12n
0041
(K12n
0041
)
A knot diagram
1
Linearized knot diagam
3 5 6 9 2 10 12 5 6 7 8 11
Solving Sequence
6,9
10 7
5,11
4 3 2 1 8 12
c
9
c
6
c
10
c
4
c
3
c
2
c
1
c
8
c
11
c
5
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h801994252u
18
− 3257723213u
17
+ ··· + 1752268067b + 1712546026,
− 6711899247u
18
+ 18127971990u
17
+ ··· + 3504536134a + 13273949042,
u
19
− 3u
18
+ ··· + u − 1i
I
u
2
= hb, −u
4
a − u
3
a + u
4
+ 2u
2
a + 2u
3
+ a
2
+ au − u
2
− a − 3u, u
5
+ u
4
− 2u
3
− u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 29 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
= h8.02 × 10
8
u
18
− 3.26 × 10
9
u
17
+ · · · + 1.75 × 10
9
b + 1.71 × 10
9
, −6.71 ×
10
9
u
18
+1.81×10
10
u
17
+· · ·+3.50×10
9
a+1.33×10
10
, u
19
−3u
18
+· · ·+u−1i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
7
=
u
−u
3
+ u
a
5
=
1.91520u
18
− 5.17272u
17
+ ··· − 4.28509u − 3.78765
−0.457689u
18
+ 1.85915u
17
+ ··· − 0.626511u − 0.977331
a
11
=
−u
2
+ 1
u
4
− 2u
2
a
4
=
1.45751u
18
− 3.31357u
17
+ ··· − 4.91160u − 4.76498
−0.457689u
18
+ 1.85915u
17
+ ··· − 0.626511u − 0.977331
a
3
=
1.45751u
18
− 3.31357u
17
+ ··· − 4.91160u − 4.76498
−0.744876u
18
+ 2.99066u
17
+ ··· − 1.02505u − 2.03630
a
2
=
0.688639u
18
− 1.66148u
17
+ ··· + 1.68347u − 4.87205
−0.257833u
18
+ 0.974337u
17
+ ··· + 0.715799u − 0.404438
a
1
=
−1.18863u
18
+ 4.77080u
17
+ ··· − 1.55380u − 4.47346
−0.835657u
18
+ 2.70331u
17
+ ··· + 4.20709u − 3.06916
a
8
=
−1.86427u
18
+ 5.34890u
17
+ ··· + 2.46844u − 0.0507078
−1.20490u
18
+ 3.02293u
17
+ ··· + 3.28483u + 1.18863
a
12
=
1.06187u
18
− 4.23698u
17
+ ··· + 1.50625u + 3.49695
0.526384u
18
− 1.72956u
17
+ ··· − 3.80190u + 2.76211
(ii) Obstruction class = −1
(iii) Cusp Shapes =
3543125147
3504536134
u
18
−
342620995
3504536134
u
17
+ ··· −
5030153644
159297097
u −
4276661878
1752268067
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
19
+ 22u
17
+ ··· − 12u − 1
c
2
, c
5
u
19
+ 6u
18
+ ··· + 4u + 1
c
3
u
19
− 6u
18
+ ··· + 21156u + 4073
c
4
, c
8
u
19
+ u
18
+ ··· − 1024u − 1024
c
6
, c
9
, c
10
u
19
− 3u
18
+ ··· + u − 1
c
7
, c
11
u
19
+ 3u
18
+ ··· − u − 1
c
12
u
19
− 13u
18
+ ··· − 13u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
19
+ 44y
18
+ ··· − 24y −1
c
2
, c
5
y
19
+ 22y
17
+ ··· − 12y −1
c
3
y
19
+ 88y
18
+ ··· − 418750764y −16589329
c
4
, c
8
y
19
+ 55y
18
+ ··· − 1048576y −1048576
c
6
, c
9
, c
10
y
19
− 35y
18
+ ··· − 13y −1
c
7
, c
11
y
19
+ 13y
18
+ ··· − 13y −1
c
12
y
19
− 11y
18
+ ··· − 13y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.469936 + 0.580822I
a = −0.046125 − 0.179882I
b = 0.481085 + 0.331499I
0.32482 + 1.95207I 0.24383 − 3.23848I
u = 0.469936 − 0.580822I
a = −0.046125 + 0.179882I
b = 0.481085 − 0.331499I
0.32482 − 1.95207I 0.24383 + 3.23848I
u = 0.203301 + 0.528628I
a = 1.67634 + 1.79943I
b = 1.52335 + 0.99869I
3.95992 − 1.78665I 6.73687 + 1.96158I
u = 0.203301 − 0.528628I
a = 1.67634 − 1.79943I
b = 1.52335 − 0.99869I
3.95992 + 1.78665I 6.73687 − 1.96158I
u = 0.008315 + 0.564548I
a = −0.357902 − 0.497669I
b = −0.424228 + 0.518164I
−0.40680 + 1.36117I −2.67817 − 4.58018I
u = 0.008315 − 0.564548I
a = −0.357902 + 0.497669I
b = −0.424228 − 0.518164I
−0.40680 − 1.36117I −2.67817 + 4.58018I
u = −0.501281 + 0.026931I
a = −1.86678 − 2.02772I
b = −0.171729 + 1.042670I
1.56074 − 3.66143I 5.39141 + 4.20256I
u = −0.501281 − 0.026931I
a = −1.86678 + 2.02772I
b = −0.171729 − 1.042670I
1.56074 + 3.66143I 5.39141 − 4.20256I
u = 1.56360
a = −0.443721
b = 1.34603
3.65542 2.34160
u = 0.111360 + 0.361359I
a = −1.08886 − 1.51643I
b = −0.369696 + 0.488600I
−0.21591 + 1.44599I −1.60179 − 5.31059I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.111360 − 0.361359I
a = −1.08886 + 1.51643I
b = −0.369696 − 0.488600I
−0.21591 − 1.44599I −1.60179 + 5.31059I
u = −1.72009 + 0.19693I
a = 0.513343 − 0.171232I
b = −1.51868 + 0.49853I
7.40108 − 4.89405I 5.57785 + 2.97654I
u = −1.72009 − 0.19693I
a = 0.513343 + 0.171232I
b = −1.51868 − 0.49853I
7.40108 + 4.89405I 5.57785 − 2.97654I
u = 2.15399 + 0.20015I
a = −0.240429 − 1.266600I
b = −0.26878 + 3.56600I
−15.5249 − 1.2175I 5.22163 + 0.77720I
u = 2.15399 − 0.20015I
a = −0.240429 + 1.266600I
b = −0.26878 − 3.56600I
−15.5249 + 1.2175I 5.22163 − 0.77720I
u = 2.15940 + 0.15889I
a = 0.037600 − 1.236710I
b = −1.03657 + 3.06900I
−15.9786 + 9.8700I 4.75713 − 4.56429I
u = 2.15940 − 0.15889I
a = 0.037600 + 1.236710I
b = −1.03657 − 3.06900I
−15.9786 − 9.8700I 4.75713 + 4.56429I
u = −2.16674 + 0.18320I
a = 0.094666 − 1.203630I
b = 0.61224 + 3.16966I
19.5194 − 4.2417I 2.18043 + 1.81116I
u = −2.16674 − 0.18320I
a = 0.094666 + 1.203630I
b = 0.61224 − 3.16966I
19.5194 + 4.2417I 2.18043 − 1.81116I
6
II. I
u
2
= hb, −u
4
a + u
4
+ · · · + a
2
− a, u
5
+ u
4
− 2u
3
− u
2
+ u − 1i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
7
=
u
−u
3
+ u
a
5
=
a
0
a
11
=
−u
2
+ 1
u
4
− 2u
2
a
4
=
a
0
a
3
=
a
−u
2
a
a
2
=
−u
4
− u
3
+ 2u
2
+ a + u − 1
−u
2
a
a
1
=
0
−u
a
8
=
1
0
a
12
=
−u
4
+ u
2
+ 1
u
4
− 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
4
a − u
4
− 2u
2
a + 2u
3
+ 3au + u
2
+ a − 5u + 2
7
(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
− u
4
− 2u
3
+ u
2
+ u + 1)
2
c
7
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
c
9
, c
10
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
c
11
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
c
12
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
8
(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
, c
9
, c
10
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
c
7
, c
11
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
c
12
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.21774
a = 0.410598 + 0.711177I
b = 0
2.40108 − 2.02988I 0.40252 + 2.76390I
u = 1.21774
a = 0.410598 − 0.711177I
b = 0
2.40108 + 2.02988I 0.40252 − 2.76390I
u = 0.309916 + 0.549911I
a = 1.58413 − 0.01647I
b = 0
0.32910 + 3.56046I −0.88631 − 6.04478I
u = 0.309916 + 0.549911I
a = −0.80632 − 1.36366I
b = 0
0.329100 − 0.499304I 3.42267 − 1.01043I
u = 0.309916 − 0.549911I
a = 1.58413 + 0.01647I
b = 0
0.32910 − 3.56046I −0.88631 + 6.04478I
u = 0.309916 − 0.549911I
a = −0.80632 + 1.36366I
b = 0
0.329100 + 0.499304I 3.42267 + 1.01043I
u = −1.41878 + 0.21917I
a = −0.252108 − 0.649344I
b = 0
5.87256 − 6.43072I 2.86519 + 5.89938I
u = −1.41878 + 0.21917I
a = −0.436295 + 0.543004I
b = 0
5.87256 − 2.37095I 4.19593 + 1.57328I
u = −1.41878 − 0.21917I
a = −0.252108 + 0.649344I
b = 0
5.87256 + 6.43072I 2.86519 − 5.89938I
u = −1.41878 − 0.21917I
a = −0.436295 − 0.543004I
b = 0
5.87256 + 2.37095I 4.19593 − 1.57328I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
5
)(u
19
+ 22u
17
+ ··· − 12u − 1)
c
2
((u
2
+ u + 1)
5
)(u
19
+ 6u
18
+ ··· + 4u + 1)
c
3
((u
2
− u + 1)
5
)(u
19
− 6u
18
+ ··· + 21156u + 4073)
c
4
, c
8
u
10
(u
19
+ u
18
+ ··· − 1024u − 1024)
c
5
((u
2
− u + 1)
5
)(u
19
+ 6u
18
+ ··· + 4u + 1)
c
6
((u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
)(u
19
− 3u
18
+ ··· + u − 1)
c
7
((u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
)(u
19
+ 3u
18
+ ··· − u − 1)
c
9
, c
10
((u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
)(u
19
− 3u
18
+ ··· + u − 1)
c
11
((u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
)(u
19
+ 3u
18
+ ··· − u − 1)
c
12
((u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
)(u
19
− 13u
18
+ ··· − 13u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
5
)(y
19
+ 44y
18
+ ··· − 24y −1)
c
2
, c
5
((y
2
+ y + 1)
5
)(y
19
+ 22y
17
+ ··· − 12y −1)
c
3
((y
2
+ y + 1)
5
)(y
19
+ 88y
18
+ ··· − 4.18751 × 10
8
y −1.65893 × 10
7
)
c
4
, c
8
y
10
(y
19
+ 55y
18
+ ··· − 1048576y −1048576)
c
6
, c
9
, c
10
((y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
)(y
19
− 35y
18
+ ··· − 13y −1)
c
7
, c
11
((y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
)(y
19
+ 13y
18
+ ··· − 13y −1)
c
12
((y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
)(y
19
− 11y
18
+ ··· − 13y −1)
12
