12a
0723
(K12a
0723
)
A knot diagram
1
Linearized knot diagam
3 8 9 10 11 12 1 2 4 5 7 6
Solving Sequence
2,9
8 3 4 10 5 11 1 7 12 6
c
8
c
2
c
3
c
9
c
4
c
10
c
1
c
7
c
11
c
6
c
5
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
22
− u
21
+ ··· − 2u + 1i
I
u
2
= hu
9
+ 3u
7
+ u
6
+ 3u
5
+ 2u
4
− u
3
+ u
2
− 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 31 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
22
− u
21
+ · · · − 2u + 1i
(i) Arc colorings
a
2
=
0
u
a
9
=
1
0
a
8
=
1
u
2
a
3
=
u
u
3
+ u
a
4
=
−u
3
u
3
+ u
a
10
=
−u
6
− u
4
+ 1
u
6
+ 2u
4
+ u
2
a
5
=
u
9
+ 2u
7
+ u
5
− 2u
3
− u
−u
9
− 3u
7
− 3u
5
+ u
a
11
=
u
12
+ 3u
10
+ 3u
8
− 2u
6
− 4u
4
− u
2
+ 1
−u
12
− 4u
10
− 6u
8
− 2u
6
+ 3u
4
+ 2u
2
a
1
=
u
3
u
5
+ u
3
+ u
a
7
=
−u
6
− u
4
+ 1
−u
8
− 2u
6
− 2u
4
a
12
=
u
14
+ 5u
12
+ 10u
10
+ 7u
8
− 4u
6
− u
5
− 8u
4
− 2u
3
− 2u
2
− u + 1
u
21
+ 7u
19
+ ··· + u
2
− 1
a
6
=
−u
15
− 4u
13
− 6u
11
+ 8u
7
+ 6u
5
− 2u
3
− 2u
u
15
+ 5u
13
+ 10u
11
+ 7u
9
− 4u
7
− 8u
5
− 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −4u
21
+ 4u
20
− 24u
19
+ 24u
18
− 64u
17
+ 68u
16
− 80u
15
+ 100u
14
− 24u
13
+ 68u
12
+
52u
11
− 16u
10
+ 36u
9
− 52u
8
− 28u
7
− 28u
6
− 32u
5
+ 8u
4
+ 4u
3
+ 4u
2
+ 4u − 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
22
+ 13u
21
+ ··· + 2u + 1
c
2
, c
8
u
22
− u
21
+ ··· − 2u + 1
c
3
, c
4
, c
5
c
7
, c
9
, c
10
u
22
− 2u
21
+ ··· − u + 2
c
6
, c
11
, c
12
u
22
− u
21
+ ··· + u
2
+ 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
22
− 7y
21
+ ··· + 2y + 1
c
2
, c
8
y
22
+ 13y
21
+ ··· + 2y + 1
c
3
, c
4
, c
5
c
7
, c
9
, c
10
y
22
− 30y
21
+ ··· + 19y + 4
c
6
, c
11
, c
12
y
22
+ 17y
21
+ ··· + 2y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.946141 + 0.011490I
−13.5904 − 5.0425I −10.78024 + 2.84693I
u = 0.946141 − 0.011490I
−13.5904 + 5.0425I −10.78024 − 2.84693I
u = 0.416424 + 0.993762I
0.95586 + 5.59232I −9.07345 − 8.52361I
u = 0.416424 − 0.993762I
0.95586 − 5.59232I −9.07345 + 8.52361I
u = −0.327821 + 1.035380I
−3.20510 − 2.90050I −16.7585 + 6.2360I
u = −0.327821 − 1.035380I
−3.20510 + 2.90050I −16.7585 − 6.2360I
u = 0.153534 + 0.829883I
−0.664834 + 0.970955I −10.56839 − 6.31245I
u = 0.153534 − 0.829883I
−0.664834 − 0.970955I −10.56839 + 6.31245I
u = −0.373530 + 0.720587I
3.67591 − 1.72367I −3.04572 + 4.67737I
u = −0.373530 − 0.720587I
3.67591 + 1.72367I −3.04572 − 4.67737I
u = −0.768463 + 0.066318I
−2.56255 + 4.06172I −9.74928 − 3.64554I
u = −0.768463 − 0.066318I
−2.56255 − 4.06172I −9.74928 + 3.64554I
u = −0.454887 + 1.179480I
−5.83119 − 8.50268I −12.8572 + 7.0300I
u = −0.454887 − 1.179480I
−5.83119 + 8.50268I −12.8572 − 7.0300I
u = 0.425814 + 1.198730I
−9.89307 + 4.30260I −17.3404 − 3.7895I
u = 0.425814 − 1.198730I
−9.89307 − 4.30260I −17.3404 + 3.7895I
u = 0.487685 + 1.287980I
−17.5204 + 10.1457I −13.8263 − 5.6856I
u = 0.487685 − 1.287980I
−17.5204 − 10.1457I −13.8263 + 5.6856I
u = −0.481775 + 1.292500I
17.8662 − 5.0843I −17.1007 + 2.8764I
u = −0.481775 − 1.292500I
17.8662 + 5.0843I −17.1007 − 2.8764I
u = 0.476877 + 0.292674I
2.80571 − 1.90068I −4.89982 + 3.73749I
u = 0.476877 − 0.292674I
2.80571 + 1.90068I −4.89982 − 3.73749I
5
II. I
u
2
= hu
9
+ 3u
7
+ u
6
+ 3u
5
+ 2u
4
− u
3
+ u
2
− 2u − 1i
(i) Arc colorings
a
2
=
0
u
a
9
=
1
0
a
8
=
1
u
2
a
3
=
u
u
3
+ u
a
4
=
−u
3
u
3
+ u
a
10
=
−u
6
− u
4
+ 1
u
6
+ 2u
4
+ u
2
a
5
=
−u
7
− u
6
− 2u
5
− 2u
4
− u
3
− u
2
+ u + 1
u
6
+ 2u
4
− u
3
+ u
2
− u − 1
a
11
=
u
7
+ 2u
5
− 2u
−u
6
− 2u
4
+ u
3
− u
2
+ u + 1
a
1
=
u
3
u
5
+ u
3
+ u
a
7
=
−u
6
− u
4
+ 1
−u
8
− 2u
6
− 2u
4
a
12
=
u
5
− u
u
7
+ u
5
− u
a
6
=
−u
4
− u
2
+ 1
−u
6
− 2u
4
− u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −14
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
9
+ 6u
8
+ 15u
7
+ 15u
6
− 5u
5
− 24u
4
− 13u
3
+ 7u
2
+ 6u − 1
c
2
, c
6
, c
8
c
11
, c
12
u
9
+ 3u
7
+ u
6
+ 3u
5
+ 2u
4
− u
3
+ u
2
− 2u − 1
c
3
, c
4
, c
5
c
7
, c
9
, c
10
(u
3
+ u
2
− 2u − 1)
3
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
9
− 6y
8
+ ··· + 50y −1
c
2
, c
6
, c
8
c
11
, c
12
y
9
+ 6y
8
+ 15y
7
+ 15y
6
− 5y
5
− 24y
4
− 13y
3
+ 7y
2
+ 6y −1
c
3
, c
4
, c
5
c
7
, c
9
, c
10
(y
3
− 5y
2
+ 6y −1)
3
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.948532
−17.6243 −14.0000
u = 0.193528 + 1.054680I
−0.704972 −14.0000
u = 0.193528 − 1.054680I
−0.704972 −14.0000
u = 0.777314
−6.34475 −14.0000
u = −0.388657 + 1.205470I
−6.34475 −14.0000
u = −0.388657 − 1.205470I
−6.34475 −14.0000
u = 0.474266 + 1.294140I
−17.6243 −14.0000
u = 0.474266 − 1.294140I
−17.6243 −14.0000
u = −0.387056
−0.704972 −14.0000
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
9
+ 6u
8
+ 15u
7
+ 15u
6
− 5u
5
− 24u
4
− 13u
3
+ 7u
2
+ 6u − 1)
· (u
22
+ 13u
21
+ ··· + 2u + 1)
c
2
, c
8
(u
9
+ 3u
7
+ ··· − 2u − 1)(u
22
− u
21
+ ··· − 2u + 1)
c
3
, c
4
, c
5
c
7
, c
9
, c
10
((u
3
+ u
2
− 2u − 1)
3
)(u
22
− 2u
21
+ ··· − u + 2)
c
6
, c
11
, c
12
(u
9
+ 3u
7
+ ··· − 2u − 1)(u
22
− u
21
+ ··· + u
2
+ 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
9
− 6y
8
+ ··· + 50y −1)(y
22
− 7y
21
+ ··· + 2y + 1)
c
2
, c
8
(y
9
+ 6y
8
+ 15y
7
+ 15y
6
− 5y
5
− 24y
4
− 13y
3
+ 7y
2
+ 6y −1)
· (y
22
+ 13y
21
+ ··· + 2y + 1)
c
3
, c
4
, c
5
c
7
, c
9
, c
10
((y
3
− 5y
2
+ 6y −1)
3
)(y
22
− 30y
21
+ ··· + 19y + 4)
c
6
, c
11
, c
12
(y
9
+ 6y
8
+ 15y
7
+ 15y
6
− 5y
5
− 24y
4
− 13y
3
+ 7y
2
+ 6y −1)
· (y
22
+ 17y
21
+ ··· + 2y + 1)
11
