11a
258
(K11a
258
)
1
Arc Sequences
4 8 1 2 11 10 3 5 7 6 9
Solving Sequence
5,11 2,6
4 1 3 10 7 9 8
c
5
c
4
c
1
c
3
c
10
c
6
c
9
c
8
c
2
, c
7
, c
11
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
4
a
3
+ 3a
2
2a + 1, u 1, b a + 1i
I
u
2
= hu
41
+ 5u
40
+ ··· 3u 1, u
40
+ 4u
39
+ ··· + 8b 1, u
40
+ 4u
39
+ ··· + 8a 1i
There are 2 irreducible components with 45 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
= ha
4
a
3
+ 3a
2
2a + 1, u 1, b a + 1i
(i) Arc colorings
a
5
=
0
1
a
11
=
a
a 1
a
2
=
1
0
a
6
=
a
2
a
2
a + 1
a
4
=
1
1
a
1
=
0
1
a
3
=
1
0
a
10
=
a
3
+ a
a
3
a
2
+ 2a 1
a
7
=
a
3
a
2
+ 2a 1
0
a
9
=
a
3
a
2
+ 2a 1
a
3
a
2
+ 2a 1
a
8
=
a
3
a
2
+ 2a 1
0
a
8
=
a
3
a
2
+ 2a 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 = 1.00000
a = 0.10488 1.55249I
b = 0.89512 1.55249I
8.43568 3.16396I 5.67855 + 1.65351I
u = 1.00000
a = 0.10488 + 1.55249I
b = 0.89512 + 1.55249I
8.43568 + 3.16396I 5.67855 1.65351I
u = 1.00000
a = 0.395123 0.506844I
b = 0.604877 0.506844I
1.43393 1.41510I 0.82145 + 5.62908I
u = 1.00000
a = 0.395123 + 0.506844I
b = 0.604877 + 0.506844I
1.43393 + 1.41510I 0.82145 5.62908I
3
II.
I
u
2
= hu
41
+5u
40
+· · ·3u1, u
40
+4u
39
+· · ·+8b1, u
40
+4u
39
+· · ·+8a1i
(i) Arc colorings
a
5
=
0
u
a
11
=
1
8
u
40
1
2
u
39
+ ··· +
17
4
u +
1
8
1
8
u
40
1
2
u
39
+ ··· +
5
4
u +
1
8
a
2
=
1
0
a
6
=
9
8
u
40
+
17
4
u
39
+ ··· 2u
7
8
5
4
u
40
+
19
4
u
39
+ ···
5
4
u 1
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
3
+ 2u
u
3
+ u
a
10
=
7.50000u
40
28.5000u
39
+ ··· + 17.5000u + 6.50000
29
4
u
40
109
4
u
39
+ ··· +
55
4
u + 6
a
7
=
13u
40
+
185
4
u
39
+ ···
85
4
u
31
4
31
4
u
40
+
53
2
u
39
+ ··· 11u
17
4
a
9
=
43
4
u
40
155
4
u
39
+ ··· +
73
4
u + 7
6.50000u
40
25.2500u
39
+ ··· + 12.2500u + 5.25000
a
8
=
43
4
u
40
155
4
u
39
+ ··· +
73
4
u + 7
53
4
u
40
+
95
2
u
39
+ ··· 22u
39
4
a
8
=
43
4
u
40
155
4
u
39
+ ··· +
73
4
u + 7
53
4
u
40
+
95
2
u
39
+ ··· 22u
39
4
(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 = 1.58265 0.18117I
a = 0.165894 1.023684I
b = 0.38748 4.17159I
19.3092 0.9583I 9.77193 + 0.58041I
u = 1.58265 + 0.18117I
a = 0.165894 + 1.023684I
b = 0.38748 + 4.17159I
19.3092 + 0.9583I 9.77193 0.58041I
u = 1.52685 0.19609I
a = 0.278069 0.548602I
b = 0.01713 1.76368I
10.55128 2.41180I 8.26396 0.10124I
u = 1.52685 + 0.19609I
a = 0.278069 + 0.548602I
b = 0.01713 + 1.76368I
10.55128 + 2.41180I 8.26396 + 0.10124I
u = 1.52575 0.31574I
a = 0.116575 + 1.039295I
b = 1.94923 + 3.91053I
17.2037 11.9877I 8.05676 + 5.99300I
u = 1.52575 + 0.31574I
a = 0.116575 1.039295I
b = 1.94923 3.91053I
17.2037 + 11.9877I 8.05676 5.99300I
u = 1.50901 0.28951I
a = 0.235486 + 0.541376I
b = 0.81785 + 2.20014I
9.13999 9.48739I 5.97929 + 7.42423I
u = 1.50901 + 0.28951I
a = 0.235486 0.541376I
b = 0.81785 2.20014I
9.13999 + 9.48739I 5.97929 7.42423I
u = 1.49556 0.25090I
a = 0.429849 0.015499I
b = 0.002514 + 0.418220I
7.31234 5.60210I 2.80476 + 2.85114I
u = 1.49556 + 0.25090I
a = 0.429849 + 0.015499I
b = 0.002514 0.418220I
7.31234 + 5.60210I 2.80476 2.85114I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.365970 0.031939I
a = 0.411373 0.351756I
b = 0.759245 0.489927I
3.50700 1.95785I 4.15911 + 3.79195I
u = 1.365970 + 0.031939I
a = 0.411373 + 0.351756I
b = 0.759245 + 0.489927I
3.50700 + 1.95785I 4.15911 3.79195I
u = 1.359664 0.109695I
a = 0.011264 1.091361I
b = 1.11338 1.47051I
9.84300 4.44580I 7.49047 + 4.00982I
u = 1.359664 + 0.109695I
a = 0.011264 + 1.091361I
b = 1.11338 + 1.47051I
9.84300 + 4.44580I 7.49047 4.00982I
u = 0.363345 0.258087I
a = 1.84934 3.01924I
b = 0.24356 1.66754I
7.29387 3.70140I 0.13489 + 3.24211I
u = 0.363345 + 0.258087I
a = 1.84934 + 3.01924I
b = 0.24356 + 1.66754I
7.29387 + 3.70140I 0.13489 3.24211I
u = 0.218730 0.254700I
a = 0.55232 2.03558I
b = 0.120038 0.786549I
0.06188 1.89506I 3.00107 + 4.96508I
u = 0.218730 + 0.254700I
a = 0.55232 + 2.03558I
b = 0.120038 + 0.786549I
0.06188 + 1.89506I 3.00107 4.96508I
u = 0.002285 0.351914I
a = 0.99770 1.12056I
b = 0.400696 0.103514I
0.739118 + 0.963294I 5.38374 5.24951I
u = 0.002285 + 0.351914I
a = 0.99770 + 1.12056I
b = 0.400696 + 0.103514I
0.739118 0.963294I 5.38374 + 5.24951I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.064994 0.588385I
a = 2.52659 0.35703I
b = 0.497557 + 0.347689I
5.44874 + 2.20051I 0.49288 3.58387I
u = 0.064994 + 0.588385I
a = 2.52659 + 0.35703I
b = 0.497557 0.347689I
5.44874 2.20051I 0.49288 + 3.58387I
u = 0.450262 0.797995I
a = 0.447834 + 0.814481I
b = 0.040046 + 1.223008I
2.77750 + 5.51756I 3.79773 7.77564I
u = 0.450262 + 0.797995I
a = 0.447834 0.814481I
b = 0.040046 1.223008I
2.77750 5.51756I 3.79773 + 7.77564I
u = 0.451556 0.680540I
a = 0.458537 + 0.469762I
b = 0.064813 + 0.604513I
0.98467 + 2.16041I 0.30534 3.36601I
u = 0.451556 + 0.680540I
a = 0.458537 0.469762I
b = 0.064813 0.604513I
0.98467 2.16041I 0.30534 + 3.36601I
u = 0.463101 0.864387I
a = 1.38884 + 0.89316I
b = 0.00016 + 1.91545I
10.75102 + 7.67961I 5.88345 5.74816I
u = 0.463101 + 0.864387I
a = 1.38884 0.89316I
b = 0.00016 1.91545I
10.75102 7.67961I 5.88345 + 5.74816I
u = 0.644526 0.652721I
a = 0.843661 0.304939I
b = 0.369469 0.080340I
3.46087 0.57126I 6.38744 + 1.36032I
u = 0.644526 + 0.652721I
a = 0.843661 + 0.304939I
b = 0.369469 + 0.080340I
3.46087 + 0.57126I 6.38744 1.36032I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.720117 0.732589I
a = 1.19187 1.03395I
b = 0.894217 0.895332I
11.55895 2.26622I 7.69126 + 0.18572I
u = 0.720117 + 0.732589I
a = 1.19187 + 1.03395I
b = 0.894217 + 0.895332I
11.55895 + 2.26622I 7.69126 0.18572I
u = 0.984199 0.167639I
a = 0.095272 + 0.438182I
b = 0.854127 + 0.635027I
1.84031 + 0.69258I 6.89864 + 1.74884I
u = 0.984199 + 0.167639I
a = 0.095272 0.438182I
b = 0.854127 0.635027I
1.84031 0.69258I 6.89864 1.74884I
u = 1.123465 0.320823I
a = 0.347957 + 1.278862I
b = 1.46601 + 2.25189I
8.62893 + 1.18234I 6.12242 + 0.09680I
u = 1.123465 + 0.320823I
a = 0.347957 1.278862I
b = 1.46601 2.25189I
8.62893 1.18234I 6.12242 0.09680I
u = 1.34522
a = 0.382415
b = 0.411122
3.09864 0.703579
u = 1.409359 0.075417I
a = 0.259656 0.542452I
b = 0.34262 2.46670I
5.20973 + 3.10308I 5.68137 4.55677I
u = 1.409359 + 0.075417I
a = 0.259656 + 0.542452I
b = 0.34262 + 2.46670I
5.20973 3.10308I 5.68137 + 4.55677I
u = 1.46107 0.09654I
a = 0.006389 1.108921I
b = 1.48020 4.84555I
13.2971 + 5.0740I 7.49177 2.86395I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.46107 + 0.09654I
a = 0.006389 + 1.108921I
b = 1.48020 + 4.84555I
13.2971 5.0740I 7.49177 + 2.86395I
8
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
4
(u
41
+ 5u
40
+ ··· 3u 1)
c
2
, c
7
u
4
(u
41
+ u
40
+ ··· + 8u + 16)
c
3
, c
4
(u + 1)
4
(u
41
+ 5u
40
+ ··· 3u 1)
c
5
, c
6
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
41
+ 2u
40
+ ··· 3u 1)
c
8
(u
4
u
3
+ u
2
+ 1)(u
41
+ 2u
40
+ ··· 20u 100)
c
9
, c
10
(u
4
u
3
+ 3u
2
2u + 1)(u
41
+ 2u
40
+ ··· 3u 1)
c
11
(u
4
u
3
+ u
2
+ 1)(u
41
+ 12u
40
+ ··· + 549u + 131)
9
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
4
(y 1)
4
(y
41
41y
40
+ ··· 13y 1)
c
2
, c
7
y
4
(y
41
+ 27y
40
+ ··· 960y 256)
c
5
, c
6
, c
9
c
10
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
41
+ 48y
40
+ ··· 7y 1)
c
8
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
41
24y
40
+ ··· 163800y 10000)
c
11
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
41
12y
40
+ ··· + 112237y 17161)
10