12a
0471
(K12a
0471
)
A knot diagram
1
Linearized knot diagam
3 6 11 8 7 2 5 1 12 4 10 9
Solving Sequence
2,7
6 3 1 5 8 9 4 12 10 11
c
6
c
2
c
1
c
5
c
7
c
8
c
4
c
12
c
9
c
11
c
3
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
40
+ 2u
39
+ ··· + 4u + 1i
I
u
2
= hu
2
− u + 1i
* 2 irreducible components of dim
C
= 0, with total 42 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
40
+ 2u
39
+ · · · + 4u + 1i
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
6
=
1
u
2
a
3
=
u
u
3
+ u
a
1
=
u
3
u
5
+ u
3
+ u
a
5
=
u
2
+ 1
u
2
a
8
=
u
4
+ u
2
+ 1
u
4
a
9
=
−u
12
− u
10
− 3u
8
− 2u
6
+ u
2
+ 1
−u
14
− 2u
12
− 5u
10
− 6u
8
− 6u
6
− 2u
4
− u
2
a
4
=
u
6
+ u
4
+ 2u
2
+ 1
u
6
+ u
2
a
12
=
u
21
+ 2u
19
+ 7u
17
+ 10u
15
+ 14u
13
+ 12u
11
+ 5u
9
− 2u
7
− 5u
5
− 2u
3
− u
u
23
+ 3u
21
+ ··· + 2u
3
+ u
a
10
=
−u
30
− 3u
28
+ ··· + 2u
2
+ 1
−u
32
− 4u
30
+ ··· − 6u
4
− 2u
2
a
11
=
u
39
+ 4u
37
+ ··· − 6u
3
− 2u
3u
39
+ 4u
38
+ ··· + 8u + 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 8u
39
+ 12u
38
+ ··· + 48u + 14
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
7
u
40
+ 8u
39
+ ··· + 4u + 1
c
2
, c
6
u
40
− 2u
39
+ ··· − 4u + 1
c
3
, c
10
u
40
+ 2u
39
+ ··· + 4u + 1
c
8
, c
9
, c
11
c
12
u
40
− 8u
39
+ ··· − 4u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
7
, c
8
, c
9
c
11
, c
12
y
40
+ 48y
39
+ ··· + 52y + 1
c
2
, c
3
, c
6
c
10
y
40
+ 8y
39
+ ··· + 4y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.007642 + 0.966456I
−11.28980 − 3.26800I −8.22072 + 2.51230I
u = −0.007642 − 0.966456I
−11.28980 + 3.26800I −8.22072 − 2.51230I
u = −0.455339 + 0.953830I
−8.76382 − 2.08770I −4.31619 + 3.32105I
u = −0.455339 − 0.953830I
−8.76382 + 2.08770I −4.31619 − 3.32105I
u = −0.424765 + 0.837410I
−1.06965 − 2.01513I −4.12618 + 3.84559I
u = −0.424765 − 0.837410I
−1.06965 + 2.01513I −4.12618 − 3.84559I
u = 0.468864 + 0.955282I
−8.59119 + 8.60337I −3.84612 − 8.10725I
u = 0.468864 − 0.955282I
−8.59119 − 8.60337I −3.84612 + 8.10725I
u = 0.547826 + 0.720762I
2.87673 + 2.07761I 8.03109 − 4.87367I
u = 0.547826 − 0.720762I
2.87673 − 2.07761I 8.03109 + 4.87367I
u = −0.054899 + 0.842893I
−2.87673 − 2.07761I −8.03109 + 4.87367I
u = −0.054899 − 0.842893I
−2.87673 + 2.07761I −8.03109 − 4.87367I
u = 0.675982 + 0.376953I
−6.76036 − 4.39632I 0.35650 + 2.56566I
u = 0.675982 − 0.376953I
−6.76036 + 4.39632I 0.35650 − 2.56566I
u = 0.570879 + 0.520043I
1.06965 − 2.01513I 4.12618 + 3.84559I
u = 0.570879 − 0.520043I
1.06965 + 2.01513I 4.12618 − 3.84559I
u = 0.897344 + 0.861664I
0.784836I 0. − 2.11264I
u = 0.897344 − 0.861664I
− 0.784836I 0. + 2.11264I
u = −0.666629 + 0.352689I
−6.87304 − 2.02249I 0.14883 + 2.38441I
u = −0.666629 − 0.352689I
−6.87304 + 2.02249I 0.14883 − 2.38441I
u = −0.903766 + 0.864696I
0.34673 + 5.67431I 0.59636 − 2.67543I
u = −0.903766 − 0.864696I
0.34673 − 5.67431I 0.59636 + 2.67543I
u = 0.873303 + 0.899277I
6.87304 + 2.02249I 0. − 2.38441I
u = 0.873303 − 0.899277I
6.87304 − 2.02249I 0. + 2.38441I
u = −0.893753 + 0.895948I
8.76382 + 2.08770I 4.31619 − 3.32105I
u = −0.893753 − 0.895948I
8.76382 − 2.08770I 4.31619 + 3.32105I
u = 0.858890 + 0.934946I
6.76036 + 4.39632I 0. − 2.56566I
u = 0.858890 − 0.934946I
6.76036 − 4.39632I 0. + 2.56566I
u = −0.350675 + 0.633429I
−0.162533 − 1.134740I −3.04903 + 5.46701I
u = −0.350675 − 0.633429I
−0.162533 + 1.134740I −3.04903 − 5.46701I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.884538 + 0.925182I
11.28980 − 3.26800I 8.22072 + 2.51230I
u = −0.884538 − 0.925182I
11.28980 + 3.26800I 8.22072 − 2.51230I
u = −0.869391 + 0.950001I
8.59119 − 8.60337I 3.84612 + 8.10725I
u = −0.869391 − 0.950001I
8.59119 + 8.60337I 3.84612 − 8.10725I
u = 0.849710 + 0.970863I
−0.34673 + 5.67431I 0. − 2.67543I
u = 0.849710 − 0.970863I
−0.34673 − 5.67431I 0. + 2.67543I
u = −0.854577 + 0.973346I
− 12.1697I 0. + 7.37185I
u = −0.854577 − 0.973346I
12.1697I 0. − 7.37185I
u = −0.376823 + 0.254532I
0.162533 − 1.134740I 3.04903 + 5.46701I
u = −0.376823 − 0.254532I
0.162533 + 1.134740I 3.04903 − 5.46701I
6
II. I
u
2
= hu
2
− u + 1i
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
6
=
1
u − 1
a
3
=
u
u − 1
a
1
=
−1
0
a
5
=
u
u − 1
a
8
=
0
−u
a
9
=
−u
−u
a
4
=
u
u
a
12
=
−u
−u + 1
a
10
=
−2u + 1
−u + 1
a
11
=
−2u + 2
−u + 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −12u + 6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
7
u
2
+ u + 1
c
3
, c
8
, c
9
c
10
, c
11
, c
12
u
2
− u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y
2
+ y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
6.08965I 0. − 10.39230I
u = 0.500000 − 0.866025I
− 6.08965I 0. + 10.39230I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
7
(u
2
+ u + 1)(u
40
+ 8u
39
+ ··· + 4u + 1)
c
2
, c
6
(u
2
+ u + 1)(u
40
− 2u
39
+ ··· − 4u + 1)
c
3
, c
10
(u
2
− u + 1)(u
40
+ 2u
39
+ ··· + 4u + 1)
c
8
, c
9
, c
11
c
12
(u
2
− u + 1)(u
40
− 8u
39
+ ··· − 4u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
7
, c
8
, c
9
c
11
, c
12
(y
2
+ y + 1)(y
40
+ 48y
39
+ ··· + 52y + 1)
c
2
, c
3
, c
6
c
10
(y
2
+ y + 1)(y
40
+ 8y
39
+ ··· + 4y + 1)
12
