12n
0186
(K12n
0186
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 10 11 3 12 7 6 1 9
Solving Sequence
5,10
6 11
3,7
2 1 4 9 12 8
c
5
c
10
c
6
c
2
c
1
c
4
c
9
c
12
c
8
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h3.22078 × 10
25
u
51
− 8.93113 × 10
25
u
50
+ ··· + 5.70770 × 10
25
b + 5.96223 × 10
25
,
5.22434 × 10
25
u
51
− 2.15422 × 10
25
u
50
+ ··· + 5.70770 × 10
25
a − 1.21095 × 10
26
, u
52
− 2u
51
+ ··· − u
2
+ 1i
I
u
2
= hb + 1, 2u
7
− u
6
− 5u
5
+ 2u
4
+ 3u
3
+ a + 2u − 1, u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 60 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
= h3.22 × 10
25
u
51
− 8.93 × 10
25
u
50
+ · · · + 5.71 × 10
25
b + 5.96 × 10
25
, 5.22 ×
10
25
u
51
−2.15×10
25
u
50
+· · ·+5.71 ×10
25
a−1.21×10
26
, u
52
−2u
51
+· · ·−u
2
+1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
−u
2
a
11
=
u
−u
3
+ u
a
3
=
−0.915316u
51
+ 0.377424u
50
+ ··· − 0.305015u + 2.12160
−0.564287u
51
+ 1.56475u
50
+ ··· + 0.657034u − 1.04459
a
7
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
−1.47960u
51
+ 1.94218u
50
+ ··· + 0.352020u + 1.07701
−0.564287u
51
+ 1.56475u
50
+ ··· + 0.657034u − 1.04459
a
1
=
−1.18721u
51
+ 3.15641u
50
+ ··· + 2.25837u − 0.599223
0.268382u
51
− 0.928704u
50
+ ··· − 0.481556u + 0.106979
a
4
=
−1.04390u
51
+ 0.780635u
50
+ ··· − 0.415024u + 1.93894
−0.616840u
51
+ 1.83391u
50
+ ··· + 0.779891u − 1.08938
a
9
=
−u
5
+ 2u
3
− u
u
7
− 3u
5
+ 2u
3
+ u
a
12
=
−0.594106u
51
+ 1.79472u
50
+ ··· + 1.62921u − 0.588090
−0.276947u
51
+ 0.473102u
50
+ ··· + 0.517256u + 0.0584887
a
8
=
0.285029u
51
− 0.390403u
50
+ ··· − 1.33955u + 0.209160
−0.492416u
51
+ 1.54733u
50
+ ··· + 0.722296u − 0.103429
(ii) Obstruction class = −1
(iii) Cusp Shapes =
1240657506593977463220736159
57076965857356338641202335
u
51
−
507731546315019686665244629
57076965857356338641202335
u
50
+
··· −
1047474987399803989773483394
57076965857356338641202335
u −
421248860824180893040358514
57076965857356338641202335
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
52
+ 15u
51
+ ··· + 154u + 1
c
2
, c
4
u
52
− 9u
51
+ ··· − 18u + 1
c
3
, c
7
u
52
− 3u
51
+ ··· − 4480u + 256
c
5
, c
6
, c
10
u
52
− 2u
51
+ ··· − u
2
+ 1
c
8
, c
12
u
52
− 2u
51
+ ··· − 4u + 1
c
9
u
52
+ 6u
51
+ ··· + 880u + 4025
c
11
u
52
− 30u
51
+ ··· − 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
52
+ 53y
51
+ ··· − 8978y + 1
c
2
, c
4
y
52
− 15y
51
+ ··· − 154y + 1
c
3
, c
7
y
52
− 51y
51
+ ··· − 6209536y + 65536
c
5
, c
6
, c
10
y
52
− 50y
51
+ ··· − 2y + 1
c
8
, c
12
y
52
− 30y
51
+ ··· − 2y + 1
c
9
y
52
− 26y
51
+ ··· − 386280850y + 16200625
c
11
y
52
− 14y
51
+ ··· − 22y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.596040 + 0.685350I
a = −0.303738 − 0.109363I
b = 1.035020 − 0.836655I
6.06449 + 5.90411I 8.42870 − 2.86873I
u = −0.596040 − 0.685350I
a = −0.303738 + 0.109363I
b = 1.035020 + 0.836655I
6.06449 − 5.90411I 8.42870 + 2.86873I
u = −0.461943 + 0.763402I
a = 0.92002 − 1.14323I
b = 1.15749 + 0.84016I
5.61921 − 10.76330I 7.42388 + 7.81134I
u = −0.461943 − 0.763402I
a = 0.92002 + 1.14323I
b = 1.15749 − 0.84016I
5.61921 + 10.76330I 7.42388 − 7.81134I
u = 0.600467 + 0.630929I
a = −0.194035 − 0.006330I
b = 0.804140 + 0.798246I
2.75283 − 0.63628I 6.47155 − 0.50182I
u = 0.600467 − 0.630929I
a = −0.194035 + 0.006330I
b = 0.804140 − 0.798246I
2.75283 + 0.63628I 6.47155 + 0.50182I
u = 0.429595 + 0.756706I
a = 1.011620 + 0.980800I
b = 1.000840 − 0.771858I
2.14152 + 5.31582I 4.93660 − 5.12952I
u = 0.429595 − 0.756706I
a = 1.011620 − 0.980800I
b = 1.000840 + 0.771858I
2.14152 − 5.31582I 4.93660 + 5.12952I
u = −0.428853 + 0.704754I
a = 1.28272 − 0.99964I
b = 0.810880 + 0.929645I
6.77557 − 0.63032I 9.03919 + 2.39060I
u = −0.428853 − 0.704754I
a = 1.28272 + 0.99964I
b = 0.810880 − 0.929645I
6.77557 + 0.63032I 9.03919 − 2.39060I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.529637 + 0.629472I
a = −0.312532 + 0.180815I
b = 0.675424 − 1.090410I
7.14701 − 3.78883I 9.80206 + 4.18447I
u = −0.529637 − 0.629472I
a = −0.312532 − 0.180815I
b = 0.675424 + 1.090410I
7.14701 + 3.78883I 9.80206 − 4.18447I
u = 0.089818 + 0.804779I
a = 0.976159 + 0.124090I
b = 0.652722 − 0.112944I
−3.05621 + 2.81915I 8.84773 − 4.69108I
u = 0.089818 − 0.804779I
a = 0.976159 − 0.124090I
b = 0.652722 + 0.112944I
−3.05621 − 2.81915I 8.84773 + 4.69108I
u = 1.160150 + 0.341361I
a = 0.444553 + 0.408247I
b = 0.621048 − 0.094499I
0.200968 + 1.333880I 0
u = 1.160150 − 0.341361I
a = 0.444553 − 0.408247I
b = 0.621048 + 0.094499I
0.200968 − 1.333880I 0
u = −1.305710 + 0.022937I
a = 0.482228 + 0.442327I
b = −1.42955 − 0.12934I
1.369850 − 0.105601I 0
u = −1.305710 − 0.022937I
a = 0.482228 − 0.442327I
b = −1.42955 + 0.12934I
1.369850 + 0.105601I 0
u = −1.31861
a = 1.09514
b = 0.151920
6.40470 0
u = 1.317970 + 0.092549I
a = 0.81132 − 1.44084I
b = −1.34968 + 0.57134I
2.18752 + 3.25685I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.317970 − 0.092549I
a = 0.81132 + 1.44084I
b = −1.34968 − 0.57134I
2.18752 − 3.25685I 0
u = −1.311760 + 0.358837I
a = 0.394809 − 0.696031I
b = 0.728756 + 0.259803I
1.32923 − 7.01296I 0
u = −1.311760 − 0.358837I
a = 0.394809 + 0.696031I
b = 0.728756 − 0.259803I
1.32923 + 7.01296I 0
u = 1.390960 + 0.114127I
a = 0.57698 − 1.72339I
b = −0.723456 + 0.697653I
3.42681 + 2.63296I 0
u = 1.390960 − 0.114127I
a = 0.57698 + 1.72339I
b = −0.723456 − 0.697653I
3.42681 − 2.63296I 0
u = 1.40302
a = −13.7171
b = −1.00767
4.91335 0
u = −1.404190 + 0.159019I
a = 0.27827 + 1.88427I
b = −0.535802 − 1.135950I
5.89367 − 6.10396I 0
u = −1.404190 − 0.159019I
a = 0.27827 − 1.88427I
b = −0.535802 + 1.135950I
5.89367 + 6.10396I 0
u = 0.289349 + 0.491612I
a = −0.104933 − 0.935937I
b = −0.665776 + 0.817316I
0.49136 + 3.75076I 5.76906 − 8.97851I
u = 0.289349 − 0.491612I
a = −0.104933 + 0.935937I
b = −0.665776 − 0.817316I
0.49136 − 3.75076I 5.76906 + 8.97851I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.43138 + 0.07294I
a = 1.036390 + 0.944754I
b = −0.270270 − 0.184770I
6.54809 − 0.25945I 0
u = −1.43138 − 0.07294I
a = 1.036390 − 0.944754I
b = −0.270270 + 0.184770I
6.54809 + 0.25945I 0
u = 1.47938 + 0.26376I
a = 0.37431 + 1.88780I
b = 0.977064 − 0.921748I
12.93400 + 4.17895I 0
u = 1.47938 − 0.26376I
a = 0.37431 − 1.88780I
b = 0.977064 + 0.921748I
12.93400 − 4.17895I 0
u = 0.355651 + 0.332212I
a = 2.25163 − 0.94682I
b = −0.586013 − 0.340416I
0.96467 − 1.11364I 8.10661 − 2.28473I
u = 0.355651 − 0.332212I
a = 2.25163 + 0.94682I
b = −0.586013 + 0.340416I
0.96467 + 1.11364I 8.10661 + 2.28473I
u = −1.48991 + 0.28027I
a = 0.08758 − 1.83114I
b = 1.125690 + 0.835388I
8.35025 − 9.10360I 0
u = −1.48991 − 0.28027I
a = 0.08758 + 1.83114I
b = 1.125690 − 0.835388I
8.35025 + 9.10360I 0
u = 1.50299 + 0.21204I
a = −1.01775 − 1.30127I
b = 0.71828 + 1.28058I
13.7566 + 6.8511I 0
u = 1.50299 − 0.21204I
a = −1.01775 + 1.30127I
b = 0.71828 − 1.28058I
13.7566 − 6.8511I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.50262 + 0.27770I
a = −0.05829 + 1.98831I
b = 1.23801 − 0.89513I
11.9860 + 14.5665I 0
u = 1.50262 − 0.27770I
a = −0.05829 − 1.98831I
b = 1.23801 + 0.89513I
11.9860 − 14.5665I 0
u = −1.51858 + 0.19470I
a = −0.846488 + 1.031250I
b = 0.703551 − 1.024970I
9.67287 − 2.30932I 0
u = −1.51858 − 0.19470I
a = −0.846488 − 1.031250I
b = 0.703551 + 1.024970I
9.67287 + 2.30932I 0
u = −0.076509 + 0.456871I
a = −1.054590 + 0.666345I
b = −1.310420 − 0.236305I
−2.05132 − 1.32678I −0.14980 + 4.04076I
u = −0.076509 − 0.456871I
a = −1.054590 − 0.666345I
b = −1.310420 + 0.236305I
−2.05132 + 1.32678I −0.14980 − 4.04076I
u = 0.452272
a = 0.656563
b = 0.108480
0.718769 13.9480
u = 1.53851 + 0.20821I
a = −1.040950 − 0.804099I
b = 0.925560 + 0.944868I
13.09740 − 2.67801I 0
u = 1.53851 − 0.20821I
a = −1.040950 + 0.804099I
b = 0.925560 − 0.944868I
13.09740 + 2.67801I 0
u = −0.202886 + 0.379947I
a = −0.30999 + 1.78499I
b = −0.885931 − 0.291667I
−1.66844 − 0.85066I −1.89079 + 2.59214I
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.202886 − 0.379947I
a = −0.30999 − 1.78499I
b = −0.885931 + 0.291667I
−1.66844 + 0.85066I −1.89079 − 2.59214I
u = −0.336802
a = 6.59478
b = −1.08789
−0.454350 34.0590
10
II. I
u
2
=
hb+1, 2u
7
−u
6
−5u
5
+2u
4
+3u
3
+a+2u−1, u
8
−u
7
−3u
6
+2u
5
+3u
4
−2u−1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
−u
2
a
11
=
u
−u
3
+ u
a
3
=
−2u
7
+ u
6
+ 5u
5
− 2u
4
− 3u
3
− 2u + 1
−1
a
7
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
−2u
7
+ u
6
+ 5u
5
− 2u
4
− 3u
3
− 2u
−1
a
1
=
−1
0
a
4
=
−2u
7
+ u
6
+ 5u
5
− 2u
4
− 3u
3
− 2u + 1
−1
a
9
=
−u
5
+ 2u
3
− u
u
7
− 3u
5
+ 2u
3
+ u
a
12
=
−u
3
+ 2u
−u
3
+ u
a
8
=
−u
2
+ 1
u
4
− 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
7
− u
6
+ 10u
5
+ 3u
4
− 6u
3
− 2u
2
− 4u − 1
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
8
c
3
, c
7
u
8
c
4
(u + 1)
8
c
5
, c
6
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
c
8
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1
c
9
u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1
c
10
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
c
11
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
c
12
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
8
c
3
, c
7
y
8
c
5
, c
6
, c
10
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
c
8
, c
12
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
c
9
, c
11
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.180120 + 0.268597I
a = −0.085690 + 0.514779I
b = −1.00000
−0.604279 − 1.131230I 1.44913 − 0.23763I
u = −1.180120 − 0.268597I
a = −0.085690 − 0.514779I
b = −1.00000
−0.604279 + 1.131230I 1.44913 + 0.23763I
u = −0.108090 + 0.747508I
a = −1.036110 + 0.260696I
b = −1.00000
−3.80435 − 2.57849I −1.70307 + 2.50491I
u = −0.108090 − 0.747508I
a = −1.036110 − 0.260696I
b = −1.00000
−3.80435 + 2.57849I −1.70307 − 2.50491I
u = 1.37100
a = −3.88842
b = −1.00000
4.85780 −9.72740
u = 1.334530 + 0.318930I
a = 0.043072 − 0.634428I
b = −1.00000
0.73474 + 6.44354I 5.13991 − 2.71216I
u = 1.334530 − 0.318930I
a = 0.043072 + 0.634428I
b = −1.00000
0.73474 − 6.44354I 5.13991 + 2.71216I
u = −0.463640
a = 2.04588
b = −1.00000
−0.799899 0.955500
14
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
52
+ 15u
51
+ ··· + 154u + 1)
c
2
((u − 1)
8
)(u
52
− 9u
51
+ ··· − 18u + 1)
c
3
, c
7
u
8
(u
52
− 3u
51
+ ··· − 4480u + 256)
c
4
((u + 1)
8
)(u
52
− 9u
51
+ ··· − 18u + 1)
c
5
, c
6
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)(u
52
− 2u
51
+ ··· − u
2
+ 1)
c
8
(u
8
+ u
7
+ ··· − 2u − 1)(u
52
− 2u
51
+ ··· − 4u + 1)
c
9
(u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
· (u
52
+ 6u
51
+ ··· + 880u + 4025)
c
10
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
52
− 2u
51
+ ··· − u
2
+ 1)
c
11
(u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
· (u
52
− 30u
51
+ ··· − 2u + 1)
c
12
(u
8
− u
7
+ ··· + 2u − 1)(u
52
− 2u
51
+ ··· − 4u + 1)
15
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
8
)(y
52
+ 53y
51
+ ··· − 8978y + 1)
c
2
, c
4
((y −1)
8
)(y
52
− 15y
51
+ ··· − 154y + 1)
c
3
, c
7
y
8
(y
52
− 51y
51
+ ··· − 6209536y + 65536)
c
5
, c
6
, c
10
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
52
− 50y
51
+ ··· − 2y + 1)
c
8
, c
12
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
52
− 30y
51
+ ··· − 2y + 1)
c
9
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
52
− 26y
51
+ ··· − 386280850y + 16200625)
c
11
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
52
− 14y
51
+ ··· − 22y + 1)
16
