11n
37
(K11n
37
)
A knot diagram
1
Linearized knot diagam
4 1 7 2 10 4 11 1 6 9 8
Solving Sequence
7,11
8 1
4,9
3 2 5 6 10
c
7
c
11
c
8
c
3
c
2
c
4
c
6
c
10
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h675u
16
− 1175u
15
+ ··· + 6869b − 641, −101u
16
− 333u
15
+ ··· + 6869a + 12165,
u
17
− 2u
16
+ ··· − u + 1i
I
u
2
= hb, u
2
+ a − u − 1, u
5
− u
4
− 2u
3
+ u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 22 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
= h675u
16
− 1175u
15
+ · · · + 6869b − 641, −101u
16
− 333u
15
+ · · · +
6869a + 12165, u
17
− 2u
16
+ · · · − u + 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u
2
a
1
=
u
−u
3
+ u
a
4
=
0.0147037u
16
+ 0.0484787u
15
+ ··· + 0.511428u − 1.77100
−0.0982676u
16
+ 0.171058u
15
+ ··· + 1.55234u + 0.0933178
a
9
=
−u
2
+ 1
u
4
− 2u
2
a
3
=
0.112971u
16
− 0.122580u
15
+ ··· − 1.04091u − 1.86432
−0.0982676u
16
+ 0.171058u
15
+ ··· + 1.55234u + 0.0933178
a
2
=
−0.0244577u
16
− 0.179648u
15
+ ··· − 0.444752u − 1.90566
0.294803u
16
− 0.513175u
15
+ ··· + 2.34299u − 0.279953
a
5
=
−0.0439656u
16
+ 0.558014u
15
+ ··· + 2.68860u + 0.741010
−1.31213u
16
+ 1.80259u
15
+ ··· − 0.866356u + 1.34678
a
6
=
0.0199447u
16
− 0.627311u
15
+ ··· − 2.46470u − 0.362644
0.896637u
16
− 1.44970u
15
+ ··· + 0.751347u − 0.887029
a
10
=
−u
5
+ 2u
3
− u
u
7
− 3u
5
+ 2u
3
+ u
a
10
=
−u
5
+ 2u
3
− u
u
7
− 3u
5
+ 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
12279
6869
u
16
+
9163
6869
u
15
+ ··· +
1304
6869
u −
3665
6869
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
17
− 6u
16
+ ··· + 11u − 1
c
2
u
17
+ 28u
16
+ ··· + 47u + 1
c
3
, c
6
u
17
+ 3u
16
+ ··· − 64u + 32
c
5
, c
9
u
17
+ 2u
16
+ ··· − u − 1
c
7
, c
8
, c
11
u
17
+ 2u
16
+ ··· − u − 1
c
10
u
17
+ 18u
15
+ ··· + u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
17
− 28y
16
+ ··· + 47y −1
c
2
y
17
− 72y
16
+ ··· + 4879y −1
c
3
, c
6
y
17
+ 33y
16
+ ··· + 8704y −1024
c
5
, c
9
y
17
+ 18y
15
+ ··· + y −1
c
7
, c
8
, c
11
y
17
− 12y
16
+ ··· + y −1
c
10
y
17
+ 36y
16
+ ··· − 3y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.956766 + 0.158240I
a = 0.07861 − 2.13628I
b = −0.242885 − 0.548785I
0.038126 − 0.592997I 0.78481 − 8.54994I
u = −0.956766 − 0.158240I
a = 0.07861 + 2.13628I
b = −0.242885 + 0.548785I
0.038126 + 0.592997I 0.78481 + 8.54994I
u = 0.963468 + 0.398041I
a = 0.625623 − 0.407887I
b = 0.41757 − 1.92272I
−1.44161 + 3.67092I −1.52651 − 6.25757I
u = 0.963468 − 0.398041I
a = 0.625623 + 0.407887I
b = 0.41757 + 1.92272I
−1.44161 − 3.67092I −1.52651 + 6.25757I
u = 0.007712 + 1.101910I
a = 0.27109 − 1.90196I
b = 0.39516 − 2.41394I
−15.6320 − 4.0811I −2.46162 + 2.01591I
u = 0.007712 − 1.101910I
a = 0.27109 + 1.90196I
b = 0.39516 + 2.41394I
−15.6320 + 4.0811I −2.46162 − 2.01591I
u = −1.15751
a = −0.708872
b = 0.345658
2.21135 4.09400
u = 1.350160 + 0.231434I
a = 0.187422 − 0.381800I
b = −0.837282 − 0.065616I
5.02190 + 3.68629I 8.19295 − 2.17087I
u = 1.350160 − 0.231434I
a = 0.187422 + 0.381800I
b = −0.837282 + 0.065616I
5.02190 − 3.68629I 8.19295 + 2.17087I
u = 0.453853 + 0.367356I
a = −1.56221 + 1.37475I
b = 1.27983 + 0.63381I
−2.81840 − 0.19128I −4.55331 − 1.19314I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.453853 − 0.367356I
a = −1.56221 − 1.37475I
b = 1.27983 − 0.63381I
−2.81840 + 0.19128I −4.55331 + 1.19314I
u = 1.36830 + 0.55869I
a = −1.25567 + 0.81024I
b = 0.85212 + 2.19396I
−11.4002 + 9.9652I 0.22842 − 4.86766I
u = 1.36830 − 0.55869I
a = −1.25567 − 0.81024I
b = 0.85212 − 2.19396I
−11.4002 − 9.9652I 0.22842 + 4.86766I
u = −1.38445 + 0.55616I
a = 1.263420 + 0.342784I
b = −0.12095 + 2.24868I
−11.30070 − 1.80882I 0.020166 + 0.772832I
u = −1.38445 − 0.55616I
a = 1.263420 − 0.342784I
b = −0.12095 − 2.24868I
−11.30070 + 1.80882I 0.020166 − 0.772832I
u = −0.223522 + 0.416926I
a = −0.753851 + 0.530995I
b = −0.416389 + 0.318127I
0.238681 − 1.150370I 3.26808 + 5.49780I
u = −0.223522 − 0.416926I
a = −0.753851 − 0.530995I
b = −0.416389 − 0.318127I
0.238681 + 1.150370I 3.26808 − 5.49780I
6
II. I
u
2
= hb, u
2
+ a − u − 1, u
5
− u
4
− 2u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u
2
a
1
=
u
−u
3
+ u
a
4
=
−u
2
+ u + 1
0
a
9
=
−u
2
+ 1
u
4
− 2u
2
a
3
=
−u
2
+ u + 1
0
a
2
=
−u
2
+ 2u + 1
−u
3
+ u
a
5
=
−u
u
3
− u
a
6
=
1
0
a
10
=
−u
4
+ u
2
+ 1
u
4
− 2u
2
a
10
=
−u
4
+ u
2
+ 1
u
4
− 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
4
− 3u
3
+ 2u
2
+ 8u + 2
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
5
c
2
, c
4
(u + 1)
5
c
3
, c
6
u
5
c
5
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
c
7
, c
8
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
c
9
u
5
− u
4
+ 2u
3
− u
2
+ u − 1
c
10
u
5
− 3u
4
+ 4u
3
− u
2
− u + 1
c
11
u
5
+ u
4
− 2u
3
− u
2
+ u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
5
c
3
, c
6
y
5
c
5
, c
9
y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1
c
7
, c
8
, c
11
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
c
10
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.21774
a = −1.70062
b = 0
0.756147 −3.75670
u = −0.309916 + 0.549911I
a = 0.896438 + 0.890762I
b = 0
−1.31583 − 1.53058I −1.49901 + 3.45976I
u = −0.309916 − 0.549911I
a = 0.896438 − 0.890762I
b = 0
−1.31583 + 1.53058I −1.49901 − 3.45976I
u = 1.41878 + 0.21917I
a = 0.453870 − 0.402731I
b = 0
4.22763 + 4.40083I 2.37737 − 5.82971I
u = 1.41878 − 0.21917I
a = 0.453870 + 0.402731I
b = 0
4.22763 − 4.40083I 2.37737 + 5.82971I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
17
− 6u
16
+ ··· + 11u − 1)
c
2
((u + 1)
5
)(u
17
+ 28u
16
+ ··· + 47u + 1)
c
3
, c
6
u
5
(u
17
+ 3u
16
+ ··· − 64u + 32)
c
4
((u + 1)
5
)(u
17
− 6u
16
+ ··· + 11u − 1)
c
5
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)(u
17
+ 2u
16
+ ··· − u − 1)
c
7
, c
8
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)(u
17
+ 2u
16
+ ··· − u − 1)
c
9
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)(u
17
+ 2u
16
+ ··· − u − 1)
c
10
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)(u
17
+ 18u
15
+ ··· + u + 1)
c
11
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)(u
17
+ 2u
16
+ ··· − u − 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
5
)(y
17
− 28y
16
+ ··· + 47y −1)
c
2
((y −1)
5
)(y
17
− 72y
16
+ ··· + 4879y −1)
c
3
, c
6
y
5
(y
17
+ 33y
16
+ ··· + 8704y −1024)
c
5
, c
9
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)(y
17
+ 18y
15
+ ··· + y −1)
c
7
, c
8
, c
11
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)(y
17
− 12y
16
+ ··· + y −1)
c
10
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)(y
17
+ 36y
16
+ ··· − 3y −1)
12
