11n
31
(K11n
31
)
A knot diagram
1
Linearized knot diagam
4 1 6 2 9 3 11 6 1 8 10
Solving Sequence
7,11 3,8
6 4 10 1 2 5 9
c
7
c
6
c
3
c
10
c
11
c
2
c
4
c
9
c
1
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h7u
10
− 8u
9
− 23u
8
+ 57u
7
− 2u
6
+ 8u
5
− 87u
4
+ 232u
3
+ 3u
2
+ 164b + 47u − 106,
38u
10
− 184u
9
+ 414u
8
− 493u
7
+ 446u
6
− 513u
5
+ 828u
4
− 650u
3
+ 397u
2
+ 82a − 395u + 350,
u
11
− 5u
10
+ 12u
9
− 16u
8
+ 16u
7
− 17u
6
+ 25u
5
− 21u
4
+ 14u
3
− 10u
2
+ 10u − 1i
I
u
2
= h−a
2
u − 2au + b − a + u, a
3
− a
2
u + 2a
2
− au − a + u − 2, u
2
+ u + 1i
I
u
3
= hb, −u
3
+ 2u
2
+ a − 2u, u
4
− u
3
+ u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 21 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
= h7u
10
− 8u
9
+ · · · + 164b − 106, 38u
10
− 184u
9
+ · · · + 82a +
350, u
11
− 5u
10
+ · · · + 10u − 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
3
=
−0.463415u
10
+ 2.24390u
9
+ ··· + 4.81707u − 4.26829
−0.0426829u
10
+ 0.0487805u
9
+ ··· − 0.286585u + 0.646341
a
8
=
1
−u
2
a
6
=
0.359756u
10
− 1.76829u
9
+ ··· − 3.79878u + 3.69512
0.0121951u
10
+ 0.164634u
9
+ ··· + 1.18902u − 0.506098
a
4
=
−0.871951u
10
+ 3.85366u
9
+ ··· + 6.35976u − 6.43902
0.884146u
10
− 3.68902u
9
+ ··· − 5.67073u + 1.43293
a
10
=
−u
u
3
+ u
a
1
=
−u
3
u
5
+ u
3
+ u
a
2
=
−0.152439u
10
+ 1.31707u
9
+ ··· + 2.76220u − 4.04878
−1.23780u
10
+ 5.41463u
9
+ ··· + 8.68902u − 0.256098
a
5
=
−0.201220u
10
+ 1.15854u
9
+ ··· + 2.00610u − 3.52439
−0.195122u
10
+ 3.11585u
9
+ ··· + 5.22561u − 0.152439
a
9
=
−u
5
− u
u
7
+ u
5
+ 2u
3
+ u
a
9
=
−u
5
− u
u
7
+ u
5
+ 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
69
82
u
10
+
319
82
u
9
−
364
41
u
8
+
460
41
u
7
−
935
82
u
6
+
476
41
u
5
−
687
41
u
4
+
520
41
u
3
−
539
41
u
2
+
673
82
u −
444
41
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
11
− 7u
10
+ ··· − 9u + 1
c
2
u
11
+ 11u
10
+ ··· + 5u + 1
c
3
, c
6
u
11
− 6u
10
+ ··· − 24u + 16
c
5
, c
8
u
11
− 2u
10
+ ··· + 96u + 64
c
7
, c
10
u
11
+ 5u
10
+ ··· + 10u + 1
c
9
, c
11
u
11
− u
10
+ ··· + 80u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
11
− 11y
10
+ ··· + 5y −1
c
2
y
11
+ 113y
10
+ ··· − 3895y −1
c
3
, c
6
y
11
+ 30y
10
+ ··· − 2240y −256
c
5
, c
8
y
11
+ 52y
10
+ ··· + 33792y −4096
c
7
, c
10
y
11
− y
10
+ ··· + 80y −1
c
9
, c
11
y
11
+ 31y
10
+ ··· + 6676y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.685415 + 0.773477I
a = −0.73400 + 2.17133I
b = −0.460151 − 0.697009I
−1.32886 − 1.52951I −7.26885 + 4.94950I
u = −0.685415 − 0.773477I
a = −0.73400 − 2.17133I
b = −0.460151 + 0.697009I
−1.32886 + 1.52951I −7.26885 − 4.94950I
u = 0.855917 + 0.653801I
a = 0.594255 + 0.858681I
b = 0.710923 − 1.191110I
4.33457 + 4.30583I −3.61862 − 3.76799I
u = 0.855917 − 0.653801I
a = 0.594255 − 0.858681I
b = 0.710923 + 1.191110I
4.33457 − 4.30583I −3.61862 + 3.76799I
u = −0.315247 + 0.806810I
a = −0.545836 + 0.331424I
b = −0.143998 + 0.360224I
−0.33628 − 1.50726I −2.98443 + 4.38710I
u = −0.315247 − 0.806810I
a = −0.545836 − 0.331424I
b = −0.143998 − 0.360224I
−0.33628 + 1.50726I −2.98443 − 4.38710I
u = 0.94424 + 1.31822I
a = −1.30445 − 1.00484I
b = −1.48748 + 2.15268I
−18.0782 + 10.5314I −5.95863 − 4.05407I
u = 0.94424 − 1.31822I
a = −1.30445 + 1.00484I
b = −1.48748 − 2.15268I
−18.0782 − 10.5314I −5.95863 + 4.05407I
u = 0.110617
a = −3.78537
b = 0.612580
−1.00288 −10.0670
u = 1.64519 + 0.99588I
a = 0.882711 + 0.513382I
b = −1.92558 − 3.89584I
−16.1660 − 1.6365I −5.13576 + 0.13357I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.64519 − 0.99588I
a = 0.882711 − 0.513382I
b = −1.92558 + 3.89584I
−16.1660 + 1.6365I −5.13576 − 0.13357I
6
II. I
u
2
= h−a
2
u − 2au + b − a + u, a
3
− a
2
u + 2a
2
− au − a + u − 2, u
2
+ u + 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
3
=
a
a
2
u + 2au + a − u
a
8
=
1
u + 1
a
6
=
a
2
u + au + a − 3u
−a
2
+ au + 3
a
4
=
−a
2
u − 2a
2
− au − 2a + 2
−a
2
u − a
2
− au − a + u + 2
a
10
=
−u
u + 1
a
1
=
−1
0
a
2
=
a
2
u + 2au + 2a − u
a
2
u + 2au + a − u
a
5
=
a
2
u + au + a − 3u
−a
2
+ au + 3
a
9
=
1
u + 1
a
9
=
1
u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4a
2
− 2au − a − 1
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
3
+ u
2
− 1)
2
c
2
, c
6
(u
3
+ u
2
+ 2u + 1)
2
c
3
(u
3
− u
2
+ 2u − 1)
2
c
4
(u
3
− u
2
+ 1)
2
c
5
, c
8
u
6
c
7
, c
11
(u
2
+ u + 1)
3
c
9
, c
10
(u
2
− u + 1)
3
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
3
− y
2
+ 2y −1)
2
c
2
, c
3
, c
6
(y
3
+ 3y
2
+ 2y −1)
2
c
5
, c
8
y
6
c
7
, c
9
, c
10
c
11
(y
2
+ y + 1)
3
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0.901916 + 0.094973I
b = −0.215080 + 1.307140I
3.02413 + 0.79824I −4.05323 − 2.24743I
u = −0.500000 + 0.866025I
a = −1.362120 + 0.277556I
b = −0.215080 − 1.307140I
3.02413 − 4.85801I −7.63258 + 5.38377I
u = −0.500000 + 0.866025I
a = −2.03980 + 0.49350I
b = −0.569840
−1.11345 − 2.02988I −15.8142 + 11.5861I
u = −0.500000 − 0.866025I
a = 0.901916 − 0.094973I
b = −0.215080 − 1.307140I
3.02413 − 0.79824I −4.05323 + 2.24743I
u = −0.500000 − 0.866025I
a = −1.362120 − 0.277556I
b = −0.215080 + 1.307140I
3.02413 + 4.85801I −7.63258 − 5.38377I
u = −0.500000 − 0.866025I
a = −2.03980 − 0.49350I
b = −0.569840
−1.11345 + 2.02988I −15.8142 − 11.5861I
10
III. I
u
3
= hb, −u
3
+ 2u
2
+ a − 2u, u
4
− u
3
+ u
2
+ 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
3
=
u
3
− 2u
2
+ 2u
0
a
8
=
1
−u
2
a
6
=
1
0
a
4
=
u
3
− 2u
2
+ 2u
0
a
10
=
−u
u
3
+ u
a
1
=
−u
3
u
3
− u
2
− 1
a
2
=
−2u
2
+ 2u
u
3
− u
2
− 1
a
5
=
u
3
−u
3
+ u
2
+ 1
a
9
=
u
2
+ 1
−u
2
a
9
=
u
2
+ 1
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
3
− 3u
2
+ 8u − 12
11
(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
6
u
4
c
5
, c
9
u
4
− u
3
+ 3u
2
− 2u + 1
c
7
u
4
− u
3
+ u
2
+ 1
c
8
, c
11
u
4
+ u
3
+ 3u
2
+ 2u + 1
c
10
u
4
+ u
3
+ u
2
+ 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
6
y
4
c
5
, c
8
, c
9
c
11
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
c
7
, c
10
y
4
+ y
3
+ 3y
2
+ 2y + 1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.351808 + 0.720342I
a = 0.59074 + 2.34806I
b = 0
−1.85594 − 1.41510I −15.1414 + 7.6022I
u = −0.351808 − 0.720342I
a = 0.59074 − 2.34806I
b = 0
−1.85594 + 1.41510I −15.1414 − 7.6022I
u = 0.851808 + 0.911292I
a = 0.409261 − 0.055548I
b = 0
5.14581 + 3.16396I −0.358581 − 1.047693I
u = 0.851808 − 0.911292I
a = 0.409261 + 0.055548I
b = 0
5.14581 − 3.16396I −0.358581 + 1.047693I
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
4
)(u
3
+ u
2
− 1)
2
(u
11
− 7u
10
+ ··· − 9u + 1)
c
2
((u + 1)
4
)(u
3
+ u
2
+ 2u + 1)
2
(u
11
+ 11u
10
+ ··· + 5u + 1)
c
3
u
4
(u
3
− u
2
+ 2u − 1)
2
(u
11
− 6u
10
+ ··· − 24u + 16)
c
4
((u + 1)
4
)(u
3
− u
2
+ 1)
2
(u
11
− 7u
10
+ ··· − 9u + 1)
c
5
u
6
(u
4
− u
3
+ 3u
2
− 2u + 1)(u
11
− 2u
10
+ ··· + 96u + 64)
c
6
u
4
(u
3
+ u
2
+ 2u + 1)
2
(u
11
− 6u
10
+ ··· − 24u + 16)
c
7
((u
2
+ u + 1)
3
)(u
4
− u
3
+ u
2
+ 1)(u
11
+ 5u
10
+ ··· + 10u + 1)
c
8
u
6
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
11
− 2u
10
+ ··· + 96u + 64)
c
9
((u
2
− u + 1)
3
)(u
4
− u
3
+ 3u
2
− 2u + 1)(u
11
− u
10
+ ··· + 80u − 1)
c
10
((u
2
− u + 1)
3
)(u
4
+ u
3
+ u
2
+ 1)(u
11
+ 5u
10
+ ··· + 10u + 1)
c
11
((u
2
+ u + 1)
3
)(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
11
− u
10
+ ··· + 80u − 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
4
)(y
3
− y
2
+ 2y −1)
2
(y
11
− 11y
10
+ ··· + 5y −1)
c
2
((y −1)
4
)(y
3
+ 3y
2
+ 2y −1)
2
(y
11
+ 113y
10
+ ··· − 3895y −1)
c
3
, c
6
y
4
(y
3
+ 3y
2
+ 2y −1)
2
(y
11
+ 30y
10
+ ··· − 2240y −256)
c
5
, c
8
y
6
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
11
+ 52y
10
+ ··· + 33792y −4096)
c
7
, c
10
((y
2
+ y + 1)
3
)(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
11
− y
10
+ ··· + 80y −1)
c
9
, c
11
((y
2
+ y + 1)
3
)(y
4
+ 5y
3
+ ··· + 2y + 1)(y
11
+ 31y
10
+ ··· + 6676y −1)
16
