12n
0249
(K12n
0249
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 10 12 11 3 1 8 7 6
Solving Sequence
7,11 3,8
4 12 6 1 10 5 2 9
c
7
c
3
c
11
c
6
c
12
c
10
c
5
c
2
c
9
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
25
+ 2u
24
+ ··· + b − 1, u
27
+ 2u
26
+ ··· + a − 2, u
28
+ 2u
27
+ ··· − 3u − 1i
I
u
2
= h−u
3
+ u
2
+ b − 2u + 1, u
4
+ 3u
2
+ a + 1, u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 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
= hu
25
+2u
24
+· · ·+b −1, u
27
+2u
26
+· · ·+a −2, u
28
+2u
27
+· · ·−3u −1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
3
=
−u
27
− 2u
26
+ ··· − 4u + 2
−u
25
− 2u
24
+ ··· + u + 1
a
8
=
1
u
2
a
4
=
−2u
27
− 4u
26
+ ··· − 2u + 3
u
27
+ 2u
26
+ ··· − 3u
2
+ 1
a
12
=
−u
u
a
6
=
u
2
+ 1
−u
2
a
1
=
−u
3
− 2u
u
3
+ u
a
10
=
u
u
3
+ u
a
5
=
−u
6
− 3u
4
+ 1
−u
8
− 4u
6
− 4u
4
− 2u
2
a
2
=
−u
27
− u
26
+ ··· − 6u + 2
−u
26
− 2u
25
+ ··· + 2u + 1
a
9
=
u
9
+ 6u
7
+ 11u
5
+ 6u
3
+ u
−u
9
− 5u
7
− 7u
5
− 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −u
27
− 2u
26
− 21u
25
− 38u
24
− 192u
23
− 314u
22
− 1007u
21
− 1483u
20
− 3363u
19
−
4430u
18
−7513u
17
−8756u
16
−11482u
15
−11630u
14
−12024u
13
−10257u
12
−8392u
11
−
5658u
10
− 3552u
9
− 1590u
8
− 624u
7
+ 24u
6
+ 115u
5
+ 133u
4
+ 47u
3
+ 22u
2
− 3u − 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
28
+ 6u
27
+ ··· + 17u + 1
c
2
, c
4
u
28
− 6u
27
+ ··· + 5u − 1
c
3
, c
8
u
28
+ u
27
+ ··· + 96u + 32
c
5
u
28
− 2u
27
+ ··· + 331u − 445
c
6
, c
7
, c
10
c
11
, c
12
u
28
− 2u
27
+ ··· + 3u − 1
c
9
u
28
+ 2u
27
+ ··· + 5u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
28
+ 38y
27
+ ··· − 17y + 1
c
2
, c
4
y
28
− 6y
27
+ ··· − 17y + 1
c
3
, c
8
y
28
− 33y
27
+ ··· − 14848y + 1024
c
5
y
28
+ 22y
27
+ ··· − 315151y + 198025
c
6
, c
7
, c
10
c
11
, c
12
y
28
+ 38y
27
+ ··· − 15y + 1
c
9
y
28
+ 34y
27
+ ··· − 15y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.152944 + 1.016320I
a = 1.019950 − 0.193834I
b = −0.647472 − 0.451263I
3.33821 − 2.49897I −1.28357 + 4.65842I
u = 0.152944 − 1.016320I
a = 1.019950 + 0.193834I
b = −0.647472 + 0.451263I
3.33821 + 2.49897I −1.28357 − 4.65842I
u = −0.068154 + 0.917237I
a = −1.70138 − 0.42865I
b = 0.734706 + 1.073090I
0.674371 + 1.046210I −3.66596 + 0.45443I
u = −0.068154 − 0.917237I
a = −1.70138 + 0.42865I
b = 0.734706 − 1.073090I
0.674371 − 1.046210I −3.66596 − 0.45443I
u = 0.285643 + 0.852384I
a = −0.007992 − 0.695316I
b = −0.322158 + 0.381569I
1.29412 − 2.63752I −1.01481 + 5.30921I
u = 0.285643 − 0.852384I
a = −0.007992 + 0.695316I
b = −0.322158 − 0.381569I
1.29412 + 2.63752I −1.01481 − 5.30921I
u = −0.329759 + 1.077170I
a = −1.84555 + 1.40687I
b = 1.245350 − 0.642218I
9.87396 + 8.39825I −2.23865 − 5.97376I
u = −0.329759 − 1.077170I
a = −1.84555 − 1.40687I
b = 1.245350 + 0.642218I
9.87396 − 8.39825I −2.23865 + 5.97376I
u = −0.274632 + 1.124510I
a = 1.67793 − 1.18741I
b = −0.957433 + 0.342031I
10.62160 + 1.12711I −1.11248 − 1.21587I
u = −0.274632 − 1.124510I
a = 1.67793 + 1.18741I
b = −0.957433 − 0.342031I
10.62160 − 1.12711I −1.11248 + 1.21587I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.540472 + 0.396254I
a = 0.247984 + 1.063200I
b = −1.162050 + 0.222187I
5.82690 − 1.64297I −5.31123 − 0.95700I
u = −0.540472 − 0.396254I
a = 0.247984 − 1.063200I
b = −1.162050 − 0.222187I
5.82690 + 1.64297I −5.31123 + 0.95700I
u = −0.578535 + 0.310326I
a = −0.10564 − 1.64155I
b = 1.025040 − 0.012062I
5.55026 + 5.30570I −6.25062 − 5.57146I
u = −0.578535 − 0.310326I
a = −0.10564 + 1.64155I
b = 1.025040 + 0.012062I
5.55026 − 5.30570I −6.25062 + 5.57146I
u = 0.487302
a = −0.909458
b = 0.360823
−1.29054 −7.57330
u = 0.303248 + 0.234862I
a = −0.89894 − 1.21637I
b = 0.057075 + 0.523211I
−0.528965 − 0.938472I −8.01967 + 7.23093I
u = 0.303248 − 0.234862I
a = −0.89894 + 1.21637I
b = 0.057075 − 0.523211I
−0.528965 + 0.938472I −8.01967 − 7.23093I
u = 0.06577 + 1.67605I
a = 0.310611 + 0.424504I
b = −0.225680 − 1.128100I
10.18720 − 3.94206I 0. + 4.56823I
u = 0.06577 − 1.67605I
a = 0.310611 − 0.424504I
b = −0.225680 + 1.128100I
10.18720 + 3.94206I 0. − 4.56823I
u = −0.01320 + 1.71171I
a = 1.60962 + 0.03543I
b = −3.88484 − 0.84278I
10.13260 + 1.33652I −3.41446 + 0.I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.01320 − 1.71171I
a = 1.60962 − 0.03543I
b = −3.88484 + 0.84278I
10.13260 − 1.33652I −3.41446 + 0.I
u = 0.03667 + 1.72969I
a = −0.912721 + 0.574767I
b = 2.22990 − 0.67323I
13.19840 − 3.25610I 0
u = 0.03667 − 1.72969I
a = −0.912721 − 0.574767I
b = 2.22990 + 0.67323I
13.19840 + 3.25610I 0
u = −0.08755 + 1.74259I
a = 2.19448 − 0.74449I
b = −5.13344 + 2.15584I
−19.5529 + 10.1409I 0
u = −0.08755 − 1.74259I
a = 2.19448 + 0.74449I
b = −5.13344 − 2.15584I
−19.5529 − 10.1409I 0
u = −0.253230
a = 3.12683
b = 0.636261
−2.04618 −0.133650
u = −0.06900 + 1.75500I
a = −2.19704 + 0.73339I
b = 5.04246 − 1.77736I
−18.5160 + 2.5730I 0
u = −0.06900 − 1.75500I
a = −2.19704 − 0.73339I
b = 5.04246 + 1.77736I
−18.5160 − 2.5730I 0
7
II.
I
u
2
= h−u
3
+ u
2
+ b − 2u + 1, u
4
+ 3u
2
+ a + 1, u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
3
=
−u
4
− 3u
2
− 1
u
3
− u
2
+ 2u − 1
a
8
=
1
u
2
a
4
=
−u
4
− 3u
2
− 1
u
3
− u
2
+ 2u − 1
a
12
=
−u
u
a
6
=
u
2
+ 1
−u
2
a
1
=
−u
3
− 2u
u
3
+ u
a
10
=
u
u
3
+ u
a
5
=
u
3
+ 2u
−u
3
− u
a
2
=
−u
4
− u
3
− 3u
2
− 2u − 1
2u
3
− u
2
+ 3u − 1
a
9
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −5u
4
+ 5u
3
− 20u
2
+ 14u − 21
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
5
c
3
, c
8
u
5
c
4
(u + 1)
5
c
5
, c
9
u
5
− u
4
+ u
2
+ u − 1
c
6
, c
7
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1
c
10
, c
11
, c
12
u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
5
c
3
, c
8
y
5
c
5
, c
9
y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1
c
6
, c
7
, c
10
c
11
, c
12
y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.233677 + 0.885557I
a = 0.827780 − 0.637683I
b = −0.340036 + 0.807849I
0.17487 − 2.21397I −7.62657 + 4.39306I
u = 0.233677 − 0.885557I
a = 0.827780 + 0.637683I
b = −0.340036 − 0.807849I
0.17487 + 2.21397I −7.62657 − 4.39306I
u = 0.416284
a = −1.54991
b = −0.268586
−2.52712 −18.4270
u = 0.05818 + 1.69128I
a = −0.552827 + 0.534136I
b = 1.47433 − 1.63485I
9.31336 − 3.33174I −6.15976 + 1.26157I
u = 0.05818 − 1.69128I
a = −0.552827 − 0.534136I
b = 1.47433 + 1.63485I
9.31336 + 3.33174I −6.15976 − 1.26157I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
28
+ 6u
27
+ ··· + 17u + 1)
c
2
((u − 1)
5
)(u
28
− 6u
27
+ ··· + 5u − 1)
c
3
, c
8
u
5
(u
28
+ u
27
+ ··· + 96u + 32)
c
4
((u + 1)
5
)(u
28
− 6u
27
+ ··· + 5u − 1)
c
5
(u
5
− u
4
+ u
2
+ u − 1)(u
28
− 2u
27
+ ··· + 331u − 445)
c
6
, c
7
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)(u
28
− 2u
27
+ ··· + 3u − 1)
c
9
(u
5
− u
4
+ u
2
+ u − 1)(u
28
+ 2u
27
+ ··· + 5u + 1)
c
10
, c
11
, c
12
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)(u
28
− 2u
27
+ ··· + 3u − 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
5
)(y
28
+ 38y
27
+ ··· − 17y + 1)
c
2
, c
4
((y −1)
5
)(y
28
− 6y
27
+ ··· − 17y + 1)
c
3
, c
8
y
5
(y
28
− 33y
27
+ ··· − 14848y + 1024)
c
5
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)(y
28
+ 22y
27
+ ··· − 315151y + 198025)
c
6
, c
7
, c
10
c
11
, c
12
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)(y
28
+ 38y
27
+ ··· − 15y + 1)
c
9
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)(y
28
+ 34y
27
+ ··· − 15y + 1)
13
