11a
157
(K11a
157
)
1
Arc Sequences
6 1 10 8 2 4 11 5 3 7 9
Solving Sequence
2,5
6 1
3,9
10 8 4 11 7
c
5
c
1
c
2
c
9
c
8
c
4
c
11
c
7
c
3
, c
6
, c
10
Representation Ideals
I =
4
\
i=1
I
u
i
2
\
j=1
I
v
j
I
u
1
= hu
2
+ u + 1, a
3
u + a
2
b bu + au + 2a 1, a
2
bu + a
2
b a
3
+ 2au + b + a u 1,
b
2
a
2
a
3
b + a
4
b
2
u + bau 2a
2
u + ba a
2
+ au b + ai
I
u
2
= h2a + u, 3b + 2u + 1, u
2
+ 2i
I
u
3
= h−a
5
+ a
2
+ b a 1, a
6
+ a
4
a
3
+ a
2
+ a + 1, a
5
+ a
4
2a
3
+ a
2
2a + 2u + 1i
I
u
4
= hu
53
4u
52
+ ··· 92u + 46, 3568u
52
14272u
51
+ ··· + 52992a 258566,
42u
52
85u
51
+ ··· + 2304b + 7550i
I
v
1
= ha, b 1, v 1i
I
v
2
= ha, v 1, b
3
b 1i
There are 6 irreducible components with 65 representations.
There are 1 irreducible components of dim
C
= 1 for 11a
157
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, a
3
u + a
2
b bu + au + 2a 1, a
2
bu + 2au + · · · + a
1, b
2
u + bau + · · · a
2
+ ai
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
u
u 1
a
3
=
0
u
a
9
=
a
b
a
10
=
a
1
15
b
4
u +
1
5
b
3
u + ··· +
1
15
a +
3
5
a
8
=
a
1
15
b
4
u +
1
5
b
3
u + ··· +
1
15
a +
3
5
a
4
=
1
15
b
4
u +
2
15
b
3
u + ···
2
5
a +
2
5
1
5
b
4
u +
1
15
b
3
u + ···
13
15
a +
1
5
a
11
=
0.133333b
4
u + 0.0666667b
3
u + ··· + 0.466667a + 0.200000
1
5
b
4
u +
1
15
b
3
u + ···
13
15
a
4
5
a
7
=
2
15
b
4
u
1
15
b
3
u + ··· +
8
15
a
1
5
0.266667b
4
u + 0.133333b
3
u + ··· + 0.933333a + 1.40000
a
7
=
2
15
b
4
u
1
15
b
3
u + ··· +
8
15
a
1
5
0.266667b
4
u + 0.133333b
3
u + ··· + 0.933333a + 1.40000
(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 = ···
a = ···
b = ···
1.64493 2.02988I 6.00000 + 3.46410I
3
II. I
u
2
= h2a + u, 3b + 2u + 1, u
2
+ 2i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
1
2
a
3
=
3
4
a
9
=
1
2
u
2
3
u
1
3
a
10
=
1
2
u 3
2
3
u
13
3
a
8
=
1
2
u
5
3
u
1
3
a
4
=
1
2
u
2
3
u
1
3
a
11
=
1
6
u
4
3
1
9
u
23
9
a
7
=
5
6
u
1
3
8
9
u
5
9
a
7
=
5
6
u
1
3
8
9
u
5
9
(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 = 1.41421I
a = 0.707107I
b = 0.333333 + 0.942809I
4.93480 0
u = 1.41421I
a = 0.707107I
b = 0.333333 0.942809I
4.93480 0
5
III.
I
u
3
= h−a
5
+a
2
+ba1, a
6
+a
4
a
3
+a
2
+a+1, a
5
+a
4
2a
3
+a
2
2a+2u+1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
1
2
a
5
1
2
a
4
+ ··· + a
1
2
a
6
=
1
2
a
5
1
2
a
4
+ ··· + a
1
2
1
2
a
5
1
2
a
4
+ ··· + a
1
2
a
1
=
1
2
a
5
+
1
2
a
4
+ ··· a +
1
2
1
2
a
5
+
1
2
a
4
+ ··· a
1
2
a
3
=
0
1
2
a
5
1
2
a
4
+ ··· + a
1
2
a
9
=
a
a
5
a
2
+ a + 1
a
10
=
a
1
2
a
5
+
1
2
a
4
1
2
a
2
+ a +
1
2
a
8
=
a
1
2
a
5
+
1
2
a
4
1
2
a
2
+ a +
1
2
a
4
=
1
2
a
5
+
1
2
a
4
1
2
a
2
+
1
2
1
2
a
5
+
1
2
a
4
1
2
a
2
+ a +
1
2
a
11
=
a
1
2
a
5
1
2
a
4
+
1
2
a
2
a
1
2
a
7
=
a
1
2
a
5
+
1
2
a
4
1
2
a
2
+ a +
1
2
a
7
=
a
1
2
a
5
+
1
2
a
4
1
2
a
2
+ a +
1
2
(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 = 0.500000 0.866025I
a = 0.467454 0.522723I
b = 0.669552 0.863143I
2.02988I 3.46410I
u = 0.500000 + 0.866025I
a = 0.467454 + 0.522723I
b = 0.669552 + 0.863143I
2.02988I 3.46410I
u = 0.500000 + 0.866025I
a = 0.420593 1.203216I
b = 1.49343 1.84400I
2.02988I 3.46410I
u = 0.500000 0.866025I
a = 0.420593 + 1.203216I
b = 1.49343 + 1.84400I
2.02988I 3.46410I
u = 0.500000 0.866025I
a = 0.888047 0.680493I
b = 0.176126 + 0.751194I
2.02988I 3.46410I
u = 0.500000 + 0.866025I
a = 0.888047 + 0.680493I
b = 0.176126 0.751194I
2.02988I 3.46410I
7
IV. I
u
4
= hu
53
4u
52
+ · · · 92u + 46, 3568u
52
14272u
51
+ · · · + 52992a
258566, 42u
52
85u
51
+ · · · + 2304b + 7550i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
+ u
2
+ 1
u
4
a
9
=
0.0673309u
52
+ 0 .269324u
51
+ ··· 5.43361u + 4.87934
0.0182292u
52
+ 0 .0368924u
51
+ ··· + 2.21875u 3.27691
a
10
=
0.0360809u
52
+ 0 .0818237u
51
+ ··· 0.160175u + 1.76215
0.0329861u
52
0.293403u
51
+ ··· + 10.6840u 8.48785
a
8
=
0.0673309u
52
+ 0 .269324u
51
+ ··· 5.43361u + 4.87934
0.0685764u
52
+ 0 .171441u
51
+ ··· 0.878472u 3.27691
a
4
=
0.0148890u
52
0.0647645u
51
+ ··· + 1.52797u + 0.135417
0.0338542u
52
0 .218750u
51
+ ··· + 6.95573u 5.48438
a
11
=
0.0122849u
52
0.0602355u
51
+ ··· + 3.26370u 3.71354
0.0312500u
52
+ 0 .0286458u
51
+ ··· + 1.52344u + 1.41146
a
7
=
0.0258530u
52
0.0409118u
51
+ ··· + 0.744150u 1.57639
0.0217014u
52
+ 0 .168837u
51
+ ··· 2.72222u + 3.70486
a
7
=
0.0258530u
52
0.0409118u
51
+ ··· + 0.744150u 1.57639
0.0217014u
52
+ 0 .168837u
51
+ ··· 2.72222u + 3.70486
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.994126 0 .446360I
a = 1.075256 0.006429I
b = 0.216342 + 0.789543I
2.22320 4.60694I 5.52820 + 4 .35998I
u = 0.994126 + 0 .446360I
a = 1.075256 + 0.006429I
b = 0.216342 0.789543I
2.22320 + 4 .60694I 5.52820 4.35998I
u = 0.932904 0 .475451I
a = 1.359195 + 0 .003369I
b = 0.269079 1.130115I
6.27152 10.90755I 7.14898 + 5 .53843I
u = 0.932904 + 0 .475451I
a = 1.359195 0 .003369I
b = 0.269079 + 1 .130115I
6.27152 + 10 .90755I 7.14898 5.53843I
u = 0.880838 0 .326928I
a = 0.813937 0 .439734I
b = 0.405908 0.650356I
7.95194 + 1 .12051I 10.90095 + 0.04798I
u = 0.880838 + 0 .326928I
a = 0.813937 + 0 .439734I
b = 0.405908 + 0.650356I
7.95194 1.12051I 10.90095 0 .04798I
u = 0.695044 1 .161578I
a = 0.150405 + 0 .956110I
b = 1.42015 + 1 .53061I
0.03277 + 10 .71238I 2.38858 7.66608I
u = 0.695044 + 1 .161578I
a = 0.150405 0 .956110I
b = 1.42015 1.53061I
0.03277 10.71238I 2.38858 + 7 .66608I
u = 0.680239 1 .136942I
a = 0.108009 1.127559I
b = 1.71258 1.88022I
4.2476 + 16 .8128I 4.77128 9.31974I
u = 0.680239 + 1 .136942I
a = 0.108009 + 1.127559I
b = 1.71258 + 1.88022I
4.2476 16 .8128I 4.77128 + 9 .31974I
9
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.626566 0.643943I
a = 0.350230 + 0 .942452I
b = 1.230952 0.339849I
1.59094 + 3 .76796I 4.39825 7.64009I
u = 0.626566 + 0 .643943I
a = 0.350230 0.942452I
b = 1.230952 + 0.339849I
1.59094 3.76796I 4.39825 + 7.64009I
u = 0.621820 1.215441I
a = 0.031961 0.693522I
b = 0.61474 1.57378I
5.26202 + 4 .45797I 7.85127 5.58566I
u = 0.621820 + 1 .215441I
a = 0.031961 + 0 .693522I
b = 0.61474 + 1.57378I
5.26202 4.45797I 7.85127 + 5.58566I
u = 0.611609 0.908410I
a = 0.545278 0.524426I
b = 1.37610 + 0.79912I
0.872134 + 1 .079724I 0.00882 + 2 .41019I
u = 0.611609 + 0 .908410I
a = 0.545278 + 0.524426I
b = 1.37610 0.79912I
0.872134 1.079724I 0.00882 2.41019I
u = 0.318662
a = 1.14613
b = 0.699608
1.00464 11.4098
u = 0.170302 1.087832I
a = 0.583420 + 0.089782I
b = 0.032088 0.702820I
3.08054 + 4 .00121I 5.97068 4.06106I
u = 0.170302 + 1 .087832I
a = 0.583420 0.089782I
b = 0.032088 + 0.702820I
3.08054 4.00121I 5.97068 + 4.06106I
u = 0.12822 1 .49963I
a = 0.183119 + 0 .615389I
b = 0.026442 + 0.889456I
4.67412 0.82373I 5.81393 + 9.12144I
10
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.12822 + 1 .49963I
a = 0.183119 0.615389I
b = 0.026442 0.889456I
4.67412 + 0 .82373I 5.81393 9.12144I
u = 0.055611 0.764069I
a = 0.408832 0.782884I
b = 0.577042 + 0.176443I
1.00556 + 1 .44727I 1.64819 5.47738I
u = 0.055611 + 0 .764069I
a = 0.408832 + 0 .782884I
b = 0.577042 0.176443I
1.00556 1 .44727I 1.64819 + 5.47738I
u = 0.013214 1.311607I
a = 0.353126 0.914959I
b = 0.107911 1.058795I
0.35444 8 .20749I 2.01966 + 6.43814I
u = 0.013214 + 1 .311607I
a = 0.353126 + 0.914959I
b = 0.107911 + 1.058795I
0.35444 + 8 .20749I 2.01966 6.43814I
u = 0.128610 1.112310I
a = 0.488746 1.089894I
b = 0.250265 0.766064I
4.15345 + 3 .73758I 2.48655 3.09856I
u = 0.128610 + 1.112310I
a = 0.488746 + 1 .089894I
b = 0.250265 + 0.766064I
4.15345 3 .73758I 2.48655 + 3.09856I
u = 0.212305 1.189433I
a = 0.271057 + 0.991682I
b = 0.264238 + 1.002203I
6.51253 1 .42478I 5.84909 + 2.88735I
u = 0.212305 + 1.189433I
a = 0.271057 0.991682I
b = 0.264238 1.002203I
6.51253 + 1 .42478I 5.84909 2.88735I
u = 0.434366 0.666743I
a = 0.691073 + 0.461492I
b = 1.24418 1 .12885I
1.88416 0 .85038I 6.25673 3.20224I
11
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.434366 + 0.666743I
a = 0.691073 0.461492I
b = 1.24418 + 1 .12885I
1.88416 + 0 .85038I 6.25673 + 3 .20224I
u = 0.496009 1.017752I
a = 0.177088 0.776519I
b = 1.19225 2 .30460I
0.67079 3.09135I 0.27355 + 5 .67301I
u = 0.496009 + 1.017752I
a = 0.177088 + 0 .776519I
b = 1.19225 + 2 .30460I
0.67079 + 3 .09135I 0.27355 5.67301I
u = 0.572191 1.085582I
a = 0.155597 + 1.061125I
b = 1.55316 + 1.54127I
4.09769 6.18849I 2.46220 + 4 .97842I
u = 0.572191 + 1.085582I
a = 0.155597 1.061125I
b = 1.55316 1.54127I
4.09769 + 6 .18849I 2.46220 4.97842I
u = 0.601851 1.072668I
a = 0.017007 1.234048I
b = 1.76091 1 .71417I
1.10548 10. 82135I 2.37098 + 9.10173I
u = 0.601851 + 1.072668I
a = 0.017007 + 1.234048I
b = 1.76091 + 1 .71417I
1.10548 + 10. 82135I 2.37098 9.10173I
u = 0.673460 1.041740I
a = 0.920212 + 0 .467454I
b = 0.28850 + 1.59821I
8.08301 10. 18318I 7.98587 + 7.31718I
u = 0.673460 + 1.041740I
a = 0.920212 0.467454I
b = 0.28850 1.59821I
8.08301 + 10. 18318I 7.98587 7.31718I
u = 0.681987 0.321709I
a = 1.47928 0.21562I
b = 0.296288 + 0.652996I
1.99983 + 1 .37832I 0.467710 0.890691I
12
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.681987 + 0.321709I
a = 1.47928 + 0. 21562I
b = 0.296288 0.652996I
1.99983 1.37832I 0.467710 + 0. 890691I
u = 0.686889 1.004729I
a = 0.571478 0.305953I
b = 0.594160 0.913334I
3.37147 6.01200I 5.83450 + 5 .48305I
u = 0.686889 + 1.004729I
a = 0.571478 + 0.305953I
b = 0.594160 + 0 .913334I
3.37147 + 6 .01200I 5.83450 5.48305I
u = 0.724250 0.436534I
a = 1.58938 + 0.00037I
b = 0.434228 1.172179I
0.72806 + 5 .75739I 4.99130 5.10418I
u = 0.724250 + 0.436534I
a = 1.58938 0.00037I
b = 0.434228 + 1.172179I
0.72806 5.75739I 4.99130 + 5 .10418I
u = 0.780684 1.053133I
a = 0.610545 0 .196516I
b = 0.271680 + 0 .276957I
5.65031 0.44478I 8.84135 1.69630I
u = 0.780684 + 1.053133I
a = 0.610545 + 0 .196516I
b = 0.271680 0.276957I
5.65031 + 0 .44478I 8.84135 + 1.69630I
u = 0.809668 0.591571I
a = 0.465982 + 1 .081772I
b = 0.59081 + 1.46918I
9.44289 + 4 .62072I 10.24286 2.08237I
u = 0.809668 + 0.591571I
a = 0.465982 1 .081772I
b = 0.59081 1.46918I
9.44289 4.62072I 10.24286 + 2 .08237I
u = 0.817792 0.679811I
a = 0.143788 0.619341I
b = 0.324370 1.087161I
4.38963 + 0 .38285I 8.33080 + 0.50210I
13
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.817792 + 0.679811I
a = 0.143788 + 0.619341I
b = 0.324370 + 1.087161I
4.38963 0 .38285I 8.33080 0.50210I
u = 0.949759 0.608985I
a = 0.408126 + 0.855632I
b = 0.783999 + 0.670084I
6.99581 5 .82913I 8.82739 + 6.38685I
u = 0.949759 + 0.608985I
a = 0.408126 0.855632I
b = 0.783999 0.670084I
6.99581 + 5 .82913I 8.82739 6.38685I
14
V. I
v
1
= ha, b 1, v 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
1
0
a
6
=
1
0
a
1
=
1
0
a
3
=
1
0
a
9
=
0
1
a
10
=
1
1
a
8
=
1
1
a
4
=
0
1
a
11
=
1
1
a
7
=
1
1
a
7
=
1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
15
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
0 0
16
VI. I
v
2
= ha, v 1, b
3
b 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
1
0
a
6
=
1
0
a
1
=
1
0
a
3
=
1
0
a
9
=
0
b
a
10
=
b
b
a
8
=
b
b
a
4
=
b
2
+ 1
b
2
a
11
=
1
b
2
a
7
=
b + 1
b
2
+ b
a
7
=
b + 1
b
2
+ b
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
17
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
2
1(vol +
1CS) Cusp shape
v = 1.00000
a = 0
b = 0.662359 0.562280I
1.64493 6.00000
v = 1.00000
a = 0
b = 0.662359 + 0.562280I
1.64493 6.00000
v = 1.00000
a = 0
b = 1.32472
1.64493 6.00000
18
VII. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
u
4
(u
2
+ 2)(1 u + u
2
)
3
(1 + u + u
2
)
6
(u
53
+ 4 u
52
+ ··· 92u 46)
c
2
u
4
(u + 2)
2
(u
2
+ u + 1)
9
(u
53
+ 24 u
52
+ ··· + 3496u 2116)
c
3
(u 1)(u + 1)
2
(u
3
u 1)(u
6
2u
4
+ u
3
+ u
2
u + 1)
2
(u
6
+ u
4
+ ··· + 2u + 1)(9u
53
+ 18 u
52
+ ··· 77u + 19)
c
4
(u 1)(u + 1)
2
(u
3
u + 1)(u
6
2u
4
u
3
+ u
2
+ u + 1)
(u
6
2u
4
+ u
3
+ u
2
u + 1)( u
6
+ u
4
+ u
3
+ u
2
+ 2 u + 1)
(9u
53
+ 18 u
52
+ ··· 9u 19)
c
5
u
4
(u
2
+ 2)( u
2
u + 1)
9
(u
53
+ 4 u
52
+ ··· 92u 46)
c
6
(u + 1)(3 u
2
2u + 1)(u
3
+ 2 u
2
+ u + 1)( u
6
2u
4
+ ··· + u + 1)
(u
6
2 u
5
5u
4
+ 5 u
3
+ 19 u
2
+ 6 u + 1)
(7u
6
+ 34 u
5
+ 52 u
4
+ 35 u
3
+ 31 u
2
3u + 1)
(16u
53
16 u
52
+ ··· + 103779u 3609)
c
7
u
7
(u 1)
3
(u + 1)
12
(u
2
+ 2)( u
53
+ 6 u
52
+ ··· 5832u 1706)
c
8
(u 1)
2
(u + 1)(u
3
u + 1)(u
6
2u
4
u
3
+ u
2
+ u + 1)
2
(u
6
+ u
4
+ ··· 2u + 1)(9u
53
+ 18 u
52
+ ··· 9u 19)
c
9
(u 1)
2
(u + 1)(u
3
u 1)(u
6
2u
4
u
3
+ u
2
+ u + 1)
(u
6
2u
4
+ u
3
+ u
2
u + 1)( u
6
+ u
4
u
3
+ u
2
2u + 1)
(9u
53
+ 18 u
52
+ ··· 77u + 19)
c
10
u
7
(u 1)
15
(u
2
+ 2)(u
53
+ 6 u
52
+ ··· 5832u 1706)
c
11
(u 1)(3 u
2
2u + 1)(u
3
u 1)( u
6
+ 4 u
5
+ ··· + 6u + 4)
(u
6
+ 4 u
5
+ 6 u
4
+ 3 u
3
u
2
u + 1)
(7u
6
50u
5
+ 124 u
4
127u
3
+ 55 u
2
12 u + 4)
(16u
53
32u
52
+ ··· 16641u + 6003)
19
VIII. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
y
4
(y + 2)
2
(y
2
+ y + 1)
9
(y
53
+ 24 y
52
+ ··· + 3496y 2116)
c
2
y
4
(y 4)
2
(y
2
+ y + 1)
9
(y
53
+ 12 y
52
+ ··· + 393508288y 4477456)
c
3
, c
9
(y 1)
3
(y
3
2y
2
+ y 1)(y
6
4y
5
+ 6 y
4
3y
3
y
2
+ y + 1)
2
(y
6
+ 2 y
5
+ 3 y
4
+ 3 y
3
y
2
2y + 1)
(81y
53
3510 y
52
+ ··· + 3421y 361)
c
4
, c
8
(y 1)
3
(y
3
2y
2
+ y 1)(y
6
4y
5
+ 6 y
4
3y
3
y
2
+ y + 1)
2
(y
6
+ 2 y
5
+ 3 y
4
+ 3 y
3
y
2
2y + 1)
(81y
53
2214 y
52
+ ··· + 7149y 361)
c
6
(y 1)(9y
2
+ 2 y + 1)(y
3
2y
2
3y 1)
(y
6
14 y
5
+ 83 y
4
189 y
3
+ 291 y
2
+ 2 y + 1)
(y
6
4 y
5
+ 6 y
4
3y
3
y
2
+ y + 1)
(49y
6
428y
5
+ 758 y
4
+ 2217 y
3
+ 1275 y
2
+ 53 y + 1)
(256y
53
+ 6016 y
52
+ ··· + 9942551577y 13024881)
c
7
, c
10
y
7
(y 1)
15
(y + 2)
2
(y
53
30y
52
+ ··· 5573800y 2910436)
c
11
(y 1)(9y
2
+ 2 y + 1)(y
3
2y
2
+ y 1)
(y
6
14 y
5
+ 71 y
4
150 y
3
+ 153 y
2
76y + 16)
(y
6
4 y
5
+ 10 y
4
11y
3
+ 19 y
2
3y + 1)
(49y
6
764y
5
+ 3446 y
4
3633y
3
+ 969 y
2
+ 296 y + 16)
(256y
53
6528y
52
+ ··· 368087463y 36036009)
20