11n
2
(K11n
2
)
A knot diagram
1
Linearized knot diagam
5 1 7 2 3 10 3 6 11 7 9
Solving Sequence
7,11
10
3,6
5 9 1 2 4 8
c
10
c
6
c
5
c
9
c
11
c
2
c
4
c
8
c
1
, c
3
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= h−2u
33
− 5u
32
+ ··· + 2b − u, −2u
33
− 4u
32
+ ··· + a + 1, u
34
+ 3u
33
+ ··· + 3u
2
− 1i
I
u
2
= h−u
2
b + b
2
+ bu − u + 1, a, u
3
− u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 40 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−2u
33
−5u
32
+· · ·+2b−u, −2u
33
−4u
32
+· · ·+a+1 , u
34
+3u
33
+· · ·+3u
2
−1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
10
=
1
−u
2
a
3
=
2u
33
+ 4u
32
+ ··· + 2u − 1
u
33
+
5
2
u
32
+ ··· − 2u
2
+
1
2
u
a
6
=
u
−u
3
+ u
a
5
=
−u
4
+ u
2
− 1
−
1
2
u
32
− u
31
+ ··· + 3u
2
+
3
2
u
a
9
=
−u
2
+ 1
−u
2
a
1
=
u
4
− u
2
+ 1
u
4
a
2
=
1
2
u
33
+ u
32
+ ··· + 2u +
1
2
1
2
u
33
+ 4u
32
+ ··· + 4u − 2
a
4
=
−2u
33
− 4u
32
+ ··· − 2u + 1
−2u
33
−
11
2
u
32
+ ··· −
5
2
u + 2
a
8
=
−u
6
+ u
4
− 2u
2
+ 1
u
8
− 2u
6
+ 2u
4
− 2u
2
a
8
=
−u
6
+ u
4
− 2u
2
+ 1
u
8
− 2u
6
+ 2u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes =
17
2
u
33
+ 19u
32
+ ··· + 10u −
25
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
34
+ 4u
33
+ ··· + 7u + 1
c
2
u
34
+ 20u
33
+ ··· − 31u + 1
c
3
, c
7
u
34
+ u
33
+ ··· − 160u − 64
c
5
u
34
− 4u
33
+ ··· + 19u + 2
c
6
, c
10
u
34
+ 3u
33
+ ··· + 3u
2
− 1
c
8
u
34
− 3u
33
+ ··· + 4u − 1
c
9
, c
11
u
34
+ 13u
33
+ ··· + 6u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
34
+ 20y
33
+ ··· − 31y + 1
c
2
y
34
− 8y
33
+ ··· − 1215y + 1
c
3
, c
7
y
34
− 35y
33
+ ··· − 29696y + 4096
c
5
y
34
− 36y
33
+ ··· − 209y + 4
c
6
, c
10
y
34
− 13y
33
+ ··· − 6y + 1
c
8
y
34
− 41y
33
+ ··· − 6y + 1
c
9
, c
11
y
34
+ 19y
33
+ ··· + 122y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.992042 + 0.199548I
a = −0.462101 − 0.975157I
b = 0.138300 − 0.172846I
−3.59663 + 0.10881I −14.08590 − 0.63240I
u = −0.992042 − 0.199548I
a = −0.462101 + 0.975157I
b = 0.138300 + 0.172846I
−3.59663 − 0.10881I −14.08590 + 0.63240I
u = −0.794834 + 0.581726I
a = 0.594117 + 0.707965I
b = 0.427182 − 0.203209I
1.44216 − 0.29929I −6.95462 + 0.76731I
u = −0.794834 − 0.581726I
a = 0.594117 − 0.707965I
b = 0.427182 + 0.203209I
1.44216 + 0.29929I −6.95462 − 0.76731I
u = −0.524940 + 0.808295I
a = 0.56013 + 1.37764I
b = −0.888548 + 0.835142I
−2.09915 − 1.64840I −5.13395 + 0.24192I
u = −0.524940 − 0.808295I
a = 0.56013 − 1.37764I
b = −0.888548 − 0.835142I
−2.09915 + 1.64840I −5.13395 − 0.24192I
u = 0.840078 + 0.614168I
a = −0.613136 − 0.577949I
b = −1.41546 + 0.81235I
1.86820 − 2.41838I −3.21586 + 3.79872I
u = 0.840078 − 0.614168I
a = −0.613136 + 0.577949I
b = −1.41546 − 0.81235I
1.86820 + 2.41838I −3.21586 − 3.79872I
u = −0.560590 + 0.879313I
a = −0.41210 − 1.50617I
b = 1.46754 − 1.00369I
−5.42766 − 6.75489I −7.92215 + 3.41714I
u = −0.560590 − 0.879313I
a = −0.41210 + 1.50617I
b = 1.46754 + 1.00369I
−5.42766 + 6.75489I −7.92215 − 3.41714I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.412943 + 0.845694I
a = −0.76752 − 1.52606I
b = 0.44847 − 1.45277I
−6.32362 + 2.72594I −8.84636 − 2.76466I
u = −0.412943 − 0.845694I
a = −0.76752 + 1.52606I
b = 0.44847 + 1.45277I
−6.32362 − 2.72594I −8.84636 + 2.76466I
u = −0.893031 + 0.587012I
a = −0.693593 − 0.583293I
b = −0.956249 − 0.192787I
1.13021 + 4.95087I −8.36503 − 5.99635I
u = −0.893031 − 0.587012I
a = −0.693593 + 0.583293I
b = −0.956249 + 0.192787I
1.13021 − 4.95087I −8.36503 + 5.99635I
u = 0.985371 + 0.556607I
a = 0.620847 + 0.887273I
b = 1.96665 − 0.76219I
−1.53468 − 5.70085I −9.72834 + 6.45202I
u = 0.985371 − 0.556607I
a = 0.620847 − 0.887273I
b = 1.96665 + 0.76219I
−1.53468 + 5.70085I −9.72834 − 6.45202I
u = 0.834354 + 0.777211I
a = −0.618925 + 0.083135I
b = −0.304799 + 1.255790I
2.63328 − 1.69138I −7.83080 + 4.78233I
u = 0.834354 − 0.777211I
a = −0.618925 − 0.083135I
b = −0.304799 − 1.255790I
2.63328 + 1.69138I −7.83080 − 4.78233I
u = 1.15996
a = 1.41319
b = 1.11298
−8.01260 −11.0100
u = 0.691950 + 0.428071I
a = 0.759172 + 0.721685I
b = 1.51068 − 1.17159I
−0.45796 + 1.44409I −6.46359 + 0.67387I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.691950 − 0.428071I
a = 0.759172 − 0.721685I
b = 1.51068 + 1.17159I
−0.45796 − 1.44409I −6.46359 − 0.67387I
u = 1.197460 + 0.057621I
a = −1.51287 − 0.16435I
b = −0.911142 + 0.347693I
−12.04690 − 5.13421I −13.68860 + 3.30024I
u = 1.197460 − 0.057621I
a = −1.51287 + 0.16435I
b = −0.911142 − 0.347693I
−12.04690 + 5.13421I −13.68860 − 3.30024I
u = 0.921897 + 0.768739I
a = 0.213500 − 0.587608I
b = −1.053180 − 0.826825I
2.37182 − 4.13713I −9.61954 + 1.32790I
u = 0.921897 − 0.768739I
a = 0.213500 + 0.587608I
b = −1.053180 + 0.826825I
2.37182 + 4.13713I −9.61954 − 1.32790I
u = −1.072580 + 0.660990I
a = −1.205520 − 0.383155I
b = −1.20314 − 2.13287I
−3.72457 + 7.16368I −7.25453 − 4.71165I
u = −1.072580 − 0.660990I
a = −1.205520 + 0.383155I
b = −1.20314 + 2.13287I
−3.72457 − 7.16368I −7.25453 + 4.71165I
u = −1.110150 + 0.615554I
a = 1.267740 + 0.575502I
b = 0.56941 + 2.11006I
−8.42620 + 2.66430I −11.45013 − 1.91985I
u = −1.110150 − 0.615554I
a = 1.267740 − 0.575502I
b = 0.56941 − 2.11006I
−8.42620 − 2.66430I −11.45013 + 1.91985I
u = −1.089780 + 0.695595I
a = 1.309000 + 0.281439I
b = 1.38267 + 2.54914I
−7.0419 + 12.5932I −9.63219 − 7.64177I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.089780 − 0.695595I
a = 1.309000 − 0.281439I
b = 1.38267 − 2.54914I
−7.0419 − 12.5932I −9.63219 + 7.64177I
u = −0.643512
a = 0.650445
b = 0.109056
−0.881314 −11.5070
u = 0.221560 + 0.275264I
a = 0.92943 + 1.46869I
b = 0.710592 − 0.584374I
−0.37760 + 1.65869I −3.04964 − 3.10072I
u = 0.221560 − 0.275264I
a = 0.92943 − 1.46869I
b = 0.710592 + 0.584374I
−0.37760 − 1.65869I −3.04964 + 3.10072I
8
II. I
u
2
= h−u
2
b + b
2
+ bu − u + 1, a, u
3
− u
2
+ 1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
10
=
1
−u
2
a
3
=
0
b
a
6
=
u
−u
2
+ u + 1
a
5
=
u
−2u
2
+ b + 2u + 1
a
9
=
−u
2
+ 1
−u
2
a
1
=
−u
u
2
− u − 1
a
2
=
u
2
b
bu + 2b
a
4
=
0
b
a
8
=
0
u
a
8
=
0
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
b − 6bu − u
2
+ 6u − 11
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
(u
2
+ u + 1)
3
c
3
, c
7
u
6
c
4
(u
2
− u + 1)
3
c
6
(u
3
+ u
2
− 1)
2
c
8
, c
11
(u
3
+ u
2
+ 2u + 1)
2
c
9
(u
3
− u
2
+ 2u − 1)
2
c
10
(u
3
− u
2
+ 1)
2
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
(y
2
+ y + 1)
3
c
3
, c
7
y
6
c
6
, c
10
(y
3
− y
2
+ 2y −1)
2
c
8
, c
9
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = 0
b = −0.818128 − 0.292480I
3.02413 − 4.85801I −2.74410 + 7.22587I
u = 0.877439 + 0.744862I
a = 0
b = 0.155769 + 0.854759I
3.02413 − 0.79824I −4.03424 − 1.64667I
u = 0.877439 − 0.744862I
a = 0
b = −0.818128 + 0.292480I
3.02413 + 4.85801I −2.74410 − 7.22587I
u = 0.877439 − 0.744862I
a = 0
b = 0.155769 − 0.854759I
3.02413 + 0.79824I −4.03424 + 1.64667I
u = −0.754878
a = 0
b = 0.662359 + 1.147240I
−1.11345 − 2.02988I −12.72167 + 5.84990I
u = −0.754878
a = 0
b = 0.662359 − 1.147240I
−1.11345 + 2.02988I −12.72167 − 5.84990I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u + 1)
3
)(u
34
+ 4u
33
+ ··· + 7u + 1)
c
2
((u
2
+ u + 1)
3
)(u
34
+ 20u
33
+ ··· − 31u + 1)
c
3
, c
7
u
6
(u
34
+ u
33
+ ··· − 160u − 64)
c
4
((u
2
− u + 1)
3
)(u
34
+ 4u
33
+ ··· + 7u + 1)
c
5
((u
2
+ u + 1)
3
)(u
34
− 4u
33
+ ··· + 19u + 2)
c
6
((u
3
+ u
2
− 1)
2
)(u
34
+ 3u
33
+ ··· + 3u
2
− 1)
c
8
((u
3
+ u
2
+ 2u + 1)
2
)(u
34
− 3u
33
+ ··· + 4u − 1)
c
9
((u
3
− u
2
+ 2u − 1)
2
)(u
34
+ 13u
33
+ ··· + 6u + 1)
c
10
((u
3
− u
2
+ 1)
2
)(u
34
+ 3u
33
+ ··· + 3u
2
− 1)
c
11
((u
3
+ u
2
+ 2u + 1)
2
)(u
34
+ 13u
33
+ ··· + 6u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y
2
+ y + 1)
3
)(y
34
+ 20y
33
+ ··· − 31y + 1)
c
2
((y
2
+ y + 1)
3
)(y
34
− 8y
33
+ ··· − 1215y + 1)
c
3
, c
7
y
6
(y
34
− 35y
33
+ ··· − 29696y + 4096)
c
5
((y
2
+ y + 1)
3
)(y
34
− 36y
33
+ ··· − 209y + 4)
c
6
, c
10
((y
3
− y
2
+ 2y −1)
2
)(y
34
− 13y
33
+ ··· − 6y + 1)
c
8
((y
3
+ 3y
2
+ 2y −1)
2
)(y
34
− 41y
33
+ ··· − 6y + 1)
c
9
, c
11
((y
3
+ 3y
2
+ 2y −1)
2
)(y
34
+ 19y
33
+ ··· + 122y + 1)
14
