12n
0236
(K12n
0236
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 10 12 11 3 5 6 7 9
Solving Sequence
5,9
10 6
3,11
2 1 4 8 7 12
c
9
c
5
c
10
c
2
c
1
c
4
c
8
c
7
c
12
c
3
, c
6
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h8.50640 × 10
25
u
28
+ 6.57905 × 10
25
u
27
+ ··· + 6.76246 × 10
26
b − 3.79834 × 10
26
,
− 8.61578 × 10
25
u
28
− 2.59629 × 10
26
u
27
+ ··· + 2.02874 × 10
27
a + 4.68843 × 10
27
,
u
29
+ 2u
28
+ ··· − 27u − 9i
I
u
2
= hb, −u
5
+ u
4
+ 3u
3
− 2u
2
+ a − 2u − 1, u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 35 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
= h8.51 × 10
25
u
28
+ 6.58 × 10
25
u
27
+ · · · + 6.76 × 10
26
b − 3.80 ×
10
26
, −8.62 × 10
25
u
28
− 2.60 × 10
26
u
27
+ · · · + 2.03 × 10
27
a + 4.69 ×
10
27
, u
29
+ 2u
28
+ · · · − 27u − 9i
(i) Arc colorings
a
5
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
6
=
u
−u
3
+ u
a
3
=
0.0424687u
28
+ 0.127975u
27
+ ··· − 7.48008u − 2.31101
−0.125788u
28
− 0.0972878u
27
+ ··· + 0.810929u + 0.561681
a
11
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
0.0424687u
28
+ 0.127975u
27
+ ··· − 7.48008u − 2.31101
−0.0837082u
28
− 0.0256521u
27
+ ··· − 0.733318u + 0.174337
a
1
=
−0.207275u
28
− 0.191732u
27
+ ··· + 2.88280u + 0.255414
−0.0856550u
28
− 0.101573u
27
+ ··· + 1.21045u + 0.659706
a
4
=
0.325774u
28
+ 0.353500u
27
+ ··· − 11.9004u − 3.50868
−0.439150u
28
− 0.368748u
27
+ ··· + 6.32577u + 3.34213
a
8
=
0.0733007u
28
+ 0.0609464u
27
+ ··· + 2.08714u − 0.768665
−0.222818u
28
− 0.235496u
27
+ ··· + 5.34101u + 1.86548
a
7
=
0.0307540u
28
+ 0.0680917u
27
+ ··· + 1.08829u − 0.755172
0.00877248u
28
+ 0.0384206u
27
+ ··· + 0.630413u − 0.299831
a
12
=
−0.121620u
28
− 0.0901590u
27
+ ··· + 1.67234u − 0.404292
−0.0856550u
28
− 0.101573u
27
+ ··· + 1.21045u + 0.659706
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
119007428794930832983364079
225415412411056701631913333
u
28
−
129789591268195455740025046
225415412411056701631913333
u
27
+
··· +
7277513301493536765918568604
225415412411056701631913333
u +
1889963069349290974137649138
225415412411056701631913333
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
29
+ 39u
28
+ ··· + 90u + 1
c
2
, c
4
u
29
− 7u
28
+ ··· − 14u + 1
c
3
, c
8
u
29
+ u
28
+ ··· − 64u + 64
c
5
, c
9
, c
10
u
29
+ 2u
28
+ ··· − 27u − 9
c
6
, c
7
, c
11
u
29
− 2u
28
+ ··· + u − 1
c
12
u
29
+ 30u
27
+ ··· + u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
29
− 91y
28
+ ··· + 8066y −1
c
2
, c
4
y
29
− 39y
28
+ ··· + 90y −1
c
3
, c
8
y
29
+ 39y
28
+ ··· + 49152y −4096
c
5
, c
9
, c
10
y
29
− 24y
28
+ ··· − 207y −81
c
6
, c
7
, c
11
y
29
+ 24y
28
+ ··· + y −1
c
12
y
29
+ 60y
28
+ ··· + y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.147151 + 0.983351I
a = −0.313005 − 1.172620I
b = 0.619200 − 0.948695I
−6.51177 + 1.38864I −2.12482 − 1.22156I
u = 0.147151 − 0.983351I
a = −0.313005 + 1.172620I
b = 0.619200 + 0.948695I
−6.51177 − 1.38864I −2.12482 + 1.22156I
u = −1.084770 + 0.203815I
a = −0.596441 + 0.550146I
b = 0.210937 + 1.119990I
−1.04955 − 1.39392I 4.85868 + 0.24433I
u = −1.084770 − 0.203815I
a = −0.596441 − 0.550146I
b = 0.210937 − 1.119990I
−1.04955 + 1.39392I 4.85868 − 0.24433I
u = −1.11805
a = −1.24644
b = 1.14578
0.572830 8.71570
u = −0.295538 + 0.824939I
a = 0.62455 + 1.90486I
b = −0.10295 + 1.91234I
−11.35050 − 1.97232I 3.18700 + 3.24359I
u = −0.295538 − 0.824939I
a = 0.62455 − 1.90486I
b = −0.10295 − 1.91234I
−11.35050 + 1.97232I 3.18700 − 3.24359I
u = −1.176090 + 0.299893I
a = 1.298590 + 0.500513I
b = −0.42404 + 1.73842I
−8.81715 − 1.90353I 3.17087 − 0.16545I
u = −1.176090 − 0.299893I
a = 1.298590 − 0.500513I
b = −0.42404 − 1.73842I
−8.81715 + 1.90353I 3.17087 + 0.16545I
u = 1.168350 + 0.466626I
a = 1.087240 − 0.320633I
b = −1.202010 − 0.277739I
−3.42349 + 3.73175I 3.28322 − 3.90780I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.168350 − 0.466626I
a = 1.087240 + 0.320633I
b = −1.202010 + 0.277739I
−3.42349 − 3.73175I 3.28322 + 3.90780I
u = 1.252040 + 0.171380I
a = 0.450233 − 0.673649I
b = −0.062273 − 1.251450I
2.49236 + 2.60548I 8.31800 − 3.45920I
u = 1.252040 − 0.171380I
a = 0.450233 + 0.673649I
b = −0.062273 + 1.251450I
2.49236 − 2.60548I 8.31800 + 3.45920I
u = −0.340740 + 0.609315I
a = 0.408720 − 0.064944I
b = −0.462808 + 0.305767I
−3.04827 − 1.57293I 6.03185 + 4.01355I
u = −0.340740 − 0.609315I
a = 0.408720 + 0.064944I
b = −0.462808 − 0.305767I
−3.04827 + 1.57293I 6.03185 − 4.01355I
u = −1.37004 + 0.47996I
a = −0.322999 − 0.704095I
b = −0.051255 − 1.357350I
−1.79428 − 6.65679I 3.36940 + 5.84463I
u = −1.37004 − 0.47996I
a = −0.322999 + 0.704095I
b = −0.051255 + 1.357350I
−1.79428 + 6.65679I 3.36940 − 5.84463I
u = 0.66485 + 1.32634I
a = −0.493095 + 1.138380I
b = 0.22459 + 2.08826I
−17.4796 + 4.1704I −1.26801 − 2.81155I
u = 0.66485 − 1.32634I
a = −0.493095 − 1.138380I
b = 0.22459 − 2.08826I
−17.4796 − 4.1704I −1.26801 + 2.81155I
u = 1.47549 + 0.19394I
a = −0.329047 + 0.005572I
b = 0.613243 + 0.013272I
2.92568 + 4.50321I 11.83396 − 2.97399I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.47549 − 0.19394I
a = −0.329047 − 0.005572I
b = 0.613243 − 0.013272I
2.92568 − 4.50321I 11.83396 + 2.97399I
u = −1.49199
a = 0.327755
b = −0.609977
6.88521 15.7090
u = 1.46992 + 0.39831I
a = −1.034790 + 0.448894I
b = 0.52841 + 1.78278I
−5.70280 + 6.55116I 6.17722 − 3.52056I
u = 1.46992 − 0.39831I
a = −1.034790 − 0.448894I
b = 0.52841 − 1.78278I
−5.70280 − 6.55116I 6.17722 + 3.52056I
u = 0.395993
a = −0.267236
b = 0.323479
0.588961 16.9080
u = −0.143752 + 0.301891I
a = −0.41284 − 2.66870I
b = −0.228813 − 0.666491I
−1.59820 − 0.73663I −0.70316 + 3.71220I
u = −0.143752 − 0.301891I
a = −0.41284 + 2.66870I
b = −0.228813 + 0.666491I
−1.59820 + 0.73663I −0.70316 − 3.71220I
u = −1.65985 + 0.54064I
a = 0.892504 + 0.457194I
b = −0.59188 + 1.84141I
−10.3510 − 11.0238I 2.19949 + 5.56380I
u = −1.65985 − 0.54064I
a = 0.892504 − 0.457194I
b = −0.59188 − 1.84141I
−10.3510 + 11.0238I 2.19949 − 5.56380I
7
II.
I
u
2
= hb, −u
5
+ u
4
+ 3u
3
− 2u
2
+ a − 2u − 1, u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
(i) Arc colorings
a
5
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
6
=
u
−u
3
+ u
a
3
=
u
5
− u
4
− 3u
3
+ 2u
2
+ 2u + 1
0
a
11
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
u
5
− u
4
− 3u
3
+ 2u
2
+ 2u + 1
−u
a
1
=
0
−u
a
4
=
u
5
− u
4
− 3u
3
+ 2u
2
+ 2u + 1
0
a
8
=
1
0
a
7
=
−u
5
+ 2u
3
+ u
u
5
− 3u
3
+ u
a
12
=
u
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
5
+ 4u + 3
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
6
c
3
, c
8
u
6
c
4
(u + 1)
6
c
5
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1
c
6
, c
7
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1
c
9
, c
10
, c
12
u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1
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
8
y
6
c
5
, c
9
, c
10
c
12
y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1
c
6
, c
7
, c
11
y
6
+ 5y
5
+ 9y
4
+ 4y
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 = −0.858925 − 1.001920I
b = 0
−4.60518 − 1.97241I 0.92955 + 2.53106I
u = −0.493180 − 0.575288I
a = −0.858925 + 1.001920I
b = 0
−4.60518 + 1.97241I 0.92955 − 2.53106I
u = 0.483672
a = 2.06752
b = 0
−0.906083 4.90820
u = 1.52087 + 0.16310I
a = 0.650045 − 0.069710I
b = 0
2.05064 + 4.59213I 1.87701 − 3.61028I
u = 1.52087 − 0.16310I
a = 0.650045 + 0.069710I
b = 0
2.05064 − 4.59213I 1.87701 + 3.61028I
u = −1.53904
a = −0.649754
b = 0
6.01515 5.47870
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
29
+ 39u
28
+ ··· + 90u + 1)
c
2
((u − 1)
6
)(u
29
− 7u
28
+ ··· − 14u + 1)
c
3
, c
8
u
6
(u
29
+ u
28
+ ··· − 64u + 64)
c
4
((u + 1)
6
)(u
29
− 7u
28
+ ··· − 14u + 1)
c
5
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
29
+ 2u
28
+ ··· − 27u − 9)
c
6
, c
7
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
29
− 2u
28
+ ··· + u − 1)
c
9
, c
10
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
29
+ 2u
28
+ ··· − 27u − 9)
c
11
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
29
− 2u
28
+ ··· + u − 1)
c
12
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
29
+ 30u
27
+ ··· + u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
29
− 91y
28
+ ··· + 8066y −1)
c
2
, c
4
((y −1)
6
)(y
29
− 39y
28
+ ··· + 90y −1)
c
3
, c
8
y
6
(y
29
+ 39y
28
+ ··· + 49152y −4096)
c
5
, c
9
, c
10
(y
6
− 7y
5
+ ··· − 5y + 1)(y
29
− 24y
28
+ ··· − 207y −81)
c
6
, c
7
, c
11
(y
6
+ 5y
5
+ ··· − 5y + 1)(y
29
+ 24y
28
+ ··· + y −1)
c
12
(y
6
− 7y
5
+ ··· − 5y + 1)(y
29
+ 60y
28
+ ··· + y −1)
13
