11a
82
(K11a
82
)
A knot diagram
1
Linearized knot diagam
5 1 9 11 2 3 10 4 7 8 6
Solving Sequence
2,5
6 1 3 7 11
4,9
8 10
c
5
c
1
c
2
c
6
c
11
c
4
c
8
c
10
c
3
, c
7
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h2u
52
+ 4u
51
+ ··· + b + 2, 2u
52
+ 2u
51
+ ··· + a + 1, u
53
+ 2u
52
+ ··· + u + 1i
I
u
2
= hu
5
− u
3
+ b + u, u
4
− u
2
+ a + u, u
6
− u
5
− u
4
+ 2u
3
− u + 1i
* 2 irreducible components of dim
C
= 0, with total 59 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
= h2u
52
+4u
51
+· · ·+b+2, 2u
52
+2u
51
+· · ·+a+1, u
53
+2u
52
+· · ·+u+1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
1
=
u
u
a
3
=
−u
3
−u
3
+ u
a
7
=
u
8
− u
6
+ u
4
+ 1
u
8
− 2u
6
+ 2u
4
a
11
=
u
3
u
5
− u
3
+ u
a
4
=
u
8
− u
6
+ u
4
+ 1
u
10
− 2u
8
+ 3u
6
− 2u
4
+ u
2
a
9
=
−2u
52
− 2u
51
+ ··· − 3u − 1
−2u
52
− 4u
51
+ ··· − 2u − 2
a
8
=
u
50
+ u
49
+ ··· + 2u
2
− 3u
−u
31
+ 7u
29
+ ··· + 2u
2
− u
a
10
=
−u
52
− u
51
+ ··· + 3u
2
− 3u
−u
52
− 2u
51
+ ··· − u − 1
a
10
=
−u
52
− u
51
+ ··· + 3u
2
− 3u
−u
52
− 2u
51
+ ··· − u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
52
− 6u
51
+ ··· + 9u
2
− 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
53
+ 2u
52
+ ··· + u + 1
c
2
u
53
+ 24u
52
+ ··· + 5u + 1
c
3
, c
8
u
53
− u
52
+ ··· − 64u − 64
c
4
, c
6
u
53
− 2u
52
+ ··· − 144u + 36
c
7
, c
9
, c
10
u
53
+ 7u
52
+ ··· − 6u − 1
c
11
u
53
+ 6u
52
+ ··· − 5u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
53
− 24y
52
+ ··· + 5y −1
c
2
y
53
+ 12y
52
+ ··· − 27y −1
c
3
, c
8
y
53
+ 39y
52
+ ··· + 8192y −4096
c
4
, c
6
y
53
− 48y
52
+ ··· + 10728y −1296
c
7
, c
9
, c
10
y
53
− 55y
52
+ ··· + 14y −1
c
11
y
53
+ 54y
51
+ ··· + 45y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.974772 + 0.241666I
a = −0.636930 − 0.032053I
b = −0.591584 + 0.107088I
−1.72966 + 0.52144I −5.77451 − 0.49909I
u = −0.974772 − 0.241666I
a = −0.636930 + 0.032053I
b = −0.591584 − 0.107088I
−1.72966 − 0.52144I −5.77451 + 0.49909I
u = 0.799486 + 0.620081I
a = 0.938129 − 0.652027I
b = 0.075820 + 0.997850I
8.47576 − 2.42942I 9.05009 + 3.27749I
u = 0.799486 − 0.620081I
a = 0.938129 + 0.652027I
b = 0.075820 − 0.997850I
8.47576 + 2.42942I 9.05009 − 3.27749I
u = −0.547871 + 0.781328I
a = −0.308221 − 0.327517I
b = −0.62380 − 2.01056I
13.9720 + 5.2869I 9.56069 − 3.45269I
u = −0.547871 − 0.781328I
a = −0.308221 + 0.327517I
b = −0.62380 + 2.01056I
13.9720 − 5.2869I 9.56069 + 3.45269I
u = 0.972428 + 0.431066I
a = −1.87141 + 0.12673I
b = −0.202339 − 1.299480I
0.32639 − 1.89843I 4.06618 + 4.82062I
u = 0.972428 − 0.431066I
a = −1.87141 − 0.12673I
b = −0.202339 + 1.299480I
0.32639 + 1.89843I 4.06618 − 4.82062I
u = −1.06609
a = 1.05441
b = 1.12898
3.31477 2.11820
u = 1.068650 + 0.060095I
a = −0.00519 − 3.08536I
b = 0.33759 − 1.38633I
1.15819 + 2.52423I 1.16527 − 3.38233I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.068650 − 0.060095I
a = −0.00519 + 3.08536I
b = 0.33759 + 1.38633I
1.15819 − 2.52423I 1.16527 + 3.38233I
u = −0.436206 + 0.813289I
a = −0.314398 + 0.221559I
b = 0.30278 + 2.64405I
13.3377 − 8.5290I 8.89957 + 3.73071I
u = −0.436206 − 0.813289I
a = −0.314398 − 0.221559I
b = 0.30278 − 2.64405I
13.3377 + 8.5290I 8.89957 − 3.73071I
u = 0.480967 + 0.776724I
a = 0.649586 − 0.094929I
b = −0.658204 + 0.307267I
8.70701 + 1.51183I 8.31702 − 0.27451I
u = 0.480967 − 0.776724I
a = 0.649586 + 0.094929I
b = −0.658204 − 0.307267I
8.70701 − 1.51183I 8.31702 + 0.27451I
u = −0.500888 + 0.763334I
a = 0.092905 + 0.528334I
b = 0.80992 + 2.38721I
6.63504 + 1.32263I 7.88405 − 2.69846I
u = −0.500888 − 0.763334I
a = 0.092905 − 0.528334I
b = 0.80992 − 2.38721I
6.63504 − 1.32263I 7.88405 + 2.69846I
u = −0.458619 + 0.779636I
a = 0.215540 − 0.450057I
b = −0.54372 − 2.68876I
6.39750 − 4.28616I 7.23871 + 3.29000I
u = −0.458619 − 0.779636I
a = 0.215540 + 0.450057I
b = −0.54372 + 2.68876I
6.39750 + 4.28616I 7.23871 − 3.29000I
u = −1.038310 + 0.379912I
a = 0.000675 + 1.082270I
b = 0.397647 + 0.928057I
−2.65480 + 1.42970I −5.16065 − 0.45006I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.038310 − 0.379912I
a = 0.000675 − 1.082270I
b = 0.397647 − 0.928057I
−2.65480 − 1.42970I −5.16065 + 0.45006I
u = −1.015080 + 0.482408I
a = −0.90550 − 1.83401I
b = −1.37700 − 0.93578I
0.80550 + 3.99450I 2.19146 − 3.84882I
u = −1.015080 − 0.482408I
a = −0.90550 + 1.83401I
b = −1.37700 + 0.93578I
0.80550 − 3.99450I 2.19146 + 3.84882I
u = 1.139850 + 0.088394I
a = 0.10640 + 3.03961I
b = −0.47640 + 1.78432I
7.92687 + 6.35005I 3.13792 − 3.33110I
u = 1.139850 − 0.088394I
a = 0.10640 − 3.03961I
b = −0.47640 − 1.78432I
7.92687 − 6.35005I 3.13792 + 3.33110I
u = 1.063420 + 0.475788I
a = 1.062240 + 0.635757I
b = −0.372939 + 1.042010I
−1.98458 − 5.32256I 0. + 8.22615I
u = 1.063420 − 0.475788I
a = 1.062240 − 0.635757I
b = −0.372939 − 1.042010I
−1.98458 + 5.32256I 0. − 8.22615I
u = −1.125350 + 0.320255I
a = 0.763783 − 0.874919I
b = 0.417595 − 1.180340I
1.92863 − 0.10640I 0
u = −1.125350 − 0.320255I
a = 0.763783 + 0.874919I
b = 0.417595 + 1.180340I
1.92863 + 0.10640I 0
u = 0.447001 + 0.697020I
a = −0.361064 + 0.077633I
b = 0.313493 − 0.141443I
2.42469 + 1.08462I 0.919907 − 0.841939I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.447001 − 0.697020I
a = −0.361064 − 0.077633I
b = 0.313493 + 0.141443I
2.42469 − 1.08462I 0.919907 + 0.841939I
u = 0.721542 + 0.368379I
a = −0.85402 + 1.37597I
b = −0.313018 − 0.601468I
1.12660 − 1.52721I 6.78225 + 4.43587I
u = 0.721542 − 0.368379I
a = −0.85402 − 1.37597I
b = −0.313018 + 0.601468I
1.12660 + 1.52721I 6.78225 − 4.43587I
u = 1.071560 + 0.575084I
a = −0.297096 + 0.333184I
b = −0.378472 − 0.297843I
0.58719 − 5.99085I 0
u = 1.071560 − 0.575084I
a = −0.297096 − 0.333184I
b = −0.378472 + 0.297843I
0.58719 + 5.99085I 0
u = −1.041330 + 0.643287I
a = 2.79132 − 0.19865I
b = 0.96495 − 1.60940I
12.49880 + 0.07857I 0
u = −1.041330 − 0.643287I
a = 2.79132 + 0.19865I
b = 0.96495 + 1.60940I
12.49880 − 0.07857I 0
u = −1.062540 + 0.617182I
a = −3.41729 + 0.35106I
b = −1.39517 + 2.15411I
4.96232 + 3.90423I 0
u = −1.062540 − 0.617182I
a = −3.41729 − 0.35106I
b = −1.39517 − 2.15411I
4.96232 − 3.90423I 0
u = 1.124890 + 0.501254I
a = −0.505347 − 0.993245I
b = 0.856758 − 0.767375I
3.12807 − 7.84635I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.124890 − 0.501254I
a = −0.505347 + 0.993245I
b = 0.856758 + 0.767375I
3.12807 + 7.84635I 0
u = 1.076360 + 0.618201I
a = 0.492452 − 0.622900I
b = 0.734818 + 0.565386I
6.93298 − 6.77646I 0
u = 1.076360 − 0.618201I
a = 0.492452 + 0.622900I
b = 0.734818 − 0.565386I
6.93298 + 6.77646I 0
u = −1.087340 + 0.612889I
a = 3.49978 − 1.02104I
b = 1.10282 − 2.74374I
4.52609 + 9.53770I 0
u = −1.087340 − 0.612889I
a = 3.49978 + 1.02104I
b = 1.10282 + 2.74374I
4.52609 − 9.53770I 0
u = −1.107260 + 0.619290I
a = −3.12658 + 1.31694I
b = −0.67475 + 2.83005I
11.3318 + 13.8883I 0
u = −1.107260 − 0.619290I
a = −3.12658 − 1.31694I
b = −0.67475 − 2.83005I
11.3318 − 13.8883I 0
u = 0.193784 + 0.702298I
a = 0.687326 + 0.512147I
b = −0.466485 − 0.753669I
5.78851 + 3.34050I 7.33924 − 3.06497I
u = 0.193784 − 0.702298I
a = 0.687326 − 0.512147I
b = −0.466485 + 0.753669I
5.78851 − 3.34050I 7.33924 + 3.06497I
u = −0.435442 + 0.365869I
a = 1.37200 + 0.49921I
b = 1.123540 − 0.275872I
2.39706 − 0.15379I 3.69002 − 1.57866I
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.435442 − 0.365869I
a = 1.37200 − 0.49921I
b = 1.123540 + 0.275872I
2.39706 + 0.15379I 3.69002 + 1.57866I
u = 0.204119 + 0.487719I
a = −0.596303 − 0.904911I
b = 0.071665 + 0.625933I
0.239475 + 1.389700I 2.08792 − 5.18976I
u = 0.204119 − 0.487719I
a = −0.596303 + 0.904911I
b = 0.071665 − 0.625933I
0.239475 − 1.389700I 2.08792 + 5.18976I
10
II. I
u
2
= hu
5
− u
3
+ b + u, u
4
− u
2
+ a + u, u
6
− u
5
− u
4
+ 2u
3
− u + 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
1
=
u
u
a
3
=
−u
3
−u
3
+ u
a
7
=
−u
3
−u
5
+ u
3
− u
a
11
=
u
3
u
5
− u
3
+ u
a
4
=
−u
3
−u
3
+ u
a
9
=
−u
4
+ u
2
− u
−u
5
+ u
3
− u
a
8
=
−u
4
+ u
2
− u
−u
5
+ u
3
− u
a
10
=
−u
4
+ u
3
+ u
2
− u
0
a
10
=
−u
4
+ u
3
+ u
2
− u
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
4
− 5u
2
+ 5u + 7
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
u
6
+ u
5
− u
4
− 2u
3
+ u + 1
c
2
, c
11
u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1
c
3
, c
8
u
6
c
5
u
6
− u
5
− u
4
+ 2u
3
− u + 1
c
7
(u + 1)
6
c
9
, c
10
(u − 1)
6
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
6
y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1
c
2
, c
11
y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1
c
3
, c
8
y
6
c
7
, c
9
, c
10
(y −1)
6
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.002190 + 0.295542I
a = 1.42918 + 0.19856I
b = 0.428243 − 0.664531I
−0.245672 + 0.924305I −0.635956 + 0.093695I
u = −1.002190 − 0.295542I
a = 1.42918 − 0.19856I
b = 0.428243 + 0.664531I
−0.245672 − 0.924305I −0.635956 − 0.093695I
u = 0.428243 + 0.664531I
a = −0.429179 + 0.198557I
b = −1.002190 − 0.295542I
3.53554 + 0.92430I 9.40317 − 0.69886I
u = 0.428243 − 0.664531I
a = −0.429179 − 0.198557I
b = −1.002190 + 0.295542I
3.53554 − 0.92430I 9.40317 + 0.69886I
u = 1.073950 + 0.558752I
a = 0.50000 − 1.37764I
b = 1.073950 − 0.558752I
1.64493 − 5.69302I 5.23279 + 4.86918I
u = 1.073950 − 0.558752I
a = 0.50000 + 1.37764I
b = 1.073950 + 0.558752I
1.64493 + 5.69302I 5.23279 − 4.86918I
14
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)(u
53
+ 2u
52
+ ··· + u + 1)
c
2
(u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)(u
53
+ 24u
52
+ ··· + 5u + 1)
c
3
, c
8
u
6
(u
53
− u
52
+ ··· − 64u − 64)
c
4
, c
6
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)(u
53
− 2u
52
+ ··· − 144u + 36)
c
5
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)(u
53
+ 2u
52
+ ··· + u + 1)
c
7
((u + 1)
6
)(u
53
+ 7u
52
+ ··· − 6u − 1)
c
9
, c
10
((u − 1)
6
)(u
53
+ 7u
52
+ ··· − 6u − 1)
c
11
(u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)(u
53
+ 6u
52
+ ··· − 5u − 1)
15
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)(y
53
− 24y
52
+ ··· + 5y −1)
c
2
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)(y
53
+ 12y
52
+ ··· − 27y −1)
c
3
, c
8
y
6
(y
53
+ 39y
52
+ ··· + 8192y −4096)
c
4
, c
6
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
· (y
53
− 48y
52
+ ··· + 10728y −1296)
c
7
, c
9
, c
10
((y −1)
6
)(y
53
− 55y
52
+ ··· + 14y −1)
c
11
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)(y
53
+ 54y
51
+ ··· + 45y −1)
16
