11a
80
(K11a
80
)
1
Arc Sequences
6 1 9 10 2 3 4 11 7 8 5
Solving Sequence
1,6
2
3,9
4 7 5 11 8 10
c
1
c
2
c
3
c
6
c
5
c
11
c
8
c
10
c
4
, c
7
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
2
+ u + 1, b, a + u + 1i
I
u
2
= hu
70
2u
69
+ ··· 5u + 1, 4.34273 × 10
22
u
69
4.12385 × 10
22
u
68
+ ··· + 3.25909 × 10
22
b + 2.18874 × 10
21
,
5.62380 × 10
21
u
69
+ 9.03622 × 10
22
u
68
+ ··· + 3.25909 × 10
22
a + 5.10475 × 10
22
i
There are 2 irreducible components with 72 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
= hu
2
+ u + 1, b, a + u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u + 1
a
3
=
u
u + 1
a
9
=
u 1
0
a
4
=
u 1
u + 1
a
7
=
1
0
a
5
=
u
u + 1
a
11
=
0
u
a
8
=
u 1
u
a
10
=
u 1
0
a
10
=
u 1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 0.500000 + 0.866025I
b = 0
1.64493 + 2.02988I 3.00000 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 0.866025I
b = 0
1.64493 2.02988I 3.00000 + 3.46410I
3
II. I
u
2
=
hu
70
2u
69
+ · · · 5u + 1, 4.34 × 10
22
u
69
4.12 × 10
22
u
68
+ · · · + 3.26 × 10
22
b +
2.19× 10
21
, 5.62×10
21
u
69
+9.04 × 10
22
u
68
+· · · +3.26 × 10
22
a + 5.10 ×10
22
i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
9
=
0.172557u
69
2.77262u
68
+ ··· + 11.2767u 1.56631
1.33250u
69
+ 1.26534u
68
+ ··· + 2.10238u 0.0671581
a
4
=
6.16000u
69
+ 7.71429u
68
+ ··· 7.97246u + 0.0318073
1.19759u
69
+ 2.98930u
68
+ ··· 11.7118u + 1.80000
a
7
=
u
5
2u
3
u
u
5
+ u
3
+ u
a
5
=
u
u
3
+ u
a
11
=
u
4
+ u
2
+ 1
u
6
2u
4
u
2
a
8
=
0.206651u
69
2.80651u
68
+ ··· + 10.5728u 2.38325
1.56679u
69
+ 1.53366u
68
+ ··· + 1.74913u 0.0331313
a
10
=
0.139304u
69
2.74008u
68
+ ··· + 11.4127u 1.65004
1.49853u
69
+ 1.59703u
68
+ ··· + 1.35674u + 0.0984985
a
10
=
0.139304u
69
2.74008u
68
+ ··· + 11.4127u 1.65004
1.49853u
69
+ 1.59703u
68
+ ··· + 1.35674u + 0.0984985
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.844269 0.045265I
a = 0.006546 0.795172I
b = 0.301564 + 0.596291I
3.62841 0.20387I 2.37460 1.94241I
u = 0.844269 + 0.045265I
a = 0.006546 + 0.795172I
b = 0.301564 0.596291I
3.62841 + 0.20387I 2.37460 + 1.94241I
u = 0.830885 0.169788I
a = 0.30991 + 1.87849I
b = 0.11322 1.44963I
2.35547 3.73849I 0.65968 + 5.52505I
u = 0.830885 + 0.169788I
a = 0.30991 1.87849I
b = 0.11322 + 1.44963I
2.35547 + 3.73849I 0.65968 5.52505I
u = 0.751358 0.033892I
a = 0.08051 4.79081I
b = 0.21619 + 3.39297I
0.708262 0.627776I 0.33072 10.74054I
u = 0.751358 + 0.033892I
a = 0.08051 + 4.79081I
b = 0.21619 3.39297I
0.708262 + 0.627776I 0.33072 + 10.74054I
u = 0.653862 0.578775I
a = 0.286152 0.181607I
b = 0.465872 + 0.855287I
4.24693 6.03843I 5.38503 + 4.49312I
u = 0.653862 + 0.578775I
a = 0.286152 + 0.181607I
b = 0.465872 0.855287I
4.24693 + 6.03843I 5.38503 4.49312I
u = 0.584393 0.923619I
a = 0.709135 0.435128I
b = 0.297602 0.783927I
3.25750 + 10.82583I 3.30151 9.69217I
u = 0.584393 + 0.923619I
a = 0.709135 + 0.435128I
b = 0.297602 + 0.783927I
3.25750 10.82583I 3.30151 + 9.69217I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.529273 1.198985I
a = 1.50544 0.47472I
b = 0.46035 + 2.07023I
5.41659 + 8.73511I 2.82151 8.44694I
u = 0.529273 + 1.198985I
a = 1.50544 + 0.47472I
b = 0.46035 2.07023I
5.41659 8.73511I 2.82151 + 8.44694I
u = 0.482173 1.223346I
a = 0.679958 0.073536I
b = 0.608018 1.105103I
7.14118 + 4.96161I 5.12928 1.33560I
u = 0.482173 + 1.223346I
a = 0.679958 + 0.073536I
b = 0.608018 + 1.105103I
7.14118 4.96161I 5.12928 + 1.33560I
u = 0.479076 0.914334I
a = 0.768566 0.402643I
b = 0.85848 + 1.33028I
0.79732 + 5.73779I 0.05244 9.67981I
u = 0.479076 + 0.914334I
a = 0.768566 + 0.402643I
b = 0.85848 1.33028I
0.79732 5.73779I 0.05244 + 9.67981I
u = 0.466698 1.191898I
a = 3.51105 + 1.02184I
b = 0.64664 4.23335I
4.04232 + 5.06949I 5.49338 + 6.43728I
u = 0.466698 + 1.191898I
a = 3.51105 1.02184I
b = 0.64664 + 4.23335I
4.04232 5.06949I 5.49338 6.43728I
u = 0.463852 0.816066I
a = 1.37056 + 1.02268I
b = 0.72412 + 1.23081I
3.32585 + 3.64623I 8.36983 8.12955I
u = 0.463852 + 0.816066I
a = 1.37056 1.02268I
b = 0.72412 1.23081I
3.32585 3.64623I 8.36983 + 8.12955I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.442791 0.496185I
a = 0.902325 + 0.530370I
b = 0.251282 1.140765I
0.30075 1.79611I 2.68640 + 3.79960I
u = 0.442791 + 0.496185I
a = 0.902325 0.530370I
b = 0.251282 + 1.140765I
0.30075 + 1.79611I 2.68640 3.79960I
u = 0.441328 1.192849I
a = 3.08795 0.28039I
b = 1.33670 + 4.17822I
4.22443 + 3.62054I 3.20189 11.55914I
u = 0.441328 + 1.192849I
a = 3.08795 + 0.28039I
b = 1.33670 4.17822I
4.22443 3.62054I 3.20189 + 11.55914I
u = 0.441288 0.716327I
a = 0.639775 + 1.198918I
b = 1.59619 0.59670I
3.62277 + 0.21985I 9.90987 0.81181I
u = 0.441288 + 0.716327I
a = 0.639775 1.198918I
b = 1.59619 + 0.59670I
3.62277 0.21985I 9.90987 + 0.81181I
u = 0.429294 1.241125I
a = 0.799242 0.081036I
b = 0.293946 + 0.593013I
7.52357 + 4.28857I 5.79489 5.52372I
u = 0.429294 + 1.241125I
a = 0.799242 + 0.081036I
b = 0.293946 0.593013I
7.52357 4.28857I 5.79489 + 5.52372I
u = 0.359060 1.223278I
a = 1.377297 0.099661I
b = 0.85035 1.49465I
6.61992 + 0.21045I 5.51156 + 1.48396I
u = 0.359060 + 1.223278I
a = 1.377297 + 0.099661I
b = 0.85035 + 1.49465I
6.61992 0.21045I 5.51156 1.48396I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.067651 1.027565I
a = 0.565294 0.847170I
b = 0.259343 0.715782I
3.59009 0.96831I 7.57977 + 1.03848I
u = 0.067651 + 1.027565I
a = 0.565294 + 0.847170I
b = 0.259343 + 0.715782I
3.59009 + 0.96831I 7.57977 1.03848I
u = 0.060029 1.146879I
a = 0.150264 + 0.654267I
b = 0.765347 + 0.497369I
1.44288 5.44607I 3.33899 + 6.15948I
u = 0.060029 + 1.146879I
a = 0.150264 0.654267I
b = 0.765347 0.497369I
1.44288 + 5.44607I 3.33899 6.15948I
u = 0.155133 0.771122I
a = 1.104604 + 0.285498I
b = 1.240371 0.415161I
0.526876 1.304498I 2.10750 + 3.09155I
u = 0.155133 + 0.771122I
a = 1.104604 0.285498I
b = 1.240371 + 0.415161I
0.526876 + 1.304498I 2.10750 3.09155I
u = 0.316117 0.579912I
a = 0.636064 0.102591I
b = 0.727553 0.209003I
0.42343 1.35672I 2.64097 + 4.74847I
u = 0.316117 + 0.579912I
a = 0.636064 + 0.102591I
b = 0.727553 + 0.209003I
0.42343 + 1.35672I 2.64097 4.74847I
u = 0.331630
a = 1.11457
b = 0.968003
2.41420 4.13218
u = 0.373837 1.243619I
a = 1.74875 0.77618I
b = 0.12259 + 2.25915I
5.26246 + 7.34394I 2.64479 3.94320I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.373837 + 1.243619I
a = 1.74875 + 0.77618I
b = 0.12259 2.25915I
5.26246 7.34394I 2.64479 + 3.94320I
u = 0.377490 0.808869I
a = 2.28345 2.04905I
b = 1.129951 + 0.695483I
1.50422 1.66288I 31.1505 27.2219I
u = 0.377490 + 0.808869I
a = 2.28345 + 2.04905I
b = 1.129951 0.695483I
1.50422 + 1.66288I 31.1505 + 27.2219I
u = 0.384830 0.888904I
a = 0.184401 + 0.011741I
b = 0.063996 0.658066I
0.32453 1.95228I 0.23973 + 3.30692I
u = 0.384830 + 0.888904I
a = 0.184401 0.011741I
b = 0.063996 + 0.658066I
0.32453 + 1.95228I 0.23973 3.30692I
u = 0.407480 1.222895I
a = 1.74718 + 0.34064I
b = 0.75491 1.56649I
8.41392 + 1.04197I 5.27908 1.54322I
u = 0.407480 + 1.222895I
a = 1.74718 0.34064I
b = 0.75491 + 1.56649I
8.41392 1.04197I 5.27908 + 1.54322I
u = 0.420042 1.181250I
a = 1.38099 + 0.86647I
b = 0.51916 2.40287I
2.80139 0.62222I 0.50443 2.35061I
u = 0.420042 + 1.181250I
a = 1.38099 0.86647I
b = 0.51916 + 2.40287I
2.80139 + 0.62222I 0.50443 + 2.35061I
u = 0.457271 1.166860I
a = 0.234574 0.097224I
b = 0.057393 0.841558I
0.90108 4.15644I 3.93686 + 3.46926I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.457271 + 1.166860I
a = 0.234574 + 0.097224I
b = 0.057393 + 0.841558I
0.90108 + 4.15644I 3.93686 3.46926I
u = 0.482887 1.183358I
a = 1.97134 + 0.18896I
b = 1.99217 + 1.72238I
2.35077 7.93022I 1.78774 + 8.95162I
u = 0.482887 + 1.183358I
a = 1.97134 0.18896I
b = 1.99217 1.72238I
2.35077 + 7.93022I 1.78774 8.95162I
u = 0.496707 1.207981I
a = 1.65080 0.37500I
b = 0.94329 + 2.66947I
7.77710 10.05507I 3.96436 + 8.18714I
u = 0.496707 + 1.207981I
a = 1.65080 + 0.37500I
b = 0.94329 2.66947I
7.77710 + 10.05507I 3.96436 8.18714I
u = 0.522937 1.211776I
a = 2.22978 0.07209I
b = 1.65844 2.70794I
4.1996 16.5161I 1.15331 + 9.73148I
u = 0.522937 + 1.211776I
a = 2.22978 + 0.07209I
b = 1.65844 + 2.70794I
4.1996 + 16.5161I 1.15331 9.73148I
u = 0.589180 1.012726I
a = 0.033082 + 0.749229I
b = 0.748943 0.049650I
2.24373 1.06051I 2.99581 4.64685I
u = 0.589180 + 1.012726I
a = 0.033082 0.749229I
b = 0.748943 + 0.049650I
2.24373 + 1.06051I 2.99581 + 4.64685I
u = 0.629527 0.806981I
a = 0.421237 + 0.148688I
b = 0.181789 0.104194I
1.04825 2.45107I 7.49304 + 7.34261I
10
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.629527 + 0.806981I
a = 0.421237 0.148688I
b = 0.181789 + 0.104194I
1.04825 + 2.45107I 7.49304 7.34261I
u = 0.643531
a = 0.463000
b = 0.708746
2.26336 7.42672
u = 0.696611 0.477869I
a = 0.785069 0.037938I
b = 0.901014 + 0.348138I
3.77878 3.85148I 5.41405 + 7.02923I
u = 0.696611 + 0.477869I
a = 0.785069 + 0.037938I
b = 0.901014 0.348138I
3.77878 + 3.85148I 5.41405 7.02923I
u = 0.741533 0.089293I
a = 1.00914 + 2.57022I
b = 0.635281 1.095359I
0.80687 + 3.39629I 5.02552 6.06179I
u = 0.741533 + 0.089293I
a = 1.00914 2.57022I
b = 0.635281 + 1.095359I
0.80687 3.39629I 5.02552 + 6.06179I
u = 0.816520 0.092061I
a = 0.19806 + 2.40885I
b = 0.52273 1.59927I
4.47780 + 5.27299I 1.13386 5.09920I
u = 0.816520 + 0.092061I
a = 0.19806 2.40885I
b = 0.52273 + 1.59927I
4.47780 5.27299I 1.13386 + 5.09920I
u = 0.851538 0.139959I
a = 0.66064 2.77011I
b = 1.04800 + 1.95322I
0.99836 + 11.50697I 1.84093 6.73084I
u = 0.851538 + 0.139959I
a = 0.66064 + 2.77011I
b = 1.04800 1.95322I
0.99836 11.50697I 1.84093 + 6.73084I
11
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)(u
70
+ 2u
69
+ ··· + 5u + 1)
c
2
(u
2
+ u + 1)(u
70
+ 38u
69
+ ··· u + 1)
c
3
(u
2
+ u + 1)(u
70
+ 2u
69
+ ··· + 23u + 107)
c
4
(u
2
+ u + 1)(u
70
+ 31u
68
+ ··· + 13u + 1)
c
5
(u
2
u + 1)(u
70
+ 2u
69
+ ··· + 5u + 1)
c
6
(u
2
+ u + 1)(u
70
+ 2u
69
+ ··· 169u + 17)
c
7
(u
2
+ u + 1)(u
70
+ 2u
69
+ ··· + u + 1)
c
8
(u + 1)
2
(u
70
+ 3u
69
+ ··· 4u + 1)
c
9
u
2
(u
70
+ 11u
69
+ ··· 4u + 4)
c
10
(u 1)
2
(u
70
+ 3u
69
+ ··· 4u + 1)
c
11
(u
2
u + 1)(u
70
+ 2u
69
+ ··· 169u + 17)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
2
+ y + 1)(y
70
+ 38y
69
+ ··· y + 1)
c
2
(y
2
+ y + 1)(y
70
10y
69
+ ··· 93y + 1)
c
3
(y
2
+ y + 1)(y
70
+ 70y
69
+ ··· + 344439y + 11449)
c
4
(y
2
+ y + 1)(y
70
+ 62y
69
+ ··· + 127y + 1)
c
6
, c
11
(y
2
+ y + 1)(y
70
58y
69
+ ··· 16865y + 289)
c
7
(y
2
+ y + 1)(y
70
10y
69
+ ··· y + 1)
c
8
(y 1)
2
(y
70
41y
69
+ ··· + 16y + 1)
c
9
y
2
(y
70
15y
69
+ ··· 200y + 16)
c
10
(y 1)
2
(y
70
41y
69
+ ··· + 16y + 1)
13