12n
0038
(K12n
0038
)
A knot diagram
1
Linearized knot diagam
3 5 6 8 2 12 11 5 6 7 10 9
Solving Sequence
2,6
5 3
1,10
9 8 4 12 7 11
c
5
c
2
c
1
c
9
c
8
c
4
c
12
c
6
c
11
c
3
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h132u
44
+ 811u
43
+ ··· + 32b + 15, −275u
44
− 1867u
43
+ ··· + 32a + 615, u
45
+ 7u
44
+ ··· − 5u − 1i
I
u
2
= h−au + 3b + 2a, a
6
− a
5
u − a
5
− 3a
4
u + 12a
3
u − 6a
3
− 9au + 18a − 27, u
2
− u + 1i
* 2 irreducible components of dim
C
= 0, with total 57 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
= h132u
44
+ 811u
43
+ · · · + 32b + 15, −275u
44
− 1867u
43
+ · · · + 32a +
615, u
45
+ 7u
44
+ · · · − 5u − 1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
5
=
1
u
2
a
3
=
u
u
3
+ u
a
1
=
u
3
u
5
+ u
3
+ u
a
10
=
8.59375u
44
+ 58.3438u
43
+ ··· − 80.9688u − 19.2188
−4.12500u
44
− 25.3438u
43
+ ··· + 14.5625u − 0.468750
a
9
=
4.46875u
44
+ 33u
43
+ ··· − 66.4063u − 19.6875
−4.12500u
44
− 25.3438u
43
+ ··· + 14.5625u − 0.468750
a
8
=
5.59375u
44
+ 40.8125u
43
+ ··· − 67.9063u − 17.5000
−5.06250u
44
− 32.0313u
43
+ ··· + 15.3750u − 0.531250
a
4
=
−u
3
u
3
+ u
a
12
=
1
32
u
43
+
3
16
u
42
+ ··· −
17
8
u +
31
32
−0.0312500u
44
− 0.218750u
43
+ ··· + 0.156250u + 0.0312500
a
7
=
−0.468750u
44
− 3.31250u
43
+ ··· + 2.71875u + 1.56250
0.468750u
44
+ 2.84375u
43
+ ··· − 0.656250u − 0.0312500
a
11
=
0.812500u
44
+ 5.40625u
43
+ ··· − 4.25000u − 0.593750
−0.468750u
44
− 2.90625u
43
+ ··· + 1.90625u + 0.0937500
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
177
16
u
44
−
1121
16
u
43
+ ··· +
1137
16
u +
95
8
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
45
+ 9u
44
+ ··· − 9u − 1
c
2
, c
5
u
45
+ 7u
44
+ ··· − 5u − 1
c
3
u
45
− 7u
44
+ ··· − 877615u − 93361
c
4
, c
8
u
45
− u
44
+ ··· + 8192u − 4096
c
6
u
45
− 9u
44
+ ··· + 203u − 37
c
7
, c
10
u
45
− 3u
44
+ ··· + 3u − 1
c
9
u
45
+ 3u
44
+ ··· + 2181u − 1201
c
11
u
45
− 23u
44
+ ··· + 3u − 1
c
12
u
45
− u
44
+ ··· + 3u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
45
+ 61y
44
+ ··· − 29y −1
c
2
, c
5
y
45
+ 9y
44
+ ··· − 9y −1
c
3
y
45
+ 113y
44
+ ··· − 304227830897y −8716276321
c
4
, c
8
y
45
− 65y
44
+ ··· + 33554432y −16777216
c
6
y
45
+ 21y
44
+ ··· + 14347y −1369
c
7
, c
10
y
45
− 23y
44
+ ··· + 3y −1
c
9
y
45
+ 17y
44
+ ··· + 9246099y −1442401
c
11
y
45
+ y
44
+ ··· + 11y −1
c
12
y
45
+ 77y
44
+ ··· + 3y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.865449 + 0.454219I
a = 0.319220 + 0.531227I
b = −0.36863 − 1.46260I
4.73623 − 2.35998I 8.35847 + 1.71921I
u = 0.865449 − 0.454219I
a = 0.319220 − 0.531227I
b = −0.36863 + 1.46260I
4.73623 + 2.35998I 8.35847 − 1.71921I
u = 0.330564 + 0.971277I
a = −1.35737 − 1.01368I
b = 0.526140 + 0.066430I
−1.15352 + 2.68830I −0.34175 − 6.47696I
u = 0.330564 − 0.971277I
a = −1.35737 + 1.01368I
b = 0.526140 − 0.066430I
−1.15352 − 2.68830I −0.34175 + 6.47696I
u = 0.622281 + 0.846945I
a = −0.079868 + 0.698124I
b = −0.493189 − 0.089540I
0.80407 + 2.44032I 0.39786 − 3.53339I
u = 0.622281 − 0.846945I
a = −0.079868 − 0.698124I
b = −0.493189 + 0.089540I
0.80407 − 2.44032I 0.39786 + 3.53339I
u = 0.755037 + 0.558570I
a = −0.0783721 − 0.0338596I
b = −0.107622 + 0.976039I
1.57242 + 1.53241I 4.30562 − 2.68000I
u = 0.755037 − 0.558570I
a = −0.0783721 + 0.0338596I
b = −0.107622 − 0.976039I
1.57242 − 1.53241I 4.30562 + 2.68000I
u = 0.867507 + 0.624607I
a = −0.382914 + 0.278991I
b = 0.56116 − 1.43284I
4.57167 + 5.62351I 7.47380 − 5.91915I
u = 0.867507 − 0.624607I
a = −0.382914 − 0.278991I
b = 0.56116 + 1.43284I
4.57167 − 5.62351I 7.47380 + 5.91915I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.180988 + 0.903067I
a = 0.74445 + 1.99166I
b = −0.201660 − 0.527398I
0.112526 − 1.116170I 0.739301 + 0.318022I
u = 0.180988 − 0.903067I
a = 0.74445 − 1.99166I
b = −0.201660 + 0.527398I
0.112526 + 1.116170I 0.739301 − 0.318022I
u = 0.482106 + 1.053890I
a = −1.78297 + 0.42437I
b = 0.516649 − 0.731743I
−0.20453 + 3.20330I 0. − 3.29559I
u = 0.482106 − 1.053890I
a = −1.78297 − 0.42437I
b = 0.516649 + 0.731743I
−0.20453 − 3.20330I 0. + 3.29559I
u = 0.601778 + 1.060200I
a = 1.28241 − 1.47156I
b = 0.110511 + 1.232230I
3.02293 − 0.08785I 0
u = 0.601778 − 1.060200I
a = 1.28241 + 1.47156I
b = 0.110511 − 1.232230I
3.02293 + 0.08785I 0
u = 0.503909 + 1.128940I
a = 2.43305 − 0.92945I
b = −0.80290 + 1.19457I
2.36444 + 7.52477I 0
u = 0.503909 − 1.128940I
a = 2.43305 + 0.92945I
b = −0.80290 − 1.19457I
2.36444 − 7.52477I 0
u = −0.084493 + 0.713012I
a = −1.01316 + 2.46838I
b = 0.821961 − 0.446891I
−0.81007 + 4.73637I −1.24888 − 6.79408I
u = −0.084493 − 0.713012I
a = −1.01316 − 2.46838I
b = 0.821961 + 0.446891I
−0.81007 − 4.73637I −1.24888 + 6.79408I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.921571 + 0.928254I
a = 0.300181 − 0.167382I
b = 0.141408 − 0.815178I
7.46078 − 0.75691I 0
u = −0.921571 − 0.928254I
a = 0.300181 + 0.167382I
b = 0.141408 + 0.815178I
7.46078 + 0.75691I 0
u = −0.904820 + 0.965905I
a = −0.790239 + 0.408383I
b = 0.376118 + 0.820565I
7.34035 − 5.98505I 0
u = −0.904820 − 0.965905I
a = −0.790239 − 0.408383I
b = 0.376118 − 0.820565I
7.34035 + 5.98505I 0
u = −1.009740 + 0.883218I
a = −0.377537 + 0.824899I
b = 1.22594 − 1.41286I
10.71020 + 1.37563I 0
u = −1.009740 − 0.883218I
a = −0.377537 − 0.824899I
b = 1.22594 + 1.41286I
10.71020 − 1.37563I 0
u = −1.034420 + 0.865842I
a = 0.664407 − 1.034480I
b = −1.61121 + 1.49060I
13.5421 + 6.5266I 0
u = −1.034420 − 0.865842I
a = 0.664407 + 1.034480I
b = −1.61121 − 1.49060I
13.5421 − 6.5266I 0
u = −0.387424 + 0.500589I
a = −1.93597 + 1.18940I
b = 1.225780 + 0.662919I
0.14562 − 6.62336I 2.53824 + 3.24872I
u = −0.387424 − 0.500589I
a = −1.93597 − 1.18940I
b = 1.225780 − 0.662919I
0.14562 + 6.62336I 2.53824 − 3.24872I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.024940 + 0.925322I
a = −0.086100 − 1.156550I
b = −0.91727 + 1.93981I
15.3464 − 2.2779I 0
u = −1.024940 − 0.925322I
a = −0.086100 + 1.156550I
b = −0.91727 − 1.93981I
15.3464 + 2.2779I 0
u = −0.128241 + 0.602764I
a = 1.33735 − 2.06079I
b = −0.899761 + 0.098160I
−2.43287 + 0.11880I −4.07087 − 1.28262I
u = −0.128241 − 0.602764I
a = 1.33735 + 2.06079I
b = −0.899761 − 0.098160I
−2.43287 − 0.11880I −4.07087 + 1.28262I
u = −0.911280 + 1.044170I
a = −1.97485 + 0.39149I
b = 1.36520 + 1.26824I
10.17040 − 8.40766I 0
u = −0.911280 − 1.044170I
a = −1.97485 − 0.39149I
b = 1.36520 − 1.26824I
10.17040 + 8.40766I 0
u = −0.907943 + 1.066410I
a = 2.33785 − 0.47875I
b = −1.69979 − 1.31179I
12.8666 − 13.6205I 0
u = −0.907943 − 1.066410I
a = 2.33785 + 0.47875I
b = −1.69979 + 1.31179I
12.8666 + 13.6205I 0
u = −0.948740 + 1.039640I
a = 1.92604 + 0.22525I
b = −1.11708 − 1.78219I
14.9563 − 4.9353I 0
u = −0.948740 − 1.039640I
a = 1.92604 − 0.22525I
b = −1.11708 + 1.78219I
14.9563 + 4.9353I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.306772 + 0.492284I
a = 1.74185 − 1.46922I
b = −1.089070 − 0.461342I
−1.91688 − 1.75082I −1.56719 − 0.68553I
u = −0.306772 − 0.492284I
a = 1.74185 + 1.46922I
b = −1.089070 + 0.461342I
−1.91688 + 1.75082I −1.56719 + 0.68553I
u = 0.455358
a = −1.30613
b = 0.610269
1.30638 7.75970
u = −0.366917 + 0.266389I
a = −1.07439 + 1.09771I
b = 0.632186 + 0.840901I
2.23991 + 0.49049I 6.10667 − 1.43657I
u = −0.366917 − 0.266389I
a = −1.07439 − 1.09771I
b = 0.632186 − 0.840901I
2.23991 − 0.49049I 6.10667 + 1.43657I
9
II. I
u
2
= h−au + 3b + 2a, −a
5
u − 3a
4
u + · · · + 18a − 27, u
2
− u + 1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
5
=
1
u − 1
a
3
=
u
u − 1
a
1
=
−1
0
a
10
=
a
1
3
au −
2
3
a
a
9
=
1
3
au +
1
3
a
1
3
au −
2
3
a
a
8
=
1
3
au +
1
3
a
1
3
au −
2
3
a
a
4
=
1
u − 1
a
12
=
−
1
3
a
2
− 1
−
1
3
a
2
u +
1
3
a
2
a
7
=
1
9
a
4
u +
1
3
a
2
u + ··· −
1
3
a
2
+ 1
−
1
9
a
4
u
a
11
=
2
9
a
4
u −
1
3
a
2
u + ··· +
1
3
a
2
− 1
−
1
9
a
4
u
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
2
27
a
5
u −
1
27
a
5
−
4
9
a
4
u −
2
9
a
3
u +
4
9
a
3
+ a
2
u −
5
3
a
2
− 2au + 2a − 3u + 6
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
6
c
2
(u
2
+ u + 1)
6
c
4
, c
8
u
12
c
6
, c
11
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
2
c
7
, c
9
, c
12
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
c
10
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
(y
2
+ y + 1)
6
c
4
, c
8
y
12
c
6
, c
11
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
c
7
, c
9
, c
10
c
12
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 0.066864 + 1.367670I
b = −0.428243 − 0.664531I
1.89061 + 1.10558I 3.50232 − 2.57477I
u = 0.500000 + 0.866025I
a = 1.217870 − 0.625927I
b = −0.428243 + 0.664531I
1.89061 + 2.95419I 7.01188 − 5.05114I
u = 0.500000 + 0.866025I
a = −1.24734 − 1.31124I
b = 1.002190 + 0.295542I
−1.89061 + 1.10558I 0.06995 − 2.75005I
u = 0.500000 + 0.866025I
a = −1.75924 − 0.42461I
b = 1.002190 − 0.295542I
−1.89061 + 2.95419I −1.81693 − 4.43387I
u = 0.500000 + 0.866025I
a = 2.09482 + 0.09194I
b = −1.073950 + 0.558752I
− 3.66314I 4.13964 + 1.97785I
u = 0.500000 + 0.866025I
a = 1.12703 + 1.76820I
b = −1.073950 − 0.558752I
7.72290I 1.09315 − 9.68468I
u = 0.500000 − 0.866025I
a = 0.066864 − 1.367670I
b = −0.428243 + 0.664531I
1.89061 − 1.10558I 3.50232 + 2.57477I
u = 0.500000 − 0.866025I
a = 1.217870 + 0.625927I
b = −0.428243 − 0.664531I
1.89061 − 2.95419I 7.01188 + 5.05114I
u = 0.500000 − 0.866025I
a = −1.24734 + 1.31124I
b = 1.002190 − 0.295542I
−1.89061 − 1.10558I 0.06995 + 2.75005I
u = 0.500000 − 0.866025I
a = −1.75924 + 0.42461I
b = 1.002190 + 0.295542I
−1.89061 − 2.95419I −1.81693 + 4.43387I
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 − 0.866025I
a = 2.09482 − 0.09194I
b = −1.073950 − 0.558752I
3.66314I 4.13964 − 1.97785I
u = 0.500000 − 0.866025I
a = 1.12703 − 1.76820I
b = −1.073950 + 0.558752I
− 7.72290I 1.09315 + 9.68468I
14
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
6
)(u
45
+ 9u
44
+ ··· − 9u − 1)
c
2
((u
2
+ u + 1)
6
)(u
45
+ 7u
44
+ ··· − 5u − 1)
c
3
((u
2
− u + 1)
6
)(u
45
− 7u
44
+ ··· − 877615u − 93361)
c
4
, c
8
u
12
(u
45
− u
44
+ ··· + 8192u − 4096)
c
5
((u
2
− u + 1)
6
)(u
45
+ 7u
44
+ ··· − 5u − 1)
c
6
((u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
2
)(u
45
− 9u
44
+ ··· + 203u − 37)
c
7
((u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
)(u
45
− 3u
44
+ ··· + 3u − 1)
c
9
((u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
)(u
45
+ 3u
44
+ ··· + 2181u − 1201)
c
10
((u
6
+ u
5
− u
4
− 2u
3
+ u + 1)
2
)(u
45
− 3u
44
+ ··· + 3u − 1)
c
11
((u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
2
)(u
45
− 23u
44
+ ··· + 3u − 1)
c
12
((u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
)(u
45
− u
44
+ ··· + 3u − 1)
15
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
6
)(y
45
+ 61y
44
+ ··· − 29y − 1)
c
2
, c
5
((y
2
+ y + 1)
6
)(y
45
+ 9y
44
+ ··· − 9y − 1)
c
3
((y
2
+ y + 1)
6
)(y
45
+ 113y
44
+ ··· − 3.04228 × 10
11
y − 8.71628 × 10
9
)
c
4
, c
8
y
12
(y
45
− 65y
44
+ ··· + 3.35544 × 10
7
y − 1.67772 × 10
7
)
c
6
((y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
)(y
45
+ 21y
44
+ ··· + 14347y − 1369)
c
7
, c
10
((y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
)(y
45
− 23y
44
+ ··· + 3y − 1)
c
9
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
· (y
45
+ 17y
44
+ ··· + 9246099y − 1442401)
c
11
((y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
)(y
45
+ y
44
+ ··· + 11y − 1)
c
12
((y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
)(y
45
+ 77y
44
+ ··· + 3y − 1)
16
