12a
0538
(K12a
0538
)
A knot diagram
1
Linearized knot diagam
3 7 8 10 11 12 2 1 4 5 6 9
Solving Sequence
2,8
7 3 4 1 9 10 5 12 6 11
c
7
c
2
c
3
c
1
c
8
c
9
c
4
c
12
c
6
c
11
c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
41
− u
40
+ ··· − 3u + 1i
* 1 irreducible components of dim
C
= 0, with total 41 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
41
− u
40
+ · · · − 3u + 1i
(i) Arc colorings
a
2
=
0
u
a
8
=
1
0
a
7
=
1
u
2
a
3
=
u
u
3
+ u
a
4
=
−u
3
u
3
+ u
a
1
=
u
3
u
5
+ u
3
+ u
a
9
=
u
8
+ u
6
+ u
4
+ 1
u
10
+ 2u
8
+ 3u
6
+ 2u
4
+ u
2
a
10
=
−u
16
− 3u
14
− 5u
12
− 4u
10
− u
8
+ 1
u
16
+ 4u
14
+ 8u
12
+ 10u
10
+ 8u
8
+ 6u
6
+ 4u
4
+ 2u
2
a
5
=
u
29
+ 6u
27
+ ··· − 2u
3
− u
−u
29
− 7u
27
+ ··· − u
3
+ u
a
12
=
u
13
+ 2u
11
+ 3u
9
+ 2u
7
+ 2u
5
+ 2u
3
+ u
u
15
+ 3u
13
+ 6u
11
+ 7u
9
+ 6u
7
+ 4u
5
+ 2u
3
+ u
a
6
=
−u
26
− 5u
24
+ ··· − u
2
+ 1
−u
28
− 6u
26
+ ··· − 8u
6
− 3u
4
a
11
=
−u
39
− 8u
37
+ ··· + 2u
3
+ 2u
−u
40
+ u
39
+ ··· − 2u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −4u
39
+4u
38
−36u
37
+36u
36
−168u
35
+172u
34
−520u
33
+548u
32
−1184u
31
+1284u
30
−
2104u
29
+ 2324u
28
− 3052u
27
+ 3364u
26
− 3756u
25
+ 4012u
24
− 4040u
23
+ 4072u
22
−
3848u
21
+3628u
20
−3236u
19
+2888u
18
−2396u
17
+2040u
16
−1584u
15
+1260u
14
−944u
13
+
676u
12
−484u
11
+312u
10
−188u
9
+100u
8
−52u
7
+8u
6
−8u
5
−12u
4
+12u
3
−8u
2
+12u−18
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
41
+ 19u
40
+ ··· + 5u − 1
c
2
, c
7
u
41
+ u
40
+ ··· − 3u − 1
c
3
u
41
− u
40
+ ··· + 7u − 5
c
4
, c
5
, c
6
c
9
, c
10
, c
11
u
41
+ u
40
+ ··· − 3u − 1
c
8
, c
12
u
41
+ 5u
40
+ ··· − 161u − 39
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
41
+ 7y
40
+ ··· + 61y −1
c
2
, c
7
y
41
+ 19y
40
+ ··· + 5y −1
c
3
y
41
− 5y
40
+ ··· + 869y −25
c
4
, c
5
, c
6
c
9
, c
10
, c
11
y
41
− 57y
40
+ ··· + 5y −1
c
8
, c
12
y
41
+ 23y
40
+ ··· − 3563y −1521
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.184893 + 0.994464I
−1.51339 − 0.94191I −13.19182 + 4.87111I
u = 0.184893 − 0.994464I
−1.51339 + 0.94191I −13.19182 − 4.87111I
u = −0.326237 + 0.909164I
−0.68099 − 1.40662I −7.42529 + 4.24722I
u = −0.326237 − 0.909164I
−0.68099 + 1.40662I −7.42529 − 4.24722I
u = −0.704783 + 0.594687I
−11.28020 − 3.60891I −10.18311 + 3.00606I
u = −0.704783 − 0.594687I
−11.28020 + 3.60891I −10.18311 − 3.00606I
u = −0.178198 + 1.087930I
−6.59012 + 2.77027I −17.1418 − 2.7652I
u = −0.178198 − 1.087930I
−6.59012 − 2.77027I −17.1418 + 2.7652I
u = 0.416484 + 1.021060I
−2.94716 + 3.14297I −16.2536 − 6.2768I
u = 0.416484 − 1.021060I
−2.94716 − 3.14297I −16.2536 + 6.2768I
u = 0.682369 + 0.551619I
−1.05424 + 2.63533I −9.41238 − 3.91934I
u = 0.682369 − 0.551619I
−1.05424 − 2.63533I −9.41238 + 3.91934I
u = 0.782799 + 0.379135I
−12.41790 − 6.21468I −11.22155 + 2.89024I
u = 0.782799 − 0.379135I
−12.41790 + 6.21468I −11.22155 − 2.89024I
u = 0.176585 + 1.127330I
−17.3002 − 3.7251I −17.4761 + 1.6666I
u = 0.176585 − 1.127330I
−17.3002 + 3.7251I −17.4761 − 1.6666I
u = −0.755772 + 0.394629I
−1.86778 + 5.03167I −10.41788 − 4.02250I
u = −0.755772 − 0.394629I
−1.86778 − 5.03167I −10.41788 + 4.02250I
u = −0.606936 + 0.980785I
−12.42340 − 1.42472I −12.03958 + 2.60670I
u = −0.606936 − 0.980785I
−12.42340 + 1.42472I −12.03958 − 2.60670I
u = −0.695732 + 0.480590I
3.22968 − 0.39612I −4.26312 + 3.61739I
u = −0.695732 − 0.480590I
3.22968 + 0.39612I −4.26312 − 3.61739I
u = 0.722948 + 0.432712I
2.97485 − 2.70484I −5.59126 + 4.77948I
u = 0.722948 − 0.432712I
2.97485 + 2.70484I −5.59126 − 4.77948I
u = 0.575889 + 1.009940I
−2.41083 + 2.23503I −11.92132 − 1.73873I
u = 0.575889 − 1.009940I
−2.41083 − 2.23503I −11.92132 + 1.73873I
u = −0.421289 + 1.102850I
−8.91985 − 3.72236I −18.4297 + 4.1960I
u = −0.421289 − 1.102850I
−8.91985 + 3.72236I −18.4297 − 4.1960I
u = −0.578382 + 1.056610I
1.52744 − 4.51839I −7.10529 + 1.88338I
u = −0.578382 − 1.056610I
1.52744 + 4.51839I −7.10529 − 1.88338I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.421891 + 1.134710I
19.4645 + 3.9367I −18.3892 − 3.6616I
u = 0.421891 − 1.134710I
19.4645 − 3.9367I −18.3892 + 3.6616I
u = 0.582220 + 1.082150I
1.06240 + 7.69869I −9.15502 − 9.24426I
u = 0.582220 − 1.082150I
1.06240 − 7.69869I −9.15502 + 9.24426I
u = −0.584962 + 1.104120I
−3.95871 − 10.11040I −13.5525 + 8.1060I
u = −0.584962 − 1.104120I
−3.95871 + 10.11040I −13.5525 − 8.1060I
u = 0.589263 + 1.117610I
−14.6038 + 11.3740I −14.2742 − 6.8674I
u = 0.589263 − 1.117610I
−14.6038 − 11.3740I −14.2742 + 6.8674I
u = 0.681978
−16.8040 −14.2280
u = −0.606927
−5.94459 −14.4600
u = 0.358849
−0.680410 −14.4230
6
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
41
+ 19u
40
+ ··· + 5u − 1
c
2
, c
7
u
41
+ u
40
+ ··· − 3u − 1
c
3
u
41
− u
40
+ ··· + 7u − 5
c
4
, c
5
, c
6
c
9
, c
10
, c
11
u
41
+ u
40
+ ··· − 3u − 1
c
8
, c
12
u
41
+ 5u
40
+ ··· − 161u − 39
7
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
41
+ 7y
40
+ ··· + 61y −1
c
2
, c
7
y
41
+ 19y
40
+ ··· + 5y −1
c
3
y
41
− 5y
40
+ ··· + 869y −25
c
4
, c
5
, c
6
c
9
, c
10
, c
11
y
41
− 57y
40
+ ··· + 5y −1
c
8
, c
12
y
41
+ 23y
40
+ ··· − 3563y −1521
8
