12n
0172
(K12n
0172
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 11 9 3 5 12 6 10 7
Solving Sequence
6,10
11 12
3,5
2 1 4 9 7 8
c
10
c
11
c
5
c
2
c
1
c
4
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
19
− 2u
18
+ ··· + b + 1, −u
19
− u
18
+ ··· + a − 1, u
20
+ 2u
19
+ ··· − 3u − 1i
I
u
2
= hu
7
+ u
5
+ 2u
3
+ u
2
+ b + u, u
6
+ u
4
+ 2u
2
+ a + u + 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 29 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
19
−2u
18
+· · ·+b+1, −u
19
−u
18
+· · ·+a−1, u
20
+2u
19
+· · ·−3u−1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
12
=
u
2
+ 1
u
2
a
3
=
u
19
+ u
18
+ ··· + u
2
+ 1
u
19
+ 2u
18
+ ··· − 4u − 1
a
5
=
u
u
3
+ u
a
2
=
2u
19
+ 2u
18
+ ··· − 4u
3
+ 1
2u
19
+ 4u
18
+ ··· − 6u − 2
a
1
=
−u
16
− 3u
14
− 7u
12
− 10u
10
− 11u
8
− 8u
6
− 4u
4
+ 1
−u
16
− 2u
14
− 4u
12
− 4u
10
− 2u
8
+ 2u
4
+ 2u
2
a
4
=
−3u
19
− 3u
18
+ ··· + u
2
+ 2u
−3u
19
− 6u
18
+ ··· + 9u + 3
a
9
=
u
4
+ u
2
+ 1
u
4
a
7
=
−u
9
− 2u
7
− 3u
5
− 2u
3
− u
−u
9
− u
7
− u
5
+ u
a
8
=
−u
8
− u
6
− u
4
+ 1
−u
10
− 2u
8
− 3u
6
− 2u
4
− u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
19
+ 2u
18
+ 12u
17
+ 2u
16
+ 32u
15
+ 51u
13
− 13u
12
+ 70u
11
−
37u
10
+ 68u
9
− 56u
8
+ 60u
7
− 63u
6
+ 33u
5
− 47u
4
+ 20u
3
− 17u
2
+ 4u − 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
20
+ 42u
19
+ ··· + 23u + 1
c
2
, c
4
u
20
− 10u
19
+ ··· + 5u − 1
c
3
, c
7
u
20
− u
19
+ ··· + 512u + 512
c
5
, c
10
u
20
+ 2u
19
+ ··· − 3u − 1
c
6
u
20
− 10u
19
+ ··· + 85u − 43
c
8
, c
12
u
20
+ 2u
19
+ ··· − 3u − 1
c
9
, c
11
u
20
− 6u
19
+ ··· + 3u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
20
− 198y
19
+ ··· − 639y + 1
c
2
, c
4
y
20
− 42y
19
+ ··· − 23y + 1
c
3
, c
7
y
20
− 57y
19
+ ··· + 1310720y + 262144
c
5
, c
10
y
20
+ 6y
19
+ ··· − 3y + 1
c
6
y
20
− 18y
19
+ ··· − 4731y + 1849
c
8
, c
12
y
20
− 42y
19
+ ··· − 3y + 1
c
9
, c
11
y
20
+ 18y
19
+ ··· − 91y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.124469 + 0.908169I
a = −0.110121 − 0.528184I
b = −0.493387 + 0.034266I
1.75893 + 1.54466I −2.08831 − 4.86880I
u = −0.124469 − 0.908169I
a = −0.110121 + 0.528184I
b = −0.493387 − 0.034266I
1.75893 − 1.54466I −2.08831 + 4.86880I
u = −0.654133 + 0.871364I
a = 0.515368 − 0.661677I
b = −0.239442 − 0.881898I
−0.94442 + 2.54047I −4.33649 − 2.91190I
u = −0.654133 − 0.871364I
a = 0.515368 + 0.661677I
b = −0.239442 + 0.881898I
−0.94442 − 2.54047I −4.33649 + 2.91190I
u = 0.783905 + 0.795880I
a = 0.264051 + 0.040665I
b = −0.174627 − 0.242030I
−3.99381 + 0.03901I −12.11070 − 0.38222I
u = 0.783905 − 0.795880I
a = 0.264051 − 0.040665I
b = −0.174627 + 0.242030I
−3.99381 − 0.03901I −12.11070 + 0.38222I
u = 0.284303 + 1.108040I
a = −1.021690 − 0.407132I
b = −0.160650 + 1.247820I
−15.4921 − 3.6755I −8.75395 + 3.01938I
u = 0.284303 − 1.108040I
a = −1.021690 + 0.407132I
b = −0.160650 − 1.247820I
−15.4921 + 3.6755I −8.75395 − 3.01938I
u = −0.903441 + 0.739223I
a = 0.70712 − 2.78319I
b = −1.41856 − 3.03717I
15.9851 − 3.3273I −13.99686 + 0.12457I
u = −0.903441 − 0.739223I
a = 0.70712 + 2.78319I
b = −1.41856 + 3.03717I
15.9851 + 3.3273I −13.99686 − 0.12457I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.806281
a = 1.15429
b = −0.930683
−19.2190 −14.0620
u = −0.803779 + 0.892292I
a = −1.76637 + 2.05694I
b = 0.41562 + 3.22944I
−6.89324 + 3.01130I −13.76983 − 2.67964I
u = −0.803779 − 0.892292I
a = −1.76637 − 2.05694I
b = 0.41562 − 3.22944I
−6.89324 − 3.01130I −13.76983 + 2.67964I
u = 0.745691 + 0.953776I
a = −0.209775 + 0.092781I
b = 0.244919 + 0.130892I
−3.50610 − 5.81808I −10.51658 + 5.66339I
u = 0.745691 − 0.953776I
a = −0.209775 − 0.092781I
b = 0.244919 − 0.130892I
−3.50610 + 5.81808I −10.51658 − 5.66339I
u = −0.784642 + 1.031280I
a = 2.58208 − 1.12211I
b = 0.86881 − 3.54329I
16.8999 + 9.5713I −12.72981 − 4.75135I
u = −0.784642 − 1.031280I
a = 2.58208 + 1.12211I
b = 0.86881 + 3.54329I
16.8999 − 9.5713I −12.72981 + 4.75135I
u = 0.216278 + 0.660670I
a = 0.396693 + 1.247630I
b = 0.738473 − 0.531917I
−1.26262 − 0.98137I −9.38815 + 0.54437I
u = 0.216278 − 0.660670I
a = 0.396693 − 1.247630I
b = 0.738473 + 0.531917I
−1.26262 + 0.98137I −9.38815 − 0.54437I
u = −0.325708
a = 1.13099
b = 0.368372
−0.688798 −14.5570
6
II. I
u
2
= hu
7
+ u
5
+ 2u
3
+ u
2
+ b + u, u
6
+ u
4
+ 2u
2
+ a + u + 1, u
9
− u
8
+
2u
7
− u
6
+ 3u
5
− u
4
+ 2u
3
+ u + 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
12
=
u
2
+ 1
u
2
a
3
=
−u
6
− u
4
− 2u
2
− u − 1
−u
7
− u
5
− 2u
3
− u
2
− u
a
5
=
u
u
3
+ u
a
2
=
−u
6
− u
4
− 2u
2
− 2u − 1
−u
7
− u
5
− 3u
3
− u
2
− 2u
a
1
=
−u
−u
3
− u
a
4
=
−u
6
− u
4
− 2u
2
− u − 1
−u
7
− u
5
− 2u
3
− u
2
− u
a
9
=
u
4
+ u
2
+ 1
u
4
a
7
=
−u
8
− u
6
− u
4
+ 1
−u
8
+ u
7
− u
6
+ 2u
5
− u
4
+ 2u
3
+ 2u + 1
a
8
=
−u
8
− u
6
− u
4
+ 1
−u
8
+ u
7
− u
6
+ 2u
5
− u
4
+ 2u
3
+ 2u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
7
− 4u
6
+ 3u
5
− 3u
4
+ 6u
3
− 3u
2
− u − 13
7
(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
+ u
6
+ 3u
5
+ u
4
+ 2u
3
+ u − 1
c
6
u
9
− 5u
8
+ 12u
7
− 15u
6
+ 9u
5
+ u
4
− 4u
3
+ 2u
2
+ u − 1
c
8
, c
12
u
9
− u
8
− 2u
7
+ 3u
6
+ u
5
− 3u
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
− u
6
+ 3u
5
− u
4
+ 2u
3
+ u + 1
c
11
u
9
− 3u
8
+ 8u
7
− 13u
6
+ 17u
5
− 17u
4
+ 12u
3
− 6u
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
7
y
9
c
5
, c
10
y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1
c
6
y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y −1
c
8
, c
12
y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1
c
9
, c
11
y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.140343 + 0.966856I
a = 0.770941 − 0.258974I
b = 0.142194 + 0.781734I
0.13850 + 2.09337I −6.69021 − 3.87975I
u = −0.140343 − 0.966856I
a = 0.770941 + 0.258974I
b = 0.142194 − 0.781734I
0.13850 − 2.09337I −6.69021 + 3.87975I
u = −0.628449 + 0.875112I
a = 0.147409 − 0.367985I
b = 0.229389 + 0.360259I
−2.26187 + 2.45442I −12.49381 − 3.35442I
u = −0.628449 − 0.875112I
a = 0.147409 + 0.367985I
b = 0.229389 − 0.360259I
−2.26187 − 2.45442I −12.49381 + 3.35442I
u = 0.796005 + 0.733148I
a = −0.24323 − 1.73417I
b = 1.07779 − 1.55873I
−6.01628 + 1.33617I −13.53709 − 1.22905I
u = 0.796005 − 0.733148I
a = −0.24323 + 1.73417I
b = 1.07779 + 1.55873I
−6.01628 − 1.33617I −13.53709 + 1.22905I
u = 0.728966 + 0.986295I
a = −1.62529 − 0.46000I
b = −0.73109 − 1.93833I
−5.24306 − 7.08493I −12.02676 + 6.64241I
u = 0.728966 − 0.986295I
a = −1.62529 + 0.46000I
b = −0.73109 + 1.93833I
−5.24306 + 7.08493I −12.02676 − 6.64241I
u = −0.512358
a = −1.09967
b = 0.563422
−2.84338 −14.5040
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
20
+ 42u
19
+ ··· + 23u + 1)
c
2
((u − 1)
9
)(u
20
− 10u
19
+ ··· + 5u − 1)
c
3
, c
7
u
9
(u
20
− u
19
+ ··· + 512u + 512)
c
4
((u + 1)
9
)(u
20
− 10u
19
+ ··· + 5u − 1)
c
5
(u
9
+ u
8
+ ··· + u − 1)(u
20
+ 2u
19
+ ··· − 3u − 1)
c
6
(u
9
− 5u
8
+ 12u
7
− 15u
6
+ 9u
5
+ u
4
− 4u
3
+ 2u
2
+ u − 1)
· (u
20
− 10u
19
+ ··· + 85u − 43)
c
8
, c
12
(u
9
− u
8
+ ··· − u + 1)(u
20
+ 2u
19
+ ··· − 3u − 1)
c
9
(u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1)
· (u
20
− 6u
19
+ ··· + 3u + 1)
c
10
(u
9
− u
8
+ ··· + u + 1)(u
20
+ 2u
19
+ ··· − 3u − 1)
c
11
(u
9
− 3u
8
+ 8u
7
− 13u
6
+ 17u
5
− 17u
4
+ 12u
3
− 6u
2
+ u + 1)
· (u
20
− 6u
19
+ ··· + 3u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
9
)(y
20
− 198y
19
+ ··· − 639y + 1)
c
2
, c
4
((y −1)
9
)(y
20
− 42y
19
+ ··· − 23y + 1)
c
3
, c
7
y
9
(y
20
− 57y
19
+ ··· + 1310720y + 262144)
c
5
, c
10
(y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1)
· (y
20
+ 6y
19
+ ··· − 3y + 1)
c
6
(y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y −1)
· (y
20
− 18y
19
+ ··· − 4731y + 1849)
c
8
, c
12
(y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1)
· (y
20
− 42y
19
+ ··· − 3y + 1)
c
9
, c
11
(y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1)
· (y
20
+ 18y
19
+ ··· − 91y + 1)
12
