12a
1242
(K12a
1242
)
A knot diagram
1
Linearized knot diagam
5 6 10 9 2 11 12 1 4 3 7 8
Solving Sequence
6,11
7 12
3,8
2 5 1 10 4 9
c
6
c
11
c
7
c
2
c
5
c
1
c
10
c
3
c
9
c
4
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−73684140u
28
− 84869082u
27
+ ··· + 145070621b − 231458335,
109671719u
28
+ 211321507u
27
+ ··· + 870423726a + 469996164, u
29
+ 2u
28
+ ··· + 3u + 3i
I
u
2
= hb − 1, a
2
− 2u + 4, u
2
− u − 1i
I
u
3
= hb + 1, a, u
2
+ u − 1i
* 3 irreducible components of dim
C
= 0, with total 35 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
= h−7.37 × 10
7
u
28
− 8.49 × 10
7
u
27
+ · · · + 1.45 × 10
8
b − 2.31 × 10
8
, 1.10 ×
10
8
u
28
+2.11×10
8
u
27
+· · · +8.70×10
8
a+4.70×10
8
, u
29
+2u
28
+· · · +3u +3i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
u
2
a
12
=
−u
−u
3
+ u
a
3
=
−0.125998u
28
− 0.242780u
27
+ ··· − 2.67848u − 0.539962
0.507919u
28
+ 0.585019u
27
+ ··· − 1.09278u + 1.59549
a
8
=
−u
2
+ 1
−u
4
+ 2u
2
a
2
=
0.381921u
28
+ 0.342239u
27
+ ··· − 3.77126u + 1.05552
0.507919u
28
+ 0.585019u
27
+ ··· − 1.09278u + 1.59549
a
5
=
−0.468112u
28
− 0.582646u
27
+ ··· + 3.90750u − 1.40739
−0.374870u
28
− 0.400564u
27
+ ··· + 1.59469u − 1.19585
a
1
=
u
3
− 2u
u
5
− 3u
3
+ u
a
10
=
0.277134u
28
+ 0.150852u
27
+ ··· + 1.79663u + 1.03838
0.00440318u
28
+ 0.00821410u
27
+ ··· + 0.0681851u + 0.279727
a
4
=
0.346692u
28
+ 0.461594u
27
+ ··· − 4.19855u + 1.50586
0.258251u
28
+ 0.340383u
27
+ ··· − 1.54868u + 1.00040
a
9
=
u
4
− 3u
2
+ 1
u
6
− 4u
4
+ 3u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes =
293684115
145070621
u
28
+
473769543
145070621
u
27
+ ··· −
27583342
145070621
u +
376643745
145070621
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
u
29
+ 3u
28
+ ··· − 4u + 11
c
3
, c
4
, c
9
c
10
u
29
+ u
28
+ ··· + 8u + 4
c
6
, c
7
, c
8
c
11
, c
12
u
29
− 2u
28
+ ··· + 3u − 3
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
y
29
− 35y
28
+ ··· + 2018y −121
c
3
, c
4
, c
9
c
10
y
29
+ 39y
28
+ ··· + 96y −16
c
6
, c
7
, c
8
c
11
, c
12
y
29
− 42y
28
+ ··· + 93y −9
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.957238 + 0.092408I
a = 0.133281 + 0.801704I
b = 0.518624 − 0.562490I
−3.64717 + 1.99860I −12.92483 − 5.25598I
u = −0.957238 − 0.092408I
a = 0.133281 − 0.801704I
b = 0.518624 + 0.562490I
−3.64717 − 1.99860I −12.92483 + 5.25598I
u = −0.417915 + 0.773611I
a = −1.69685 + 0.96816I
b = 1.64439 − 0.07500I
−14.7963 + 2.4842I −12.51838 − 2.49231I
u = −0.417915 − 0.773611I
a = −1.69685 − 0.96816I
b = 1.64439 + 0.07500I
−14.7963 − 2.4842I −12.51838 + 2.49231I
u = 1.18567
a = 0.637942
b = 1.42557
−7.33093 −11.5170
u = 1.210040 + 0.155277I
a = 0.148791 + 1.323010I
b = −0.663260 − 0.808481I
−11.90960 − 2.74446I −13.57354 + 3.19351I
u = 1.210040 − 0.155277I
a = 0.148791 − 1.323010I
b = −0.663260 + 0.808481I
−11.90960 + 2.74446I −13.57354 − 3.19351I
u = −1.196500 + 0.243718I
a = −0.237053 − 0.864265I
b = −1.50253 + 0.09750I
−10.33150 + 4.22769I −14.4747 − 4.4209I
u = −1.196500 − 0.243718I
a = −0.237053 + 0.864265I
b = −1.50253 − 0.09750I
−10.33150 − 4.22769I −14.4747 + 4.4209I
u = 0.759489
a = −0.494566
b = −0.342129
−1.45016 −4.67260
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.212910 + 0.458406I
a = −0.378766 − 1.282880I
b = 1.67070 + 0.24034I
19.6011 − 6.7273I −15.0335 + 3.7545I
u = 1.212910 − 0.458406I
a = −0.378766 + 1.282880I
b = 1.67070 − 0.24034I
19.6011 + 6.7273I −15.0335 − 3.7545I
u = 0.393984 + 0.502310I
a = 1.01292 + 1.12566I
b = −1.344740 − 0.017303I
−5.21965 − 1.67900I −11.30545 + 4.50169I
u = 0.393984 − 0.502310I
a = 1.01292 − 1.12566I
b = −1.344740 + 0.017303I
−5.21965 + 1.67900I −11.30545 − 4.50169I
u = −0.405673 + 0.323850I
a = 1.81380 − 2.24627I
b = −0.657758 + 0.300683I
−6.62321 + 1.10353I −8.65226 − 6.29840I
u = −0.405673 − 0.323850I
a = 1.81380 + 2.24627I
b = −0.657758 − 0.300683I
−6.62321 − 1.10353I −8.65226 + 6.29840I
u = −0.372797
a = 0.875809
b = 1.09755
−2.21609 2.51900
u = −1.64751
a = −0.250551
b = −0.668321
−9.96220 −4.00000
u = 0.181930 + 0.272469I
a = −0.761066 − 1.146960I
b = 0.244593 + 0.276761I
−0.167538 − 0.756994I −5.06839 + 9.13574I
u = 0.181930 − 0.272469I
a = −0.761066 + 1.146960I
b = 0.244593 − 0.276761I
−0.167538 + 0.756994I −5.06839 − 9.13574I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.70546 + 0.00477I
a = 0.148318 − 0.582342I
b = 0.663162 + 0.664949I
−13.14750 − 2.29645I 0. + 3.83143I
u = 1.70546 − 0.00477I
a = 0.148318 + 0.582342I
b = 0.663162 − 0.664949I
−13.14750 + 2.29645I 0. − 3.83143I
u = −1.78389
a = 0.547557
b = 1.64670
−18.2348 −12.5570
u = 1.78805 + 0.06048I
a = −0.376146 + 0.573320I
b = −1.66502 − 0.19299I
18.2495 − 5.5663I 0
u = 1.78805 − 0.06048I
a = −0.376146 − 0.573320I
b = −1.66502 + 0.19299I
18.2495 + 5.5663I 0
u = −1.79084 + 0.03907I
a = −0.016087 − 0.967046I
b = −0.686725 + 1.113520I
16.5545 + 3.6131I 0
u = −1.79084 − 0.03907I
a = −0.016087 + 0.967046I
b = −0.686725 − 1.113520I
16.5545 − 3.6131I 0
u = −1.79469 + 0.12774I
a = 0.050769 + 0.974515I
b = 1.69889 − 0.39866I
8.82781 + 9.35694I 0
u = −1.79469 − 0.12774I
a = 0.050769 − 0.974515I
b = 1.69889 + 0.39866I
8.82781 − 9.35694I 0
7
II. I
u
2
= hb − 1, a
2
− 2u + 4, u
2
− u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
u + 1
a
12
=
−u
−u − 1
a
3
=
a
1
a
8
=
−u
−u
a
2
=
a + 1
1
a
5
=
−a
−1
a
1
=
1
0
a
10
=
2u − 2
−au + u
a
4
=
−a
−au − a − 1
a
9
=
0
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −16
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u + 1)
4
c
3
, c
4
, c
9
c
10
(u
2
+ 2)
2
c
5
(u − 1)
4
c
6
, c
7
, c
8
(u
2
− u − 1)
2
c
11
, c
12
(u
2
+ u − 1)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
4
c
3
, c
4
, c
9
c
10
(y + 2)
4
c
6
, c
7
, c
8
c
11
, c
12
(y
2
− 3y + 1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 2.28825I
b = 1.00000
−7.56670 −16.0000
u = −0.618034
a = − 2.28825I
b = 1.00000
−7.56670 −16.0000
u = 1.61803
a = 0.874032I
b = 1.00000
−15.4624 −16.0000
u = 1.61803
a = − 0.874032I
b = 1.00000
−15.4624 −16.0000
11
III. I
u
3
= hb + 1, a, u
2
+ u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
−u + 1
a
12
=
−u
−u + 1
a
3
=
0
−1
a
8
=
u
u
a
2
=
−1
−1
a
5
=
0
−1
a
1
=
−1
0
a
10
=
0
u
a
4
=
0
−1
a
9
=
0
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −18
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
c
3
, c
4
, c
9
c
10
u
2
c
5
(u + 1)
2
c
6
, c
7
, c
8
u
2
+ u − 1
c
11
, c
12
u
2
− u − 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
2
c
3
, c
4
, c
9
c
10
y
2
c
6
, c
7
, c
8
c
11
, c
12
y
2
− 3y + 1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 0
b = −1.00000
−2.63189 −18.0000
u = −1.61803
a = 0
b = −1.00000
−10.5276 −18.0000
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
2
)(u + 1)
4
(u
29
+ 3u
28
+ ··· − 4u + 11)
c
3
, c
4
, c
9
c
10
u
2
(u
2
+ 2)
2
(u
29
+ u
28
+ ··· + 8u + 4)
c
5
((u − 1)
4
)(u + 1)
2
(u
29
+ 3u
28
+ ··· − 4u + 11)
c
6
, c
7
, c
8
((u
2
− u − 1)
2
)(u
2
+ u − 1)(u
29
− 2u
28
+ ··· + 3u − 3)
c
11
, c
12
(u
2
− u − 1)(u
2
+ u − 1)
2
(u
29
− 2u
28
+ ··· + 3u − 3)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
((y −1)
6
)(y
29
− 35y
28
+ ··· + 2018y −121)
c
3
, c
4
, c
9
c
10
y
2
(y + 2)
4
(y
29
+ 39y
28
+ ··· + 96y −16)
c
6
, c
7
, c
8
c
11
, c
12
((y
2
− 3y + 1)
3
)(y
29
− 42y
28
+ ··· + 93y −9)
17
