12n
0218
(K12n
0218
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 11 9 3 12 6 5 10 8
Solving Sequence
6,11 3,5
2 1 4 10 12 9 7 8
c
5
c
2
c
1
c
4
c
10
c
11
c
9
c
6
c
8
c
3
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
22
− 2u
21
+ ··· + b − 1, −u
22
− u
21
+ ··· + a − 1, u
23
+ 2u
22
+ ··· + 2u + 1i
I
u
2
= hu
8
− 2u
6
+ u
5
+ 2u
4
− u
3
+ b + u, u
7
− 2u
5
+ u
4
+ 2u
3
− u
2
+ a + u,
u
9
− u
8
− 2u
7
+ 3u
6
+ u
5
− 3u
4
+ 2u
3
− u + 1i
* 2 irreducible components of dim
C
= 0, with total 32 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
22
−2u
21
+· · ·+b−1, −u
22
−u
21
+· · ·+a−1, u
23
+2u
22
+· · ·+2u+1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
u
22
+ u
21
+ ··· + 2u + 1
u
22
+ 2u
21
+ ··· + 2u + 1
a
5
=
1
u
2
a
2
=
2u
22
+ 2u
21
+ ··· + 3u + 1
2u
22
+ 3u
21
+ ··· + 3u + 2
a
1
=
−u
19
+ 4u
17
− 8u
15
+ 8u
13
− 5u
11
+ 2u
9
− 2u
7
− u
3
−u
21
+ 5u
19
− 13u
17
+ 20u
15
− 20u
13
+ 11u
11
− u
9
− 4u
7
+ u
5
+ u
3
− u
a
4
=
3u
22
+ 3u
21
+ ··· + 5u + 2
3u
22
+ 4u
21
+ ··· + 5u + 3
a
10
=
−u
−u
3
+ u
a
12
=
u
3
u
5
− u
3
+ u
a
9
=
−u
3
−u
3
+ u
a
7
=
u
6
− u
4
+ 1
u
6
− 2u
4
+ u
2
a
8
=
−u
11
+ 2u
9
− 2u
7
− u
3
−u
13
+ 3u
11
− 5u
9
+ 4u
7
− 2u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 10u
22
+ 11u
21
−52u
20
−76u
19
+ 117u
18
+ 231u
17
−102u
16
−390u
15
−69u
14
+ 358u
13
+
268u
12
−113u
11
−261u
10
−92u
9
+98u
8
+84u
7
+19u
6
+10u
5
−11u
4
−33u
3
−u
2
+12u+11
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
23
+ 44u
22
+ ··· + 46u + 1
c
2
, c
4
u
23
− 10u
22
+ ··· + 10u − 1
c
3
, c
7
u
23
− u
22
+ ··· + 512u − 512
c
5
, c
10
u
23
− 2u
22
+ ··· + 2u − 1
c
6
, c
9
u
23
− 6u
22
+ ··· + 30u − 7
c
8
, c
12
u
23
+ 24u
21
+ ··· + 2u − 1
c
11
u
23
− 12u
22
+ ··· + 2u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
23
− 168y
22
+ ··· + 3538y −1
c
2
, c
4
y
23
− 44y
22
+ ··· + 46y −1
c
3
, c
7
y
23
+ 57y
22
+ ··· + 2621440y −262144
c
5
, c
10
y
23
− 12y
22
+ ··· + 2y −1
c
6
, c
9
y
23
+ 12y
22
+ ··· + 410y −49
c
8
, c
12
y
23
+ 48y
22
+ ··· + 2y −1
c
11
y
23
+ 24y
21
+ ··· − 2y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.793555 + 0.695238I
a = 0.27094 − 1.56138I
b = 0.563681 − 0.478675I
19.5669 − 2.6314I −2.85971 + 2.82212I
u = −0.793555 − 0.695238I
a = 0.27094 + 1.56138I
b = 0.563681 + 0.478675I
19.5669 + 2.6314I −2.85971 − 2.82212I
u = 1.022100 + 0.407841I
a = 1.44074 + 0.54244I
b = −0.140040 + 1.113520I
0.05293 + 1.76634I 1.02070 − 2.91905I
u = 1.022100 − 0.407841I
a = 1.44074 − 0.54244I
b = −0.140040 − 1.113520I
0.05293 − 1.76634I 1.02070 + 2.91905I
u = −0.260502 + 0.851460I
a = −0.101202 + 0.883554I
b = 2.59298 − 0.76044I
−16.8734 + 5.0391I −2.11769 − 1.80422I
u = −0.260502 − 0.851460I
a = −0.101202 − 0.883554I
b = 2.59298 + 0.76044I
−16.8734 − 5.0391I −2.11769 + 1.80422I
u = −1.072240 + 0.511021I
a = 0.05623 + 2.46640I
b = 1.31169 + 1.69266I
−0.80606 − 4.86361I 0.58399 + 4.58744I
u = −1.072240 − 0.511021I
a = 0.05623 − 2.46640I
b = 1.31169 − 1.69266I
−0.80606 + 4.86361I 0.58399 − 4.58744I
u = −1.151200 + 0.397103I
a = 0.378436 − 0.814303I
b = −0.061499 − 0.968355I
4.03412 − 1.87941I 8.76933 + 1.17253I
u = −1.151200 − 0.397103I
a = 0.378436 + 0.814303I
b = −0.061499 + 0.968355I
4.03412 + 1.87941I 8.76933 − 1.17253I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.669270 + 0.402839I
a = 0.378364 − 0.834298I
b = 0.169751 + 0.377973I
−1.04289 + 1.55239I −1.64388 − 5.32889I
u = 0.669270 − 0.402839I
a = 0.378364 + 0.834298I
b = 0.169751 − 0.377973I
−1.04289 − 1.55239I −1.64388 + 5.32889I
u = −0.769798
a = 0.600191
b = 0.484933
1.02417 10.8970
u = 1.151500 + 0.497191I
a = −0.371757 − 0.631221I
b = 0.523099 − 0.584172I
3.31594 + 6.22870I 6.08610 − 5.76635I
u = 1.151500 − 0.497191I
a = −0.371757 + 0.631221I
b = 0.523099 + 0.584172I
3.31594 − 6.22870I 6.08610 + 5.76635I
u = 1.233100 + 0.274609I
a = −2.75962 + 0.02500I
b = −1.94681 − 1.66946I
−12.07890 − 1.46711I 2.75909 − 0.30473I
u = 1.233100 − 0.274609I
a = −2.75962 − 0.02500I
b = −1.94681 + 1.66946I
−12.07890 + 1.46711I 2.75909 + 0.30473I
u = 0.152344 + 0.678389I
a = 0.370495 + 0.387642I
b = −0.142963 − 0.519552I
0.49322 − 1.74871I 2.97942 + 3.32574I
u = 0.152344 − 0.678389I
a = 0.370495 − 0.387642I
b = −0.142963 + 0.519552I
0.49322 + 1.74871I 2.97942 − 3.32574I
u = −1.179040 + 0.569666I
a = −1.36258 − 2.90160I
b = −3.23838 − 0.97939I
−14.1285 − 10.2729I 0.79116 + 5.29982I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.179040 − 0.569666I
a = −1.36258 + 2.90160I
b = −3.23838 + 0.97939I
−14.1285 + 10.2729I 0.79116 − 5.29982I
u = −0.386868 + 0.554823I
a = −1.100140 − 0.404039I
b = −1.37398 + 0.73498I
−2.78460 + 0.52794I −2.81721 + 0.26390I
u = −0.386868 − 0.554823I
a = −1.100140 + 0.404039I
b = −1.37398 − 0.73498I
−2.78460 − 0.52794I −2.81721 − 0.26390I
7
II. I
u
2
= hu
8
− 2u
6
+ u
5
+ 2u
4
− u
3
+ b + u, u
7
− 2u
5
+ u
4
+ 2u
3
− u
2
+ a +
u, u
9
− u
8
− 2u
7
+ 3u
6
+ u
5
− 3u
4
+ 2u
3
− u + 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
−u
7
+ 2u
5
− u
4
− 2u
3
+ u
2
− u
−u
8
+ 2u
6
− u
5
− 2u
4
+ u
3
− u
a
5
=
1
u
2
a
2
=
−u
7
+ 2u
5
− u
4
− 2u
3
+ u
2
− u − 1
−u
8
+ 2u
6
− u
5
− 2u
4
+ u
3
− u
2
− u
a
1
=
−1
−u
2
a
4
=
−u
7
+ 2u
5
− u
4
− 2u
3
+ u
2
− u
−u
8
+ 2u
6
− u
5
− 2u
4
+ u
3
− u
a
10
=
−u
−u
3
+ u
a
12
=
u
3
u
5
− u
3
+ u
a
9
=
−u
3
−u
3
+ u
a
7
=
u
6
− u
4
+ 1
u
6
− 2u
4
+ u
2
a
8
=
u
6
− u
4
+ 1
u
6
− 2u
4
+ u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
8
− 2u
7
+ u
6
+ 4u
5
− 3u
4
− 6u
3
+ u
2
− u − 2
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
9
c
3
, c
7
u
9
c
4
(u + 1)
9
c
5
u
9
− u
8
− 2u
7
+ 3u
6
+ u
5
− 3u
4
+ 2u
3
− u + 1
c
6
u
9
− 3u
8
+ 8u
7
− 13u
6
+ 17u
5
− 17u
4
+ 12u
3
− 6u
2
+ u + 1
c
8
u
9
− u
8
+ 2u
7
− u
6
+ 3u
5
− u
4
+ 2u
3
+ u + 1
c
9
u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1
c
10
u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1
c
11
u
9
− 5u
8
+ 12u
7
− 15u
6
+ 9u
5
+ u
4
− 4u
3
+ 2u
2
+ u − 1
c
12
u
9
+ u
8
+ 2u
7
+ u
6
+ 3u
5
+ u
4
+ 2u
3
+ u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
9
c
3
, c
7
y
9
c
5
, c
10
y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1
c
6
, c
9
y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1
c
8
, c
12
y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1
c
11
y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y −1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.772920 + 0.510351I
a = −0.628748 − 1.040710I
b = −0.390818 − 0.846696I
−3.42837 + 2.09337I −2.59545 − 4.13635I
u = 0.772920 − 0.510351I
a = −0.628748 + 1.040710I
b = −0.390818 + 0.846696I
−3.42837 − 2.09337I −2.59545 + 4.13635I
u = −0.825933
a = 1.66309
b = 0.134499
−0.446489 0.580470
u = −1.173910 + 0.391555I
a = 1.321020 + 0.175437I
b = 0.779205 − 0.999551I
2.72642 − 1.33617I 3.11790 + 0.38556I
u = −1.173910 − 0.391555I
a = 1.321020 − 0.175437I
b = 0.779205 + 0.999551I
2.72642 + 1.33617I 3.11790 − 0.38556I
u = 0.141484 + 0.739668I
a = 0.081981 + 0.728244I
b = −1.195640 − 0.366692I
−1.02799 − 2.45442I −1.02595 + 3.19656I
u = 0.141484 − 0.739668I
a = 0.081981 − 0.728244I
b = −1.195640 + 0.366692I
−1.02799 + 2.45442I −1.02595 − 3.19656I
u = 1.172470 + 0.500383I
a = 0.89420 − 1.47834I
b = 1.74000 − 0.61288I
1.95319 + 7.08493I 2.21327 − 6.71575I
u = 1.172470 − 0.500383I
a = 0.89420 + 1.47834I
b = 1.74000 + 0.61288I
1.95319 − 7.08493I 2.21327 + 6.71575I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
23
+ 44u
22
+ ··· + 46u + 1)
c
2
((u − 1)
9
)(u
23
− 10u
22
+ ··· + 10u − 1)
c
3
, c
7
u
9
(u
23
− u
22
+ ··· + 512u − 512)
c
4
((u + 1)
9
)(u
23
− 10u
22
+ ··· + 10u − 1)
c
5
(u
9
− u
8
+ ··· − u + 1)(u
23
− 2u
22
+ ··· + 2u − 1)
c
6
(u
9
− 3u
8
+ 8u
7
− 13u
6
+ 17u
5
− 17u
4
+ 12u
3
− 6u
2
+ u + 1)
· (u
23
− 6u
22
+ ··· + 30u − 7)
c
8
(u
9
− u
8
+ ··· + u + 1)(u
23
+ 24u
21
+ ··· + 2u − 1)
c
9
(u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1)
· (u
23
− 6u
22
+ ··· + 30u − 7)
c
10
(u
9
+ u
8
+ ··· − u − 1)(u
23
− 2u
22
+ ··· + 2u − 1)
c
11
(u
9
− 5u
8
+ 12u
7
− 15u
6
+ 9u
5
+ u
4
− 4u
3
+ 2u
2
+ u − 1)
· (u
23
− 12u
22
+ ··· + 2u − 1)
c
12
(u
9
+ u
8
+ ··· + u − 1)(u
23
+ 24u
21
+ ··· + 2u − 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
9
)(y
23
− 168y
22
+ ··· + 3538y −1)
c
2
, c
4
((y −1)
9
)(y
23
− 44y
22
+ ··· + 46y −1)
c
3
, c
7
y
9
(y
23
+ 57y
22
+ ··· + 2621440y −262144)
c
5
, c
10
(y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1)
· (y
23
− 12y
22
+ ··· + 2y −1)
c
6
, c
9
(y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1)
· (y
23
+ 12y
22
+ ··· + 410y −49)
c
8
, c
12
(y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1)
· (y
23
+ 48y
22
+ ··· + 2y −1)
c
11
(y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y −1)
· (y
23
+ 24y
21
+ ··· − 2y −1)
13
