12n
0042
(K12n
0042
)
A knot diagram
1
Linearized knot diagam
3 5 6 9 2 12 11 5 6 7 8 10
Solving Sequence
2,6
5 3
1,10
9 4 8 12 7 11
c
5
c
2
c
1
c
9
c
4
c
8
c
12
c
6
c
11
c
3
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h18u
38
+ 107u
37
+ ··· + 16b − 35, −35u
38
− 228u
37
+ ··· + 16a − 167, u
39
+ 6u
38
+ ··· + 6u + 1i
I
u
2
= h−au + b, a
5
+ a
4
u − a
4
− 2a
3
u + a
2
+ au − a − u, u
2
− u + 1i
* 2 irreducible components of dim
C
= 0, with total 49 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
= h18u
38
+ 107u
37
+ · · · + 16b − 35, −35u
38
− 228u
37
+ · · · + 16a −
167, u
39
+ 6u
38
+ · · · + 6u + 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
=
2.18750u
38
+ 14.2500u
37
+ ··· + 54.5000u + 10.4375
−1.12500u
38
− 6.68750u
37
+ ··· + 2.68750u + 2.18750
a
9
=
3.31250u
38
+ 20.9375u
37
+ ··· + 51.8125u + 8.25000
−1.12500u
38
− 6.68750u
37
+ ··· + 2.68750u + 2.18750
a
4
=
−u
3
u
3
+ u
a
8
=
3.18750u
38
+ 21.3125u
37
+ ··· + 64.1875u + 11.5000
−3.12500u
38
− 15.5625u
37
+ ··· − 3.93750u + 1.06250
a
12
=
−0.0625000u
37
− 0.312500u
36
+ ··· − 2.31250u + 0.937500
0.0625000u
38
+ 0.312500u
37
+ ··· + 2.31250u
2
+ 0.0625000u
a
7
=
0.875000u
38
+ 5.18750u
37
+ ··· + 5.56250u + 1.93750
0.687500u
37
+ 3.43750u
36
+ ··· + 3.93750u + 0.812500
a
11
=
−
3
4
u
38
− 4u
37
+ ··· −
17
8
u −
1
4
1
8
u
38
+
1
16
u
37
+ ··· −
35
16
u −
11
16
(ii) Obstruction class = −1
(iii) Cusp Shapes =
227
16
u
38
+
1349
16
u
37
+ ··· +
1989
16
u +
25
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
39
+ 8u
38
+ ··· − 10u − 1
c
2
, c
5
u
39
+ 6u
38
+ ··· + 6u + 1
c
3
u
39
− 6u
38
+ ··· + 227832u + 23497
c
4
, c
8
u
39
+ u
38
+ ··· + 2048u + 1024
c
6
u
39
− 9u
38
+ ··· + 179u − 17
c
7
, c
10
, c
11
u
39
+ 3u
38
+ ··· − 3u + 1
c
9
u
39
− 3u
38
+ ··· − 3u + 1
c
12
u
39
+ 11u
38
+ ··· + 267u + 73
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
39
+ 52y
38
+ ··· + 58y −1
c
2
, c
5
y
39
+ 8y
38
+ ··· − 10y −1
c
3
y
39
+ 96y
38
+ ··· − 14329729890y −552109009
c
4
, c
8
y
39
+ 55y
38
+ ··· − 5242880y −1048576
c
6
y
39
+ 9y
38
+ ··· + 2665y −289
c
7
, c
10
, c
11
y
39
− 35y
38
+ ··· − 3y −1
c
9
y
39
− 67y
38
+ ··· − 3y −1
c
12
y
39
− 7y
38
+ ··· − 292543y −5329
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.217775 + 0.986965I
a = 0.478937 − 0.405328I
b = −0.504345 − 0.384424I
−5.08086 + 0.15570I −9.83989 − 1.69370I
u = 0.217775 − 0.986965I
a = 0.478937 + 0.405328I
b = −0.504345 + 0.384424I
−5.08086 − 0.15570I −9.83989 + 1.69370I
u = 0.795838 + 0.548145I
a = 0.160432 + 0.477438I
b = 0.134027 − 0.467904I
2.99441 + 1.56903I 2.04971 − 2.63058I
u = 0.795838 − 0.548145I
a = 0.160432 − 0.477438I
b = 0.134027 + 0.467904I
2.99441 − 1.56903I 2.04971 + 2.63058I
u = 0.395368 + 0.848396I
a = −0.328402 + 0.067571I
b = 0.187167 + 0.251900I
−0.32291 + 1.65676I −2.57784 − 5.09388I
u = 0.395368 − 0.848396I
a = −0.328402 − 0.067571I
b = 0.187167 − 0.251900I
−0.32291 − 1.65676I −2.57784 + 5.09388I
u = 0.835550 + 0.682012I
a = −0.122808 − 0.434535I
b = −0.193746 + 0.446832I
−1.08176 + 5.00495I −4.00000 − 5.49460I
u = 0.835550 − 0.682012I
a = −0.122808 + 0.434535I
b = −0.193746 − 0.446832I
−1.08176 − 5.00495I −4.00000 + 5.49460I
u = 0.820354 + 0.374203I
a = −0.141671 − 0.553737I
b = −0.090989 + 0.507274I
−0.79445 − 1.82967I −2.60045 + 1.37386I
u = 0.820354 − 0.374203I
a = −0.141671 + 0.553737I
b = −0.090989 − 0.507274I
−0.79445 + 1.82967I −2.60045 − 1.37386I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.516539 + 1.059320I
a = 0.121118 − 0.354987I
b = −0.438608 + 0.055062I
1.18484 + 3.42225I 0. − 3.84715I
u = 0.516539 − 1.059320I
a = 0.121118 + 0.354987I
b = −0.438608 − 0.055062I
1.18484 − 3.42225I 0. + 3.84715I
u = 0.639199 + 1.006420I
a = −0.023637 + 0.351005I
b = 0.368368 − 0.200573I
−2.22349 + 0.53849I −5.53531 + 1.24212I
u = 0.639199 − 1.006420I
a = −0.023637 − 0.351005I
b = 0.368368 + 0.200573I
−2.22349 − 0.53849I −5.53531 − 1.24212I
u = 0.463738 + 1.131070I
a = −0.157364 + 0.415273I
b = 0.542679 − 0.014588I
−3.39936 + 6.68540I −5.94355 − 5.90487I
u = 0.463738 − 1.131070I
a = −0.157364 − 0.415273I
b = 0.542679 + 0.014588I
−3.39936 − 6.68540I −5.94355 + 5.90487I
u = −0.146263 + 0.696408I
a = 1.54768 − 0.36171I
b = −0.025530 − 1.130720I
−6.52633 + 3.36713I −11.83061 − 4.84786I
u = −0.146263 − 0.696408I
a = 1.54768 + 0.36171I
b = −0.025530 + 1.130720I
−6.52633 − 3.36713I −11.83061 + 4.84786I
u = −0.887416 + 0.936488I
a = −1.41323 − 1.17323I
b = −2.35284 + 0.28232I
2.09421 − 3.28881I 0
u = −0.887416 − 0.936488I
a = −1.41323 + 1.17323I
b = −2.35284 − 0.28232I
2.09421 + 3.28881I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.018770 + 0.851930I
a = 1.42653 + 0.88370I
b = 2.20615 − 0.31501I
7.27711 + 5.72796I 0
u = −1.018770 − 0.851930I
a = 1.42653 − 0.88370I
b = 2.20615 + 0.31501I
7.27711 − 5.72796I 0
u = −1.012200 + 0.890886I
a = −1.38271 − 0.92662I
b = −2.22509 + 0.29392I
12.39950 + 1.53189I 0
u = −1.012200 − 0.890886I
a = −1.38271 + 0.92662I
b = −2.22509 − 0.29392I
12.39950 − 1.53189I 0
u = −0.364738 + 0.539930I
a = −2.16669 − 0.02235I
b = −0.802344 + 1.161710I
−5.82107 − 5.39582I −8.47886 + 1.18765I
u = −0.364738 − 0.539930I
a = −2.16669 + 0.02235I
b = −0.802344 − 1.161710I
−5.82107 + 5.39582I −8.47886 − 1.18765I
u = −0.985908 + 0.935010I
a = 1.34032 + 0.99776I
b = 2.25434 − 0.26951I
10.13770 − 2.96345I 0
u = −0.985908 − 0.935010I
a = 1.34032 − 0.99776I
b = 2.25434 + 0.26951I
10.13770 + 2.96345I 0
u = −0.939008 + 1.004180I
a = 1.25206 + 1.11303I
b = 2.29337 − 0.21215I
9.89948 − 4.09070I 0
u = −0.939008 − 1.004180I
a = 1.25206 − 1.11303I
b = 2.29337 + 0.21215I
9.89948 + 4.09070I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.894077 + 1.060480I
a = 1.15539 + 1.20734I
b = 2.31337 − 0.14581I
6.5826 − 12.7219I 0
u = −0.894077 − 1.060480I
a = 1.15539 − 1.20734I
b = 2.31337 + 0.14581I
6.5826 + 12.7219I 0
u = −0.918184 + 1.042580I
a = −1.18777 − 1.16067I
b = −2.30068 + 0.17262I
11.8898 − 8.5952I 0
u = −0.918184 − 1.042580I
a = −1.18777 + 1.16067I
b = −2.30068 − 0.17262I
11.8898 + 8.5952I 0
u = −0.304253 + 0.441278I
a = 2.04672 + 0.15438I
b = 0.690845 − 0.856203I
−0.29697 − 2.23720I −4.18982 + 2.52656I
u = −0.304253 − 0.441278I
a = 2.04672 − 0.15438I
b = 0.690845 + 0.856203I
−0.29697 + 2.23720I −4.18982 − 2.52656I
u = −0.035517 + 0.529752I
a = −1.44264 − 0.17191I
b = −0.142307 + 0.758134I
−0.933124 + 0.964799I −7.52834 − 5.01190I
u = −0.035517 − 0.529752I
a = −1.44264 + 0.17191I
b = −0.142307 − 0.758134I
−0.933124 − 0.964799I −7.52834 + 5.01190I
u = −0.356068
a = −2.32453
b = −0.827691
−1.93664 −5.10000
8
II. I
u
2
= h−au + b, a
5
+ a
4
u − a
4
− 2a
3
u + a
2
+ au − a − u, 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
au
a
9
=
−au + a
au
a
4
=
1
u − 1
a
8
=
−au + a
au
a
12
=
−a
2
u − 1
−a
2
u + a
2
a
7
=
−a
4
+ a
2
u − a
2
+ 1
−a
4
u
a
11
=
a
4
u − a
4
− a
2
u + a
2
− 1
−a
4
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = a
4
u − a
4
− 4a
3
− 5a
2
u + 5a
2
+ 3au + a − 4u − 6
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
5
c
2
(u
2
+ u + 1)
5
c
4
, c
8
u
10
c
6
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
c
7
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
c
9
, c
12
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
c
10
, c
11
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
(y
2
+ y + 1)
5
c
4
, c
8
y
10
c
6
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
c
7
, c
10
, c
11
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
c
9
, c
12
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = −0.881753 − 0.117510I
b = −0.339110 − 0.822375I
−0.329100 + 0.499304I −2.53179 + 1.09027I
u = 0.500000 + 0.866025I
a = 0.542643 + 0.704866I
b = −0.339110 + 0.822375I
−0.32910 + 3.56046I −5.04069 − 7.43801I
u = 0.500000 + 0.866025I
a = 0.383413 − 0.664091I
b = 0.766826
−2.40108 + 2.02988I −6.62546 − 4.42764I
u = 0.500000 + 0.866025I
a = −0.811514 − 0.994721I
b = 0.455697 − 1.200150I
−5.87256 − 2.37095I −6.60498 − 0.29447I
u = 0.500000 + 0.866025I
a = 1.267210 + 0.205431I
b = 0.455697 + 1.200150I
−5.87256 + 6.43072I −9.19707 − 7.98272I
u = 0.500000 − 0.866025I
a = −0.881753 + 0.117510I
b = −0.339110 + 0.822375I
−0.329100 − 0.499304I −2.53179 − 1.09027I
u = 0.500000 − 0.866025I
a = 0.542643 − 0.704866I
b = −0.339110 − 0.822375I
−0.32910 − 3.56046I −5.04069 + 7.43801I
u = 0.500000 − 0.866025I
a = 0.383413 + 0.664091I
b = 0.766826
−2.40108 − 2.02988I −6.62546 + 4.42764I
u = 0.500000 − 0.866025I
a = −0.811514 + 0.994721I
b = 0.455697 + 1.200150I
−5.87256 + 2.37095I −6.60498 + 0.29447I
u = 0.500000 − 0.866025I
a = 1.267210 − 0.205431I
b = 0.455697 − 1.200150I
−5.87256 − 6.43072I −9.19707 + 7.98272I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
5
)(u
39
+ 8u
38
+ ··· − 10u − 1)
c
2
((u
2
+ u + 1)
5
)(u
39
+ 6u
38
+ ··· + 6u + 1)
c
3
((u
2
− u + 1)
5
)(u
39
− 6u
38
+ ··· + 227832u + 23497)
c
4
, c
8
u
10
(u
39
+ u
38
+ ··· + 2048u + 1024)
c
5
((u
2
− u + 1)
5
)(u
39
+ 6u
38
+ ··· + 6u + 1)
c
6
((u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
)(u
39
− 9u
38
+ ··· + 179u − 17)
c
7
((u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
)(u
39
+ 3u
38
+ ··· − 3u + 1)
c
9
((u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
)(u
39
− 3u
38
+ ··· − 3u + 1)
c
10
, c
11
((u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
)(u
39
+ 3u
38
+ ··· − 3u + 1)
c
12
((u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
)(u
39
+ 11u
38
+ ··· + 267u + 73)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
5
)(y
39
+ 52y
38
+ ··· + 58y −1)
c
2
, c
5
((y
2
+ y + 1)
5
)(y
39
+ 8y
38
+ ··· − 10y −1)
c
3
((y
2
+ y + 1)
5
)(y
39
+ 96y
38
+ ··· − 1.43297 × 10
10
y −5.52109 × 10
8
)
c
4
, c
8
y
10
(y
39
+ 55y
38
+ ··· − 5242880y −1048576)
c
6
((y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
)(y
39
+ 9y
38
+ ··· + 2665y −289)
c
7
, c
10
, c
11
((y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
)(y
39
− 35y
38
+ ··· − 3y −1)
c
9
((y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
)(y
39
− 67y
38
+ ··· − 3y −1)
c
12
((y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
)(y
39
− 7y
38
+ ··· − 292543y −5329)
14
