12n
0105
(K12n
0105
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 9 4 11 12 5 7 8 10
Solving Sequence
7,11
8 12
4,9
3 6 5 2 10 1
c
7
c
11
c
8
c
3
c
6
c
5
c
2
c
10
c
12
c
1
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h22873134u
26
− 91748515u
25
+ ··· + 37746988b − 44822939,
63338341u
26
− 295194910u
25
+ ··· + 37746988a − 66189548, u
27
− 5u
26
+ ··· − 6u + 1i
I
u
2
= hb, 2u
5
− u
4
− 7u
3
+ u
2
+ a + 5u + 4, u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
I
u
3
= h−a
2
+ b − 3a − 1, a
3
+ 3a
2
+ 2a + 1, u
2
+ u − 1i
* 3 irreducible components of dim
C
= 0, with total 39 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
= h2.29 × 10
7
u
26
− 9.17 × 10
7
u
25
+ · · · + 3.77 × 10
7
b − 4.48 × 10
7
, 6.33 ×
10
7
u
26
−2.95×10
8
u
25
+· · · +3.77×10
7
a−6.62×10
7
, u
27
−5u
26
+· · · − 6u +1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
u
2
a
12
=
−u
−u
3
+ u
a
4
=
−1.67797u
26
+ 7.82036u
25
+ ··· − 8.45077u + 1.75351
−0.605959u
26
+ 2.43062u
25
+ ··· − 2.79918u + 1.18746
a
9
=
−u
2
+ 1
−u
4
+ 2u
2
a
3
=
−2.28393u
26
+ 10.2510u
25
+ ··· − 11.2499u + 2.94096
−0.605959u
26
+ 2.43062u
25
+ ··· − 2.79918u + 1.18746
a
6
=
−0.101693u
26
+ 0.926944u
25
+ ··· − 7.17642u + 3.14845
1.45363u
26
− 5.69060u
25
+ ··· + 5.80702u − 1.50691
a
5
=
0.758454u
26
− 2.47127u
25
+ ··· − 3.68028u + 1.91598
0.144041u
26
− 0.819382u
25
+ ··· + 0.700822u − 0.562543
a
2
=
−1.05084u
26
+ 5.94352u
25
+ ··· − 11.3894u + 3.35900
−0.144041u
26
+ 0.819382u
25
+ ··· − 0.700822u + 0.562543
a
10
=
u
u
a
1
=
u
5
− 2u
3
− u
u
5
− 3u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
725958293
18873494
u
26
+
3202886145
18873494
u
25
+ ··· −
5326606613
18873494
u +
1076937225
18873494
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
27
+ u
26
+ ··· + 514u + 1
c
2
, c
4
u
27
− 9u
26
+ ··· + 20u + 1
c
3
, c
6
u
27
− 3u
26
+ ··· + 128u + 64
c
5
, c
9
u
27
+ 2u
26
+ ··· − 352u − 64
c
7
, c
8
, c
10
c
11
u
27
+ 5u
26
+ ··· − 6u − 1
c
12
u
27
+ u
26
+ ··· + 500u − 89
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
27
+ 59y
26
+ ··· + 262978y −1
c
2
, c
4
y
27
− y
26
+ ··· + 514y −1
c
3
, c
6
y
27
+ 45y
26
+ ··· + 180224y −4096
c
5
, c
9
y
27
+ 40y
26
+ ··· + 103424y −4096
c
7
, c
8
, c
10
c
11
y
27
− 29y
26
+ ··· − 14y −1
c
12
y
27
+ 67y
26
+ ··· − 314794y −7921
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.514269 + 0.943772I
a = 0.96114 + 1.49139I
b = −0.33697 − 2.32589I
15.0681 − 1.4972I −9.94344 − 0.11775I
u = −0.514269 − 0.943772I
a = 0.96114 − 1.49139I
b = −0.33697 + 2.32589I
15.0681 + 1.4972I −9.94344 + 0.11775I
u = −0.664233 + 0.874955I
a = −1.06704 − 1.64357I
b = −0.62816 + 2.16426I
14.6013 + 7.4637I −10.63048 − 4.31713I
u = −0.664233 − 0.874955I
a = −1.06704 + 1.64357I
b = −0.62816 − 2.16426I
14.6013 − 7.4637I −10.63048 + 4.31713I
u = −1.182480 + 0.163891I
a = 0.485579 + 0.426627I
b = 0.14520 − 1.74397I
1.12392 + 3.54626I −14.5183 − 3.2040I
u = −1.182480 − 0.163891I
a = 0.485579 − 0.426627I
b = 0.14520 + 1.74397I
1.12392 − 3.54626I −14.5183 + 3.2040I
u = 1.261340 + 0.096236I
a = 0.969318 + 0.141624I
b = 0.808086 − 0.842084I
−4.50481 − 1.41612I −14.8457 + 0.7290I
u = 1.261340 − 0.096236I
a = 0.969318 − 0.141624I
b = 0.808086 + 0.842084I
−4.50481 + 1.41612I −14.8457 − 0.7290I
u = −0.153216 + 0.715283I
a = −0.15544 − 1.50896I
b = −0.68505 + 1.29368I
3.93766 − 0.32566I −7.68828 − 0.01937I
u = −0.153216 − 0.715283I
a = −0.15544 + 1.50896I
b = −0.68505 − 1.29368I
3.93766 + 0.32566I −7.68828 + 0.01937I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.543053 + 0.346105I
a = 0.952357 − 0.112943I
b = 0.116929 − 1.071230I
2.02138 + 3.53683I −8.00351 − 9.60393I
u = −0.543053 − 0.346105I
a = 0.952357 + 0.112943I
b = 0.116929 + 1.071230I
2.02138 − 3.53683I −8.00351 + 9.60393I
u = 1.348160 + 0.304621I
a = −0.990189 + 0.519594I
b = −1.27433 − 0.79621I
−0.78811 − 3.39068I −12.00000 + 2.97054I
u = 1.348160 − 0.304621I
a = −0.990189 − 0.519594I
b = −1.27433 + 0.79621I
−0.78811 + 3.39068I −12.00000 − 2.97054I
u = 0.552139
a = −7.52398
b = −0.222467
−2.46059 −111.160
u = 1.53250 + 0.12438I
a = 0.477076 + 0.429045I
b = 0.181555 + 0.768118I
−4.88486 − 5.37353I −12.0000 + 7.7726I
u = 1.53250 − 0.12438I
a = 0.477076 − 0.429045I
b = 0.181555 − 0.768118I
−4.88486 + 5.37353I −12.0000 − 7.7726I
u = −1.55351 + 0.07692I
a = −0.278760 + 0.128293I
b = 0.464875 + 0.727258I
−7.33821 + 0.72358I −12.00000 + 0.I
u = −1.55351 − 0.07692I
a = −0.278760 − 0.128293I
b = 0.464875 − 0.727258I
−7.33821 − 0.72358I −12.00000 + 0.I
u = 0.440522
a = −0.842102
b = 0.209937
−0.703245 −13.8830
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.60962
a = −3.04132
b = −0.471477
−10.1232 −65.7520
u = 1.56503 + 0.38318I
a = 0.979548 − 0.112812I
b = 0.04694 + 2.26665I
8.39568 − 3.40964I −12.00000 + 0.I
u = 1.56503 − 0.38318I
a = 0.979548 + 0.112812I
b = 0.04694 − 2.26665I
8.39568 + 3.40964I −12.00000 + 0.I
u = 1.61847 + 0.30353I
a = −1.37457 + 0.38272I
b = −0.75828 − 1.91136I
7.09877 − 11.89130I −12.00000 + 0.I
u = 1.61847 − 0.30353I
a = −1.37457 − 0.38272I
b = −0.75828 + 1.91136I
7.09877 + 11.89130I −12.00000 + 0.I
u = 0.093749 + 0.195668I
a = −1.25531 + 1.76548I
b = 0.661199 + 0.033436I
−0.945949 + 0.075456I −9.99859 + 1.12534I
u = 0.093749 − 0.195668I
a = −1.25531 − 1.76548I
b = 0.661199 − 0.033436I
−0.945949 − 0.075456I −9.99859 − 1.12534I
7
II.
I
u
2
= hb, 2u
5
− u
4
− 7u
3
+ u
2
+ a + 5u + 4, u
6
− u
5
− 3u
4
+ 2 u
3
+ 2 u
2
+ u − 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
u
2
a
12
=
−u
−u
3
+ u
a
4
=
−2u
5
+ u
4
+ 7u
3
− u
2
− 5u − 4
0
a
9
=
−u
2
+ 1
−u
4
+ 2u
2
a
3
=
−2u
5
+ u
4
+ 7u
3
− u
2
− 5u − 4
0
a
6
=
1
0
a
5
=
−u
5
+ 2u
3
+ u
−u
5
+ 3u
3
− u
a
2
=
−u
5
+ u
4
+ 5u
3
− u
2
− 6u − 4
u
5
− 3u
3
+ u
a
10
=
u
u
a
1
=
u
5
− 2u
3
− u
u
5
− 3u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 10u
5
− 6u
4
− 30u
3
+ 5u
2
+ 17u + 7
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 = −0.493180 + 0.575288I
a = 0.631845 − 0.143944I
b = 0
1.31531 + 1.97241I −10.05095 − 2.83524I
u = −0.493180 − 0.575288I
a = 0.631845 + 0.143944I
b = 0
1.31531 − 1.97241I −10.05095 + 2.83524I
u = 0.483672
a = −5.85846
b = 0
−2.38379 12.9340
u = 1.52087 + 0.16310I
a = 0.453123 + 0.323434I
b = 0
−5.34051 − 4.59213I −15.4320 + 0.4465I
u = 1.52087 − 0.16310I
a = 0.453123 − 0.323434I
b = 0
−5.34051 + 4.59213I −15.4320 − 0.4465I
u = −1.53904
a = −1.31147
b = 0
−9.30502 −17.9680
11
III. I
u
3
= h−a
2
+ b − 3a − 1, a
3
+ 3a
2
+ 2a + 1, u
2
+ u − 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u + 1
a
12
=
−u
−u + 1
a
4
=
a
a
2
+ 3a + 1
a
9
=
u
u
a
3
=
a
2
+ 4a + 1
a
2
+ 3a + 1
a
6
=
a + 2
a + 2
a
5
=
a + 2
a + 2
a
2
=
2a + 2
a + 2
a
10
=
u
u
a
1
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −a
2
u + 3au + a + 3u − 9
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
4
(u
3
− u
2
+ 1)
2
c
5
, c
9
u
6
c
6
(u
3
+ u
2
+ 2u + 1)
2
c
7
, c
8
(u
2
+ u − 1)
3
c
10
, c
11
, c
12
(u
2
− u − 1)
3
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
4
(y
3
− y
2
+ 2y −1)
2
c
5
, c
9
y
6
c
7
, c
8
, c
10
c
11
, c
12
(y
2
− 3y + 1)
3
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −0.337641 + 0.562280I
b = −0.215080 + 1.307140I
2.03717 + 2.82812I −7.98462 + 1.83947I
u = 0.618034
a = −0.337641 − 0.562280I
b = −0.215080 − 1.307140I
2.03717 − 2.82812I −7.98462 − 1.83947I
u = 0.618034
a = −2.32472
b = −0.569840
−2.10041 −17.1210
u = −1.61803
a = −0.337641 + 0.562280I
b = −0.215080 + 1.307140I
−5.85852 + 2.82812I −12.87990 − 2.78145I
u = −1.61803
a = −0.337641 − 0.562280I
b = −0.215080 − 1.307140I
−5.85852 − 2.82812I −12.87990 + 2.78145I
u = −1.61803
a = −2.32472
b = −0.569840
−9.99610 3.85000
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
3
− u
2
+ 2u − 1)
2
(u
27
+ u
26
+ ··· + 514u + 1)
c
2
((u − 1)
6
)(u
3
+ u
2
− 1)
2
(u
27
− 9u
26
+ ··· + 20u + 1)
c
3
u
6
(u
3
− u
2
+ 2u − 1)
2
(u
27
− 3u
26
+ ··· + 128u + 64)
c
4
((u + 1)
6
)(u
3
− u
2
+ 1)
2
(u
27
− 9u
26
+ ··· + 20u + 1)
c
5
u
6
(u
6
+ u
5
+ ··· + u − 1)(u
27
+ 2u
26
+ ··· − 352u − 64)
c
6
u
6
(u
3
+ u
2
+ 2u + 1)
2
(u
27
− 3u
26
+ ··· + 128u + 64)
c
7
, c
8
(u
2
+ u − 1)
3
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)
· (u
27
+ 5u
26
+ ··· − 6u − 1)
c
9
u
6
(u
6
− u
5
+ ··· − u − 1)(u
27
+ 2u
26
+ ··· − 352u − 64)
c
10
, c
11
(u
2
− u − 1)
3
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)
· (u
27
+ 5u
26
+ ··· − 6u − 1)
c
12
(u
2
− u − 1)
3
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)
· (u
27
+ u
26
+ ··· + 500u − 89)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
3
+ 3y
2
+ 2y −1)
2
(y
27
+ 59y
26
+ ··· + 262978y −1)
c
2
, c
4
((y −1)
6
)(y
3
− y
2
+ 2y −1)
2
(y
27
− y
26
+ ··· + 514y −1)
c
3
, c
6
y
6
(y
3
+ 3y
2
+ 2y −1)
2
(y
27
+ 45y
26
+ ··· + 180224y −4096)
c
5
, c
9
y
6
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1)
· (y
27
+ 40y
26
+ ··· + 103424y −4096)
c
7
, c
8
, c
10
c
11
(y
2
− 3y + 1)
3
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
· (y
27
− 29y
26
+ ··· − 14y −1)
c
12
(y
2
− 3y + 1)
3
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1)
· (y
27
+ 67y
26
+ ··· − 314794y −7921)
17
