11a
241
(K11a
241
)
1
Arc Sequences
7 1 11 10 8 9 2 6 3 4 5
Solving Sequence
5,8
6 9
7,10
4 11 1 3 2
c
5
c
8
c
6
c
4
c
10
c
11
c
3
c
2
c
1
, c
7
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
3
+ b
2
+ 2b + 1, a + 1, u 1i
I
u
2
= hu
50
4u
49
+ ··· 3u 1, u
12
+ 5u
10
+ 2u
9
9u
8
8u
7
+ 4u
6
+ 10u
5
+ 6u
4
2u
3
5u
2
+ a 2u 1,
u
49
3u
48
+ ··· + 4b + 1i
There are 2 irreducible components with 53 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
= hb
3
+ b
2
+ 2b + 1, a + 1, u 1i
(i) Arc colorings
a
5
=
1
0
a
8
=
0
1
a
6
=
1
1
a
9
=
1
0
a
7
=
0
1
a
10
=
1
b
a
4
=
b + 1
b
2
a
11
=
b
2
b 1
b
2
b 1
a
1
=
0
b
2
b 1
a
3
=
0
b
2
b 1
a
2
=
0
b
2
b 1
a
2
=
0
b
2
b 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 = 1.00000
b = 0.569840
2.75839 14.6948
u = 1.00000
a = 1.00000
b = 0.215080 1.307141I
1.37919 2.82812I 10.15260 + 3.54173I
u = 1.00000
a = 1.00000
b = 0.215080 + 1.307141I
1.37919 + 2.82812I 10.15260 3.54173I
3
II.
I
u
2
= hu
50
4u
49
+· · ·3u1, u
12
+5u
10
+· · ·+a1, u
49
3u
48
+· · ·+4b+1i
(i) Arc colorings
a
5
=
1
0
a
8
=
0
u
a
6
=
1
u
2
a
9
=
u
u
3
+ u
a
7
=
u
2
+ 1
u
4
+ 2u
2
a
10
=
u
12
5u
10
2u
9
+ 9u
8
+ 8u
7
4u
6
10u
5
6u
4
+ 2u
3
+ 5u
2
+ 2u + 1
1
4
u
49
+
3
4
u
48
+ ··· + u
1
4
a
4
=
1
4
u
49
3
4
u
48
+ ··· u +
5
4
3u
49
+
17
2
u
48
+ ···
19
2
u
5
2
a
11
=
3
4
u
49
+
11
4
u
48
+ ···
11
2
u
1
4
7.25000u
49
+ 19.2500u
48
+ ··· 20.5000u 5.75000
a
1
=
13
2
u
49
33
2
u
48
+ ··· + 15u +
11
2
7.25000u
49
+ 19.2500u
48
+ ··· 20.5000u 5.75000
a
3
=
11
2
u
49
+
29
2
u
48
+ ··· 14u
5
2
10.7500u
49
+ 28.7500u
48
+ ··· 30.5000u 8.25000
a
2
=
7
2
u
49
21
2
u
48
+ ··· + 11u +
9
2
18.7500u
49
+ 50.7500u
48
+ ··· 52.5000u 14.2500
a
2
=
7
2
u
49
21
2
u
48
+ ··· + 11u +
9
2
18.7500u
49
+ 50.7500u
48
+ ··· 52.5000u 14.2500
(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 = 1.347802 0.244464I
a = 2.76858 0.61561I
b = 0.331579 1.330071I
0.80897 7.30656I 11.87888 + 5.13251I
u = 1.347802 + 0.244464I
a = 2.76858 + 0.61561I
b = 0.331579 + 1.330071I
0.80897 + 7.30656I 11.87888 5.13251I
u = 1.341701 0.206608I
a = 1.86713 + 0.80037I
b = 0.772297 0.099116I
5.29863 3.31697I 16.7691 + 3.0814I
u = 1.341701 + 0.206608I
a = 1.86713 0.80037I
b = 0.772297 + 0.099116I
5.29863 + 3.31697I 16.7691 3.0814I
u = 1.338188 0.148593I
a = 0.19104 + 1.90685I
b = 0.304563 + 1.171450I
2.05727 + 0.60926I 13.40155 0.92964I
u = 1.338188 + 0.148593I
a = 0.19104 1.90685I
b = 0.304563 1.171450I
2.05727 0.60926I 13.40155 + 0.92964I
u = 1.178281 0.322665I
a = 1.48378 0.70190I
b = 0.019632 + 1.351750I
3.75475 1.24423I 7.62293 + 0.40296I
u = 1.178281 + 0.322665I
a = 1.48378 + 0.70190I
b = 0.019632 1.351750I
3.75475 + 1.24423I 7.62293 0.40296I
u = 1.139666 0.147758I
a = 0.894279 0.021486I
b = 0.166424 + 0.369815I
1.55529 0.78493I 9.19094 + 1.36537I
u = 1.139666 + 0.147758I
a = 0.894279 + 0.021486I
b = 0.166424 0.369815I
1.55529 + 0.78493I 9.19094 1.36537I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.969038 0.498459I
a = 0.480236 + 0.039738I
b = 0.317517 1.315332I
0.04307 + 4.61787I 12.77680 3.96714I
u = 0.969038 + 0.498459I
a = 0.480236 0.039738I
b = 0.317517 + 1.315332I
0.04307 4.61787I 12.77680 + 3.96714I
u = 0.889209 0.489632I
a = 0.914872 0.265987I
b = 0.748495 0.067917I
4.29681 + 0.76442I 18.3162 1.2723I
u = 0.889209 + 0.489632I
a = 0.914872 + 0.265987I
b = 0.748495 + 0.067917I
4.29681 0.76442I 18.3162 + 1.2723I
u = 0.781241 0.521632I
a = 1.48070 0.17158I
b = 0.300001 + 1.216562I
0.78921 3.02934I 14.4745 + 3.5215I
u = 0.781241 + 0.521632I
a = 1.48070 + 0.17158I
b = 0.300001 1.216562I
0.78921 + 3.02934I 14.4745 3.5215I
u = 0.314976 0.745791I
a = 0.083118 + 0.257952I
b = 0.292411 1.122108I
0.61352 1.42597I 11.17419 + 2.49743I
u = 0.314976 + 0.745791I
a = 0.083118 0.257952I
b = 0.292411 + 1.122108I
0.61352 + 1.42597I 11.17419 2.49743I
u = 0.268160 0.812232I
a = 0.602597 0.643600I
b = 0.777098 + 0.128997I
2.36749 5.37835I 14.0557 + 6.0904I
u = 0.268160 + 0.812232I
a = 0.602597 + 0.643600I
b = 0.777098 0.128997I
2.36749 + 5.37835I 14.0557 6.0904I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.254741
a = 0.765895
b = 0.299163
0.616490 16.5514
u = 0.241290 0.847148I
a = 1.61512 1.15526I
b = 0.332926 + 1.345877I
2.27588 9.39250I 9.20146 + 7.63091I
u = 0.241290 + 0.847148I
a = 1.61512 + 1.15526I
b = 0.332926 1.345877I
2.27588 + 9.39250I 9.20146 7.63091I
u = 0.171490 0.615665I
a = 0.002971 + 0.651556I
b = 0.202172 0.562210I
1.12881 2.03777I 7.78527 + 5.62795I
u = 0.171490 + 0.615665I
a = 0.002971 0.651556I
b = 0.202172 + 0.562210I
1.12881 + 2.03777I 7.78527 5.62795I
u = 0.079525 0.749141I
a = 0.02801 + 2.28150I
b = 0.031726 1.385862I
7.10349 2.64910I 4.03358 + 3.41831I
u = 0.079525 + 0.749141I
a = 0.02801 2.28150I
b = 0.031726 + 1.385862I
7.10349 + 2.64910I 4.03358 3.41831I
u = 0.091742 0.486520I
a = 0.623257 1.176060I
b = 0.666165 + 0.120010I
0.711738 + 0.697273I 10.71279 1.15101I
u = 0.091742 + 0.486520I
a = 0.623257 + 1.176060I
b = 0.666165 0.120010I
0.711738 0.697273I 10.71279 + 1.15101I
u = 0.131513 0.599202I
a = 2.32061 1.43271I
b = 0.287032 + 1.337147I
3.88325 + 4.20193I 5.74987 2.95437I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.131513 + 0.599202I
a = 2.32061 + 1.43271I
b = 0.287032 1.337147I
3.88325 4.20193I 5.74987 + 2.95437I
u = 0.224526 0.262956I
a = 1.38836 1.23755I
b = 0.214240 1.257321I
2.72213 2.30998I 4.83095 + 2.73383I
u = 0.224526 + 0.262956I
a = 1.38836 + 1.23755I
b = 0.214240 + 1.257321I
2.72213 + 2.30998I 4.83095 2.73383I
u = 1.255562 0.159319I
a = 0.765886 + 0.233068I
b = 0.237505 1.383800I
0.55189 1.61966I 13.09729 0.91495I
u = 1.255562 + 0.159319I
a = 0.765886 0.233068I
b = 0.237505 + 1.383800I
0.55189 + 1.61966I 13.09729 + 0.91495I
u = 1.316220 0.301977I
a = 1.085195 0.703146I
b = 0.06826 + 1.41694I
2.73057 + 6.43368I 9.43687 5.75913I
u = 1.316220 + 0.301977I
a = 1.085195 + 0.703146I
b = 0.06826 1.41694I
2.73057 6.43368I 9.43687 + 5.75913I
u = 1.325178 0.170535I
a = 0.867882 0.049722I
b = 0.625043 0.306332I
4.77278 + 1.48125I 17.3142 0.2721I
u = 1.325178 + 0.170535I
a = 0.867882 + 0.049722I
b = 0.625043 + 0.306332I
4.77278 1.48125I 17.3142 + 0.2721I
u = 1.356679 0.239831I
a = 0.904726 0.022505I
b = 0.396406 + 0.624218I
3.70738 + 5.14926I 14.3632 6.3732I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.356679 + 0.239831I
a = 0.904726 + 0.022505I
b = 0.396406 0.624218I
3.70738 5.14926I 14.3632 + 6.3732I
u = 1.41244 0.34853I
a = 2.36904 0.06189I
b = 0.350812 1.359942I
2.97930 + 13.70142I 12.8277 8.3385I
u = 1.41244 + 0.34853I
a = 2.36904 + 0.06189I
b = 0.350812 + 1.359942I
2.97930 13.70142I 12.8277 + 8.3385I
u = 1.41770 0.32777I
a = 1.45674 + 0.74032I
b = 0.815626 0.148361I
7.73338 + 9.49487I 17.3734 6.5886I
u = 1.41770 + 0.32777I
a = 1.45674 0.74032I
b = 0.815626 + 0.148361I
7.73338 9.49487I 17.3734 + 6.5886I
u = 1.41793 0.29464I
a = 0.063774 + 1.003663I
b = 0.367759 + 1.080262I
4.88671 + 5.18577I 14.7099 2.7132I
u = 1.41793 + 0.29464I
a = 0.063774 1.003663I
b = 0.367759 1.080262I
4.88671 5.18577I 14.7099 + 2.7132I
u = 1.49175 0.03659I
a = 1.42650 1.15493I
b = 0.380094 1.257643I
8.44132 + 4.36522I 16.7897 3.3870I
u = 1.49175 + 0.03659I
a = 1.42650 + 1.15493I
b = 0.380094 + 1.257643I
8.44132 4.36522I 16.7897 + 3.3870I
u = 1.49338
a = 1.77503
b = 0.835547
12.3381 20.6748
9
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
7
u
3
(u
50
+ u
49
+ ··· + 20u + 8)
c
2
u
3
(u
50
+ 21u
49
+ ··· + 592u + 64)
c
3
, c
4
, c
10
(u
3
u
2
+ 2u 1)(u
50
+ 2u
49
+ ··· + 2u 1)
c
5
, c
6
, c
8
(u + 1)
3
(u
50
+ 4u
49
+ ··· + 3u 1)
c
9
, c
11
(u
3
u
2
+ 1)(u
50
+ 2u
49
+ ··· 92u 17)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
7
y
3
(y
50
21y
49
+ ··· 592y + 64)
c
2
y
3
(y
50
+ 11y
49
+ ··· 19712y + 4096)
c
3
, c
4
, c
10
(y
3
+ 3y
2
+ 2y 1)(y
50
+ 42y
49
+ ··· 2y + 1)
c
5
, c
6
, c
8
(y 1)
3
(y
50
44y
49
+ ··· 3y + 1)
c
9
, c
11
(y
3
y
2
+ 2y 1)(y
50
30y
49
+ ··· + 70y + 289)
11