11a
18
(K11a
18
)
1
Arc Sequences
4 1 6 2 9 3 11 5 8 7 10
Solving Sequence
5,9 2,6
4 1 3 7 8 10 11
c
5
c
4
c
1
c
2
c
6
c
8
c
9
c
11
c
3
, c
7
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= ha
6
a
5
a
4
+ 2a
3
a + 1, u 1, a
5
a
4
a
3
+ 2a
2
+ b a 1i
I
u
2
= hu
3
+ u
2
1, a, u
2
+ b u 1i
I
u
3
= hu
72
+ 8u
71
+ ··· + 12u + 1,
1.92088 × 10
24
u
71
1.57701 × 10
24
u
70
+ ··· + 1.93314 × 10
24
a + 1.04791 × 10
25
,
1.63603 × 10
25
u
71
+ 1.29840 × 10
26
u
70
+ ··· + 9.66569 × 10
23
b + 2.67997 × 10
25
i
There are 3 irreducible components with 81 representations.
1
The knot diagram image is adapter from “C. Livingston and A. H. Moore, KnotInfo: Table of Knot
Invariants, http://www.indiana.edu/ knotinfo”
1
I. I
u
1
= ha
6
a
5
a
4
+ 2a
3
a + 1, u 1, a
5
a
4
a
3
+ 2a
2
+ b a 1i
(i) Arc colorings
a
5
=
0
1
a
9
=
a
a
5
+ a
4
+ a
3
2a
2
+ a + 1
a
2
=
1
0
a
6
=
a
2
a
2
a
4
=
1
1
a
1
=
0
1
a
3
=
1
1
a
7
=
a
2
a
2
a
8
=
a
a
5
+ a
4
+ a
3
2a
2
+ 1
a
10
=
a
3
+ a
a
5
+ a
4
+ a
3
2a
2
+ 1
a
11
=
a
5
a
4
2a
3
+ a
2
+ a 1
a
2
a
11
=
a
5
a
4
2a
3
+ a
2
+ a 1
a
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = a
5
+ 3a
4
a
3
a
2
+ 4a 8
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.002193 0.295542I
b = 1.92017 0.02483I
3.53554 0.92430I 10.88169 + 1.11590I
u = 1.00000
a = 1.002193 + 0.295542I
b = 1.92017 + 0.02483I
3.53554 + 0.92430I 10.88169 1.11590I
u = 1.00000
a = 0.428243 0.664531I
b = 1.113439 + 0.398729I
0.245672 0.924305I 6.22669 0.83820I
u = 1.00000
a = 0.428243 + 0.664531I
b = 1.113439 0.398729I
0.245672 + 0.924305I 6.22669 + 0.83820I
u = 1.00000
a = 1.073950 0.558752I
b = 1.80674 0.17750I
1.64493 + 5.69302I 8.89162 7.09196I
u = 1.00000
a = 1.073950 + 0.558752I
b = 1.80674 + 0.17750I
1.64493 5.69302I 8.89162 + 7.09196I
3
II. I
u
2
= hu
3
+ u
2
1, a, u
2
+ b u 1i
(i) Arc colorings
a
5
=
0
u
a
9
=
0
u
2
+ u + 1
a
2
=
1
0
a
6
=
0
u
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
u
2
+ u 1
a
7
=
u
2
1
u
2
a
8
=
0
u
2
+ u + 1
a
10
=
0
u
2
+ u + 1
a
11
=
u
2
+ 1
u + 1
a
11
=
u
2
+ 1
u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
+ 4
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.877439 0.744862I
a = 0
b = 0.337641 + 0.562280I
4.66906 2.82812I 4.21508 + 1.30714I
u = 0.877439 + 0.744862I
a = 0
b = 0.337641 0.562280I
4.66906 + 2.82812I 4.21508 1.30714I
u = 0.754878
a = 0
b = 2.32472
0.531480 4.56984
5
III. I
u
3
=
hu
72
+8u
71
+· · ·+12u+1, 1.92×10
24
u
71
1.58×10
24
u
70
+· · ·+1.93×10
24
a+
1.05 × 10
25
, 1.64 × 10
25
u
71
+ 1.30 × 10
26
u
70
+ · · · + 9.67 × 10
23
b + 2.68 × 10
25
i
(i) Arc colorings
a
5
=
0
u
a
9
=
0.993656u
71
+ 0.815776u
70
+ ··· 90.0472u 5.42075
16.9262u
71
134.330u
70
+ ··· 305.927u 27.7266
a
2
=
1
0
a
6
=
9.96754u
71
+ 77.3907u
70
+ ··· + 144.678u + 13.4041
13.1525u
71
91.1616u
70
+ ··· 56.3419u 4.13750
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
u
2
+ 1
u
4
a
7
=
6.33059u
71
+ 57.4604u
70
+ ··· + 171.919u + 16.6797
12.8177u
71
+ 99.3697u
70
+ ··· + 219.573u + 19.9553
a
8
=
0.993656u
71
+ 0.815776u
70
+ ··· 90.0472u 5.42075
26.9614u
71
204.250u
70
+ ··· 390.535u 34.8601
a
10
=
25.3677u
71
+ 160.911u
70
+ ··· + 48.5429u + 7.43421
16.3920u
71
135.134u
70
+ ··· 343.210u 31.0010
a
11
=
8.51903u
71
+ 61.2402u
70
+ ··· + 60.8917u + 10.3248
23.7775u
71
177.686u
70
+ ··· 317.855u 28.0992
a
11
=
8.51903u
71
+ 61.2402u
70
+ ··· + 60.8917u + 10.3248
23.7775u
71
177.686u
70
+ ··· 317.855u 28.0992
(ii) Obstruction class = 1
(iii) Cusp Shapes =
61209740507681817230593115
966569388932529699723664
u
71
221532067967751911466074591
483284694466264849861832
u
70
+
···
749827309103295309695025749
966569388932529699723664
u
32331816260469680314082073
483284694466264849861832
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.165241 0.683428I
a = 0.995396 0.096870I
b = 2.79312 + 0.05869I
4.5389 16.0300I 2.95349 + 10 .09848I
u = 1.165241 + 0 .683428I
a = 0.995396 + 0.096870I
b = 2.79312 0.05869I
4.5389 + 16 .0300I 2.95349 10.09848I
u = 1.140802 0.662832I
a = 0.974472 + 0 .048829I
b = 2.83146 0. 00973I
1.48181 10.73104I 6.00773 + 7 .14911I
u = 1.140802 + 0 .662832I
a = 0.974472 0 .048829I
b = 2.83146 + 0. 00973I
1.48181 + 10.73104I 6.00773 7.14911I
u = 1.118621 0.680094I
a = 0.058388 + 0.926694I
b = 0.101204 + 0.742633I
5.82177 9.49602I 1.51728 + 6 .55107I
u = 1.118621 + 0 .680094I
a = 0.058388 0.926694I
b = 0.101204 0.742633I
5.82177 + 9.49602I 1.51728 6.55107I
u = 1.096879 0.686747I
a = 0.885404 0.052652I
b = 2.95792 + 0.03786I
6.23840 6.61734I 1.39327 + 5 .53156I
u = 1.096879 + 0 .686747I
a = 0.885404 + 0.052652I
b = 2.95792 0.03786I
6.23840 + 6.61734I 1.39327 5.53156I
u = 1.074128 0.521226I
a = 1.035124 0 .223044I
b = 2.67382 + 0. 25539I
3.02672 8.64627I 7.57936 + 9 .09630I
u = 1.074128 + 0 .521226I
a = 1.035124 + 0 .223044I
b = 2.67382 0. 25539I
3.02672 + 8.64627I 7.57936 9.09630I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.061924 0.763944I
a = 0.175884 + 0.773351I
b = 0.198967 + 0.607383I
6.72982 1.38866I 0.46791 1.55400I
u = 1.061924 + 0.763944I
a = 0.175884 0.773351I
b = 0.198967 0.607383I
6.72982 + 1.38866I 0.46791 + 1.55400I
u = 1.058858 0.675792I
a = 0.000499 0.826534I
b = 0.051069 0.657389I
3.04313 4.90017I 4.09415 + 2.80236I
u = 1.058858 + 0.675792I
a = 0.000499 + 0.826534I
b = 0.051069 + 0.657389I
3.04313 + 4.90017I 4.09415 2.80236I
u = 1.012759 0.471533I
a = 1.033467 + 0.377729I
b = 2.52125 0.35315I
3.84415 3.26268I 8.70986 + 4.01928I
u = 1.012759 + 0.471533I
a = 1.033467 0.377729I
b = 2.52125 + 0.35315I
3.84415 + 3.26268I 8.70986 4.01928I
u = 0.927726 0.549859I
a = 0.418674 0.780274I
b = 0.284406 0.612163I
1.23521 4.19775I 3.49766 + 6.68711I
u = 0.927726 + 0.549859I
a = 0.418674 + 0.780274I
b = 0.284406 + 0.612163I
1.23521 + 4.19775I 3.49766 6.68711I
u = 0.876096 0.806268I
a = 0.366768 0.276719I
b = 0.313702 0.177902I
4.16423 3.00649I 10.90678 + 5 .59644I
u = 0.876096 + 0.806268I
a = 0.366768 + 0.276719I
b = 0.313702 + 0.177902I
4.16423 + 3.00649I 10.90678 5.59644I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.845544 0.559163I
a = 0.643178 0.575010I
b = 2.23468 + 1.15341I
3.01523 2.24466I 3.71428 + 2 .62096I
u = 0.845544 + 0 .559163I
a = 0.643178 + 0 .575010I
b = 2.23468 1.15341I
3.01523 + 2 .24466I 3.71428 2.62096I
u = 0.825916 0.351549I
a = 1.032632 + 0.859491I
b = 1.87494 0.31361I
2.85223 0.05705I 5.94816 + 3 .77217I
u = 0.825916 + 0 .351549I
a = 1.032632 0.859491I
b = 1.87494 + 0 .31361I
2.85223 + 0 .05705I 5.94816 3.77217I
u = 0.779374 0.488319I
a = 0.799156 + 0 .715713I
b = 0.542087 + 0.539958I
1.78878 0.06007I 1.43743 + 0 .72520I
u = 0.779374 + 0 .488319I
a = 0.799156 0.715713I
b = 0.542087 0.539958I
1.78878 + 0 .06007I 1.43743 0.72520I
u = 0.764659 0.241751I
a = 1.15534 1 .08884I
b = 1.63745 + 0.02266I
1.00613 + 5 .17017I 1.04226 1.11128I
u = 0.764659 + 0 .241751I
a = 1.15534 + 1 .08884I
b = 1.63745 0.02266I
1.00613 5.17017I 1.04226 + 1 .11128I
u = 0.632470 0.934244I
a = 0.967416 + 0.260579I
b = 0.787219 + 0 .037927I
8.03882 4.81398I 0.98382 + 5.72253I
u = 0.632470 + 0 .934244I
a = 0.967416 0.260579I
b = 0.787219 0.037927I
8.03882 + 4 .81398I 0.98382 5.72253I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.575072 0.844979I
a = 1.032786 0.117739I
b = 0.782184 + 0.090080I
4.51528 0.75882I 2.00847 + 2 .21103I
u = 0.575072 + 0.844979I
a = 1.032786 + 0.117739I
b = 0.782184 0.090080I
4.51528 + 0 .75882I 2.00847 2.21103I
u = 0.529936 0.889148I
a = 0.178925 + 1.097738I
b = 0.213686 0.923317I
7.96250 + 0 .80203I 1.32154 0.61190I
u = 0.529936 + 0.889148I
a = 0.178925 1.097738I
b = 0.213686 + 0.923317I
7.96250 0.80203I 1.32154 + 0.61190I
u = 0.494612 0.902976I
a = 1.142258 + 0.196155I
b = 0.890679 0.085199I
7.71847 + 3 .66806I 1.23654 1.87350I
u = 0.494612 + 0.902976I
a = 1.142258 0.196155I
b = 0.890679 + 0.085199I
7.71847 3.66806I 1.23654 + 1.87350I
u = 0.443742 0.906716I
a = 0.203389 1.200623I
b = 0.161705 + 0.684630I
3.59461 + 4 .95936I 3.13446 3.17768I
u = 0.443742 + 0.906716I
a = 0.203389 + 1.200623I
b = 0.161705 0.684630I
3.59461 4.95936I 3.13446 + 3 .17768I
u = 0.442787 0.968153I
a = 0.267710 + 1.199319I
b = 0.289615 0.615259I
6.75363 + 10.01730I 0.17382 6.23023I
u = 0.442787 + 0.968153I
a = 0.267710 1.199319I
b = 0.289615 + 0.615259I
6.75363 10.01730I 0.17382 + 6 .23023I
10
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.120567 0.643601I
a = 0.19318 1.63070I
b = 0.322179 + 0.205154I
0.63477 + 4 .40212I 4.16070 6.41943I
u = 0.120567 + 0 .643601I
a = 0.19318 + 1.63070I
b = 0.322179 0.205154I
0.63477 4.40212I 4.16070 + 6 .41943I
u = 0.098616 0.127628I
a = 4.08292 + 0 .62979I
b = 0.574788 + 0.458498I
1.73034 + 0.74165I 3.53259 1.03757I
u = 0.098616 + 0 .127628I
a = 4.08292 0 .62979I
b = 0.574788 0.458498I
1.73034 0.74165I 3.53259 + 1.03757I
u = 0.151161 0.484816I
a = 0.64042 + 1 .67606I
b = 0.397431 0.013459I
1.57298 + 0 .11426I 7.05869 0.49031I
u = 0.151161 + 0.484816I
a = 0.64042 1 .67606I
b = 0.397431 + 0 .013459I
1.57298 0.11426I 7.05869 + 0 .49031I
u = 0.620181
a = 0.611212
b = 0.380543
0.987420 10.0942
u = 0.699638 0.493461I
a = 0.324192 + 1 .020600I
b = 0.584538 + 0 .096124I
0.602714 0.211764I 6.49349 0.29601I
u = 0.699638 + 0.493461I
a = 0.324192 1.020600I
b = 0.584538 0.096124I
0.602714 + 0 .211764I 6.49349 + 0.29601I
u = 0.718716 0.653541I
a = 0.248985 1.175059I
b = 0.562527 0.216488I
2.23267 4.45748I 2.81509 + 3 .29885I
11
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.718716 + 0.653541I
a = 0.248985 + 1.175059I
b = 0.562527 + 0 .216488I
2.23267 + 4.45748I 2.81509 3.29885I
u = 0.822483 0.540736I
a = 1.041117 + 0.173903I
b = 2.02631 0.98997I
2.80142 + 0.88223I 2.02170 2.91032I
u = 0.822483 + 0.540736I
a = 1.041117 0.173903I
b = 2.02631 + 0.98997I
2.80142 0.88223I 2.02170 + 2 .91032I
u = 0.877965 0.552533I
a = 0.113901 1.047556I
b = 0.725333 0 .185577I
2.61393 + 3.52046I 2.33746 3.83061I
u = 0.877965 + 0.552533I
a = 0.113901 + 1.047556I
b = 0.725333 + 0 .185577I
2.61393 3.52046I 2.33746 + 3 .83061I
u = 0.888684
a = 0.377440
b = 4.47694
0.330921 46.0796
u = 0.944057 0.552796I
a = 1.046228 0 .045506I
b = 2.12039 + 0. 87214I
1.37174 + 4 .55999I 7.56588 4.82929I
u = 0.944057 + 0.552796I
a = 1.046228 + 0 .045506I
b = 2.12039 0. 87214I
1.37174 4.55999I 7.56588 + 4 .82929I
u = 0.949003 0.622115I
a = 1.114422 + 0.041442I
b = 2.07183 0.82795I
1.52562 + 9.43889I 3.96108 7.86356I
u = 0.949003 + 0.622115I
a = 1.114422 0.041442I
b = 2.07183 + 0.82795I
1.52562 9.43889I 3.96108 + 7 .86356I
12
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.994888 0.199419I
a = 0.075815 + 0.622960I
b = 1.039649 0.344746I
0.802882 + 0.774259I 7.90864 + 1.29464I
u = 0.994888 + 0. 199419I
a = 0.075815 0.622960I
b = 1.039649 + 0.344746I
0.802882 0.774259I 7.90864 1.29464I
u = 1.110362 0.388556I
a = 0.895860 + 0.157967I
b = 2.29646 + 0.70345I
4.30882 + 3.51764I 10.31752 4.45263I
u = 1.110362 + 0. 388556I
a = 0.895860 0.157967I
b = 2.29646 0.70345I
4.30882 3.51764I 10.31752 + 4.45263I
u = 1.175902 0.292913I
a = 0.814424 0.265263I
b = 2.30809 0.55748I
4.53906 1.05261I 10.58932 + 3.82745I
u = 1.175902 + 0. 292913I
a = 0.814424 + 0.265263I
b = 2.30809 + 0.55748I
4.53906 + 1.05261I 10.58932 3.82745I
u = 1.266648 0.028494I
a = 0.537635 0.589841I
b = 1.87590 0.13818I
1.26979 1.25057I 0.108352 + 1.105957I
u = 1.266648 + 0. 028494I
a = 0.537635 + 0.589841I
b = 1.87590 + 0. 13818I
1.26979 + 1.25057I 0.108352 1.105957I
u = 1.274689 0.104537I
a = 0.670913 0.483797I
b = 2.09556 0.29667I
2.48615 2.12650I 6.43169 + 3 .64338I
u = 1.274689 + 0. 104537I
a = 0.670913 + 0.483797I
b = 2.09556 + 0.29667I
2.48615 + 2.12650I 6.43169 3.64338I
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.346383 0.089859I
a = 0.710126 + 0 .562358I
b = 1.99927 + 0.36411I
0.24731 6.78521I 3.13164 + 6 .33016I
u = 1.346383 + 0.089859I
a = 0.710126 0.562358I
b = 1.99927 0.36411I
0.24731 + 6 .78521I 3.13164 6.33016I
13
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
6
(u
3
+ u
2
1)(u
72
+ 8u
71
+ ··· + 12u + 1)
c
2
(u + 1)
6
(u
3
+ u
2
+ 2u + 1)(u
72
+ 32u
71
+ ··· 8u + 1)
c
3
, c
6
u
6
(u
3
u
2
+ 2u 1)(u
72
+ 2u
71
+ ··· 128u 64)
c
4
(u + 1)
6
(u
3
+ u
2
1)(u
72
+ 8u
71
+ ··· + 12u + 1)
c
5
u
3
(u
6
+ u
5
+ ··· + u + 1)(u
72
+ 2u
71
+ ··· + 20u + 8)
c
7
(u + 1)
3
(u
6
u
5
+ ··· u + 1)(u
72
+ 5u
71
+ ··· 12u 1)
c
8
u
3
(u
6
u
5
+ ··· u + 1)(u
72
+ 2u
71
+ ··· + 20u + 8)
c
9
u
3
(u
6
+ 3u
5
+ ··· + u + 1)(u
72
+ 24u
71
+ ··· + 1872u + 64)
c
10
(u + 1)
3
(u
6
+ u
5
+ ··· + u + 1)(u
72
+ 5u
71
+ ··· 12u 1)
c
11
(u + 1)
3
(u
6
3u
5
+ 5u
4
4u
3
+ 2u
2
u + 1)
(u
72
+ 39u
71
+ ··· + 52u + 1)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
6
(y
3
y
2
+ 2y 1)(y
72
32y
71
+ ··· + 8y + 1)
c
2
(y 1)
6
(y
3
+ 3y
2
+ 2y 1)(y
72
+ 24y
71
+ ··· + 2568y + 1)
c
3
, c
6
y
6
(y
3
+ 3y
2
+ 2y 1)(y
72
+ 42y
71
+ ··· + 73728y + 4096)
c
5
, c
8
y
3
(y
6
3y
5
+ ··· y + 1)(y
72
24y
71
+ ··· 1872y + 64)
c
7
(y 1)
3
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
(y
72
39y
71
+ ··· 52y + 1)
c
9
y
3
(y
6
+ y
5
+ ··· + 3y + 1)(y
72
+ 44y
71
+ ··· 601344y + 4096)
c
10
(y 1)
3
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
(y
72
39y
71
+ ··· 52y + 1)
c
11
(y 1)
3
(y
6
+ y
5
+ ··· + 3y + 1)(y
72
7y
71
+ ··· 2176y + 1)
15