12n
0025
(K12n
0025
)
A knot diagram
1
Linearized knot diagam
3 5 6 7 2 10 5 12 11 6 9 8
Solving Sequence
6,10
7
2,11
5 4 3 1 9 12 8
c
6
c
10
c
5
c
4
c
3
c
1
c
9
c
11
c
8
c
2
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
11
+ 2u
10
+ 2u
9
+ 3u
7
+ 6u
6
+ 6u
5
− 2u
4
− 3u
3
− 3u
2
+ 2b,
u
11
+ 2u
10
+ u
9
− 2u
8
+ u
7
+ 6u
6
+ 3u
5
− 8u
4
− 9u
3
− u
2
+ 2a + 3u + 1,
u
13
+ 3u
12
+ 5u
11
+ 4u
10
+ 6u
9
+ 11u
8
+ 17u
7
+ 12u
6
+ 6u
5
+ u
2
+ u + 1i
I
u
2
= hu
3
a + 2u
2
a + u
3
− au + 2u
2
+ 8b − 3a − u − 3, −u
2
a − u
3
+ a
2
+ au + 3u
2
− 3u + 2, u
4
− u
3
+ u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 21 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
= hu
11
+2u
10
+· · ·−3u
2
+2b, u
11
+2u
10
+· · ·+2a+1, u
13
+3u
12
+· · ·+u+1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
7
=
1
−u
2
a
2
=
−
1
2
u
11
− u
10
+ ··· −
3
2
u −
1
2
−
1
2
u
11
− u
10
+ ··· +
3
2
u
3
+
3
2
u
2
a
11
=
u
u
a
5
=
−
1
2
u
9
−
5
2
u
5
−
3
2
u −
1
2
1
2
u
11
+ u
10
+ ··· +
3
2
u
3
+
3
2
u
2
a
4
=
−u
10
−
3
2
u
9
+ ··· −
3
2
u −
1
2
u
12
+
3
2
u
11
+ ··· +
3
2
u
3
+
3
2
u
2
a
3
=
u
12
+
3
2
u
11
+ ··· −
3
2
u −
1
2
u
12
+
3
2
u
11
+ ··· +
3
2
u
3
+
3
2
u
2
a
1
=
−u
9
− 3u
5
− u
−u
9
− u
7
− 3u
5
− 2u
3
− u
a
9
=
u
3
u
3
+ u
a
12
=
u
5
+ u
u
5
+ u
3
+ u
a
8
=
u
7
+ 2u
3
u
7
+ u
5
+ 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
9
2
u
12
+ 11u
11
+ 15u
10
+
11
2
u
9
+
33
2
u
8
+ 35u
7
+ 50u
6
+
25
2
u
5
−
11
2
u
4
−
27
2
u
3
+ 3u
2
+
15
2
u +
7
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
13
− u
12
+ ··· + 3u − 1
c
2
, c
5
u
13
+ 5u
12
+ ··· − 5u − 1
c
3
u
13
− 5u
12
+ ··· − 4501u − 833
c
4
, c
7
u
13
+ u
12
+ ··· − 1152u − 256
c
6
, c
10
u
13
− 3u
12
+ 5u
11
− 4u
10
+ 6u
9
− 11u
8
+ 17u
7
− 12u
6
+ 6u
5
− u
2
+ u − 1
c
8
, c
9
, c
11
c
12
u
13
+ u
12
+ ··· − u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
13
+ 35y
12
+ ··· − 129y −1
c
2
, c
5
y
13
− y
12
+ ··· + 3y −1
c
3
y
13
+ 47y
12
+ ··· − 4829293y −693889
c
4
, c
7
y
13
− 45y
12
+ ··· − 212992y −65536
c
6
, c
10
y
13
+ y
12
+ ··· − y −1
c
8
, c
9
, c
11
c
12
y
13
+ 25y
12
+ ··· − y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.440311 + 0.759939I
a = 0.796729 + 0.648672I
b = −0.267976 + 0.211150I
0.01165 − 1.74285I 0.51102 + 3.65273I
u = −0.440311 − 0.759939I
a = 0.796729 − 0.648672I
b = −0.267976 − 0.211150I
0.01165 + 1.74285I 0.51102 − 3.65273I
u = −0.748468
a = 0.262818
b = 0.939012
1.70389 6.48230
u = −0.082679 + 0.714751I
a = 1.41033 − 2.09260I
b = 0.247845 − 0.688505I
−1.32993 − 1.48407I −5.25082 + 4.83992I
u = −0.082679 − 0.714751I
a = 1.41033 + 2.09260I
b = 0.247845 + 0.688505I
−1.32993 + 1.48407I −5.25082 − 4.83992I
u = 0.939578 + 0.989792I
a = −0.103034 − 0.771838I
b = −1.151440 − 0.150255I
8.78280 + 3.51047I 4.53999 − 2.23004I
u = 0.939578 − 0.989792I
a = −0.103034 + 0.771838I
b = −1.151440 + 0.150255I
8.78280 − 3.51047I 4.53999 + 2.23004I
u = 0.532801 + 0.327453I
a = −1.35777 + 1.25278I
b = 0.616504 + 0.991708I
0.55073 + 2.69724I 3.82886 − 1.91679I
u = 0.532801 − 0.327453I
a = −1.35777 − 1.25278I
b = 0.616504 − 0.991708I
0.55073 − 2.69724I 3.82886 + 1.91679I
u = −0.99353 + 1.07384I
a = 0.40970 + 1.86567I
b = −1.14718 + 1.28471I
−14.0646 − 8.3921I 3.62108 + 3.78756I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.99353 − 1.07384I
a = 0.40970 − 1.86567I
b = −1.14718 − 1.28471I
−14.0646 + 8.3921I 3.62108 − 3.78756I
u = −1.08162 + 0.98798I
a = −0.787365 − 0.595530I
b = −1.26726 − 1.23564I
−13.71940 + 0.78911I 4.00870 + 0.19018I
u = −1.08162 − 0.98798I
a = −0.787365 + 0.595530I
b = −1.26726 + 1.23564I
−13.71940 − 0.78911I 4.00870 − 0.19018I
6
II.
I
u
2
= hu
3
a+u
3
+· · ·−3a−3, −u
2
a−u
3
+a
2
+au+3u
2
−3u+2, u
4
−u
3
+u
2
+1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
7
=
1
−u
2
a
2
=
a
−
1
8
u
3
a −
1
8
u
3
+ ··· +
3
8
a +
3
8
a
11
=
u
u
a
5
=
1
8
u
3
a +
1
8
u
3
+ ··· +
5
8
a −
3
8
−
1
8
u
3
a −
1
8
u
3
+ ··· +
3
8
a −
5
8
a
4
=
1
8
u
3
a +
1
8
u
3
+ ··· +
5
8
a −
3
8
−
1
8
u
3
a −
1
8
u
3
+ ··· +
3
8
a −
5
8
a
3
=
−u
2
+ a + u − 1
−
1
8
u
3
a −
1
8
u
3
+ ··· +
3
8
a −
5
8
a
1
=
−1
0
a
9
=
u
3
u
3
+ u
a
12
=
−u
2
− 1
u
3
− u
2
− 1
a
8
=
1
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
1
2
u
3
a + 2u
2
a +
1
2
u
3
−
3
2
au − 3u
2
−
1
2
a +
5
2
u +
1
2
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
4
c
2
(u
2
+ u + 1)
4
c
4
, c
7
u
8
c
6
(u
4
− u
3
+ u
2
+ 1)
2
c
8
, c
9
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
10
(u
4
+ u
3
+ u
2
+ 1)
2
c
11
, c
12
(u
4
+ u
3
+ 3u
2
+ 2u + 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)
4
c
4
, c
7
y
8
c
6
, c
10
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
c
8
, c
9
, c
11
c
12
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.351808 + 0.720342I
a = 1.04112 + 1.08095I
b = 0.500000 + 0.866025I
−0.211005 + 0.614778I 2.20786 + 0.04655I
u = −0.351808 + 0.720342I
a = −1.08443 − 2.30813I
b = 0.500000 − 0.866025I
−0.21101 − 3.44499I −2.55284 + 7.82341I
u = −0.351808 − 0.720342I
a = 1.04112 − 1.08095I
b = 0.500000 − 0.866025I
−0.211005 − 0.614778I 2.20786 − 0.04655I
u = −0.351808 − 0.720342I
a = −1.08443 + 2.30813I
b = 0.500000 + 0.866025I
−0.21101 + 3.44499I −2.55284 − 7.82341I
u = 0.851808 + 0.911292I
a = 0.076953 − 0.582938I
b = 0.500000 − 0.866025I
6.79074 + 1.13408I 2.75261 − 0.95911I
u = 0.851808 + 0.911292I
a = −1.03364 + 1.22414I
b = 0.500000 + 0.866025I
6.79074 + 5.19385I 2.09237 − 4.44058I
u = 0.851808 − 0.911292I
a = 0.076953 + 0.582938I
b = 0.500000 + 0.866025I
6.79074 − 1.13408I 2.75261 + 0.95911I
u = 0.851808 − 0.911292I
a = −1.03364 − 1.22414I
b = 0.500000 − 0.866025I
6.79074 − 5.19385I 2.09237 + 4.44058I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
4
)(u
13
− u
12
+ ··· + 3u − 1)
c
2
((u
2
+ u + 1)
4
)(u
13
+ 5u
12
+ ··· − 5u − 1)
c
3
((u
2
− u + 1)
4
)(u
13
− 5u
12
+ ··· − 4501u − 833)
c
4
, c
7
u
8
(u
13
+ u
12
+ ··· − 1152u − 256)
c
5
((u
2
− u + 1)
4
)(u
13
+ 5u
12
+ ··· − 5u − 1)
c
6
(u
4
− u
3
+ u
2
+ 1)
2
· (u
13
− 3u
12
+ 5u
11
− 4u
10
+ 6u
9
− 11u
8
+ 17u
7
− 12u
6
+ 6u
5
− u
2
+ u − 1)
c
8
, c
9
((u
4
− u
3
+ 3u
2
− 2u + 1)
2
)(u
13
+ u
12
+ ··· − u − 1)
c
10
(u
4
+ u
3
+ u
2
+ 1)
2
· (u
13
− 3u
12
+ 5u
11
− 4u
10
+ 6u
9
− 11u
8
+ 17u
7
− 12u
6
+ 6u
5
− u
2
+ u − 1)
c
11
, c
12
((u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
)(u
13
+ u
12
+ ··· − u − 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
4
)(y
13
+ 35y
12
+ ··· − 129y −1)
c
2
, c
5
((y
2
+ y + 1)
4
)(y
13
− y
12
+ ··· + 3y −1)
c
3
((y
2
+ y + 1)
4
)(y
13
+ 47y
12
+ ··· − 4829293y −693889)
c
4
, c
7
y
8
(y
13
− 45y
12
+ ··· − 212992y −65536)
c
6
, c
10
((y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
)(y
13
+ y
12
+ ··· − y −1)
c
8
, c
9
, c
11
c
12
((y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
)(y
13
+ 25y
12
+ ··· − y −1)
12
