12n
0119
(K12n
0119
)
A knot diagram
1
Linearized knot diagam
3 5 6 2 9 3 12 10 5 8 7 11
Solving Sequence
5,9 3,6
7 10 2 1 4 8 11 12
c
5
c
6
c
9
c
2
c
1
c
4
c
8
c
10
c
12
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h6027u
16
+ 12005u
15
+ ··· + 18155b + 3034, 2046u
16
+ 7175u
15
+ ··· + 18155a − 15643,
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
= hb + 1, u
8
+ 2u
7
+ 3u
6
+ 3u
5
+ 4u
4
+ 4u
3
+ 3u
2
+ a + 2u + 1, u
9
+ u
8
+ 2u
7
+ u
6
+ 3u
5
+ u
4
+ 2u
3
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 26 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
= h6027u
16
+ 12005u
15
+ · · · + 18155b + 3034, 2046u
16
+ 7175u
15
+
· · · + 18155a − 15643, u
17
+ 2u
16
+ · · · − u − 1i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
u
a
3
=
−0.112696u
16
− 0.395208u
15
+ ··· − 0.119581u + 0.861636
−0.331975u
16
− 0.661250u
15
+ ··· − 0.611787u − 0.167116
a
6
=
1
u
2
a
7
=
−0.297108u
16
− 0.860094u
15
+ ··· − 0.406169u + 0.544313
−0.309777u
16
− 0.234922u
15
+ ··· − 0.512476u + 0.102561
a
10
=
−u
u
a
2
=
−0.444671u
16
− 1.05646u
15
+ ··· − 0.731369u + 0.694519
−0.331975u
16
− 0.661250u
15
+ ··· − 0.611787u − 0.167116
a
1
=
−0.389535u
16
− 1.25558u
15
+ ··· − 0.355660u + 0.912751
0.0924263u
16
+ 0.395483u
15
+ ··· − 0.0505095u − 0.368438
a
4
=
−0.120848u
16
− 0.427706u
15
+ ··· − 0.448857u + 0.864335
−0.283063u
16
− 0.466263u
15
+ ··· − 0.636133u − 0.183310
a
8
=
−u
3
u
3
+ u
a
11
=
−u
5
− u
u
5
+ u
3
+ u
a
12
=
−0.802534u
16
− 0.874966u
15
+ ··· − 0.821261u + 0.711650
0.114128u
16
− 0.545029u
15
+ ··· + 0.609860u − 0.0377857
(ii) Obstruction class = −1
(iii) Cusp Shapes =
25781
18155
u
16
+
6669
3631
u
15
+ ··· +
4791
18155
u −
139178
18155
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
17
+ 40u
16
+ ··· + 501u + 1
c
2
, c
4
u
17
− 10u
16
+ ··· + 25u − 1
c
3
, c
6
u
17
+ 3u
16
+ ··· + 512u + 512
c
5
, c
9
u
17
+ 2u
16
+ ··· − u − 1
c
7
, c
11
u
17
− 2u
16
+ ··· − u − 1
c
8
, c
10
u
17
+ 18u
15
+ ··· + 5u + 1
c
12
u
17
+ 12u
16
+ ··· + 5u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
17
− 244y
16
+ ··· + 241285y −1
c
2
, c
4
y
17
− 40y
16
+ ··· + 501y −1
c
3
, c
6
y
17
+ 57y
16
+ ··· + 6553600y −262144
c
5
, c
9
y
17
+ 18y
15
+ ··· + 5y −1
c
7
, c
11
y
17
− 12y
16
+ ··· + 5y −1
c
8
, c
10
y
17
+ 36y
16
+ ··· + 17y −1
c
12
y
17
− 12y
16
+ ··· + 81y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.01926
a = 1.09396
b = −2.50941
−7.51678 −13.0910
u = 0.814269 + 0.697849I
a = −0.135222 + 1.183260I
b = −0.021919 − 1.343220I
−3.61218 − 4.89234I −8.33842 + 5.53487I
u = 0.814269 − 0.697849I
a = −0.135222 − 1.183260I
b = −0.021919 + 1.343220I
−3.61218 + 4.89234I −8.33842 − 5.53487I
u = −0.151212 + 0.886118I
a = 0.259000 − 0.151859I
b = 0.317232 + 0.130128I
1.33808 + 1.89910I 2.88769 − 4.28758I
u = −0.151212 − 0.886118I
a = 0.259000 + 0.151859I
b = 0.317232 − 0.130128I
1.33808 − 1.89910I 2.88769 + 4.28758I
u = −0.524511 + 0.603470I
a = 0.421833 − 0.890237I
b = −0.173796 + 0.509907I
−0.23007 + 1.50880I −2.26409 − 4.36176I
u = −0.524511 − 0.603470I
a = 0.421833 + 0.890237I
b = −0.173796 − 0.509907I
−0.23007 − 1.50880I −2.26409 + 4.36176I
u = 0.413009 + 0.524274I
a = 2.66953 + 0.96171I
b = −0.655191 + 0.306962I
−3.04819 + 0.77610I −9.75157 + 2.68802I
u = 0.413009 − 0.524274I
a = 2.66953 − 0.96171I
b = −0.655191 − 0.306962I
−3.04819 − 0.77610I −9.75157 − 2.68802I
u = 0.568174
a = −1.49974
b = −1.19167
−2.38424 −2.38490
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.471408
a = 1.15792
b = 0.0451202
−1.25348 −8.31380
u = −1.08781 + 1.09421I
a = −1.21553 + 1.53450I
b = 2.42560 − 0.51435I
15.4466 + 9.8759I −8.70686 − 4.52620I
u = −1.08781 − 1.09421I
a = −1.21553 − 1.53450I
b = 2.42560 + 0.51435I
15.4466 − 9.8759I −8.70686 + 4.52620I
u = 1.09696 + 1.09725I
a = −1.31679 − 1.21871I
b = 2.40736 + 0.21866I
19.7302 − 4.0499I −6.15248 + 1.91448I
u = 1.09696 − 1.09725I
a = −1.31679 + 1.21871I
b = 2.40736 − 0.21866I
19.7302 + 4.0499I −6.15248 − 1.91448I
u = −1.09946 + 1.09842I
a = −1.55889 + 0.97165I
b = 2.52869 + 0.07532I
15.4312 − 1.7937I −8.77960 + 0.70466I
u = −1.09946 − 1.09842I
a = −1.55889 − 0.97165I
b = 2.52869 − 0.07532I
15.4312 + 1.7937I −8.77960 − 0.70466I
6
II.
I
u
2
= hb +1, u
8
+ 2u
7
+ · · · + a + 1, u
9
+ u
8
+ 2u
7
+ u
6
+ 3u
5
+ u
4
+ 2u
3
+ u − 1i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
u
a
3
=
−u
8
− 2u
7
− 3u
6
− 3u
5
− 4u
4
− 4u
3
− 3u
2
− 2u − 1
−1
a
6
=
1
u
2
a
7
=
1
u
2
a
10
=
−u
u
a
2
=
−u
8
− 2u
7
− 3u
6
− 3u
5
− 4u
4
− 4u
3
− 3u
2
− 2u − 2
−1
a
1
=
−1
0
a
4
=
−u
8
− 2u
7
− 3u
6
− 3u
5
− 4u
4
− 4u
3
− 3u
2
− 2u − 1
−1
a
8
=
−u
3
u
3
+ u
a
11
=
−u
5
− u
u
5
+ u
3
+ u
a
12
=
−u
7
− 2u
5
− 2u
3
− 2u
u
8
+ u
7
+ u
6
+ 2u
5
+ u
4
+ 2u
3
+ 2u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
8
− 8u
7
− 12u
6
− 11u
5
− 18u
4
− 17u
3
− 15u
2
− 6u − 16
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
9
c
3
, c
6
u
9
c
4
(u + 1)
9
c
5
u
9
+ u
8
+ 2u
7
+ u
6
+ 3u
5
+ u
4
+ 2u
3
+ u − 1
c
7
u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1
c
8
u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1
c
9
u
9
− u
8
+ 2u
7
− u
6
+ 3u
5
− u
4
+ 2u
3
+ u + 1
c
10
u
9
− 3u
8
+ 8u
7
− 13u
6
+ 17u
5
− 17u
4
+ 12u
3
− 6u
2
+ u + 1
c
11
u
9
− u
8
− 2u
7
+ 3u
6
+ u
5
− 3u
4
+ 2u
3
− u + 1
c
12
u
9
+ 5u
8
+ 12u
7
+ 15u
6
+ 9u
5
− u
4
− 4u
3
− 2u
2
+ u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
9
c
3
, c
6
y
9
c
5
, c
9
y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1
c
7
, c
11
y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1
c
8
, c
10
y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1
c
12
y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y −1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.140343 + 0.966856I
a = 1.004430 + 0.297869I
b = −1.00000
0.13850 − 2.09337I −5.16894 + 4.06115I
u = 0.140343 − 0.966856I
a = 1.004430 − 0.297869I
b = −1.00000
0.13850 + 2.09337I −5.16894 − 4.06115I
u = 0.628449 + 0.875112I
a = 0.275254 + 0.816341I
b = −1.00000
−2.26187 − 2.45442I −4.66498 + 3.27944I
u = 0.628449 − 0.875112I
a = 0.275254 − 0.816341I
b = −1.00000
−2.26187 + 2.45442I −4.66498 − 3.27944I
u = −0.796005 + 0.733148I
a = −0.070080 − 0.850995I
b = −1.00000
−6.01628 − 1.33617I −9.21174 + 0.80685I
u = −0.796005 − 0.733148I
a = −0.070080 + 0.850995I
b = −1.00000
−6.01628 + 1.33617I −9.21174 − 0.80685I
u = −0.728966 + 0.986295I
a = 0.195086 − 0.635552I
b = −1.00000
−5.24306 + 7.08493I −7.33806 − 6.93476I
u = −0.728966 − 0.986295I
a = 0.195086 + 0.635552I
b = −1.00000
−5.24306 − 7.08493I −7.33806 + 6.93476I
u = 0.512358
a = −3.80937
b = −1.00000
−2.84338 −27.2330
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
17
+ 40u
16
+ ··· + 501u + 1)
c
2
((u − 1)
9
)(u
17
− 10u
16
+ ··· + 25u − 1)
c
3
, c
6
u
9
(u
17
+ 3u
16
+ ··· + 512u + 512)
c
4
((u + 1)
9
)(u
17
− 10u
16
+ ··· + 25u − 1)
c
5
(u
9
+ u
8
+ ··· + u − 1)(u
17
+ 2u
16
+ ··· − u − 1)
c
7
(u
9
+ u
8
+ ··· − u − 1)(u
17
− 2u
16
+ ··· − u − 1)
c
8
(u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1)
· (u
17
+ 18u
15
+ ··· + 5u + 1)
c
9
(u
9
− u
8
+ ··· + u + 1)(u
17
+ 2u
16
+ ··· − u − 1)
c
10
(u
9
− 3u
8
+ 8u
7
− 13u
6
+ 17u
5
− 17u
4
+ 12u
3
− 6u
2
+ u + 1)
· (u
17
+ 18u
15
+ ··· + 5u + 1)
c
11
(u
9
− u
8
+ ··· − u + 1)(u
17
− 2u
16
+ ··· − u − 1)
c
12
(u
9
+ 5u
8
+ 12u
7
+ 15u
6
+ 9u
5
− u
4
− 4u
3
− 2u
2
+ u + 1)
· (u
17
+ 12u
16
+ ··· + 5u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
9
)(y
17
− 244y
16
+ ··· + 241285y −1)
c
2
, c
4
((y −1)
9
)(y
17
− 40y
16
+ ··· + 501y −1)
c
3
, c
6
y
9
(y
17
+ 57y
16
+ ··· + 6553600y −262144)
c
5
, c
9
(y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1)
· (y
17
+ 18y
15
+ ··· + 5y −1)
c
7
, c
11
(y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1)
· (y
17
− 12y
16
+ ··· + 5y −1)
c
8
, c
10
(y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1)
· (y
17
+ 36y
16
+ ··· + 17y −1)
c
12
(y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y −1)
· (y
17
− 12y
16
+ ··· + 81y −1)
12
