12n
0476
(K12n
0476
)
A knot diagram
1
Linearized knot diagam
3 6 8 9 2 11 12 5 4 6 7 8
Solving Sequence
6,11
7 12
3,8
4 2 1 5 10 9
c
6
c
11
c
7
c
3
c
2
c
1
c
5
c
10
c
9
c
4
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−399369286162u
41
− 497609046536u
40
+ ··· + 724090717741b − 733512526727,
− 1720004901137u
41
+ 81273516863u
40
+ ··· + 4344544306446a − 9063190310470,
u
42
+ 2u
41
+ ··· − u + 3i
I
u
2
= hb − 1, a
2
− 2a − 2u + 5, u
2
− u − 1i
I
u
3
= hb + 1, a + 1, u
2
+ u − 1i
* 3 irreducible components of dim
C
= 0, with total 48 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−3.99×10
11
u
41
−4.98×10
11
u
40
+· · ·+7.24×10
11
b−7.34×10
11
, −1.72×
10
12
u
41
+8.13×10
10
u
40
+· · ·+4.34×10
12
a−9.06×10
12
, u
42
+2u
41
+· · ·−u+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
=
0.395900u
41
− 0.0187070u
40
+ ··· − 2.07593u + 2.08611
0.551546u
41
+ 0.687219u
40
+ ··· − 1.66740u + 1.01301
a
8
=
−u
2
+ 1
−u
4
+ 2u
2
a
4
=
0.816087u
41
+ 0.338018u
40
+ ··· − 4.06896u + 3.36970
0.759348u
41
+ 0.839252u
40
+ ··· − 2.61706u + 1.37583
a
2
=
0.947446u
41
+ 0.668512u
40
+ ··· − 3.74333u + 3.09912
0.551546u
41
+ 0.687219u
40
+ ··· − 1.66740u + 1.01301
a
1
=
u
3
− 2u
u
5
− 3u
3
+ u
a
5
=
0.835547u
41
+ 0.380630u
40
+ ··· + 6.27378u + 0.289814
0.900701u
41
+ 0.654885u
40
+ ··· + 0.388683u + 1.42313
a
10
=
u
u
a
9
=
0.988524u
41
+ 0.535525u
40
+ ··· − 1.74947u + 4.21201
0.136440u
41
+ 0.143354u
40
+ ··· + 0.793437u + 1.35900
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
784004418607
724090717741
u
41
+
2116932802003
724090717741
u
40
+ ··· −
5911721578314
724090717741
u −
3580528018791
724090717741
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
42
+ 17u
41
+ ··· + 1720u + 121
c
2
, c
5
u
42
+ 3u
41
+ ··· + 24u + 11
c
3
u
42
+ u
41
+ ··· − 160u + 100
c
4
, c
8
, c
9
u
42
− u
41
+ ··· − 32u
2
+ 4
c
6
, c
7
, c
10
c
11
, c
12
u
42
− 2u
41
+ ··· + u + 3
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
42
+ 23y
41
+ ··· + 258748y + 14641
c
2
, c
5
y
42
− 17y
41
+ ··· − 1720y + 121
c
3
y
42
− 23y
41
+ ··· − 179200y + 10000
c
4
, c
8
, c
9
y
42
+ 37y
41
+ ··· − 256y + 16
c
6
, c
7
, c
10
c
11
, c
12
y
42
− 48y
41
+ ··· − 115y + 9
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.049280 + 0.091551I
a = 0.412704 − 0.129591I
b = 0.776169 + 0.605347I
−4.77432 − 2.23397I −10.46787 + 3.24456I
u = −1.049280 − 0.091551I
a = 0.412704 + 0.129591I
b = 0.776169 − 0.605347I
−4.77432 + 2.23397I −10.46787 − 3.24456I
u = 0.673657 + 0.613325I
a = 0.46793 − 1.91116I
b = 1.100550 + 0.724909I
−1.23257 − 9.56169I −8.38946 + 7.95988I
u = 0.673657 − 0.613325I
a = 0.46793 + 1.91116I
b = 1.100550 − 0.724909I
−1.23257 + 9.56169I −8.38946 − 7.95988I
u = −0.576900 + 0.629034I
a = −0.25109 − 1.86774I
b = −0.957258 + 0.776390I
3.58322 + 5.14982I −3.56640 − 6.23803I
u = −0.576900 − 0.629034I
a = −0.25109 + 1.86774I
b = −0.957258 − 0.776390I
3.58322 − 5.14982I −3.56640 + 6.23803I
u = 0.553857 + 0.616758I
a = −0.988096 + 0.680405I
b = 0.572226 − 0.890720I
0.35890 − 3.56775I −5.83599 + 3.90838I
u = 0.553857 − 0.616758I
a = −0.988096 − 0.680405I
b = 0.572226 + 0.890720I
0.35890 + 3.56775I −5.83599 − 3.90838I
u = 0.786427
a = −0.785424
b = −0.488406
−1.56406 −3.83480
u = −0.415353 + 0.654172I
a = 0.983300 + 0.831773I
b = −0.801220 − 0.815689I
4.05957 − 0.81400I −1.93866 − 0.14480I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.415353 − 0.654172I
a = 0.983300 − 0.831773I
b = −0.801220 + 0.815689I
4.05957 + 0.81400I −1.93866 + 0.14480I
u = 0.430526 + 0.624525I
a = −0.00031 − 1.73658I
b = 0.721316 + 0.800712I
0.718852 − 0.648201I −5.19720 + 2.33460I
u = 0.430526 − 0.624525I
a = −0.00031 + 1.73658I
b = 0.721316 − 0.800712I
0.718852 + 0.648201I −5.19720 − 2.33460I
u = 0.289860 + 0.689463I
a = −0.902818 + 0.978781I
b = 0.995362 − 0.745920I
−0.10093 + 5.18138I −5.92756 − 3.03712I
u = 0.289860 − 0.689463I
a = −0.902818 − 0.978781I
b = 0.995362 + 0.745920I
−0.10093 − 5.18138I −5.92756 + 3.03712I
u = −1.333380 + 0.055601I
a = 0.083600 − 0.453941I
b = 0.713786 + 0.763955I
−4.90307 − 2.25991I 0
u = −1.333380 − 0.055601I
a = 0.083600 + 0.453941I
b = 0.713786 − 0.763955I
−4.90307 + 2.25991I 0
u = 0.476375 + 0.392459I
a = −0.604894 + 0.926643I
b = −1.261030 + 0.067596I
−6.08478 − 1.41154I −9.66825 + 4.90149I
u = 0.476375 − 0.392459I
a = −0.604894 − 0.926643I
b = −1.261030 − 0.067596I
−6.08478 + 1.41154I −9.66825 − 4.90149I
u = −0.475739 + 0.258622I
a = 1.77841 − 2.91372I
b = −0.798648 + 0.217030I
−6.83314 + 0.96606I −8.00205 − 7.45219I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.475739 − 0.258622I
a = 1.77841 + 2.91372I
b = −0.798648 − 0.217030I
−6.83314 − 0.96606I −8.00205 + 7.45219I
u = 1.45883 + 0.17891I
a = 0.245004 − 0.259927I
b = −0.587321 + 0.900636I
−1.96643 − 2.13199I 0
u = 1.45883 − 0.17891I
a = 0.245004 + 0.259927I
b = −0.587321 − 0.900636I
−1.96643 + 2.13199I 0
u = −1.48695 + 0.16260I
a = 0.560834 + 1.090920I
b = 0.940796 − 0.731217I
−5.49810 + 3.38316I 0
u = −1.48695 − 0.16260I
a = 0.560834 − 1.090920I
b = 0.940796 + 0.731217I
−5.49810 − 3.38316I 0
u = 1.51893
a = 1.19880
b = 1.30296
−8.88024 0
u = −1.53094 + 0.09497I
a = −1.171830 − 0.180431I
b = −1.351610 − 0.153172I
−12.84770 + 3.06597I 0
u = −1.53094 − 0.09497I
a = −1.171830 + 0.180431I
b = −1.351610 + 0.153172I
−12.84770 − 3.06597I 0
u = 1.53459 + 0.05929I
a = 0.06391 + 1.58702I
b = −0.843652 − 0.499599I
−13.66930 − 2.02433I 0
u = 1.53459 − 0.05929I
a = 0.06391 − 1.58702I
b = −0.843652 + 0.499599I
−13.66930 + 2.02433I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.53944 + 0.19194I
a = −0.385692 − 0.184391I
b = 0.433918 + 0.972193I
−6.56394 + 6.51432I 0
u = −1.53944 − 0.19194I
a = −0.385692 + 0.184391I
b = 0.433918 − 0.972193I
−6.56394 − 6.51432I 0
u = 1.55140 + 0.19874I
a = −0.84494 + 1.14482I
b = −1.088960 − 0.737531I
−3.47009 − 8.18357I 0
u = 1.55140 − 0.19874I
a = −0.84494 − 1.14482I
b = −1.088960 + 0.737531I
−3.47009 + 8.18357I 0
u = −0.425520
a = 1.66207
b = 1.09128
−2.26322 5.01380
u = 0.242324 + 0.345213I
a = −0.599361 − 1.192230I
b = 0.361519 + 0.367349I
−0.227692 − 0.948273I −4.31648 + 7.21437I
u = 0.242324 − 0.345213I
a = −0.599361 + 1.192230I
b = 0.361519 − 0.367349I
−0.227692 + 0.948273I −4.31648 − 7.21437I
u = −1.59476 + 0.19372I
a = 1.02426 + 1.19954I
b = 1.179170 − 0.691260I
−8.8337 + 12.5829I 0
u = −1.59476 − 0.19372I
a = 1.02426 − 1.19954I
b = 1.179170 + 0.691260I
−8.8337 − 12.5829I 0
u = −1.64206
a = −0.939257
b = −0.789120
−10.0738 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.67244 + 0.02610I
a = 0.894316 − 0.090456I
b = 0.836517 − 0.423614I
−14.0854 + 1.7540I 0
u = 1.67244 − 0.02610I
a = 0.894316 + 0.090456I
b = 0.836517 + 0.423614I
−14.0854 − 1.7540I 0
9
II. I
u
2
= hb − 1, a
2
− 2a − 2u + 5, 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
4
=
−au + u + 1
−au − a + u + 2
a
2
=
a + 1
1
a
1
=
1
0
a
5
=
−a
−1
a
10
=
u
u
a
9
=
−au + 2u − 2
−au
(ii) Obstruction class = 1
(iii) Cusp Shapes = −16
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
4
c
2
(u + 1)
4
c
3
, c
4
, c
8
c
9
(u
2
+ 2)
2
c
6
, c
7
(u
2
− u − 1)
2
c
10
, c
11
, c
12
(u
2
+ u − 1)
2
11
(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
8
c
9
(y + 2)
4
c
6
, c
7
, c
10
c
11
, c
12
(y
2
− 3y + 1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 1.00000 + 2.28825I
b = 1.00000
−7.56670 −16.0000
u = −0.618034
a = 1.00000 − 2.28825I
b = 1.00000
−7.56670 −16.0000
u = 1.61803
a = 1.000000 + 0.874032I
b = 1.00000
−15.4624 −16.0000
u = 1.61803
a = 1.000000 − 0.874032I
b = 1.00000
−15.4624 −16.0000
13
III. I
u
3
= hb + 1, a + 1, 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
=
−1
−1
a
8
=
u
u
a
4
=
−1
−1
a
2
=
−2
−1
a
1
=
−1
0
a
5
=
−1
−1
a
10
=
u
u
a
9
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −18
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
c
3
, c
4
, c
8
c
9
u
2
c
5
(u + 1)
2
c
6
, c
7
u
2
+ u − 1
c
10
, c
11
, c
12
u
2
− u − 1
15
(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
8
c
9
y
2
c
6
, c
7
, c
10
c
11
, c
12
y
2
− 3y + 1
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −1.00000
b = −1.00000
−2.63189 −18.0000
u = −1.61803
a = −1.00000
b = −1.00000
−10.5276 −18.0000
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
42
+ 17u
41
+ ··· + 1720u + 121)
c
2
((u − 1)
2
)(u + 1)
4
(u
42
+ 3u
41
+ ··· + 24u + 11)
c
3
u
2
(u
2
+ 2)
2
(u
42
+ u
41
+ ··· − 160u + 100)
c
4
, c
8
, c
9
u
2
(u
2
+ 2)
2
(u
42
− u
41
+ ··· − 32u
2
+ 4)
c
5
((u − 1)
4
)(u + 1)
2
(u
42
+ 3u
41
+ ··· + 24u + 11)
c
6
, c
7
((u
2
− u − 1)
2
)(u
2
+ u − 1)(u
42
− 2u
41
+ ··· + u + 3)
c
10
, c
11
, c
12
(u
2
− u − 1)(u
2
+ u − 1)
2
(u
42
− 2u
41
+ ··· + u + 3)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
42
+ 23y
41
+ ··· + 258748y + 14641)
c
2
, c
5
((y −1)
6
)(y
42
− 17y
41
+ ··· − 1720y + 121)
c
3
y
2
(y + 2)
4
(y
42
− 23y
41
+ ··· − 179200y + 10000)
c
4
, c
8
, c
9
y
2
(y + 2)
4
(y
42
+ 37y
41
+ ··· − 256y + 16)
c
6
, c
7
, c
10
c
11
, c
12
((y
2
− 3y + 1)
3
)(y
42
− 48y
41
+ ··· − 115y + 9)
19
