12n
0103
(K12n
0103
)
A knot diagram
1
Linearized knot diagam
3 5 6 2 9 3 11 12 5 7 8 10
Solving Sequence
5,9 3,6
7 10 11 2 1 4 12 8
c
5
c
6
c
9
c
10
c
2
c
1
c
4
c
12
c
8
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h2.50755 × 10
20
u
28
− 4.57339 × 10
20
u
27
+ ··· + 6.70474 × 10
20
b + 2.31276 × 10
20
,
4.30865 × 10
20
u
28
− 1.13718 × 10
21
u
27
+ ··· + 6.70474 × 10
20
a + 6.48791 × 10
20
, u
29
− 2u
28
+ ··· − u + 1i
I
u
2
= hb + 1, u
5
+ 2u
4
+ 4u
3
+ 5u
2
+ a + 4u + 3, u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 35 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
= h2.51 × 10
20
u
28
− 4.57 × 10
20
u
27
+ · · · + 6.70 × 10
20
b + 2.31 × 10
20
, 4.31 ×
10
20
u
28
−1.14×10
21
u
27
+· · ·+6.70×10
20
a+6.49×10
20
, u
29
−2u
28
+· · ·−u+1i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
u
a
3
=
−0.642627u
28
+ 1.69608u
27
+ ··· + 4.76582u − 0.967660
−0.373996u
28
+ 0.682112u
27
+ ··· + 1.36157u − 0.344944
a
6
=
1
u
2
a
7
=
−0.525413u
28
+ 0.870263u
27
+ ··· + 1.86350u + 0.104816
−0.165523u
28
+ 0.372194u
27
+ ··· + 0.644339u + 0.119793
a
10
=
−u
u
a
11
=
0.00550978u
28
− 0.113964u
27
+ ··· − 2.05097u − 0.744280
−0.0294122u
28
+ 0.103634u
27
+ ··· + 0.884693u + 0.0789002
a
2
=
−1.01662u
28
+ 2.37819u
27
+ ··· + 6.12739u − 1.31260
−0.373996u
28
+ 0.682112u
27
+ ··· + 1.36157u − 0.344944
a
1
=
−0.582562u
28
+ 1.02418u
27
+ ··· + 2.16299u + 0.0440466
0.0571491u
28
− 0.153913u
27
+ ··· − 0.299489u + 0.0607697
a
4
=
−0.715822u
28
+ 1.86209u
27
+ ··· + 5.07394u − 0.901780
−0.390136u
28
+ 0.726358u
27
+ ··· + 1.45438u − 0.364566
a
12
=
−0.690936u
28
+ 1.24246u
27
+ ··· + 2.50784u + 0.224610
0.165523u
28
− 0.372194u
27
+ ··· − 0.644339u − 0.119793
a
8
=
0.0239024u
28
+ 0.0103299u
27
+ ··· + 1.16628u + 0.665380
−0.0294122u
28
+ 0.103634u
27
+ ··· + 0.884693u + 0.0789002
(ii) Obstruction class = −1
(iii) Cusp Shapes =
478078822365653903278
134094852819026608583
u
28
−
5561400214755760811553
670474264095133042915
u
27
+ ··· −
2275460989738397018314
134094852819026608583
u −
5831311641101896181097
670474264095133042915
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
29
+ 39u
28
+ ··· + 2258u + 1
c
2
, c
4
u
29
− 7u
28
+ ··· − 54u + 1
c
3
, c
6
u
29
+ 5u
28
+ ··· + 384u + 64
c
5
, c
9
u
29
+ 2u
28
+ ··· − u − 1
c
7
, c
8
, c
10
c
11
u
29
+ 2u
28
+ ··· + 5u + 1
c
12
u
29
− 12u
28
+ ··· + 3529u + 937
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
29
− 91y
28
+ ··· + 4903026y −1
c
2
, c
4
y
29
− 39y
28
+ ··· + 2258y −1
c
3
, c
6
y
29
+ 39y
28
+ ··· + 212992y −4096
c
5
, c
9
y
29
+ 30y
27
+ ··· + 13y −1
c
7
, c
8
, c
10
c
11
y
29
− 36y
28
+ ··· + 13y −1
c
12
y
29
− 36y
28
+ ··· + 62137329y −877969
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.867238 + 0.470147I
a = 0.237339 + 1.389930I
b = −0.92240 − 1.35617I
−12.12860 − 4.62991I −15.7252 + 5.0837I
u = 0.867238 − 0.470147I
a = 0.237339 − 1.389930I
b = −0.92240 + 1.35617I
−12.12860 + 4.62991I −15.7252 − 5.0837I
u = −0.915025
a = 0.602737
b = −2.04399
−14.6155 −18.8040
u = 0.160840 + 1.087590I
a = 0.0855322 + 0.1114450I
b = 0.493211 − 0.128104I
2.18324 − 1.77578I −1.23792 + 3.54893I
u = 0.160840 − 1.087590I
a = 0.0855322 − 0.1114450I
b = 0.493211 + 0.128104I
2.18324 + 1.77578I −1.23792 − 3.54893I
u = −0.766809 + 0.460777I
a = 0.217501 − 1.292190I
b = −0.793934 + 1.006770I
−3.36571 + 3.55459I −14.7590 − 7.4317I
u = −0.766809 − 0.460777I
a = 0.217501 + 1.292190I
b = −0.793934 − 1.006770I
−3.36571 − 3.55459I −14.7590 + 7.4317I
u = 0.807312
a = 0.103435
b = −1.67794
−5.54567 −19.0490
u = −0.452011 + 1.154640I
a = −0.089699 − 0.245433I
b = 0.631471 + 0.354822I
−4.66125 + 4.01059I −6.69076 − 1.05481I
u = −0.452011 − 1.154640I
a = −0.089699 + 0.245433I
b = 0.631471 − 0.354822I
−4.66125 − 4.01059I −6.69076 + 1.05481I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.580536 + 0.452475I
a = 0.450270 + 1.338160I
b = −0.597116 − 0.490346I
−0.77928 − 1.44092I −6.71227 + 4.83159I
u = 0.580536 − 0.452475I
a = 0.450270 − 1.338160I
b = −0.597116 + 0.490346I
−0.77928 + 1.44092I −6.71227 − 4.83159I
u = −0.734655
a = 1.16704
b = 0.195873
−7.96223 −11.3580
u = 0.317549 + 0.579846I
a = 3.74066 + 0.93387I
b = −0.910186 + 0.367717I
−10.62150 + 0.95783I −12.17772 + 5.24325I
u = 0.317549 − 0.579846I
a = 3.74066 − 0.93387I
b = −0.910186 − 0.367717I
−10.62150 − 0.95783I −12.17772 − 5.24325I
u = −0.355366 + 0.430210I
a = 2.58524 − 2.38862I
b = −0.804622 − 0.092789I
−2.40653 − 0.39885I −18.5948 − 3.1258I
u = −0.355366 − 0.430210I
a = 2.58524 + 2.38862I
b = −0.804622 + 0.092789I
−2.40653 + 0.39885I −18.5948 + 3.1258I
u = −0.465770
a = −2.93524
b = −1.09023
−2.20812 4.71460
u = 1.12576 + 1.06472I
a = −0.564004 − 1.170310I
b = 1.78727 + 0.43721I
18.7699 − 11.3822I −14.5597 + 4.9176I
u = 1.12576 − 1.06472I
a = −0.564004 + 1.170310I
b = 1.78727 − 0.43721I
18.7699 + 11.3822I −14.5597 − 4.9176I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.14151 + 1.07839I
a = −0.559145 + 1.007550I
b = 1.72676 − 0.31572I
−11.7546 + 8.6101I −13.1323 − 5.7611I
u = −1.14151 − 1.07839I
a = −0.559145 − 1.007550I
b = 1.72676 + 0.31572I
−11.7546 − 8.6101I −13.1323 + 5.7611I
u = 1.10890 + 1.13111I
a = −0.839564 − 0.595930I
b = 1.78058 − 0.11270I
18.9482 + 3.1990I −14.9528 − 0.8134I
u = 1.10890 − 1.13111I
a = −0.839564 + 0.595930I
b = 1.78058 + 0.11270I
18.9482 − 3.1990I −14.9528 + 0.8134I
u = 1.14879 + 1.10311I
a = −0.592413 − 0.839025I
b = 1.68896 + 0.17655I
−8.95243 − 4.17612I −10.28879 + 2.37984I
u = 1.14879 − 1.10311I
a = −0.592413 + 0.839025I
b = 1.68896 − 0.17655I
−8.95243 + 4.17612I −10.28879 − 2.37984I
u = −1.13279 + 1.12536I
a = −0.701441 + 0.703233I
b = 1.71752 − 0.02970I
−11.62970 − 0.31954I −13.41116 + 1.33191I
u = −1.13279 − 1.12536I
a = −0.701441 − 0.703233I
b = 1.71752 + 0.02970I
−11.62970 + 0.31954I −13.41116 − 1.33191I
u = 0.385892
a = 1.12146
b = 0.0212520
−0.763627 −13.0190
7
II.
I
u
2
= hb+1, u
5
+2u
4
+4u
3
+5u
2
+a+4u +3, u
6
+u
5
+3u
4
+2u
3
+2u
2
+u−1i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
u
a
3
=
−u
5
− 2u
4
− 4u
3
− 5u
2
− 4u − 3
−1
a
6
=
1
u
2
a
7
=
1
u
2
a
10
=
−u
u
a
11
=
−u
3
− 2u
−u
5
− u
3
+ u
a
2
=
−u
5
− 2u
4
− 4u
3
− 5u
2
− 4u − 4
−1
a
1
=
−1
0
a
4
=
−u
5
− 2u
4
− 4u
3
− 5u
2
− 4u − 3
−1
a
12
=
−u
2
− 1
u
2
a
8
=
u
5
+ 2u
3
+ u
−u
5
− u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −7u
5
− 15u
4
− 29u
3
− 33u
2
− 28u − 32
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
6
c
3
, c
6
u
6
c
4
(u + 1)
6
c
5
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
7
, c
8
u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
9
, c
12
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1
c
10
, c
11
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
6
y
6
c
5
, c
9
, c
12
y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1
c
7
, c
8
, c
10
c
11
y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.873214
a = −1.31147
b = −1.00000
−9.30502 −18.5710
u = 0.138835 + 1.234450I
a = 0.631845 + 0.143944I
b = −1.00000
1.31531 − 1.97241I −11.10050 + 4.53432I
u = 0.138835 − 1.234450I
a = 0.631845 − 0.143944I
b = −1.00000
1.31531 + 1.97241I −11.10050 − 4.53432I
u = −0.408802 + 1.276380I
a = 0.453123 − 0.323434I
b = −1.00000
−5.34051 + 4.59213I −13.7303 − 5.9632I
u = −0.408802 − 1.276380I
a = 0.453123 + 0.323434I
b = −1.00000
−5.34051 − 4.59213I −13.7303 + 5.9632I
u = 0.413150
a = −5.85846
b = −1.00000
−2.38379 −51.7680
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
29
+ 39u
28
+ ··· + 2258u + 1)
c
2
((u − 1)
6
)(u
29
− 7u
28
+ ··· − 54u + 1)
c
3
, c
6
u
6
(u
29
+ 5u
28
+ ··· + 384u + 64)
c
4
((u + 1)
6
)(u
29
− 7u
28
+ ··· − 54u + 1)
c
5
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
29
+ 2u
28
+ ··· − u − 1)
c
7
, c
8
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
29
+ 2u
28
+ ··· + 5u + 1)
c
9
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
29
+ 2u
28
+ ··· − u − 1)
c
10
, c
11
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
29
+ 2u
28
+ ··· + 5u + 1)
c
12
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
29
− 12u
28
+ ··· + 3529u + 937)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
29
− 91y
28
+ ··· + 4903026y −1)
c
2
, c
4
((y −1)
6
)(y
29
− 39y
28
+ ··· + 2258y −1)
c
3
, c
6
y
6
(y
29
+ 39y
28
+ ··· + 212992y −4096)
c
5
, c
9
(y
6
+ 5y
5
+ ··· − 5y + 1)(y
29
+ 30y
27
+ ··· + 13y −1)
c
7
, c
8
, c
10
c
11
(y
6
− 7y
5
+ ··· − 5y + 1)(y
29
− 36y
28
+ ··· + 13y −1)
c
12
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1)
· (y
29
− 36y
28
+ ··· + 62137329y −877969)
13
