11a
64
(K11a
64
)
1
Arc Sequences
5 1 9 2 4 11 10 3 8 6 7
Solving Sequence
1,5
2 3
4,7
11 6 10 8 9
c
1
c
2
c
4
c
11
c
6
c
10
c
7
c
9
c
3
, c
5
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
2
+ u + 1, b 1, a + u + 1i
I
u
2
= hu
50
+ 2u
49
+ ··· 3u + 1, u
48
u
47
+ ··· + b 1, u
49
+ u
48
+ ··· + a 1i
There are 2 irreducible components with 52 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 1, a + u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u + 1
a
3
=
u
u + 1
a
4
=
u
u + 1
a
7
=
u 1
1
a
11
=
u
1
a
6
=
1
0
a
10
=
u 1
1
a
8
=
u 1
1
a
9
=
u 1
1
a
9
=
u 1
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 = 0.500000 0.866025I
a = 0.500000 + 0.866025I
b = 1.00000
1.64493 + 2.02988I 9.00000 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 0.866025I
b = 1.00000
1.64493 2.02988I 9.00000 + 3.46410I
3
II.
I
u
2
= hu
50
+2u
49
+· · · 3u + 1, u
48
u
47
+· · · +b 1, u
49
+u
48
+· · · +a 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
4
=
u
u
3
+ u
a
7
=
u
49
u
48
+ ··· 5u + 1
u
48
+ u
47
+ ··· 2u + 1
a
11
=
u
49
2u
48
+ ··· 2u + 1
u
48
+ u
47
+ ··· 3u + 1
a
6
=
u
3
u
5
+ u
3
+ u
a
10
=
u
49
3u
48
+ ··· + 26u
3
+ 2u
u
48
+ u
47
+ ··· 3u + 1
a
8
=
u
49
u
48
+ ··· 3u + 1
u
48
+ u
47
+ ··· 2u + 1
a
9
=
u
49
3u
48
+ ··· 2u
2
+ 3u
u
49
+ u
48
+ ··· 3u + 1
a
9
=
u
49
3u
48
+ ··· 2u
2
+ 3u
u
49
+ u
48
+ ··· 3u + 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 = 0.820332 0.598366I
a = 1.72642 + 0.48517I
b = 1.352761 + 0.362996I
3.29225 8.34439I 11.22860 + 4.83765I
u = 0.820332 + 0.598366I
a = 1.72642 0.48517I
b = 1.352761 0.362996I
3.29225 + 8.34439I 11.22860 4.83765I
u = 0.784288 0.784815I
a = 2.49385 0.28996I
b = 1.41586 + 0.11171I
9.73666 1.34570I 15.5702 + 0.7599I
u = 0.784288 + 0.784815I
a = 2.49385 + 0.28996I
b = 1.41586 0.11171I
9.73666 + 1.34570I 15.5702 0.7599I
u = 0.774646 0.583619I
a = 0.1120235 + 0.0271994I
b = 0.133262 0.831170I
1.38173 4.04307I 7.01868 + 3.20265I
u = 0.774646 + 0.583619I
a = 0.1120235 0.0271994I
b = 0.133262 + 0.831170I
1.38173 + 4.04307I 7.01868 3.20265I
u = 0.744548 0.937493I
a = 2.34902 1.59118I
b = 1.41536 0.14641I
9.27302 + 7.09927I 14.4198 6.3991I
u = 0.744548 + 0.937493I
a = 2.34902 + 1.59118I
b = 1.41536 + 0.14641I
9.27302 7.09927I 14.4198 + 6.3991I
u = 0.691553 0.810916I
a = 0.302555 + 0.293069I
b = 0.566666 0.530676I
3.39308 + 0.60483I 12.63177 + 0.83622I
u = 0.691553 + 0.810916I
a = 0.302555 0.293069I
b = 0.566666 + 0.530676I
3.39308 0.60483I 12.63177 0.83622I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.691332 1.051380I
a = 1.50080 2.19587I
b = 1.356027 0.382707I
1.9302 + 14.0131I 9.10827 9.32145I
u = 0.691332 + 1.051380I
a = 1.50080 + 2.19587I
b = 1.356027 + 0.382707I
1.9302 14.0131I 9.10827 + 9.32145I
u = 0.687541 0.895098I
a = 1.017637 + 0.499800I
b = 0.490724 + 0.589200I
3.13688 + 4.70114I 11.25136 7.35452I
u = 0.687541 + 0.895098I
a = 1.017637 0.499800I
b = 0.490724 0.589200I
3.13688 4.70114I 11.25136 + 7.35452I
u = 0.682574 0.595506I
a = 1.22843 + 0.73443I
b = 1.096203 + 0.388771I
1.57629 + 0.36486I 10.25677 + 0.07653I
u = 0.682574 + 0.595506I
a = 1.22843 0.73443I
b = 1.096203 0.388771I
1.57629 0.36486I 10.25677 0.07653I
u = 0.671404 1.040633I
a = 1.074443 + 0.524871I
b = 0.126757 + 0.868356I
2.73323 + 9.52065I 5.03643 7.69857I
u = 0.671404 + 1.040633I
a = 1.074443 0.524871I
b = 0.126757 0.868356I
2.73323 9.52065I 5.03643 + 7.69857I
u = 0.649765 1.016232I
a = 0.147703 + 0.611241I
b = 1.136100 0.443889I
0.35735 + 4.82997I 8.07489 4.80231I
u = 0.649765 + 1.016232I
a = 0.147703 0.611241I
b = 1.136100 + 0.443889I
0.35735 4.82997I 8.07489 + 4.80231I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.215855 0.483721I
a = 1.24120 + 2.09218I
b = 1.087887 + 0.165328I
1.40881 + 0.77400I 6.94609 + 0.76417I
u = 0.215855 + 0.483721I
a = 1.24120 2.09218I
b = 1.087887 0.165328I
1.40881 0.77400I 6.94609 0.76417I
u = 0.015708 1.080761I
a = 0.40973 + 1.39311I
b = 1.224772 + 0.385024I
3.57195 + 1.39688I 3.31825 0.66691I
u = 0.015708 + 1.080761I
a = 0.40973 1.39311I
b = 1.224772 0.385024I
3.57195 1.39688I 3.31825 + 0.66691I
u = 0.030065 1.108866I
a = 0.032879 1.296561I
b = 0.047274 0.835322I
7.20082 2.98868I 0.09209 + 2.99503I
u = 0.030065 + 1.108866I
a = 0.032879 + 1.296561I
b = 0.047274 + 0.835322I
7.20082 + 2.98868I 0.09209 2.99503I
u = 0.069277 1.129561I
a = 0.353052 + 1.270343I
b = 1.302713 + 0.373085I
2.98615 7.33040I 4.59712 + 5.87500I
u = 0.069277 + 1.129561I
a = 0.353052 1.270343I
b = 1.302713 0.373085I
2.98615 + 7.33040I 4.59712 5.87500I
u = 0.203905 0.774988I
a = 0.339779 0.707399I
b = 0.104654 0.432324I
1.18724 1.58310I 1.45548 + 5.77798I
u = 0.203905 + 0.774988I
a = 0.339779 + 0.707399I
b = 0.104654 + 0.432324I
1.18724 + 1.58310I 1.45548 5.77798I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.294861
a = 0.407744
b = 0.375797
0.739246 14.0544
u = 0.330969 1.011959I
a = 0.103921 + 1.108893I
b = 1.260390 + 0.078461I
2.80671 3.20550I 10.87412 + 5.48169I
u = 0.330969 + 1.011959I
a = 0.103921 1.108893I
b = 1.260390 0.078461I
2.80671 + 3.20550I 10.87412 5.48169I
u = 0.568393 0.860366I
a = 0.881581 + 0.266099I
b = 0.263398 + 0.099445I
0.39231 2.25929I 1.99740 + 3.42645I
u = 0.568393 + 0.860366I
a = 0.881581 0.266099I
b = 0.263398 0.099445I
0.39231 + 2.25929I 1.99740 3.42645I
u = 0.587482 1.054521I
a = 0.122117 + 0.706179I
b = 1.241067 0.324695I
0.243412 + 0.350536I 7.73133 1.27441I
u = 0.587482 + 1.054521I
a = 0.122117 0.706179I
b = 1.241067 + 0.324695I
0.243412 0.350536I 7.73133 + 1.27441I
u = 0.630329 1.035100I
a = 1.103981 + 0.520432I
b = 0.039568 + 0.776582I
3.45504 3.62076I 3.32917 + 2.61098I
u = 0.630329 + 1.035100I
a = 1.103981 0.520432I
b = 0.039568 0.776582I
3.45504 + 3.62076I 3.32917 2.61098I
u = 0.640142
a = 2.08837
b = 1.32705
5.84071 16.2554
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.666955 1.024889I
a = 1.65644 2.46648I
b = 1.297339 0.337432I
0.72091 7.64132I 8.02664 + 5.65658I
u = 0.666955 + 1.024889I
a = 1.65644 + 2.46648I
b = 1.297339 + 0.337432I
0.72091 + 7.64132I 8.02664 5.65658I
u = 0.697130 0.856379I
a = 3.37809 1.13298I
b = 1.311800 0.024858I
5.15830 2.67468I 12.04075 + 3.18893I
u = 0.697130 + 0.856379I
a = 3.37809 + 1.13298I
b = 1.311800 + 0.024858I
5.15830 + 2.67468I 12.04075 3.18893I
u = 0.700858 0.501693I
a = 0.128943 0.081096I
b = 0.015463 0.734978I
1.96381 1.49641I 5.67052 + 2.83851I
u = 0.700858 + 0.501693I
a = 0.128943 + 0.081096I
b = 0.015463 + 0.734978I
1.96381 + 1.49641I 5.67052 2.83851I
u = 0.735015 0.393449I
a = 1.51775 + 0.59793I
b = 1.290933 + 0.313969I
2.12444 5.28818I 11.12195 + 5.66232I
u = 0.735015 + 0.393449I
a = 1.51775 0.59793I
b = 1.290933 0.313969I
2.12444 + 5.28818I 11.12195 5.66232I
u = 0.741668 0.605033I
a = 1.94725 + 0.82754I
b = 1.274810 + 0.306801I
1.95830 + 2.25536I 10.04739 0.82477I
u = 0.741668 + 0.605033I
a = 1.94725 0.82754I
b = 1.274810 0.306801I
1.95830 2.25536I 10.04739 + 0.82477I
9
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)(u
50
+ 2u
49
+ ··· 3u + 1)
c
2
, c
5
(u
2
+ u + 1)(u
50
+ 18u
49
+ ··· + u + 1)
c
3
, c
8
u
2
(u
50
+ u
49
+ ··· + 4u 4)
c
4
(u
2
u + 1)(u
50
+ 2u
49
+ ··· 3u + 1)
c
6
(u + 1)
2
(u
50
+ 3u
49
+ ··· + 9u
2
1)
c
7
u
2
(u
50
+ 15u
49
+ ··· + 104u + 16)
c
9
u
2
(u
50
+ 15u
49
+ ··· + 104u + 16)
c
10
(u 1)
2
(u
50
+ 3u
49
+ ··· + 9u
2
1)
c
11
(u 1)
2
(u
50
+ 3u
49
+ ··· + 9u
2
1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
2
+ y + 1)(y
50
+ 18y
49
+ ··· + y + 1)
c
2
, c
5
(y
2
+ y + 1)(y
50
+ 30y
49
+ ··· 119y + 1)
c
3
, c
8
y
2
(y
50
15y
49
+ ··· 104y + 16)
c
6
(y 1)
2
(y
50
41y
49
+ ··· 18y + 1)
c
7
y
2
(y
50
+ 37y
49
+ ··· 3360y + 256)
c
9
y
2
(y
50
+ 37y
49
+ ··· 3360y + 256)
c
10
(y 1)
2
(y
50
41y
49
+ ··· 18y + 1)
c
11
(y 1)
2
(y
50
41y
49
+ ··· 18y + 1)
11