12n
0161
(K12n
0161
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 9 11 3 1 7 12 6 10
Solving Sequence
6,11 3,7
4 8 12 10 1 9 5 2
c
6
c
3
c
7
c
11
c
10
c
12
c
9
c
5
c
2
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
52
+ u
51
+ ··· + b + u, −u
52
+ u
51
+ ··· + a + 5u, u
54
− 2u
53
+ ··· + 4u
2
− 1i
I
u
2
= h−u
5
− u
3
+ b − u + 1, u
7
+ u
6
+ 2u
5
+ u
4
+ 2u
3
+ u
2
+ a + u, u
9
+ u
8
+ 2u
7
+ u
6
+ 3u
5
+ u
4
+ 2u
3
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 63 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
52
+u
51
+· · ·+b+u, −u
52
+u
51
+· · ·+a+5u, u
54
−2u
53
+· · ·+4u
2
−1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
u
52
− u
51
+ ··· + 6u
2
− 5u
u
52
− u
51
+ ··· + 3u
2
− u
a
7
=
1
u
2
a
4
=
−u
53
+ 3u
52
+ ··· − 6u − 1
u
53
+ u
52
+ ··· + 2u
2
− 2u
a
8
=
u
17
+ 2u
15
+ 5u
13
+ 6u
11
+ 7u
9
+ 6u
7
+ 4u
5
+ 2u
3
+ u
−u
17
− 3u
15
− 7u
13
− 10u
11
− 11u
9
− 8u
7
− 4u
5
+ u
a
12
=
−u
u
a
10
=
−u
3
u
3
+ u
a
1
=
−u
5
− u
u
5
+ u
3
+ u
a
9
=
u
5
+ u
u
7
+ u
5
+ 2u
3
+ u
a
5
=
−u
12
− u
10
− 3u
8
− 2u
6
− 2u
4
− u
2
+ 1
−u
14
− 2u
12
− 5u
10
− 6u
8
− 6u
6
− 4u
4
− u
2
a
2
=
u
50
− u
49
+ ··· − 5u + 1
u
52
− u
51
+ ··· − 5u
3
+ 3u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
53
+ 2u
52
+ ··· + 4u − 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
54
+ 14u
53
+ ··· + 12u + 1
c
2
, c
4
u
54
− 10u
53
+ ··· + 8u − 1
c
3
, c
7
u
54
− u
53
+ ··· − 1024u + 512
c
5
u
54
+ 2u
53
+ ··· + 220u − 200
c
6
, c
11
u
54
+ 2u
53
+ ··· + 4u
2
− 1
c
8
u
54
− 2u
53
+ ··· − 2u + 1
c
9
u
54
− 10u
53
+ ··· + 676u − 61
c
10
, c
12
u
54
− 18u
53
+ ··· + 8u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
54
+ 62y
53
+ ··· − 72y + 1
c
2
, c
4
y
54
− 14y
53
+ ··· − 12y + 1
c
3
, c
7
y
54
− 57y
53
+ ··· − 5767168y + 262144
c
5
y
54
+ 2y
53
+ ··· + 341200y + 40000
c
6
, c
11
y
54
+ 18y
53
+ ··· − 8y + 1
c
8
y
54
+ 62y
53
+ ··· − 8y + 1
c
9
y
54
+ 10y
53
+ ··· − 22900y + 3721
c
10
, c
12
y
54
+ 38y
53
+ ··· − 68y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.593814 + 0.806287I
a = −0.636999 − 0.227987I
b = −0.1217130 − 0.0332567I
−0.33427 − 1.89104I −1.02934 + 3.26612I
u = 0.593814 − 0.806287I
a = −0.636999 + 0.227987I
b = −0.1217130 + 0.0332567I
−0.33427 + 1.89104I −1.02934 − 3.26612I
u = 0.180121 + 0.965813I
a = −0.594859 − 0.700416I
b = 0.136216 + 0.368678I
1.26301 − 2.44603I 2.73199 + 4.98223I
u = 0.180121 − 0.965813I
a = −0.594859 + 0.700416I
b = 0.136216 − 0.368678I
1.26301 + 2.44603I 2.73199 − 4.98223I
u = 0.071641 + 1.019550I
a = 1.27503 − 1.16413I
b = −0.851885 + 0.413364I
3.28922 − 2.50118I 2.52265 + 4.62453I
u = 0.071641 − 1.019550I
a = 1.27503 + 1.16413I
b = −0.851885 − 0.413364I
3.28922 + 2.50118I 2.52265 − 4.62453I
u = −0.760882 + 0.685041I
a = −0.333764 − 0.729244I
b = −0.91421 + 1.34330I
−2.42886 − 2.49711I −6.09592 + 3.26971I
u = −0.760882 − 0.685041I
a = −0.333764 + 0.729244I
b = −0.91421 − 1.34330I
−2.42886 + 2.49711I −6.09592 − 3.26971I
u = 0.803356 + 0.644099I
a = −0.363676 + 0.484008I
b = −2.27231 − 0.36526I
4.20288 + 1.73783I −3.96774 + 0.13211I
u = 0.803356 − 0.644099I
a = −0.363676 − 0.484008I
b = −2.27231 + 0.36526I
4.20288 − 1.73783I −3.96774 − 0.13211I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.743941 + 0.723947I
a = 1.296630 + 0.497606I
b = 0.351768 + 1.145760I
−4.60943 + 0.53524I −6.65010 + 0.78439I
u = 0.743941 − 0.723947I
a = 1.296630 − 0.497606I
b = 0.351768 − 1.145760I
−4.60943 − 0.53524I −6.65010 − 0.78439I
u = −0.040432 + 0.958127I
a = −2.37271 + 0.02401I
b = 1.54595 + 0.64385I
0.680231 + 1.026360I 0.362891 + 0.577482I
u = −0.040432 − 0.958127I
a = −2.37271 − 0.02401I
b = 1.54595 − 0.64385I
0.680231 − 1.026360I 0.362891 − 0.577482I
u = −0.714465 + 0.763494I
a = 0.138140 + 1.250370I
b = 1.70446 − 1.06690I
−3.72278 + 1.64945I −9.08237 − 1.96376I
u = −0.714465 − 0.763494I
a = 0.138140 − 1.250370I
b = 1.70446 + 1.06690I
−3.72278 − 1.64945I −9.08237 + 1.96376I
u = 0.823105 + 0.665323I
a = 0.319376 − 0.932123I
b = 2.69399 + 0.67083I
3.08774 + 8.72867I −5.48413 − 4.26141I
u = 0.823105 − 0.665323I
a = 0.319376 + 0.932123I
b = 2.69399 − 0.67083I
3.08774 − 8.72867I −5.48413 + 4.26141I
u = −0.097324 + 1.084370I
a = 3.37839 − 1.05528I
b = −2.41040 + 0.52269I
10.44700 + 1.25812I 2.80140 − 1.05629I
u = −0.097324 − 1.084370I
a = 3.37839 + 1.05528I
b = −2.41040 − 0.52269I
10.44700 − 1.25812I 2.80140 + 1.05629I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.128512 + 1.081900I
a = −3.37735 + 0.83410I
b = 2.46629 − 0.51176I
9.60149 + 8.44587I 1.50673 − 5.85664I
u = −0.128512 − 1.081900I
a = −3.37735 − 0.83410I
b = 2.46629 + 0.51176I
9.60149 − 8.44587I 1.50673 + 5.85664I
u = −0.807760 + 0.739295I
a = −0.520537 + 0.077016I
b = 0.673789 + 0.625242I
−5.15070 − 1.55728I −3.41228 + 1.62313I
u = −0.807760 − 0.739295I
a = −0.520537 − 0.077016I
b = 0.673789 − 0.625242I
−5.15070 + 1.55728I −3.41228 − 1.62313I
u = −0.506885 + 0.991107I
a = −1.80520 + 1.33240I
b = 0.96224 + 1.21089I
7.37339 − 2.01731I 0
u = −0.506885 − 0.991107I
a = −1.80520 − 1.33240I
b = 0.96224 − 1.21089I
7.37339 + 2.01731I 0
u = 0.624887 + 0.943323I
a = −0.228701 + 0.787647I
b = −0.307430 + 0.090678I
0.16355 − 2.91161I 0. + 2.36931I
u = 0.624887 − 0.943323I
a = −0.228701 − 0.787647I
b = −0.307430 − 0.090678I
0.16355 + 2.91161I 0. − 2.36931I
u = −0.548486 + 1.001280I
a = 2.24692 − 1.41567I
b = −1.39112 − 1.33428I
7.75921 + 5.16031I 0. − 5.15990I
u = −0.548486 − 1.001280I
a = 2.24692 + 1.41567I
b = −1.39112 + 1.33428I
7.75921 − 5.16031I 0. + 5.15990I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.782876 + 0.849138I
a = −0.281362 + 1.292200I
b = −0.282478 − 0.823905I
−0.09216 − 5.90377I −6.51038 + 5.50131I
u = 0.782876 − 0.849138I
a = −0.281362 − 1.292200I
b = −0.282478 + 0.823905I
−0.09216 + 5.90377I −6.51038 − 5.50131I
u = −0.684869 + 0.949371I
a = −1.65615 + 1.47578I
b = 2.17505 + 0.72111I
−3.14902 + 3.72291I −7.13132 + 0.I
u = −0.684869 − 0.949371I
a = −1.65615 − 1.47578I
b = 2.17505 − 0.72111I
−3.14902 − 3.72291I −7.13132 + 0.I
u = 0.767085 + 0.893425I
a = 0.565950 − 0.681295I
b = −0.571422 + 0.670697I
0.0463916 + 0.0789452I 0
u = 0.767085 − 0.893425I
a = 0.565950 + 0.681295I
b = −0.571422 − 0.670697I
0.0463916 − 0.0789452I 0
u = 0.697552 + 0.975593I
a = −1.103840 − 0.722587I
b = 0.62795 − 1.51314I
−3.84507 − 6.03561I 0
u = 0.697552 − 0.975593I
a = −1.103840 + 0.722587I
b = 0.62795 + 1.51314I
−3.84507 + 6.03561I 0
u = −0.697282 + 0.998483I
a = 1.81767 − 0.37021I
b = −1.27625 − 1.22230I
−1.48579 + 8.04265I 0
u = −0.697282 − 0.998483I
a = 1.81767 + 0.37021I
b = −1.27625 + 1.22230I
−1.48579 − 8.04265I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.737707 + 0.986539I
a = 0.264849 + 0.912543I
b = 0.567709 − 0.785725I
−4.39392 + 7.36741I 0
u = −0.737707 − 0.986539I
a = 0.264849 − 0.912543I
b = 0.567709 + 0.785725I
−4.39392 − 7.36741I 0
u = 0.702152 + 1.027660I
a = 1.34427 + 2.48138I
b = −2.60875 + 0.42340I
5.35703 − 7.40628I 0
u = 0.702152 − 1.027660I
a = 1.34427 − 2.48138I
b = −2.60875 − 0.42340I
5.35703 + 7.40628I 0
u = 0.717612 + 1.027020I
a = −1.72080 − 2.58367I
b = 3.11897 − 0.58149I
4.1859 − 14.5074I 0
u = 0.717612 − 1.027020I
a = −1.72080 + 2.58367I
b = 3.11897 + 0.58149I
4.1859 + 14.5074I 0
u = −0.649599 + 0.312516I
a = −0.395839 + 0.699565I
b = −1.50211 + 0.45460I
5.93552 − 0.76797I −3.85312 − 0.11331I
u = −0.649599 − 0.312516I
a = −0.395839 − 0.699565I
b = −1.50211 − 0.45460I
5.93552 + 0.76797I −3.85312 + 0.11331I
u = −0.660388 + 0.239204I
a = 0.392626 − 1.167170I
b = 1.385750 − 0.144674I
5.30723 + 6.14183I −5.10684 − 4.83754I
u = −0.660388 − 0.239204I
a = 0.392626 + 1.167170I
b = 1.385750 + 0.144674I
5.30723 − 6.14183I −5.10684 + 4.83754I
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.570438
a = −0.597054
b = 0.412635
−1.74370 −4.75930
u = 0.401684 + 0.251203I
a = −0.718595 − 0.849362I
b = −0.003567 + 0.541793I
−0.534257 − 1.150530I −6.04520 + 5.80427I
u = 0.401684 − 0.251203I
a = −0.718595 + 0.849362I
b = −0.003567 − 0.541793I
−0.534257 + 1.150530I −6.04520 − 5.80427I
u = −0.320915
a = 2.73811
b = 0.794388
−2.14124 −1.56380
10
II. I
u
2
= h−u
5
− u
3
+ b − u + 1, u
7
+ u
6
+ 2u
5
+ u
4
+ 2u
3
+ u
2
+ a + u, u
9
+
u
8
+ 2u
7
+ u
6
+ 3u
5
+ u
4
+ 2u
3
+ u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
−u
7
− u
6
− 2u
5
− u
4
− 2u
3
− u
2
− u
u
5
+ u
3
+ u − 1
a
7
=
1
u
2
a
4
=
−u
7
− u
6
− 2u
5
− u
4
− 2u
3
− u
2
− u
u
5
+ u
3
+ u − 1
a
8
=
1
u
2
a
12
=
−u
u
a
10
=
−u
3
u
3
+ u
a
1
=
−u
5
− u
u
5
+ u
3
+ u
a
9
=
u
5
+ u
u
7
+ u
5
+ 2u
3
+ u
a
5
=
u
5
+ u
−u
5
− u
3
− u
a
2
=
−u
7
− u
6
− 3u
5
− u
4
− 2u
3
− u
2
− 2u
2u
5
+ 2u
3
+ 2u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
7
− 4u
6
− 5u
5
− 5u
4
− 10u
3
− 5u
2
− u − 11
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
9
c
3
, c
7
u
9
c
4
(u + 1)
9
c
5
, c
8
u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1
c
6
u
9
+ u
8
+ 2u
7
+ u
6
+ 3u
5
+ u
4
+ 2u
3
+ u − 1
c
9
u
9
+ 5u
8
+ 12u
7
+ 15u
6
+ 9u
5
− u
4
− 4u
3
− 2u
2
+ u + 1
c
10
u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1
c
11
u
9
− u
8
+ 2u
7
− u
6
+ 3u
5
− u
4
+ 2u
3
+ u + 1
c
12
u
9
− 3u
8
+ 8u
7
− 13u
6
+ 17u
5
− 17u
4
+ 12u
3
− 6u
2
+ u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y − 1)
9
c
3
, c
7
y
9
c
5
, c
8
y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y − 1
c
6
, c
11
y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y − 1
c
9
y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y − 1
c
10
, c
12
y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y − 1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.140343 + 0.966856I
a = 0.900982 − 0.594909I
b = −0.663053 + 0.788921I
0.13850 − 2.09337I −4.27981 + 4.44592I
u = 0.140343 − 0.966856I
a = 0.900982 + 0.594909I
b = −0.663053 − 0.788921I
0.13850 + 2.09337I −4.27981 − 4.44592I
u = 0.628449 + 0.875112I
a = 0.249476 + 1.304240I
b = −1.52709 − 0.20930I
−2.26187 − 2.45442I −4.16203 + 2.47153I
u = 0.628449 − 0.875112I
a = 0.249476 − 1.304240I
b = −1.52709 + 0.20930I
−2.26187 + 2.45442I −4.16203 − 2.47153I
u = −0.796005 + 0.733148I
a = −0.766570 + 0.255687I
b = 0.224752 + 0.919301I
−6.01628 − 1.33617I −13.03110 + 0.17445I
u = −0.796005 − 0.733148I
a = −0.766570 − 0.255687I
b = 0.224752 − 0.919301I
−6.01628 + 1.33617I −13.03110 − 0.17445I
u = −0.728966 + 0.986295I
a = 0.721488 + 0.307914I
b = 0.124310 − 1.173370I
−5.24306 + 7.08493I −11.12684 − 5.18429I
u = −0.728966 − 0.986295I
a = 0.721488 − 0.307914I
b = 0.124310 + 1.173370I
−5.24306 − 7.08493I −11.12684 + 5.18429I
u = 0.512358
a = −1.21075
b = −0.317835
−2.84338 −14.8000
14
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
54
+ 14u
53
+ ··· + 12u + 1)
c
2
((u − 1)
9
)(u
54
− 10u
53
+ ··· + 8u − 1)
c
3
, c
7
u
9
(u
54
− u
53
+ ··· − 1024u + 512)
c
4
((u + 1)
9
)(u
54
− 10u
53
+ ··· + 8u − 1)
c
5
(u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1)
· (u
54
+ 2u
53
+ ··· + 220u − 200)
c
6
(u
9
+ u
8
+ ··· + u − 1)(u
54
+ 2u
53
+ ··· + 4u
2
− 1)
c
8
(u
9
+ u
8
+ ··· − u − 1)(u
54
− 2u
53
+ ··· − 2u + 1)
c
9
(u
9
+ 5u
8
+ 12u
7
+ 15u
6
+ 9u
5
− u
4
− 4u
3
− 2u
2
+ u + 1)
· (u
54
− 10u
53
+ ··· + 676u − 61)
c
10
(u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1)
· (u
54
− 18u
53
+ ··· + 8u + 1)
c
11
(u
9
− u
8
+ ··· + u + 1)(u
54
+ 2u
53
+ ··· + 4u
2
− 1)
c
12
(u
9
− 3u
8
+ 8u
7
− 13u
6
+ 17u
5
− 17u
4
+ 12u
3
− 6u
2
+ u + 1)
· (u
54
− 18u
53
+ ··· + 8u + 1)
15
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y − 1)
9
)(y
54
+ 62y
53
+ ··· − 72y + 1)
c
2
, c
4
((y − 1)
9
)(y
54
− 14y
53
+ ··· − 12y + 1)
c
3
, c
7
y
9
(y
54
− 57y
53
+ ··· − 5767168y + 262144)
c
5
(y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y − 1)
· (y
54
+ 2y
53
+ ··· + 341200y + 40000)
c
6
, c
11
(y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y − 1)
· (y
54
+ 18y
53
+ ··· − 8y + 1)
c
8
(y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y − 1)
· (y
54
+ 62y
53
+ ··· − 8y + 1)
c
9
(y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y − 1)
· (y
54
+ 10y
53
+ ··· − 22900y + 3721)
c
10
, c
12
(y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y − 1)
· (y
54
+ 38y
53
+ ··· − 68y + 1)
16
