11a
109
(K11a
109
)
1
Arc Sequences
5 1 10 9 2 11 3 4 8 6 7
Solving Sequence
2,5
6
1,9
4 8 10 11 3 7
c
5
c
1
c
4
c
8
c
9
c
10
c
3
c
7
c
2
, c
6
, c
11
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hu 1, a, b 1i
I
u
2
= ha
4
2a
2
+ 2, u + 1, a
3
+ a
2
+ b + 3a 1i
I
u
3
= hu
18
6u
16
+ ··· u + 1, u
16
4u
14
+ 6u
12
6u
10
+ 6u
8
4u
6
+ 2u
4
+ b,
u
14
+ 5u
12
10u
10
+ 11u
8
8u
6
+ 4u
4
2u
2
+ a + 1i
I
u
4
= hu
47
2u
46
+ ··· 5u 5, 2.13332 × 10
25
u
46
+ 1.32520 × 10
25
u
45
+ ··· + 7.07391 × 10
25
b 3.16964 × 10
25
,
3.87852 × 10
25
u
46
+ 1.30869 × 10
26
u
45
+ ··· + 3.53695 × 10
26
a + 6.44082 × 10
26
i
There are 4 irreducible components with 70 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 1, a, b 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
1
a
6
=
1
1
a
1
=
1
1
a
9
=
0
1
a
4
=
0
1
a
8
=
0
1
a
10
=
0
1
a
11
=
1
0
a
3
=
0
1
a
7
=
0
1
a
7
=
0
1
(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 = 1.00000
a = 0
b = 1.00000
0 0
3
II. I
u
2
= ha
4
2a
2
+ 2, u + 1, a
3
+ a
2
+ b + 3a 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
1
a
6
=
1
1
a
1
=
1
1
a
9
=
a
a
3
a
2
3a + 1
a
4
=
a
2
a
3
+ a
2
a + 1
a
8
=
a
3
+ a
2a
3
2a 1
a
10
=
a
3
a
a
3
+ a + 1
a
11
=
a
3
a + 1
a
3
+ a
a
3
=
0
1
a
7
=
a
3
+ a
a
3
a 1
a
7
=
a
3
+ a
a
3
a 1
(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 = 1.00000
a = 1.098684 0.455090I
b = 2.65246 1.18850I
2.46740 3.66386I 4.00000 + 4.00000I
u = 1.00000
a = 1.098684 + 0.455090I
b = 2.65246 + 1.18850I
2.46740 + 3.66386I 4.00000 4.00000I
u = 1.00000
a = 1.098684 0.455090I
b = 2.65246 + 0.81150I
2.46740 + 3.66386I 4.00000 4.00000I
u = 1.00000
a = 1.098684 + 0.455090I
b = 2.65246 0.81150I
2.46740 3.66386I 4.00000 + 4.00000I
5
III. I
u
3
= hu
18
6u
16
+ · · · u + 1, u
16
4u
14
+ 6u
12
6u
10
+ 6u
8
4u
6
+
2u
4
+ b, u
14
+ 5u
12
+ · · · + a + 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
1
u
2
a
9
=
u
14
5u
12
+ 10u
10
11u
8
+ 8u
6
4u
4
+ 2u
2
1
u
16
+ 4u
14
6u
12
+ 6u
10
6u
8
+ 4u
6
2u
4
a
4
=
u
8
3u
6
+ 3u
4
2u
2
+ 1
u
8
+ 2u
6
a
8
=
u
5
2u
3
+ u
u
7
+ u
5
+ u
a
10
=
u
2
1
u
2
a
11
=
1
0
a
3
=
u
2
+ 1
u
4
a
7
=
0
u
a
7
=
0
u
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.355855 0.124178I
a = 0.748065 + 0.480615I
b = 2.94473 + 0.48998I
5.69302I 5.51057I
u = 1.355855 + 0.124178I
a = 0.748065 0.480615I
b = 2.94473 0.48998I
5.69302I 5.51057I
u = 1.212368 0.197153I
a = 0.843964 + 0.106529I
b = 3.60849 + 0.19177I
1.89061 0.92430I 3.71672 + 0.79423I
u = 1.212368 + 0.197153I
a = 0.843964 0.106529I
b = 3.60849 0.19177I
1.89061 + 0.92430I 3.71672 0.79423I
u = 0.783020 0.545597I
a = 0.029911 + 0.827835I
b = 0.959369 0.349265I
1.89061 0.92430I 3.71672 + 0.79423I
u = 0.783020 + 0.545597I
a = 0.029911 0.827835I
b = 0.959369 + 0.349265I
1.89061 + 0.92430I 3.71672 0.79423I
u = 0.546610 0.614343I
a = 0.949915 0.148109I
b = 0.327289 0.649379I
1.89061 + 0.92430I 3.71672 0.79423I
u = 0.546610 + 0.614343I
a = 0.949915 + 0.148109I
b = 0.327289 + 0.649379I
1.89061 0.92430I 3.71672 + 0.79423I
u = 0.223621 0.567170I
a = 1.05393 1.35147I
b = 0.259714 0.407223I
1.89061 + 0.92430I 3.71672 0.79423I
u = 0.223621 + 0.567170I
a = 1.05393 + 1.35147I
b = 0.259714 + 0.407223I
1.89061 0.92430I 3.71672 + 0.79423I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.479853 0.699604I
a = 0.17289 + 1.41649I
b = 0.613639 + 0.892486I
5.69302I 5.51057I
u = 0.479853 + 0.699604I
a = 0.17289 1.41649I
b = 0.613639 0.892486I
5.69302I 5.51057I
u = 0.876002 0.575426I
a = 1.149120 + 0.116988I
b = 2.41047 1.23528I
5.69302I 5.51057I
u = 0.876002 + 0.575426I
a = 1.149120 0.116988I
b = 2.41047 + 1.23528I
5.69302I 5.51057I
u = 0.988747 0.370017I
a = 0.790968 0.594908I
b = 1.34691 + 1.12718I
1.89061 0.92430I 3.71672 + 0.79423I
u = 0.988747 + 0.370017I
a = 0.790968 + 0.594908I
b = 1.34691 1.12718I
1.89061 + 0.92430I 3.71672 0.79423I
u = 1.329630 0.068746I
a = 0.346990 0.481846I
b = 0.513947 + 0.688083I
1.89061 0.92430I 3.71672 + 0.79423I
u = 1.329630 + 0.068746I
a = 0.346990 + 0.481846I
b = 0.513947 0.688083I
1.89061 + 0.92430I 3.71672 0.79423I
8
IV. I
u
4
=
hu
47
2u
46
+· · ·5u5, 2.13×10
25
u
46
+1.33×10
25
u
45
+· · ·+7.07×10
25
b
3.17× 10
25
, 3.88×10
25
u
46
+1.31 ×10
26
u
45
+· · · +3.54 ×10
26
a + 6.44× 10
26
i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
1
u
2
a
9
=
0.109657u
46
0.370004u
45
+ ··· + 2.29818u 1.82101
0.301576u
46
0.187337u
45
+ ··· + 0.698107u + 0.448075
a
4
=
0.174139u
46
0.0273104u
45
+ ··· + 0.576171u 0.797149
0.0647978u
46
0.174501u
45
+ ··· + 2.69114u + 0.507856
a
8
=
0.396967u
46
0.812539u
45
+ ··· + 9.00630u + 1.24000
0.177926u
46
0.0720309u
45
+ ··· + 0.509939u 0.373748
a
10
=
0.139947u
46
0.0508260u
45
+ ··· + 2.41001u 1.69291
0.637159u
46
0.496634u
45
+ ··· + 0.404491u + 0.477987
a
11
=
0.214699u
46
0.724996u
45
+ ··· + 7.13089u + 0.541906
0.282512u
46
+ 0.177536u
45
+ ··· 4.31638u 1.75683
a
3
=
u
2
+ 1
u
4
a
7
=
0.351366u
46
0.985244u
45
+ ··· + 10.2865u + 2.55956
0.295597u
46
+ 0.0459638u
45
+ ··· 1.61540u 1.07350
a
7
=
0.351366u
46
0.985244u
45
+ ··· + 10.2865u + 2.55956
0.295597u
46
+ 0.0459638u
45
+ ··· 1.61540u 1.07350
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
9
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.148181 0.688109I
a = 0.080150 + 0.655942I
b = 0.440411 1.028840I
6.36842 9.89029I 3.34606 + 5.56719I
u = 1.148181 + 0.688109I
a = 0.080150 0.655942I
b = 0.440411 + 1.028840I
6.36842 + 9.89029I 3.34606 5.56719I
u = 1.088143 0.292285I
a = 1.034103 0.056568I
b = 3.15639 + 0.31445I
5.47077 3.68113I 9.62046 + 4.56447I
u = 1.088143 + 0.292285I
a = 1.034103 + 0.056568I
b = 3.15639 0.31445I
5.47077 + 3.68113I 9.62046 4.56447I
u = 1.088097 0.178123I
a = 0.922010 0.605617I
b = 2.65568 0.48273I
4.58829 + 3.84650I 8.76666 3.56046I
u = 1.088097 + 0.178123I
a = 0.922010 + 0.605617I
b = 2.65568 + 0.48273I
4.58829 3.84650I 8.76666 + 3.56046I
u = 1.071626 0.732285I
a = 0.717650 0.180252I
b = 1.324654 0.248787I
7.88170 7.13370I 4.69775 + 5.86187I
u = 1.071626 + 0.732285I
a = 0.717650 + 0.180252I
b = 1.324654 + 0.248787I
7.88170 + 7.13370I 4.69775 5.86187I
u = 1.025602 0.594700I
a = 0.048941 0.698553I
b = 0.759700 + 0.891691I
0.49152 5.74739I 0.29552 + 5.57964I
u = 1.025602 + 0.594700I
a = 0.048941 + 0.698553I
b = 0.759700 0.891691I
0.49152 + 5.74739I 0.29552 5.57964I
10
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.929993 0.511244I
a = 1.151507 0.206529I
b = 2.84688 0.90982I
1.23516 6.21145I 1.74487 + 7.02826I
u = 0.929993 + 0.511244I
a = 1.151507 + 0.206529I
b = 2.84688 + 0.90982I
1.23516 + 6.21145I 1.74487 7.02826I
u = 0.896677 0.582973I
a = 0.774367 + 0.215147I
b = 1.054559 + 0.751665I
1.50796 3.62695I 2.13655 + 6.26888I
u = 0.896677 + 0.582973I
a = 0.774367 0.215147I
b = 1.054559 0.751665I
1.50796 + 3.62695I 2.13655 6.26888I
u = 0.805471 0.476797I
a = 0.821430 + 1.062937I
b = 2.11281 + 0.84025I
0.77532 + 2.18171I 0.087129 + 0.127753I
u = 0.805471 + 0.476797I
a = 0.821430 1.062937I
b = 2.11281 0.84025I
0.77532 2.18171I 0.087129 0.127753I
u = 0.596301 0.917514I
a = 0.278057 0.803541I
b = 0.435163 + 0.048454I
9.33252 + 1.08584I 6.61100 0.78668I
u = 0.596301 + 0.917514I
a = 0.278057 + 0.803541I
b = 0.435163 0.048454I
9.33252 1.08584I 6.61100 + 0.78668I
u = 0.466771 0.950234I
a = 0.836171 0.017324I
b = 0.222547 + 0.233097I
8.45114 + 3.90837I 6.00297 1.23296I
u = 0.466771 + 0.950234I
a = 0.836171 + 0.017324I
b = 0.222547 0.233097I
8.45114 3.90837I 6.00297 + 1.23296I
11
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.312493 0.174261I
a = 0.47650 + 1.72046I
b = 0.668065 + 0.440198I
1.43130 0.33753I 6.95713 + 1.01845I
u = 0.312493 + 0.174261I
a = 0.47650 1.72046I
b = 0.668065 0.440198I
1.43130 + 0.33753I 6.95713 1.01845I
u = 0.390678 0.441527I
a = 1.73021 + 0.74761I
b = 0.340284 0.647603I
0.11745 + 4.26570I 1.86221 7.53589I
u = 0.390678 + 0.441527I
a = 1.73021 0.74761I
b = 0.340284 + 0.647603I
0.11745 4.26570I 1.86221 + 7.53589I
u = 0.422418 0.972261I
a = 0.101390 1.183947I
b = 0.44401 1.34279I
6.36874 9.11603I 3.03425 + 5.54417I
u = 0.422418 + 0.972261I
a = 0.101390 + 1.183947I
b = 0.44401 + 1.34279I
6.36874 + 9.11603I 3.03425 5.54417I
u = 0.448301 0.828589I
a = 0.687794 + 0.976545I
b = 0.858176 + 1.098554I
3.39185 1.64022I 0.179783 + 0.219910I
u = 0.448301 + 0.828589I
a = 0.687794 0.976545I
b = 0.858176 1.098554I
3.39185 + 1.64022I 0.179783 0.219910I
u = 0.663150 0.901843I
a = 0.981095 0.394024I
b = 1.72777 0.01848I
8.04432 + 4.08182I 4.48123 4.68553I
u = 0.663150 + 0.901843I
a = 0.981095 + 0.394024I
b = 1.72777 + 0.01848I
8.04432 4.08182I 4.48123 + 4.68553I
12
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.792757 0.587153I
a = 0.480661 1.051031I
b = 0.258030 0.275829I
0.241182 1.057236I 0.289576 0.557042I
u = 0.792757 + 0.587153I
a = 0.480661 + 1.051031I
b = 0.258030 + 0.275829I
0.241182 + 1.057236I 0.289576 + 0.557042I
u = 0.831768
a = 0.799419
b = 0.680761
1.34703 7.25788
u = 0.851604 0.554884I
a = 0.459211 0.585545I
b = 0.044513 + 0.738267I
2.42833 + 2.22486I 2.96572 3.27842I
u = 0.851604 + 0.554884I
a = 0.459211 + 0.585545I
b = 0.044513 0.738267I
2.42833 2.22486I 2.96572 + 3.27842I
u = 1.013143 0.212457I
a = 0.285013 + 0.539428I
b = 0.156238 0.625453I
2.04844 + 0.43724I 5.38341 0.85631I
u = 1.013143 + 0.212457I
a = 0.285013 0.539428I
b = 0.156238 + 0.625453I
2.04844 0.43724I 5.38341 + 0.85631I
u = 1.020128 0.751054I
a = 0.308971 + 0.846951I
b = 0.394858 + 0.401948I
6.94963 + 1.98085I 3.48623 0.28252I
u = 1.020128 + 0.751054I
a = 0.308971 0.846951I
b = 0.394858 0.401948I
6.94963 1.98085I 3.48623 + 0.28252I
u = 1.055238 0.511956I
a = 0.762076 + 0.577759I
b = 2.33049 0.48844I
4.03460 + 3.27231I 6.93849 3.87386I
13
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.055238 + 0.511956I
a = 0.762076 0.577759I
b = 2.33049 + 0.48844I
4.03460 3.27231I 6.93849 + 3.87386I
u = 1.063996 0.607981I
a = 1.019616 0.043957I
b = 3.32088 + 0.87652I
1.69804 + 10.75148I 2.97142 9.27459I
u = 1.063996 + 0.607981I
a = 1.019616 + 0.043957I
b = 3.32088 0.87652I
1.69804 10.75148I 2.97142 + 9.27459I
u = 1.118201 0.639814I
a = 0.712700 0.575906I
b = 2.73705 0.06629I
1.38638 + 7.13549I 2.46560 4.47635I
u = 1.118201 + 0.639814I
a = 0.712700 + 0.575906I
b = 2.73705 + 0.06629I
1.38638 7.13549I 2.46560 + 4.47635I
u = 1.173855 0.678590I
a = 0.941620 + 0.040473I
b = 3.60158 0.36509I
4.0697 + 15.1191I 0.26102 9.40367I
u = 1.173855 + 0.678590I
a = 0.941620 0.040473I
b = 3.60158 + 0.36509I
4.0697 15.1191I 0.26102 + 9.40367I
14
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)(u + 1)
4
(u
18
6u
16
+ ··· + u + 1)(u
47
+ 2u
46
+ ··· 5u + 5)
c
2
(u + 1)
5
(u
18
+ 12u
17
+ ··· + u + 1)(u
47
+ 18u
46
+ ··· + 445u + 25)
c
3
u(u
4
+ 2u
2
+ 2)(u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)
3
(u
47
6u
46
+ ··· + 736u 128)
c
4
(u)(u
4
2u
2
+ 2)(1 + u 2u
3
u
4
+ u
5
+ u
6
)
3
(u
47
2u
46
+ ··· + 4u 2)
c
5
(u 1)
4
(u + 1)(u
18
6u
16
+ ··· + u + 1)(u
47
+ 2u
46
+ ··· 5u + 5)
c
6
(u 1)
4
(u + 1)(u
18
6u
16
+ ··· u + 1)(u
47
+ 2u
46
+ ··· + 23u 5)
c
7
u(u
4
+ 2u
2
+ 2)(u
6
+ u
5
u
4
2u
3
+ u + 1)
3
(u
47
2u
46
+ ··· 3652u + 3866)
c
8
(u)(u
4
2u
2
+ 2)(1 + u 2u
3
u
4
+ u
5
+ u
6
)
3
(u
47
2u
46
+ ··· + 4u 2)
c
9
u(u
2
+ 2u + 2)
2
(u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)
3
(u
47
+ 22u
46
+ ··· + 8u + 4)
c
10
(u 1)(u + 1)
4
(u
18
6u
16
+ ··· u + 1)(u
47
+ 2u
46
+ ··· + 23u 5)
c
11
(u 1)(u + 1)
4
(u
18
6u
16
+ ··· u + 1)(u
47
+ 2u
46
+ ··· + 23u 5)
15
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y 1)
5
(y
18
12y
17
+ ··· y + 1)(y
47
18y
46
+ ··· + 445y 25)
c
2
(y 1)
5
(y
18
12y
17
+ ··· + 7y + 1)
(y
47
+ 30y
46
+ ··· 49175y 625)
c
3
y(y
2
+ 2y + 2)
2
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
3
(y
47
+ 10y
46
+ ··· 154624y 16384)
c
4
y(y
2
2y + 2)
2
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
3
(y
47
22y
46
+ ··· + 8y 4)
c
6
, c
10
(y 1)
5
(y
18
12y
17
+ ··· y + 1)(y
47
50y
46
+ ··· 211y 25)
c
7
y(y
2
+ 2y + 2)
2
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
3
(y
47
14y
46
+ ··· + 245188856y 14945956)
c
8
y(y
2
2y + 2)
2
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
3
(y
47
22y
46
+ ··· + 8y 4)
c
9
y(y
2
+ 4)
2
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
3
(y
47
+ 6y
46
+ ··· 96y 16)
c
11
(y 1)
5
(y
18
12y
17
+ ··· y + 1)(y
47
50y
46
+ ··· 211y 25)
16