11n
79
(K11n
79
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 11 10 4 1 7 6 9
Solving Sequence
1,4
2
5,8
9 3 7 10 6 11
c
1
c
4
c
8
c
3
c
7
c
9
c
6
c
11
c
2
, c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
10
− u
9
+ 9u
8
+ 4u
7
− 32u
6
+ u
5
+ 46u
4
− 13u
3
− 12u
2
+ 4b − 5u,
u
10
+ 2u
9
− 8u
8
− 13u
7
+ 28u
6
+ 31u
5
− 47u
4
− 33u
3
+ 25u
2
+ 4a + 17u + 5,
u
11
+ 5u
10
+ 2u
9
− 21u
8
− 15u
7
+ 43u
6
+ 30u
5
− 30u
4
− 18u
3
− 12u
2
− 1i
I
u
2
= hb
4
+ b
3
+ b
2
+ 1, a, u − 1i
* 2 irreducible components of dim
C
= 0, with total 15 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−u
10
−u
9
+· · ·+4b−5u, u
10
+2u
9
+· · ·+4a+5, u
11
+5u
10
+· · ·−12u
2
−1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
8
=
−
1
4
u
10
−
1
2
u
9
+ ··· −
17
4
u −
5
4
1
4
u
10
+
1
4
u
9
+ ··· + 3u
2
+
5
4
u
a
9
=
−
1
4
u
9
−
1
4
u
8
+ ··· − 3u −
5
4
1
4
u
10
+
1
4
u
9
+ ··· + 3u
2
+
5
4
u
a
3
=
−u
2
+ 1
u
2
a
7
=
−
1
4
u
10
−
1
2
u
9
+ ··· −
17
4
u −
5
4
−u
10
−
13
4
u
9
+ ··· + u +
3
4
a
10
=
−
3
4
u
10
−
13
4
u
9
+ ··· −
3
4
u −
1
2
−
1
2
u
10
−
3
2
u
9
+ ··· + 8u
2
+
1
2
a
6
=
1
4
u
10
+
5
4
u
9
+ ··· −
17
4
u −
1
2
−
1
8
u
10
−
1
4
u
9
+ ··· +
11
8
u −
1
8
a
11
=
1
8
u
10
+
1
2
u
9
+ ··· −
15
8
u +
15
8
1
8
u
10
+
1
2
u
9
+ ··· −
7
8
u −
1
8
a
11
=
1
8
u
10
+
1
2
u
9
+ ··· −
15
8
u +
15
8
1
8
u
10
+
1
2
u
9
+ ··· −
7
8
u −
1
8
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
5
4
u
10
−
31
4
u
9
−
29
4
u
8
+
63
2
u
7
+ 39u
6
−
271
4
u
5
− 71u
4
+
211
4
u
3
+
65
2
u
2
+
91
4
u +
1
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
11
− 5u
10
+ ··· + 12u
2
+ 1
c
2
u
11
+ 21u
10
+ ··· − 24u + 1
c
3
, c
7
u
11
− u
10
+ ··· + 8u + 16
c
5
, c
6
, c
9
c
10
u
11
− 2u
10
+ ··· + 2u − 1
c
8
, c
11
u
11
+ 12u
9
+ 38u
7
− 2u
6
+ 14u
5
− 12u
4
+ 13u
3
− u
2
+ 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
11
− 21y
10
+ ··· − 24y −1
c
2
y
11
− 73y
10
+ ··· + 168y −1
c
3
, c
7
y
11
+ 27y
10
+ ··· + 320y −256
c
5
, c
6
, c
9
c
10
y
11
+ 12y
10
+ ··· + 2y −1
c
8
, c
11
y
11
+ 24y
10
+ ··· + 2y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.22744
a = −0.642449
b = 0.354025
−2.29341 −5.45610
u = 1.28076 + 0.60275I
a = 0.935338 + 0.483017I
b = −0.499697 − 0.386367I
1.86679 − 1.71507I −2.68555 + 1.25777I
u = 1.28076 − 0.60275I
a = 0.935338 − 0.483017I
b = −0.499697 + 0.386367I
1.86679 + 1.71507I −2.68555 − 1.25777I
u = −0.286174 + 0.444607I
a = 1.20455 + 0.84468I
b = 0.542271 − 0.749372I
6.37335 − 2.43510I 2.89338 + 1.98880I
u = −0.286174 − 0.444607I
a = 1.20455 − 0.84468I
b = 0.542271 + 0.749372I
6.37335 + 2.43510I 2.89338 − 1.98880I
u = 0.069460 + 0.264957I
a = −1.23197 − 1.54081I
b = −0.189666 + 0.452281I
−0.058810 − 0.998414I −1.10999 + 6.77459I
u = 0.069460 − 0.264957I
a = −1.23197 + 1.54081I
b = −0.189666 − 0.452281I
−0.058810 + 0.998414I −1.10999 − 6.77459I
u = −2.04088 + 0.26755I
a = −0.131689 − 1.255590I
b = −0.04048 + 2.41762I
−10.35590 + 6.75197I −2.99345 − 2.75276I
u = −2.04088 − 0.26755I
a = −0.131689 + 1.255590I
b = −0.04048 − 2.41762I
−10.35590 − 6.75197I −2.99345 + 2.75276I
u = −2.13689 + 0.09549I
a = 0.044998 + 1.293010I
b = 0.01056 − 2.42567I
−17.2404 + 2.6821I −5.87634 − 2.38377I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −2.13689 − 0.09549I
a = 0.044998 − 1.293010I
b = 0.01056 + 2.42567I
−17.2404 − 2.6821I −5.87634 + 2.38377I
6
II. I
u
2
= hb
4
+ b
3
+ b
2
+ 1, a, u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
−1
0
a
8
=
0
b
a
9
=
b
b
a
3
=
0
1
a
7
=
0
b
a
10
=
b
b
3
+ b
a
6
=
b
3
b
3
+ b
2
+ 1
a
11
=
b
2
+ 1
b
2
a
11
=
b
2
+ 1
b
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4b
2
− 3b − 5
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
4
c
2
, c
4
(u + 1)
4
c
3
, c
7
u
4
c
5
, c
6
u
4
− u
3
+ 3u
2
− 2u + 1
c
8
u
4
− u
3
+ u
2
+ 1
c
9
, c
10
u
4
+ u
3
+ 3u
2
+ 2u + 1
c
11
u
4
+ u
3
+ u
2
+ 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
7
y
4
c
5
, c
6
, c
9
c
10
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
c
8
, c
11
y
4
+ y
3
+ 3y
2
+ 2y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0
b = 0.351808 + 0.720342I
−1.85594 + 1.41510I −4.47493 − 4.18840I
u = 1.00000
a = 0
b = 0.351808 − 0.720342I
−1.85594 − 1.41510I −4.47493 + 4.18840I
u = 1.00000
a = 0
b = −0.851808 + 0.911292I
5.14581 − 3.16396I −2.02507 + 3.47609I
u = 1.00000
a = 0
b = −0.851808 − 0.911292I
5.14581 + 3.16396I −2.02507 − 3.47609I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
4
)(u
11
− 5u
10
+ ··· + 12u
2
+ 1)
c
2
((u + 1)
4
)(u
11
+ 21u
10
+ ··· − 24u + 1)
c
3
, c
7
u
4
(u
11
− u
10
+ ··· + 8u + 16)
c
4
((u + 1)
4
)(u
11
− 5u
10
+ ··· + 12u
2
+ 1)
c
5
, c
6
(u
4
− u
3
+ 3u
2
− 2u + 1)(u
11
− 2u
10
+ ··· + 2u − 1)
c
8
(u
4
− u
3
+ u
2
+ 1)(u
11
+ 12u
9
+ ··· − u
2
+ 1)
c
9
, c
10
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
11
− 2u
10
+ ··· + 2u − 1)
c
11
(u
4
+ u
3
+ u
2
+ 1)(u
11
+ 12u
9
+ ··· − u
2
+ 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
4
)(y
11
− 21y
10
+ ··· − 24y −1)
c
2
((y −1)
4
)(y
11
− 73y
10
+ ··· + 168y −1)
c
3
, c
7
y
4
(y
11
+ 27y
10
+ ··· + 320y −256)
c
5
, c
6
, c
9
c
10
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)(y
11
+ 12y
10
+ ··· + 2y −1)
c
8
, c
11
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
11
+ 24y
10
+ ··· + 2y −1)
12
