12a
0850
(K12a
0850
)
A knot diagram
1
Linearized knot diagam
4 5 9 10 3 11 12 1 2 6 7 8
Solving Sequence
7,12
8
1,4
2 9 3 11 6 5 10
c
7
c
12
c
1
c
8
c
3
c
11
c
6
c
5
c
10
c
2
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−412178201122641u
45
− 775553559035752u
44
+ ··· + 353733604752049b + 614435998813861,
402182189901230u
45
+ 1007252468993735u
44
+ ··· + 353733604752049a − 1539279368053536,
u
46
+ 2u
45
+ ··· + u + 1i
I
u
2
= hb + 1, a − u − 1, u
2
+ u − 1i
* 2 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−4.12×10
14
u
45
−7.76×10
14
u
44
+· · ·+3.54×10
14
b+6.14×10
14
, 4.02×
10
14
u
45
+1.01×10
15
u
44
+· · ·+3.54×10
14
a−1.54×10
15
, u
46
+2u
45
+· · ·+u+1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
8
=
1
u
2
a
1
=
−u
−u
3
+ u
a
4
=
−1.13696u
45
− 2.84749u
44
+ ··· + 10.2682u + 4.35152
1.16522u
45
+ 2.19248u
44
+ ··· − 8.68849u − 1.73700
a
2
=
1.82251u
45
+ 1.39735u
44
+ ··· − 4.08553u + 0.311253
2.24768u
45
− 0.402662u
44
+ ··· + 1.51126u + 1.82251
a
9
=
−u
2
+ 1
−u
4
+ 2u
2
a
3
=
−2.46826u
45
− 5.24035u
44
+ ··· + 19.2748u + 6.20584
−0.0909725u
45
+ 1.39965u
44
+ ··· − 6.03410u − 1.46820
a
11
=
u
u
a
6
=
−u
2
+ 1
−u
2
a
5
=
−1.96200u
45
− 4.63888u
44
+ ··· + 17.5120u + 6.75902
0.311421u
45
+ 1.20112u
44
+ ··· − 5.48102u − 0.962037
a
10
=
−u
3
+ 2u
−u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
473682658173898
353733604752049
u
45
−
99045102071581
353733604752049
u
44
+ ··· +
10709047957584936
353733604752049
u +
2653235649051613
353733604752049
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
46
− 7u
45
+ ··· − 28u + 4
c
2
, c
5
u
46
+ 3u
45
+ ··· + 32u + 1
c
3
u
46
+ 23u
44
+ ··· − 59u − 1
c
4
u
46
+ 2u
45
+ ··· − 21u − 1
c
6
, c
7
, c
8
c
10
, c
11
, c
12
u
46
− 2u
45
+ ··· − u + 1
c
9
u
46
+ 2u
45
+ ··· + u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
46
− 15y
45
+ ··· − 328y + 16
c
2
, c
5
y
46
− 25y
45
+ ··· − 588y + 1
c
3
y
46
+ 46y
45
+ ··· − 3437y + 1
c
4
y
46
+ 30y
45
+ ··· − 445y + 1
c
6
, c
7
, c
8
c
10
, c
11
, c
12
y
46
− 66y
45
+ ··· − 9y + 1
c
9
y
46
− 10y
45
+ ··· − 9y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.767158 + 0.433545I
a = −0.152356 + 0.237919I
b = −0.608935 + 0.607878I
−0.20687 + 1.83894I −8.26616 − 3.43927I
u = 0.767158 − 0.433545I
a = −0.152356 − 0.237919I
b = −0.608935 − 0.607878I
−0.20687 − 1.83894I −8.26616 + 3.43927I
u = 1.14595
a = 1.66543
b = −0.824680
−1.55865 0
u = 1.191040 + 0.091569I
a = 0.269705 − 0.115985I
b = −0.231211 + 1.314110I
−3.16460 − 3.49316I 0
u = 1.191040 − 0.091569I
a = 0.269705 + 0.115985I
b = −0.231211 − 1.314110I
−3.16460 + 3.49316I 0
u = −1.224310 + 0.038292I
a = 0.57689 − 2.38554I
b = −0.16096 − 1.67792I
−4.65165 + 0.65229I 0
u = −1.224310 − 0.038292I
a = 0.57689 + 2.38554I
b = −0.16096 + 1.67792I
−4.65165 − 0.65229I 0
u = −0.594658 + 0.493055I
a = 0.569482 + 0.019769I
b = 1.216190 − 0.260596I
0.76638 + 8.84991I −5.93176 − 9.09785I
u = −0.594658 − 0.493055I
a = 0.569482 − 0.019769I
b = 1.216190 + 0.260596I
0.76638 − 8.84991I −5.93176 + 9.09785I
u = −1.223150 + 0.246640I
a = 0.481344 + 0.605666I
b = −0.206918 + 0.060427I
−6.68187 + 3.83644I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.223150 − 0.246640I
a = 0.481344 − 0.605666I
b = −0.206918 − 0.060427I
−6.68187 − 3.83644I 0
u = 1.280350 + 0.145473I
a = −1.341390 + 0.223322I
b = 0.232936 + 0.197353I
−8.55312 − 5.24786I 0
u = 1.280350 − 0.145473I
a = −1.341390 − 0.223322I
b = 0.232936 − 0.197353I
−8.55312 + 5.24786I 0
u = 1.280250 + 0.244696I
a = 1.346480 − 0.364424I
b = −0.045681 + 0.143330I
−5.29495 − 11.44320I 0
u = 1.280250 − 0.244696I
a = 1.346480 + 0.364424I
b = −0.045681 − 0.143330I
−5.29495 + 11.44320I 0
u = −0.605806 + 0.310686I
a = −0.880404 + 0.351414I
b = −1.112130 + 0.350962I
−2.39123 + 3.64692I −10.43839 − 7.63926I
u = −0.605806 − 0.310686I
a = −0.880404 − 0.351414I
b = −1.112130 − 0.350962I
−2.39123 − 3.64692I −10.43839 + 7.63926I
u = 0.454796 + 0.454478I
a = 0.539747 − 0.277523I
b = 0.450806 − 0.330065I
−1.27121 − 1.36834I −10.65988 + 5.78093I
u = 0.454796 − 0.454478I
a = 0.539747 + 0.277523I
b = 0.450806 + 0.330065I
−1.27121 + 1.36834I −10.65988 − 5.78093I
u = −1.377040 + 0.092167I
a = −0.832099 − 0.286390I
b = −0.267749 − 0.034492I
−7.56750 + 0.10369I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.377040 − 0.092167I
a = −0.832099 + 0.286390I
b = −0.267749 + 0.034492I
−7.56750 − 0.10369I 0
u = −0.073726 + 0.603893I
a = −0.64203 + 1.26805I
b = 0.0927429 − 0.0646432I
2.34175 − 5.27392I −2.70147 + 5.21952I
u = −0.073726 − 0.603893I
a = −0.64203 − 1.26805I
b = 0.0927429 + 0.0646432I
2.34175 + 5.27392I −2.70147 − 5.21952I
u = 0.550313
a = 0.180040
b = −0.495725
−0.914439 −10.8210
u = −0.408387 + 0.270742I
a = −0.10890 + 2.30267I
b = 0.546367 + 0.446835I
2.00550 + 2.33057I −1.71776 − 9.37052I
u = −0.408387 − 0.270742I
a = −0.10890 − 2.30267I
b = 0.546367 − 0.446835I
2.00550 − 2.33057I −1.71776 + 9.37052I
u = −1.54354
a = −0.486260
b = −0.308213
−7.66046 0
u = 0.446300 + 0.095003I
a = 0.75867 − 2.98128I
b = −0.36758 + 1.78388I
0.836196 − 0.200718I 20.7380 − 14.2709I
u = 0.446300 − 0.095003I
a = 0.75867 + 2.98128I
b = −0.36758 − 1.78388I
0.836196 + 0.200718I 20.7380 + 14.2709I
u = 0.077400 + 0.391993I
a = 1.30428 − 0.97124I
b = 0.045336 − 0.233987I
−0.394888 − 1.327150I −5.21037 + 3.91369I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.077400 − 0.391993I
a = 1.30428 + 0.97124I
b = 0.045336 + 0.233987I
−0.394888 + 1.327150I −5.21037 − 3.91369I
u = −0.186943 + 0.264216I
a = 1.76910 + 1.31460I
b = 1.040340 − 0.428302I
2.61344 − 0.38254I 2.30820 − 3.85448I
u = −0.186943 − 0.264216I
a = 1.76910 − 1.31460I
b = 1.040340 + 0.428302I
2.61344 + 0.38254I 2.30820 + 3.85448I
u = −1.77945
a = −3.67500
b = −6.90609
−12.3107 0
u = −1.78677 + 0.02018I
a = −0.70395 − 1.63402I
b = −1.35526 − 2.52294I
−14.1039 + 3.9673I 0
u = −1.78677 − 0.02018I
a = −0.70395 + 1.63402I
b = −1.35526 + 2.52294I
−14.1039 − 3.9673I 0
u = 1.79497 + 0.00903I
a = −1.32414 − 1.96060I
b = −2.65985 − 4.75153I
−15.7848 − 0.8615I 0
u = 1.79497 − 0.00903I
a = −1.32414 + 1.96060I
b = −2.65985 + 4.75153I
−15.7848 + 0.8615I 0
u = 1.79436 + 0.06450I
a = −1.58447 + 0.56774I
b = −3.05444 + 1.14510I
−17.7122 − 5.2449I 0
u = 1.79436 − 0.06450I
a = −1.58447 − 0.56774I
b = −3.05444 − 1.14510I
−17.7122 + 5.2449I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.80654 + 0.03660I
a = 2.93105 − 0.19688I
b = 5.72048 − 0.32024I
19.5334 + 6.0912I 0
u = −1.80654 − 0.03660I
a = 2.93105 + 0.19688I
b = 5.72048 + 0.32024I
19.5334 − 6.0912I 0
u = −1.80661 + 0.06306I
a = −2.75336 − 0.16817I
b = −5.50064 − 0.25278I
−16.6222 + 12.8685I 0
u = −1.80661 − 0.06306I
a = −2.75336 + 0.16817I
b = −5.50064 + 0.25278I
−16.6222 − 12.8685I 0
u = 1.82069 + 0.02719I
a = 1.93426 − 0.40849I
b = 3.99351 − 0.87975I
−19.3942 − 0.7197I 0
u = 1.82069 − 0.02719I
a = 1.93426 + 0.40849I
b = 3.99351 + 0.87975I
−19.3942 + 0.7197I 0
9
II. I
u
2
= hb + 1, a − u − 1, u
2
+ u − 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
8
=
1
−u + 1
a
1
=
−u
−u + 1
a
4
=
u + 1
−1
a
2
=
−u
−u + 1
a
9
=
u
u
a
3
=
u + 2
0
a
11
=
u
u
a
6
=
u
u − 1
a
5
=
2u + 2
u − 1
a
10
=
1
−u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 1
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
2
c
2
(u + 1)
2
c
3
, c
4
, c
10
c
11
, c
12
u
2
− u − 1
c
5
(u − 1)
2
c
6
, c
7
, c
8
c
9
u
2
+ u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
2
c
2
, c
5
(y −1)
2
c
3
, c
4
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y
2
− 3y + 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 1.61803
b = −1.00000
0.657974 1.00000
u = −1.61803
a = −0.618034
b = −1.00000
−7.23771 1.00000
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
2
(u
46
− 7u
45
+ ··· − 28u + 4)
c
2
((u + 1)
2
)(u
46
+ 3u
45
+ ··· + 32u + 1)
c
3
(u
2
− u − 1)(u
46
+ 23u
44
+ ··· − 59u − 1)
c
4
(u
2
− u − 1)(u
46
+ 2u
45
+ ··· − 21u − 1)
c
5
((u − 1)
2
)(u
46
+ 3u
45
+ ··· + 32u + 1)
c
6
, c
7
, c
8
(u
2
+ u − 1)(u
46
− 2u
45
+ ··· − u + 1)
c
9
(u
2
+ u − 1)(u
46
+ 2u
45
+ ··· + u + 1)
c
10
, c
11
, c
12
(u
2
− u − 1)(u
46
− 2u
45
+ ··· − u + 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
2
(y
46
− 15y
45
+ ··· − 328y + 16)
c
2
, c
5
((y −1)
2
)(y
46
− 25y
45
+ ··· − 588y + 1)
c
3
(y
2
− 3y + 1)(y
46
+ 46y
45
+ ··· − 3437y + 1)
c
4
(y
2
− 3y + 1)(y
46
+ 30y
45
+ ··· − 445y + 1)
c
6
, c
7
, c
8
c
10
, c
11
, c
12
(y
2
− 3y + 1)(y
46
− 66y
45
+ ··· − 9y + 1)
c
9
(y
2
− 3y + 1)(y
46
− 10y
45
+ ··· − 9y + 1)
15
