u
5
− u
4
− 2u
3
+ u + 1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
7
=
1
u
2
a
11
=
−u
−u
3
+ u
a
1
=
−u
3
−u
3
+ u
a
4
=
u
4
+ u
3
− u
2
− u
u
3
− u − 1
a
2
=
u
4
− u
2
− u
−1
a
3
=
u
4
+ u
3
− u
2
− u
u
3
− u − 1
a
9
=
u
3
u
5
− u
3
+ u
a
5
=
u
3
u
3
− u
a
8
=
1
u
2
a
8
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
4
− 3u
2
− 3u − 3
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
6
c
2
, c
4
(u + 1)
6
c
3
, c
7
u
6
c
5
, c
8
, c
10
u
6
− u
5
− u
4
+ 2u
3
− u + 1
c
6
u
6
+ u
5
− u
4
− 2u
3
+ u + 1
c
9
, c
11
u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
7
y
6
c
5
, c
6
, c
8
c
10
y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1
c
9
, c
11
y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.002190 + 0.295542I
a = −0.685196 + 1.063260I
b = −1.258210 + 0.569162I
−3.53554 − 0.92430I −6.79748 + 1.68215I
u = 1.002190 − 0.295542I
a = −0.685196 − 1.063260I
b = −1.258210 − 0.569162I
−3.53554 + 0.92430I −6.79748 − 1.68215I
u = −0.428243 + 0.664531I
a = 0.917982 + 0.270708I
b = −0.082955 − 0.592379I
0.245672 − 0.924305I −1.96974 + 0.88960I
u = −0.428243 − 0.664531I
a = 0.917982 − 0.270708I
b = −0.082955 + 0.592379I
0.245672 + 0.924305I −1.96974 − 0.88960I
u = −1.073950 + 0.558752I
a = −0.732786 + 0.381252I
b = −0.158836 + 1.200140I
−1.64493 + 5.69302I −5.23279 − 6.15196I
u = −1.073950 − 0.558752I
a = −0.732786 − 0.381252I
b = −0.158836 − 1.200140I
−1.64493 − 5.69302I −5.23279 + 6.15196I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
23
− 7u
22
+ ··· − 7u + 1)
c
2
((u + 1)
6
)(u
23
+ 35u
22
+ ··· + 11u + 1)
c
3
, c
7
u
6
(u
23
− u
22
+ ··· + 128u + 64)
c
4
((u + 1)
6
)(u
23
− 7u
22
+ ··· − 7u + 1)
c
5
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)(u
23
− 2u
22
+ ··· − 108u − 36)
c
6
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)(u
23
+ 2u
22
+ ··· − 2u − 1)
c
8
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)(u
23
+ 24u
21
+ ··· + 2u + 1)
c
9
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)(u
23
+ 12u
22
+ ··· + 2u + 1)
c
10
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)(u
23
+ 2u
22
+ ··· − 2u − 1)
c
11
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)(u
23
+ 6u
22
+ ··· − 18u − 7)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
6
)(y
23
− 35y
22
+ ··· + 11y −1)
c
2
((y −1)
6
)(y
23
− 87y
22
+ ··· + 15y −1)
c
3
, c
7
y
6
(y
23
+ 39y
22
+ ··· + 28672y −4096)
c
5
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
· (y
23
+ 12y
22
+ ··· + 10296y −1296)
c
6
, c
10
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)(y
23
− 12y
22
+ ··· + 2y −1)
c
8
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)(y
23
+ 48y
22
+ ··· + 2y −1)
c
9
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)(y
23
+ 24y
21
+ ··· + 6y −1)
c
11
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)(y
23
+ 12y
22
+ ··· + 282y −49)
13
