12n
0327
(K12n
0327
)
A knot diagram
1
Linearized knot diagam
3 6 9 7 2 11 3 12 7 5 1 9
Solving Sequence
8,12 3,9
4 1 7 5 11 6 2 10
c
8
c
3
c
12
c
7
c
4
c
11
c
6
c
2
c
10
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h7.90917 × 10
51
u
45
+ 2.18534 × 10
51
u
44
+ ··· + 2.95768 × 10
53
b + 5.20715 × 10
52
,
− 7.60793 × 10
53
u
45
− 9.21427 × 10
53
u
44
+ ··· + 2.07037 × 10
54
a + 1.08529 × 10
54
, u
46
+ u
45
+ ··· + 20u − 7i
I
u
2
= h−u
15
+ u
14
+ ··· + b + 1, −2u
15
− 3u
14
+ ··· + a + 4,
u
16
− 5u
14
− u
13
+ 13u
12
+ 3u
11
− 23u
10
− 6u
9
+ 29u
8
+ 8u
7
− 26u
6
− 8u
5
+ 16u
4
+ 4u
3
− 6u
2
− u + 1i
* 2 irreducible components of dim
C
= 0, with total 62 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
= h7.91×10
51
u
45
+2.19×10
51
u
44
+· · ·+2.96×10
53
b+5.21×10
52
, −7.61×
10
53
u
45
−9.21×10
53
u
44
+· · ·+2.07×10
54
a+1.09×10
54
, u
46
+u
45
+· · ·+20u−7i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
3
=
0.367466u
45
+ 0.445053u
44
+ ··· − 0.961201u − 0.524199
−0.0267412u
45
− 0.00738870u
44
+ ··· − 0.488769u − 0.176055
a
9
=
1
−u
2
a
4
=
0.365109u
45
+ 0.654340u
44
+ ··· − 0.429450u − 0.157144
0.164232u
45
− 0.120965u
44
+ ··· + 3.76063u − 1.65757
a
1
=
u
−u
3
+ u
a
7
=
−0.236300u
45
+ 0.371650u
44
+ ··· − 14.5049u + 8.55430
0.359878u
45
+ 0.0463945u
44
+ ··· + 4.16574u − 1.59970
a
5
=
−0.296182u
45
+ 0.163568u
44
+ ··· + 3.77995u − 0.792151
0.358221u
45
− 0.341011u
44
+ ··· + 8.83971u − 4.10151
a
11
=
−u
3
u
5
− u
3
+ u
a
6
=
−0.0116433u
45
+ 0.300125u
44
+ ··· − 9.16821u + 6.98444
0.144819u
45
+ 0.138605u
44
+ ··· − 1.99873u + 0.550517
a
2
=
0.797806u
45
+ 0.285789u
44
+ ··· + 9.18124u + 0.589049
−0.315736u
45
+ 0.0205995u
44
+ ··· − 5.78048u + 0.658747
a
10
=
−0.185941u
45
− 0.174269u
44
+ ··· + 9.41906u − 2.70922
−0.0739068u
45
− 0.292990u
44
+ ··· + 0.996475u − 1.61521
(ii) Obstruction class = −1
(iii) Cusp Shapes = 1.28396u
45
+ 1.62893u
44
+ ··· − 44.8335u + 13.8636
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
46
+ 15u
45
+ ··· + 2321u + 49
c
2
, c
5
u
46
+ 3u
45
+ ··· − 33u + 7
c
3
, c
10
u
46
+ u
45
+ ··· + 19u + 1
c
4
u
46
+ 5u
45
+ ··· − 2895u − 209
c
6
u
46
− 2u
45
+ ··· + 15u − 19
c
7
u
46
− 3u
45
+ ··· − 113405u + 77291
c
8
, c
12
u
46
− u
45
+ ··· − 20u − 7
c
9
u
46
+ 6u
45
+ ··· − 75u + 9
c
11
u
46
− 33u
45
+ ··· − 1366u + 49
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
46
+ 53y
45
+ ··· − 1065437y + 2401
c
2
, c
5
y
46
− 15y
45
+ ··· − 2321y + 49
c
3
, c
10
y
46
− 63y
45
+ ··· − 121y + 1
c
4
y
46
− 67y
45
+ ··· − 133885y + 43681
c
6
y
46
+ 2y
45
+ ··· − 985y + 361
c
7
y
46
− 71y
45
+ ··· − 125034513563y + 5973898681
c
8
, c
12
y
46
− 33y
45
+ ··· − 1366y + 49
c
9
y
46
− 82y
45
+ ··· + 5697y + 81
c
11
y
46
− 29y
45
+ ··· + 39654y + 2401
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.020210 + 0.218826I
a = 1.96292 + 1.78837I
b = −1.39819 − 1.19639I
2.78540 − 4.43305I 7.31683 + 2.46379I
u = −1.020210 − 0.218826I
a = 1.96292 − 1.78837I
b = −1.39819 + 1.19639I
2.78540 + 4.43305I 7.31683 − 2.46379I
u = 0.095870 + 1.057170I
a = 0.449277 − 0.228270I
b = 2.15949 − 0.25287I
10.80880 + 1.13108I 8.33919 + 0.09433I
u = 0.095870 − 1.057170I
a = 0.449277 + 0.228270I
b = 2.15949 + 0.25287I
10.80880 − 1.13108I 8.33919 − 0.09433I
u = −0.746943 + 0.560155I
a = 0.238979 − 0.456580I
b = −1.019220 + 0.072002I
2.29880 + 1.44879I 7.06855 + 0.81086I
u = −0.746943 − 0.560155I
a = 0.238979 + 0.456580I
b = −1.019220 − 0.072002I
2.29880 − 1.44879I 7.06855 − 0.81086I
u = 1.052910 + 0.350962I
a = 0.652736 + 0.849236I
b = 0.00161 − 1.51789I
0.32230 + 4.17094I 4.61272 − 9.33404I
u = 1.052910 − 0.350962I
a = 0.652736 − 0.849236I
b = 0.00161 + 1.51789I
0.32230 − 4.17094I 4.61272 + 9.33404I
u = −0.750943 + 0.440135I
a = 1.60823 + 1.02092I
b = −0.351616 + 1.000540I
−1.61798 − 2.21982I 4.07754 + 3.82774I
u = −0.750943 − 0.440135I
a = 1.60823 − 1.02092I
b = −0.351616 − 1.000540I
−1.61798 + 2.21982I 4.07754 − 3.82774I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.17301
a = −2.66494
b = 1.80680
9.52036 8.51710
u = −1.068340 + 0.517725I
a = 0.464086 + 1.138280I
b = −0.566866 − 0.471411I
3.83816 − 4.69784I 11.69749 + 4.30932I
u = −1.068340 − 0.517725I
a = 0.464086 − 1.138280I
b = −0.566866 + 0.471411I
3.83816 + 4.69784I 11.69749 − 4.30932I
u = 0.870642 + 0.823142I
a = −0.126114 + 0.358538I
b = 0.623928 + 0.125399I
−4.07204 + 3.05319I 15.3891 − 6.8553I
u = 0.870642 − 0.823142I
a = −0.126114 − 0.358538I
b = 0.623928 − 0.125399I
−4.07204 − 3.05319I 15.3891 + 6.8553I
u = −0.086683 + 1.203980I
a = −0.374986 + 0.103292I
b = −2.30710 − 0.06443I
10.00560 + 7.40008I 7.39350 − 4.31642I
u = −0.086683 − 1.203980I
a = −0.374986 − 0.103292I
b = −2.30710 + 0.06443I
10.00560 − 7.40008I 7.39350 + 4.31642I
u = −1.226800 + 0.055153I
a = −1.27654 − 0.97559I
b = 0.91044 + 1.53643I
4.08578 + 2.48503I 10.70236 − 2.61783I
u = −1.226800 − 0.055153I
a = −1.27654 + 0.97559I
b = 0.91044 − 1.53643I
4.08578 − 2.48503I 10.70236 + 2.61783I
u = −1.226440 + 0.133188I
a = −0.021931 − 0.272138I
b = −0.127594 − 0.708761I
2.26273 − 2.31358I 11.62542 + 2.67684I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.226440 − 0.133188I
a = −0.021931 + 0.272138I
b = −0.127594 + 0.708761I
2.26273 + 2.31358I 11.62542 − 2.67684I
u = −1.106930 + 0.595666I
a = −0.835543 − 0.024113I
b = −0.293003 − 1.317840I
−0.37143 − 2.13318I 6.00000 + 0.I
u = −1.106930 − 0.595666I
a = −0.835543 + 0.024113I
b = −0.293003 + 1.317840I
−0.37143 + 2.13318I 6.00000 + 0.I
u = −0.283256 + 0.684205I
a = 0.210975 − 0.269709I
b = −0.682833 + 0.139746I
1.63018 + 0.14156I 8.28809 + 0.51848I
u = −0.283256 − 0.684205I
a = 0.210975 + 0.269709I
b = −0.682833 − 0.139746I
1.63018 − 0.14156I 8.28809 − 0.51848I
u = −0.019529 + 0.725158I
a = −0.777308 + 0.478111I
b = 0.500963 − 0.256778I
0.47573 − 4.85542I 5.18911 + 6.50934I
u = −0.019529 − 0.725158I
a = −0.777308 − 0.478111I
b = 0.500963 + 0.256778I
0.47573 + 4.85542I 5.18911 − 6.50934I
u = 1.260380 + 0.241271I
a = 1.265940 − 0.312252I
b = −1.138130 + 0.326314I
6.24098 + 2.89108I 0
u = 1.260380 − 0.241271I
a = 1.265940 + 0.312252I
b = −1.138130 − 0.326314I
6.24098 − 2.89108I 0
u = 0.640221 + 0.191594I
a = −1.51549 + 0.94495I
b = 0.776010 − 0.543436I
−1.81559 + 1.36574I 0.37681 − 4.32154I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.640221 − 0.191594I
a = −1.51549 − 0.94495I
b = 0.776010 + 0.543436I
−1.81559 − 1.36574I 0.37681 + 4.32154I
u = 1.309540 + 0.386533I
a = −1.175660 + 0.376325I
b = 1.196350 − 0.295467I
4.61382 + 9.03597I 0
u = 1.309540 − 0.386533I
a = −1.175660 − 0.376325I
b = 1.196350 + 0.295467I
4.61382 − 9.03597I 0
u = 0.626188
a = 1.42934
b = 1.02581
7.47324 21.6980
u = −1.37352 + 0.48037I
a = −1.81578 − 1.08493I
b = 2.71556 − 0.36369I
15.4278 − 6.5327I 0
u = −1.37352 − 0.48037I
a = −1.81578 + 1.08493I
b = 2.71556 + 0.36369I
15.4278 + 6.5327I 0
u = −0.538688
a = −0.314299
b = −0.326196
0.769410 13.1770
u = 1.34301 + 0.58680I
a = −1.59637 + 1.02056I
b = 2.15155 + 1.01181I
14.6331 + 4.7800I 0
u = 1.34301 − 0.58680I
a = −1.59637 − 1.02056I
b = 2.15155 − 1.01181I
14.6331 − 4.7800I 0
u = −1.39068 + 0.61150I
a = 1.57695 + 1.08397I
b = −2.52625 + 0.71884I
14.1038 − 13.8171I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.39068 − 0.61150I
a = 1.57695 − 1.08397I
b = −2.52625 − 0.71884I
14.1038 + 13.8171I 0
u = 1.51006 + 0.50481I
a = 1.54874 − 0.74984I
b = −2.47250 − 0.79950I
15.1375 − 1.1980I 0
u = 1.51006 − 0.50481I
a = 1.54874 + 0.74984I
b = −2.47250 + 0.79950I
15.1375 + 1.1980I 0
u = 1.61183
a = 1.65979
b = −2.42384
11.0760 0
u = 0.281467 + 0.135960I
a = −3.23232 − 0.74415I
b = 0.306108 + 0.706255I
−1.71245 − 1.30539I −1.21304 + 2.73809I
u = 0.281467 − 0.135960I
a = −3.23232 + 0.74415I
b = 0.306108 − 0.706255I
−1.71245 + 1.30539I −1.21304 − 2.73809I
9
II.
I
u
2
= h−u
15
+u
14
+· · ·+b+1, −2u
15
−3u
14
+· · ·+a+4, u
16
−5u
14
+· · ·−u+1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
3
=
2u
15
+ 3u
14
+ ··· − 3u − 4
u
15
− u
14
+ ··· − 4u − 1
a
9
=
1
−u
2
a
4
=
4u
15
+ 4u
14
+ ··· − 6u − 8
u
15
− u
14
+ ··· − u
2
− 3u
a
1
=
u
−u
3
+ u
a
7
=
2u
15
+ u
14
+ ··· + u + 4
−3u
15
− 2u
14
+ ··· + 2u + 3
a
5
=
−5u
15
− u
14
+ ··· + 9u + 9
u
15
− 4u
13
+ ··· + u + 1
a
11
=
−u
3
u
5
− u
3
+ u
a
6
=
2u
15
+ u
14
+ ··· + u + 4
−3u
15
− 2u
14
+ ··· + 2u + 4
a
2
=
3u
15
+ 6u
14
+ ··· + 5u − 8
−2u
15
− 3u
14
+ ··· − 3u + 3
a
10
=
−2u
15
+ 11u
13
+ ··· − 5u − 1
6u
15
+ 3u
14
+ ··· − 5u − 9
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
15
− u
14
− 21u
13
− u
12
+ 59u
11
+ 11u
10
− 107u
9
− 31u
8
+
137u
7
+ 47u
6
− 125u
5
− 54u
4
+ 74u
3
+ 30u
2
− 23u − 3
10
(iv) u-Polynomials at the component
11
Crossings u-Polynomials at each crossing
c
1
u
16
− 8u
15
+ ··· − 16u + 1
c
2
u
16
+ 4u
15
+ ··· + 4u + 1
c
3
u
16
− 10u
14
+ ··· − 2u − 1
c
4
u
16
+ 5u
13
+ ··· − 8u + 1
c
5
u
16
− 4u
15
+ ··· − 4u + 1
c
6
u
16
− u
15
+ ··· − 4u
2
− 1
c
7
u
16
+ 4u
14
+ ··· − 6u + 1
c
8
u
16
− 5u
14
+ ··· − u + 1
c
9
u
16
− 7u
15
+ ··· + 16u − 1
c
10
u
16
− 10u
14
+ ··· + 2u − 1
c
11
u
16
+ 10u
15
+ ··· + 13u + 1
c
12
u
16
− 5u
14
+ ··· + u + 1
12
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
16
+ 20y
15
+ ··· − 52y + 1
c
2
, c
5
y
16
− 8y
15
+ ··· − 16y + 1
c
3
, c
10
y
16
− 20y
15
+ ··· + 4y + 1
c
4
y
16
− 12y
14
+ ··· − 20y + 1
c
6
y
16
+ 9y
15
+ ··· + 8y + 1
c
7
y
16
+ 8y
15
+ ··· − 10y + 1
c
8
, c
12
y
16
− 10y
15
+ ··· − 13y + 1
c
9
y
16
+ 9y
15
+ ··· − 26y + 1
c
11
y
16
+ 2y
15
+ ··· − 17y + 1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.772761 + 0.712653I
a = 1.12093 + 0.97459I
b = 0.174212 + 0.983327I
−0.10256 − 4.89171I 5.59970 + 5.98315I
u = −0.772761 − 0.712653I
a = 1.12093 − 0.97459I
b = 0.174212 − 0.983327I
−0.10256 + 4.89171I 5.59970 − 5.98315I
u = 1.026470 + 0.385848I
a = −0.41213 + 1.88514I
b = 0.64749 − 1.56601I
2.91384 + 5.58512I 6.71356 − 9.57258I
u = 1.026470 − 0.385848I
a = −0.41213 − 1.88514I
b = 0.64749 + 1.56601I
2.91384 − 5.58512I 6.71356 + 9.57258I
u = 0.868992 + 0.775777I
a = −0.370213 + 0.101152I
b = 0.267354 + 0.100433I
−4.40041 + 2.92387I −6.18250 + 1.00458I
u = 0.868992 − 0.775777I
a = −0.370213 − 0.101152I
b = 0.267354 − 0.100433I
−4.40041 − 2.92387I −6.18250 − 1.00458I
u = −1.153510 + 0.323030I
a = −0.758240 + 0.176409I
b = 0.125626 − 1.307330I
0.78724 − 3.14561I 8.37696 + 4.04477I
u = −1.153510 − 0.323030I
a = −0.758240 − 0.176409I
b = 0.125626 + 1.307330I
0.78724 + 3.14561I 8.37696 − 4.04477I
u = 0.730829 + 0.328251I
a = −0.94332 − 1.95374I
b = 0.722249 + 1.146800I
1.78640 − 2.52540I 4.45953 + 3.01445I
u = 0.730829 − 0.328251I
a = −0.94332 + 1.95374I
b = 0.722249 − 1.146800I
1.78640 + 2.52540I 4.45953 − 3.01445I
15
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.004110 + 0.721375I
a = −0.824924 − 0.586632I
b = −0.144368 − 1.128410I
0.607371 − 0.610177I 7.63791 − 0.29957I
u = −1.004110 − 0.721375I
a = −0.824924 + 0.586632I
b = −0.144368 + 1.128410I
0.607371 + 0.610177I 7.63791 + 0.29957I
u = −0.698430 + 0.203647I
a = 1.97025 − 0.90358I
b = −0.148546 + 0.648408I
−1.042730 + 0.875084I 7.78189 + 1.85913I
u = −0.698430 − 0.203647I
a = 1.97025 + 0.90358I
b = −0.148546 − 0.648408I
−1.042730 − 0.875084I 7.78189 − 1.85913I
u = 0.491820
a = −2.36858
b = −1.05815
7.14832 −3.64500
u = 1.51322
a = 1.80388
b = −2.22988
11.4926 17.8710
16
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
16
− 8u
15
+ ··· − 16u + 1)(u
46
+ 15u
45
+ ··· + 2321u + 49)
c
2
(u
16
+ 4u
15
+ ··· + 4u + 1)(u
46
+ 3u
45
+ ··· − 33u + 7)
c
3
(u
16
− 10u
14
+ ··· − 2u − 1)(u
46
+ u
45
+ ··· + 19u + 1)
c
4
(u
16
+ 5u
13
+ ··· − 8u + 1)(u
46
+ 5u
45
+ ··· − 2895u − 209)
c
5
(u
16
− 4u
15
+ ··· − 4u + 1)(u
46
+ 3u
45
+ ··· − 33u + 7)
c
6
(u
16
− u
15
+ ··· − 4u
2
− 1)(u
46
− 2u
45
+ ··· + 15u − 19)
c
7
(u
16
+ 4u
14
+ ··· − 6u + 1)(u
46
− 3u
45
+ ··· − 113405u + 77291)
c
8
(u
16
− 5u
14
+ ··· − u + 1)(u
46
− u
45
+ ··· − 20u − 7)
c
9
(u
16
− 7u
15
+ ··· + 16u − 1)(u
46
+ 6u
45
+ ··· − 75u + 9)
c
10
(u
16
− 10u
14
+ ··· + 2u − 1)(u
46
+ u
45
+ ··· + 19u + 1)
c
11
(u
16
+ 10u
15
+ ··· + 13u + 1)(u
46
− 33u
45
+ ··· − 1366u + 49)
c
12
(u
16
− 5u
14
+ ··· + u + 1)(u
46
− u
45
+ ··· − 20u − 7)
17
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
16
+ 20y
15
+ ··· − 52y + 1)(y
46
+ 53y
45
+ ··· − 1065437y + 2401)
c
2
, c
5
(y
16
− 8y
15
+ ··· − 16y + 1)(y
46
− 15y
45
+ ··· − 2321y + 49)
c
3
, c
10
(y
16
− 20y
15
+ ··· + 4y + 1)(y
46
− 63y
45
+ ··· − 121y + 1)
c
4
(y
16
− 12y
14
+ ··· − 20y + 1)(y
46
− 67y
45
+ ··· − 133885y + 43681)
c
6
(y
16
+ 9y
15
+ ··· + 8y + 1)(y
46
+ 2y
45
+ ··· − 985y + 361)
c
7
(y
16
+ 8y
15
+ ··· − 10y + 1)
· (y
46
− 71y
45
+ ··· − 125034513563y + 5973898681)
c
8
, c
12
(y
16
− 10y
15
+ ··· − 13y + 1)(y
46
− 33y
45
+ ··· − 1366y + 49)
c
9
(y
16
+ 9y
15
+ ··· − 26y + 1)(y
46
− 82y
45
+ ··· + 5697y + 81)
c
11
(y
16
+ 2y
15
+ ··· − 17y + 1)(y
46
− 29y
45
+ ··· + 39654y + 2401)
18
