12n
0120
(K12n
0120
)
A knot diagram
1
Linearized knot diagam
3 5 6 2 10 3 12 11 5 9 7 8
Solving Sequence
5,9
10
3,6
7 11 2 1 4 8 12
c
9
c
5
c
6
c
10
c
2
c
1
c
4
c
8
c
12
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h15121u
16
− 24215u
15
+ ··· + 18155b − 26228, 15121u
16
− 24215u
15
+ ··· + 18155a − 26228,
u
17
− 2u
16
+ 2u
15
+ 7u
13
− 14u
12
+ 14u
11
− 2u
10
+ 7u
9
− 14u
8
+ 14u
7
− 12u
6
− 2u
2
− u + 1i
I
u
2
= h−u
7
+ u
6
+ u
5
− 2u
4
− u
3
+ 2u
2
+ b + u − 2, −u
7
+ u
6
+ u
5
− 2u
4
− u
3
+ 2u
2
+ a − 2,
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 25 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
= h15121u
16
− 24215u
15
+ · · · + 18155b − 26228, 15121u
16
−
24215u
15
+ · · · + 18155a − 26228, u
17
− 2u
16
+ · · · − u + 1i
(i) Arc colorings
a
5
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
3
=
−0.832884u
16
+ 1.33379u
15
+ ··· + 1.24985u + 1.44467
−0.832884u
16
+ 1.33379u
15
+ ··· + 2.24985u + 1.44467
a
6
=
u
−u
3
+ u
a
7
=
0.102561u
16
+ 0.104654u
15
+ ··· − 0.0345359u − 0.615037
0.0863674u
16
+ 0.0881300u
15
+ ··· − 0.0290829u − 0.623189
a
11
=
−u
2
+ 1
−u
2
a
2
=
−0.832884u
16
+ 1.33379u
15
+ ··· + 1.24985u + 1.44467
−0.830185u
16
+ 1.33655u
15
+ ··· + 1.74894u + 1.11270
a
1
=
0.368438u
16
− 0.644451u
15
+ ··· + 0.212669u − 0.317929
0.471000u
16
− 0.539796u
15
+ ··· + 0.178133u − 0.932966
a
4
=
−0.816690u
16
+ 1.35032u
15
+ ··· + 1.24440u + 1.45282
−0.719526u
16
+ 1.44946u
15
+ ··· + 2.21168u + 1.50174
a
8
=
u
4
− u
2
+ 1
u
4
a
12
=
−0.167337u
16
− 0.170752u
15
+ ··· + 0.0563481u + 0.582429
−0.140347u
16
− 0.143211u
15
+ ··· + 0.0472597u − 0.737318
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
116556
18155
u
16
+
32086
3631
u
15
+ ··· +
186341
18155
u +
375193
18155
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
17
+ 37u
16
+ ··· + 129u + 1
c
2
, c
4
u
17
− 9u
16
+ ··· − 9u + 1
c
3
, c
6
u
17
+ 3u
16
+ ··· + 1664u − 256
c
5
, c
9
u
17
+ 2u
16
+ ··· − u − 1
c
7
, c
11
, c
12
u
17
+ 2u
16
+ ··· − u + 1
c
8
, c
10
u
17
+ 18u
15
+ ··· + 5u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
17
− 169y
16
+ ··· + 16813y −1
c
2
, c
4
y
17
− 37y
16
+ ··· + 129y −1
c
3
, c
6
y
17
+ 51y
16
+ ··· + 835584y −65536
c
5
, c
9
y
17
+ 18y
15
+ ··· + 5y −1
c
7
, c
11
, c
12
y
17
− 12y
16
+ ··· + 5y −1
c
8
, c
10
y
17
+ 36y
16
+ ··· + 17y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.01926
a = −0.421975
b = 0.597288
5.87185 17.0960
u = −0.814269 + 0.697849I
a = 0.772298 − 0.496323I
b = −0.041970 + 0.201526I
1.96725 − 4.89234I 8.35716 + 5.36349I
u = −0.814269 − 0.697849I
a = 0.772298 + 0.496323I
b = −0.041970 − 0.201526I
1.96725 + 4.89234I 8.35716 − 5.36349I
u = 0.151212 + 0.886118I
a = −0.24895 − 2.10381I
b = −0.097735 − 1.217690I
−2.98302 + 1.89910I 0.81624 − 3.73789I
u = 0.151212 − 0.886118I
a = −0.24895 + 2.10381I
b = −0.097735 + 1.217690I
−2.98302 − 1.89910I 0.81624 + 3.73789I
u = 0.524511 + 0.603470I
a = −0.232214 − 0.919977I
b = 0.292297 − 0.316506I
−1.41487 + 1.50880I 1.51941 − 3.79939I
u = 0.524511 − 0.603470I
a = −0.232214 + 0.919977I
b = 0.292297 + 0.316506I
−1.41487 − 1.50880I 1.51941 + 3.79939I
u = −0.413009 + 0.524274I
a = −0.410879 + 0.002963I
b = −0.823888 + 0.527237I
1.40326 + 0.77610I 7.08751 + 0.48404I
u = −0.413009 − 0.524274I
a = −0.410879 − 0.002963I
b = −0.823888 − 0.527237I
1.40326 − 0.77610I 7.08751 − 0.48404I
u = −0.568174
a = 0.153918
b = −0.414256
0.739304 14.0850
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.471408
a = 2.32178
b = 2.79318
−0.391449 30.6130
u = 1.08781 + 1.09421I
a = −0.81831 + 1.23483I
b = 0.26950 + 2.32904I
−17.0915 + 9.8759I 4.86349 − 4.59062I
u = 1.08781 − 1.09421I
a = −0.81831 − 1.23483I
b = 0.26950 − 2.32904I
−17.0915 − 9.8759I 4.86349 + 4.59062I
u = −1.09696 + 1.09725I
a = 0.98973 + 1.18652I
b = −0.10723 + 2.28377I
18.1033 − 4.0499I 2.18495 + 1.91746I
u = −1.09696 − 1.09725I
a = 0.98973 − 1.18652I
b = −0.10723 − 2.28377I
18.1033 + 4.0499I 2.18495 − 1.91746I
u = 1.09946 + 1.09842I
a = −1.07854 + 1.01904I
b = 0.02092 + 2.11746I
−17.0762 − 1.7937I 4.77435 + 0.71535I
u = 1.09946 − 1.09842I
a = −1.07854 − 1.01904I
b = 0.02092 − 2.11746I
−17.0762 + 1.7937I 4.77435 − 0.71535I
6
II. I
u
2
= h−u
7
+ u
6
+ u
5
− 2u
4
− u
3
+ 2u
2
+ b + u − 2, −u
7
+ u
6
+ u
5
− 2u
4
−
u
3
+ 2u
2
+ a − 2, u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1i
(i) Arc colorings
a
5
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
3
=
u
7
− u
6
− u
5
+ 2u
4
+ u
3
− 2u
2
+ 2
u
7
− u
6
− u
5
+ 2u
4
+ u
3
− 2u
2
− u + 2
a
6
=
u
−u
3
+ u
a
7
=
u
−u
3
+ u
a
11
=
−u
2
+ 1
−u
2
a
2
=
u
7
− u
6
− u
5
+ 2u
4
+ u
3
− 2u
2
+ 2
u
7
− u
6
− u
5
+ 2u
4
+ u
3
− 2u
2
− 2u + 2
a
1
=
0
−u
a
4
=
u
7
− u
6
− u
5
+ 2u
4
+ u
3
− 2u
2
+ 2
u
7
− u
6
− u
5
+ 2u
4
+ u
3
− 2u
2
− u + 2
a
8
=
u
4
− u
2
+ 1
u
4
a
12
=
−u
6
+ u
4
− 2u
2
+ 1
u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u
2
− 2u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
7
+ 2u
6
− 4u
4
− 3u
3
+ u
2
+ 1
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
8
c
3
, c
6
u
8
c
4
(u + 1)
8
c
5
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1
c
7
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
c
8
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
c
9
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
c
10
u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1
c
11
, c
12
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
8
c
3
, c
6
y
8
c
5
, c
9
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
c
7
, c
11
, c
12
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
c
8
, c
10
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.570868 + 0.730671I
a = 0.663977 − 0.849844I
b = 0.09311 − 1.58052I
−0.604279 − 1.131230I 5.26238 + 0.22273I
u = 0.570868 − 0.730671I
a = 0.663977 + 0.849844I
b = 0.09311 + 1.58052I
−0.604279 + 1.131230I 5.26238 − 0.22273I
u = −0.855237 + 0.665892I
a = −0.727959 − 0.566792I
b = 0.127279 − 1.232690I
−3.80435 − 2.57849I 2.12884 + 3.87967I
u = −0.855237 − 0.665892I
a = −0.727959 + 0.566792I
b = 0.127279 + 1.232690I
−3.80435 + 2.57849I 2.12884 − 3.87967I
u = −1.09818
a = −0.910598
b = 0.187581
4.85780 7.72210
u = 1.031810 + 0.655470I
a = 0.690511 − 0.438656I
b = −0.341297 − 1.094130I
0.73474 + 6.44354I 7.14098 − 6.66742I
u = 1.031810 − 0.655470I
a = 0.690511 + 0.438656I
b = −0.341297 + 1.094130I
0.73474 − 6.44354I 7.14098 + 6.66742I
u = 0.603304
a = 1.65754
b = 1.05424
−0.799899 0.213560
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
17
+ 37u
16
+ ··· + 129u + 1)
c
2
((u − 1)
8
)(u
17
− 9u
16
+ ··· − 9u + 1)
c
3
, c
6
u
8
(u
17
+ 3u
16
+ ··· + 1664u − 256)
c
4
((u + 1)
8
)(u
17
− 9u
16
+ ··· − 9u + 1)
c
5
(u
8
+ u
7
+ ··· − 2u − 1)(u
17
+ 2u
16
+ ··· − u − 1)
c
7
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)(u
17
+ 2u
16
+ ··· − u + 1)
c
8
(u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
· (u
17
+ 18u
15
+ ··· + 5u − 1)
c
9
(u
8
− u
7
+ ··· + 2u − 1)(u
17
+ 2u
16
+ ··· − u − 1)
c
10
(u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
· (u
17
+ 18u
15
+ ··· + 5u − 1)
c
11
, c
12
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
17
+ 2u
16
+ ··· − u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
8
)(y
17
− 169y
16
+ ··· + 16813y −1)
c
2
, c
4
((y −1)
8
)(y
17
− 37y
16
+ ··· + 129y −1)
c
3
, c
6
y
8
(y
17
+ 51y
16
+ ··· + 835584y −65536)
c
5
, c
9
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
17
+ 18y
15
+ ··· + 5y −1)
c
7
, c
11
, c
12
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
17
− 12y
16
+ ··· + 5y −1)
c
8
, c
10
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
17
+ 36y
16
+ ··· + 17y −1)
12
