12n
0562
(K12n
0562
)
A knot diagram
1
Linearized knot diagam
3 7 9 12 11 2 4 12 1 7 5 4
Solving Sequence
4,12
5
1,9
3 8 7 2 6 11 10
c
4
c
12
c
3
c
8
c
7
c
2
c
6
c
11
c
10
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−1863653922u
24
− 2022363077u
23
+ ··· + 3360966496b + 12865105183,
1528807327u
24
+ 2234282167u
23
+ ··· + 3360966496a + 6708798419, u
25
+ u
24
+ ··· − 13u + 1i
I
u
2
= h−u
3
+ b − 2u, u
2
a + a
2
+ u
2
+ 2a + 3, u
4
+ 3u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 33 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−1.86 × 10
9
u
24
− 2.02 × 10
9
u
23
+ · · · + 3.36 × 10
9
b + 1.29 × 10
10
, 1.53 ×
10
9
u
24
+2.23×10
9
u
23
+· · · +3.36×10
9
a+6.71×10
9
, u
25
+u
24
+· · ·−13u +1i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
u
a
5
=
1
−u
2
a
1
=
u
u
a
9
=
−0.454871u
24
− 0.664774u
23
+ ··· − 25.1625u − 1.99609
0.554499u
24
+ 0.601721u
23
+ ··· + 24.9344u − 3.82780
a
3
=
3.46960u
24
+ 3.91721u
23
+ ··· + 154.960u − 24.4260
0.0868909u
24
+ 0.140131u
23
+ ··· + 2.56515u + 0.501418
a
8
=
−0.454871u
24
− 0.664774u
23
+ ··· − 25.1625u − 1.99609
0.508672u
24
+ 0.570274u
23
+ ··· + 22.6605u − 3.61790
a
7
=
0.0538011u
24
− 0.0944999u
23
+ ··· − 2.50201u − 5.61399
0.508672u
24
+ 0.570274u
23
+ ··· + 22.6605u − 3.61790
a
2
=
1.70781u
24
+ 1.96070u
23
+ ··· + 78.9836u − 7.58147
−0.391234u
24
− 0.397877u
23
+ ··· − 17.0453u + 3.25795
a
6
=
−u
2
− 1
u
4
+ 2u
2
a
11
=
−u
u
3
+ u
a
10
=
−0.491571u
24
− 0.694228u
23
+ ··· − 27.4957u − 1.73897
0.517800u
24
+ 0.572267u
23
+ ··· + 22.6011u − 3.57068
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
2040219475
840241624
u
24
−
4044991761
1680483248
u
23
+ ··· −
48458587153
420120812
u +
40726820499
1680483248
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
25
+ 17u
24
+ ··· − 99u + 25
c
2
, c
6
u
25
− u
24
+ ··· − u + 5
c
3
u
25
− u
24
+ ··· + 4u + 4
c
4
, c
5
, c
11
c
12
u
25
+ u
24
+ ··· − 13u + 1
c
7
u
25
+ 3u
24
+ ··· − 64u + 16
c
8
u
25
− 3u
24
+ ··· + u − 5
c
9
u
25
+ 3u
24
+ ··· − 508u − 284
c
10
u
25
− 3u
24
+ ··· − 88u + 16
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
25
− 13y
24
+ ··· + 31101y −625
c
2
, c
6
y
25
− 17y
24
+ ··· − 99y −25
c
3
y
25
+ 3y
24
+ ··· − 88y −16
c
4
, c
5
, c
11
c
12
y
25
+ 39y
24
+ ··· + 81y −1
c
7
y
25
− 59y
24
+ ··· − 128y −256
c
8
y
25
+ 43y
24
+ ··· − 199y −25
c
9
y
25
− 33y
24
+ ··· + 788576y −80656
c
10
y
25
+ 47y
24
+ ··· − 11232y −256
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.624991 + 0.595786I
a = 0.484646 − 1.009420I
b = 0.941749 + 0.518744I
−3.66314 − 3.94550I −5.29300 + 5.55988I
u = −0.624991 − 0.595786I
a = 0.484646 + 1.009420I
b = 0.941749 − 0.518744I
−3.66314 + 3.94550I −5.29300 − 5.55988I
u = 0.252585 + 0.824264I
a = −0.240808 − 0.235376I
b = 0.233701 + 1.203520I
−0.748773 − 0.682544I −5.47786 − 0.77855I
u = 0.252585 − 0.824264I
a = −0.240808 + 0.235376I
b = 0.233701 − 1.203520I
−0.748773 + 0.682544I −5.47786 + 0.77855I
u = −0.056975 + 0.796798I
a = 0.57589 − 1.49170I
b = 0.245174 − 0.378805I
−0.91521 + 2.31525I −6.76575 − 3.60061I
u = −0.056975 − 0.796798I
a = 0.57589 + 1.49170I
b = 0.245174 + 0.378805I
−0.91521 − 2.31525I −6.76575 + 3.60061I
u = 0.190393 + 1.306700I
a = 1.118360 + 0.139326I
b = 0.842110 − 0.769571I
−5.78601 + 2.83024I −3.27261 − 3.01183I
u = 0.190393 − 1.306700I
a = 1.118360 − 0.139326I
b = 0.842110 + 0.769571I
−5.78601 − 2.83024I −3.27261 + 3.01183I
u = −0.400919 + 1.321610I
a = −1.216630 + 0.430422I
b = −1.04154 − 0.95993I
−9.75869 − 7.65326I −6.00769 + 5.46151I
u = −0.400919 − 1.321610I
a = −1.216630 − 0.430422I
b = −1.04154 + 0.95993I
−9.75869 + 7.65326I −6.00769 − 5.46151I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.609656
a = 0.373523
b = −0.820762
−2.05187 −4.04220
u = 0.309869 + 0.431283I
a = −0.661655 − 0.723351I
b = −0.316822 + 0.492066I
0.029101 + 1.081130I 0.32410 − 6.32395I
u = 0.309869 − 0.431283I
a = −0.661655 + 0.723351I
b = −0.316822 − 0.492066I
0.029101 − 1.081130I 0.32410 + 6.32395I
u = −0.08247 + 1.56386I
a = −0.673650 + 0.474452I
b = −0.655196 − 0.330676I
−9.03687 + 1.50948I −7.50220 − 1.53531I
u = −0.08247 − 1.56386I
a = −0.673650 − 0.474452I
b = −0.655196 + 0.330676I
−9.03687 − 1.50948I −7.50220 + 1.53531I
u = 0.13440 + 1.57758I
a = −0.296006 − 0.170433I
b = −0.570998 − 1.077920I
−9.03805 + 1.00676I −6.89835 + 0.I
u = 0.13440 − 1.57758I
a = −0.296006 + 0.170433I
b = −0.570998 + 1.077920I
−9.03805 − 1.00676I −6.89835 + 0.I
u = 0.05831 + 1.83206I
a = −1.330010 + 0.291426I
b = −1.11436 + 1.10386I
−17.4985 + 4.0971I −3.59802 + 0.I
u = 0.05831 − 1.83206I
a = −1.330010 − 0.291426I
b = −1.11436 − 1.10386I
−17.4985 − 4.0971I −3.59802 + 0.I
u = −0.12155 + 1.83051I
a = 1.333980 + 0.150763I
b = 1.04153 + 1.31415I
18.2454 − 10.2132I −5.59802 + 4.65205I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.12155 − 1.83051I
a = 1.333980 − 0.150763I
b = 1.04153 − 1.31415I
18.2454 + 10.2132I −5.59802 − 4.65205I
u = 0.138522 + 0.029459I
a = −6.47837 − 1.35588I
b = −0.074953 + 0.990969I
1.72086 + 2.04571I 7.83287 − 4.05455I
u = 0.138522 − 0.029459I
a = −6.47837 + 1.35588I
b = −0.074953 − 0.990969I
1.72086 − 2.04571I 7.83287 + 4.05455I
u = 0.00765 + 1.88332I
a = 1.197500 + 0.358483I
b = 1.37998 + 0.95343I
16.9141 + 1.4662I −6.72237 + 0.I
u = 0.00765 − 1.88332I
a = 1.197500 − 0.358483I
b = 1.37998 − 0.95343I
16.9141 − 1.4662I −6.72237 + 0.I
7
II. I
u
2
= h−u
3
+ b − 2u, u
2
a + a
2
+ u
2
+ 2a + 3, u
4
+ 3u
2
+ 1i
(i) Arc colorings
a
4
=
1
0
a
12
=
0
u
a
5
=
1
−u
2
a
1
=
u
u
a
9
=
a
u
3
+ 2u
a
3
=
−u
3
a − 2au + 1
1
a
8
=
a
u
2
a + u
3
+ 2u
a
7
=
u
2
a + u
3
+ a + 2u
u
2
a + u
3
+ 2u
a
2
=
−u
3
a − u
2
a − u
3
− 2au − a − 2u
−u
2
a − a + u
a
6
=
u
2
+ 1
u
2
+ 1
a
11
=
−u
u
3
+ u
a
10
=
u
2
a + u
3
+ a + u
u
2
a + 2u
3
+ 3u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
a − 4a − 4
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)
4
c
2
, c
6
, c
8
(u
4
− u
2
+ 1)
2
c
3
(u
2
+ 1)
4
c
4
, c
5
, c
11
c
12
(u
4
+ 3u
2
+ 1)
2
c
7
u
8
− 2u
7
+ 9u
6
− 8u
5
+ 17u
4
− 16u
3
+ 4u
2
+ 4
c
9
u
8
+ 4u
7
− 14u
5
− 7u
4
+ 14u
3
+ 22u
2
+ 12u + 4
c
10
(u − 1)
8
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
2
+ y + 1)
4
c
2
, c
6
, c
8
(y
2
− y + 1)
4
c
3
(y + 1)
8
c
4
, c
5
, c
11
c
12
(y
2
+ 3y + 1)
4
c
7
y
8
+ 14y
7
+ 83y
6
+ 186y
5
+ 113y
4
− 48y
3
+ 152y
2
+ 32y + 16
c
9
y
8
− 16y
7
+ 98y
6
− 264y
5
+ 353y
4
− 168y
3
+ 92y
2
+ 32y + 16
c
10
(y −1)
8
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034I
a = −0.80902 + 1.40126I
b = 1.000000I
0.65797 + 2.02988I −2.00000 − 3.46410I
u = 0.618034I
a = −0.80902 − 1.40126I
b = 1.000000I
0.65797 − 2.02988I −2.00000 + 3.46410I
u = − 0.618034I
a = −0.80902 + 1.40126I
b = − 1.000000I
0.65797 + 2.02988I −2.00000 − 3.46410I
u = − 0.618034I
a = −0.80902 − 1.40126I
b = − 1.000000I
0.65797 − 2.02988I −2.00000 + 3.46410I
u = 1.61803I
a = 0.309017 + 0.535233I
b = − 1.000000I
−7.23771 − 2.02988I −2.00000 + 3.46410I
u = 1.61803I
a = 0.309017 − 0.535233I
b = − 1.000000I
−7.23771 + 2.02988I −2.00000 − 3.46410I
u = − 1.61803I
a = 0.309017 + 0.535233I
b = 1.000000I
−7.23771 − 2.02988I −2.00000 + 3.46410I
u = − 1.61803I
a = 0.309017 − 0.535233I
b = 1.000000I
−7.23771 + 2.02988I −2.00000 − 3.46410I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
4
)(u
25
+ 17u
24
+ ··· − 99u + 25)
c
2
, c
6
((u
4
− u
2
+ 1)
2
)(u
25
− u
24
+ ··· − u + 5)
c
3
((u
2
+ 1)
4
)(u
25
− u
24
+ ··· + 4u + 4)
c
4
, c
5
, c
11
c
12
((u
4
+ 3u
2
+ 1)
2
)(u
25
+ u
24
+ ··· − 13u + 1)
c
7
(u
8
− 2u
7
+ 9u
6
− 8u
5
+ 17u
4
− 16u
3
+ 4u
2
+ 4)
· (u
25
+ 3u
24
+ ··· − 64u + 16)
c
8
((u
4
− u
2
+ 1)
2
)(u
25
− 3u
24
+ ··· + u − 5)
c
9
(u
8
+ 4u
7
− 14u
5
− 7u
4
+ 14u
3
+ 22u
2
+ 12u + 4)
· (u
25
+ 3u
24
+ ··· − 508u − 284)
c
10
((u − 1)
8
)(u
25
− 3u
24
+ ··· − 88u + 16)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
4
)(y
25
− 13y
24
+ ··· + 31101y −625)
c
2
, c
6
((y
2
− y + 1)
4
)(y
25
− 17y
24
+ ··· − 99y −25)
c
3
((y + 1)
8
)(y
25
+ 3y
24
+ ··· − 88y −16)
c
4
, c
5
, c
11
c
12
((y
2
+ 3y + 1)
4
)(y
25
+ 39y
24
+ ··· + 81y −1)
c
7
(y
8
+ 14y
7
+ 83y
6
+ 186y
5
+ 113y
4
− 48y
3
+ 152y
2
+ 32y + 16)
· (y
25
− 59y
24
+ ··· − 128y −256)
c
8
((y
2
− y + 1)
4
)(y
25
+ 43y
24
+ ··· − 199y −25)
c
9
(y
8
− 16y
7
+ 98y
6
− 264y
5
+ 353y
4
− 168y
3
+ 92y
2
+ 32y + 16)
· (y
25
− 33y
24
+ ··· + 788576y −80656)
c
10
((y −1)
8
)(y
25
+ 47y
24
+ ··· − 11232y −256)
13
