12n
0724
(K12n
0724
)
A knot diagram
1
Linearized knot diagam
4 6 7 9 2 3 11 12 4 1 9 8
Solving Sequence
8,12 4,9
5 1 2 11 7 3 6 10
c
8
c
4
c
12
c
1
c
11
c
7
c
3
c
6
c
10
c
2
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
34
− 2u
33
+ ··· + 2b + 2, 3u
35
+ 9u
34
+ ··· + 2a + 12, u
36
+ 3u
35
+ ··· + 5u + 1i
I
u
2
= h−u
2
a + b, −u
2
a + a
2
− u
2
− a + u − 2, u
3
− u
2
+ 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 42 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−u
34
−2u
33
+· · ·+2b+2, 3u
35
+9u
34
+· · ·+2a+12, u
36
+3u
35
+· · ·+5u+1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
4
=
−
3
2
u
35
−
9
2
u
34
+ ··· −
13
2
u − 6
1
2
u
34
+ u
33
+ ··· +
1
2
u − 1
a
9
=
1
u
2
a
5
=
−
3
2
u
35
−
9
2
u
34
+ ··· −
15
2
u − 7
−
1
2
u
34
− u
33
+ ··· +
1
2
u − 1
a
1
=
−u
u
a
2
=
−
1
2
u
35
−
3
2
u
34
+ ··· + 12u
2
−
15
2
u
1
2
u
34
+ u
33
+ ··· +
9
2
u
2
+
1
2
u
a
11
=
u
u
3
+ u
a
7
=
−u
4
− u
2
+ 1
−u
6
− 2u
4
− u
2
a
3
=
−2u
35
− 6u
34
+ ··· −
25
2
u −
17
2
−
3
2
u
35
− 2u
34
+ ··· +
3
2
u −
1
2
a
6
=
u
35
+ 3u
34
+ ··· +
19
2
u +
3
2
−
1
2
u
35
− 2u
34
+ ··· −
3
2
u −
1
2
a
10
=
u
5
+ 2u
3
+ u
−u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
35
−
5
2
u
34
+ ··· − 12u −
21
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
36
− 4u
35
+ ··· + 4u + 1
c
2
, c
3
, c
5
c
6
u
36
+ 4u
35
+ ··· − 2u + 1
c
4
, c
9
u
36
+ u
35
+ ··· + 32u + 64
c
7
u
36
+ 3u
35
+ ··· − 905u + 241
c
8
, c
11
, c
12
u
36
− 3u
35
+ ··· − 5u + 1
c
10
u
36
− 5u
35
+ ··· + 9u + 3
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
36
+ 44y
35
+ ··· − 26y + 1
c
2
, c
3
, c
5
c
6
y
36
− 40y
35
+ ··· − 26y + 1
c
4
, c
9
y
36
− 35y
35
+ ··· − 82944y + 4096
c
7
y
36
+ 15y
35
+ ··· − 1047493y + 58081
c
8
, c
11
, c
12
y
36
+ 35y
35
+ ··· − 29y + 1
c
10
y
36
+ 35y
35
+ ··· − 885y + 9
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.584918 + 0.622459I
a = −0.393281 + 0.608399I
b = −1.024860 + 0.631550I
−1.06074 − 3.44814I −9.28468 + 0.39535I
u = −0.584918 − 0.622459I
a = −0.393281 − 0.608399I
b = −1.024860 − 0.631550I
−1.06074 + 3.44814I −9.28468 − 0.39535I
u = −0.750111 + 0.395247I
a = 0.51873 − 1.46690I
b = 0.823106 − 0.363879I
−1.86419 + 7.98474I −10.81435 − 5.76584I
u = −0.750111 − 0.395247I
a = 0.51873 + 1.46690I
b = 0.823106 + 0.363879I
−1.86419 − 7.98474I −10.81435 + 5.76584I
u = −0.704933 + 0.458093I
a = −0.514731 + 1.219340I
b = −0.898045 + 0.419560I
5.07187 + 4.59949I −7.09710 − 5.90387I
u = −0.704933 − 0.458093I
a = −0.514731 − 1.219340I
b = −0.898045 − 0.419560I
5.07187 − 4.59949I −7.09710 + 5.90387I
u = −0.650250 + 0.527342I
a = 0.486347 − 0.943180I
b = 0.959920 − 0.500736I
5.33026 − 0.08250I −6.15458 − 0.14164I
u = −0.650250 − 0.527342I
a = 0.486347 + 0.943180I
b = 0.959920 + 0.500736I
5.33026 + 0.08250I −6.15458 + 0.14164I
u = 0.781619
a = −0.928457
b = 0.540395
−7.32938 −12.6770
u = 0.333725 + 1.215310I
a = −0.313201 − 0.777608I
b = −0.067999 + 0.938160I
−3.58469 − 4.02994I −8.71251 + 3.82957I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.333725 − 1.215310I
a = −0.313201 + 0.777608I
b = −0.067999 − 0.938160I
−3.58469 + 4.02994I −8.71251 − 3.82957I
u = 0.239600 + 1.278530I
a = 0.185138 + 0.409163I
b = 0.007593 − 0.541465I
2.51540 − 3.15546I 0.55293 + 6.85711I
u = 0.239600 − 1.278530I
a = 0.185138 − 0.409163I
b = 0.007593 + 0.541465I
2.51540 + 3.15546I 0.55293 − 6.85711I
u = 0.570266 + 0.400726I
a = −0.877489 − 0.756539I
b = 0.297801 + 0.607076I
−6.37924 − 1.83035I −12.43917 + 3.59444I
u = 0.570266 − 0.400726I
a = −0.877489 + 0.756539I
b = 0.297801 − 0.607076I
−6.37924 + 1.83035I −12.43917 − 3.59444I
u = −0.096640 + 1.314140I
a = −1.86833 − 0.54003I
b = 2.39929 + 1.89454I
−5.49210 + 1.78793I −8.00000 + 1.75055I
u = −0.096640 − 1.314140I
a = −1.86833 + 0.54003I
b = 2.39929 − 1.89454I
−5.49210 − 1.78793I −8.00000 − 1.75055I
u = 0.003093 + 1.351590I
a = 1.346750 + 0.048804I
b = −1.75931 − 0.78922I
2.91594 + 0.49680I −6.76015 − 1.46543I
u = 0.003093 − 1.351590I
a = 1.346750 − 0.048804I
b = −1.75931 + 0.78922I
2.91594 − 0.49680I −6.76015 + 1.46543I
u = 0.632707
a = 0.514711
b = −0.261536
−1.47548 −4.26650
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.118075 + 1.391920I
a = −0.902722 + 0.302362I
b = 1.248310 + 0.060515I
4.66390 − 2.61771I −2.92989 + 4.33183I
u = 0.118075 − 1.391920I
a = −0.902722 − 0.302362I
b = 1.248310 − 0.060515I
4.66390 + 2.61771I −2.92989 − 4.33183I
u = 0.20804 + 1.44313I
a = 0.670803 − 0.683664I
b = −1.072410 + 0.518782I
−0.46527 − 4.68513I −8.00000 + 0.I
u = 0.20804 − 1.44313I
a = 0.670803 + 0.683664I
b = −1.072410 − 0.518782I
−0.46527 + 4.68513I −8.00000 + 0.I
u = −0.28295 + 1.47214I
a = −2.04519 + 0.99112I
b = 3.54834 − 1.01341I
4.15419 + 11.75140I 0
u = −0.28295 − 1.47214I
a = −2.04519 − 0.99112I
b = 3.54834 + 1.01341I
4.15419 − 11.75140I 0
u = −0.25376 + 1.49060I
a = 2.05918 − 0.92299I
b = −3.47531 + 0.84038I
11.38500 + 8.10134I 0
u = −0.25376 − 1.49060I
a = 2.05918 + 0.92299I
b = −3.47531 − 0.84038I
11.38500 − 8.10134I 0
u = −0.16804 + 1.50765I
a = 1.98331 − 0.75551I
b = −3.17021 + 0.44936I
5.88626 − 0.82923I 0
u = −0.16804 − 1.50765I
a = 1.98331 + 0.75551I
b = −3.17021 − 0.44936I
5.88626 + 0.82923I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.21821 + 1.50283I
a = −2.04552 + 0.85247I
b = 3.36643 − 0.66614I
11.93180 + 3.06807I 0
u = −0.21821 − 1.50283I
a = −2.04552 − 0.85247I
b = 3.36643 + 0.66614I
11.93180 − 3.06807I 0
u = −0.419044
a = 3.28668
b = 1.05760
−9.57946 −1.90810
u = 0.352645 + 0.225056I
a = 0.666608 + 0.999666I
b = −0.054962 − 0.463386I
−0.508387 − 0.873575I −8.67347 + 7.93671I
u = 0.352645 − 0.225056I
a = 0.666608 − 0.999666I
b = −0.054962 + 0.463386I
−0.508387 + 0.873575I −8.67347 − 7.93671I
u = −0.226552
a = −2.78574
b = −0.591839
−1.26761 −6.33540
8
II. I
u
2
= h−u
2
a + b, −u
2
a + a
2
− u
2
− a + u − 2, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
4
=
a
u
2
a
a
9
=
1
u
2
a
5
=
a
u
2
a
a
1
=
−u
u
a
2
=
−u
2
− a − u − 1
−u
2
a + 2u − 1
a
11
=
u
u
2
− u + 1
a
7
=
u
−u
a
3
=
−au + 2a
u
2
a + au − a
a
6
=
au − u
2
− 2a − 1
−u
2
a − au + a + u − 1
a
10
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
2
a − au − 5u
2
+ 3u − 20
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
(u
2
+ u − 1)
3
c
4
, c
9
u
6
c
5
, c
6
(u
2
− u − 1)
3
c
7
, c
10
(u
3
+ u
2
− 1)
2
c
8
(u
3
− u
2
+ 2u − 1)
2
c
11
, c
12
(u
3
+ u
2
+ 2u + 1)
2
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
(y
2
− 3y + 1)
3
c
4
, c
9
y
6
c
7
, c
10
(y
3
− y
2
+ 2y −1)
2
c
8
, c
11
, c
12
(y
3
+ 3y
2
+ 2y −1)
2
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = −1.071720 + 0.909787I
b = 1.27003 − 2.11500I
−5.85852 − 2.82812I −10.89327 + 4.43024I
u = 0.215080 + 1.307140I
a = 0.409360 − 0.347508I
b = −0.485107 + 0.807858I
2.03717 − 2.82812I −11.10015 − 0.15818I
u = 0.215080 − 1.307140I
a = −1.071720 − 0.909787I
b = 1.27003 + 2.11500I
−5.85852 + 2.82812I −10.89327 − 4.43024I
u = 0.215080 − 1.307140I
a = 0.409360 + 0.347508I
b = −0.485107 − 0.807858I
2.03717 + 2.82812I −11.10015 + 0.15818I
u = 0.569840
a = −0.818721
b = −0.265853
−2.10041 −19.1820
u = 0.569840
a = 2.14344
b = 0.696013
−9.99610 −21.8310
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u − 1)
3
)(u
36
− 4u
35
+ ··· + 4u + 1)
c
2
, c
3
((u
2
+ u − 1)
3
)(u
36
+ 4u
35
+ ··· − 2u + 1)
c
4
, c
9
u
6
(u
36
+ u
35
+ ··· + 32u + 64)
c
5
, c
6
((u
2
− u − 1)
3
)(u
36
+ 4u
35
+ ··· − 2u + 1)
c
7
((u
3
+ u
2
− 1)
2
)(u
36
+ 3u
35
+ ··· − 905u + 241)
c
8
((u
3
− u
2
+ 2u − 1)
2
)(u
36
− 3u
35
+ ··· − 5u + 1)
c
10
((u
3
+ u
2
− 1)
2
)(u
36
− 5u
35
+ ··· + 9u + 3)
c
11
, c
12
((u
3
+ u
2
+ 2u + 1)
2
)(u
36
− 3u
35
+ ··· − 5u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
− 3y + 1)
3
)(y
36
+ 44y
35
+ ··· − 26y + 1)
c
2
, c
3
, c
5
c
6
((y
2
− 3y + 1)
3
)(y
36
− 40y
35
+ ··· − 26y + 1)
c
4
, c
9
y
6
(y
36
− 35y
35
+ ··· − 82944y + 4096)
c
7
((y
3
− y
2
+ 2y −1)
2
)(y
36
+ 15y
35
+ ··· − 1047493y + 58081)
c
8
, c
11
, c
12
((y
3
+ 3y
2
+ 2y −1)
2
)(y
36
+ 35y
35
+ ··· − 29y + 1)
c
10
((y
3
− y
2
+ 2y −1)
2
)(y
36
+ 35y
35
+ ··· − 885y + 9)
14
