11a
133
(K11a
133
)
1
Arc Sequences
5 1 8 7 2 10 4 3 11 6 9
Solving Sequence
6,10
7
2,11
5 1 3 4 9 8
c
6
c
10
c
5
c
1
c
2
c
4
c
9
c
8
c
3
, c
7
, c
11
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hb
2
+ 3, u 1, b + 2a 1i
I
u
2
= hb
4
+ 4b
2
+ 12b + 19, u + 1, 2b
3
+ b
2
16b + 30a 31i
I
u
3
= hu
45
+ 3u
44
+ ··· 10u + 3,
1.76440 × 10
28
u
44
1.37529 × 10
29
u
43
+ ··· + 5.50017 × 10
29
b 8.96128 × 10
29
,
2.06193 × 10
30
u
44
6.01232 × 10
30
u
43
+ ··· + 3.30010 × 10
30
a + 6.27804 × 10
30
i
There are 3 irreducible components with 51 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
2
+ 3, u 1, b + 2a 1i
(i) Arc colorings
a
6
=
0
1
a
10
=
1
2
b +
1
2
b
a
7
=
1
2
b
1
2
1
2
b
1
2
a
2
=
1
0
a
11
=
1
2
b +
1
2
1
2
b
1
2
a
5
=
1
1
a
1
=
0
1
a
3
=
1
1
a
4
=
1
1
a
9
=
1
2
b
1
2
1
2
b
1
2
a
8
=
1
2
b
1
2
1
2
b
1
2
a
8
=
1
2
b
1
2
1
2
b
1
2
(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 = 0.500000 0.866025I
b = 1.73205I
1.64493 2.02988I 12.00000 + 3.46410I
u = 1.00000
a = 0.500000 + 0.866025I
b = 1.73205I
1.64493 + 2.02988I 12.00000 3.46410I
3
II. I
u
2
= hb
4
+ 4b
2
+ 12b + 19, u + 1, 2b
3
+ b
2
16b + 30a 31i
(i) Arc colorings
a
6
=
0
1
a
10
=
0.0666667b
3
0.0333333b
2
+ 0.533333b + 1.03333
b
a
7
=
0.0666667b
3
+ 0.0333333b
2
0.533333b 0.0333333
0.0333333b
3
0.266667b
2
0.233333b + 0.266667
a
2
=
1
0
a
11
=
0.0666667b
3
0.0333333b
2
+ 0.533333b + 1.03333
0.0666667b
3
+ 0.0333333b
2
+ 0.466667b 1.03333
a
5
=
1
1
a
1
=
0
1
a
3
=
1
1
a
4
=
0.133333b
3
0.0666667b
2
+ 0.0666667b + 0.0666667
0.133333b
3
0.0666667b
2
+ 0.0666667b 1.93333
a
9
=
0.0666667b
3
0.0333333b
2
+ 0.533333b + 0.0333333
0.0333333b
3
+ 0.266667b
2
+ 0.233333b 0.266667
a
8
=
0.0333333b
3
+ 0.266667b
2
+ 0.233333b 0.266667
0.133333b
3
+ 0.566667b
2
0.0666667b 0.566667
a
8
=
0.0333333b
3
+ 0.266667b
2
+ 0.233333b 0.266667
0.133333b
3
+ 0.566667b
2
0.0666667b 0.566667
(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 = 0.500000 0.866025I
b = 1.22474 1.02494I
3.28987 2.02988I 6.00000 + 3.46410I
u = 1.00000
a = 0.500000 + 0.866025I
b = 1.22474 + 1.02494I
3.28987 + 2.02988I 6.00000 3.46410I
u = 1.00000
a = 0.500000 0.866025I
b = 1.22474 2.43916I
3.28987 2.02988I 6.00000 + 3.46410I
u = 1.00000
a = 0.500000 + 0.866025I
b = 1.22474 + 2.43916I
3.28987 + 2.02988I 6.00000 3.46410I
5
III. I
u
3
=
hu
45
+3u
44
+· · ·10u+3, 1.76×10
28
u
44
1.38×10
29
u
43
+· · ·+5.50×10
29
b
8.96× 10
29
, 2.06× 10
30
u
44
6.01 ×10
30
u
43
+· · · +3.30 ×10
30
a + 6.28× 10
30
i
(i) Arc colorings
a
6
=
0
u
a
10
=
0.624808u
44
+ 1.82186u
43
+ ··· + 4.91181u 1.90238
0.0320791u
44
+ 0.250046u
43
+ ··· 2.98495u + 1.62927
a
7
=
0.0537447u
44
0.344680u
43
+ ··· 5.12163u + 0.267775
0.533105u
44
+ 1.36999u
43
+ ··· + 2.83094u 1.11756
a
2
=
1
0
a
11
=
0.624808u
44
+ 1.82186u
43
+ ··· + 4.91181u 1.90238
0.0136042u
44
+ 0.0158962u
43
+ ··· 0.584872u + 1.47158
a
5
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
u
2
+ 1
u
4
a
4
=
0.462814u
44
+ 1.36316u
43
+ ··· + 7.23893u 4.30830
0.159389u
44
+ 0.403194u
43
+ ··· 0.0731152u + 1.63223
a
9
=
0.856420u
44
+ 2.54555u
43
+ ··· + 0.345192u + 1.03435
0.300455u
44
+ 0.234787u
43
+ ··· 5.03047u + 2.08984
a
8
=
0.461630u
44
+ 1.36094u
43
+ ··· 0.131102u + 1.45161
0.0625802u
44
0.0313115u
43
+ ··· 5.31461u + 1.99257
a
8
=
0.461630u
44
+ 1.36094u
43
+ ··· 0.131102u + 1.45161
0.0625802u
44
0.0313115u
43
+ ··· 5.31461u + 1.99257
(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 = 1.216732 0.046184I
a = 0.549190 + 0.690086I
b = 0.17320 + 1.89525I
0.78716 + 2.48122I 3.04347 4.76589I
u = 1.216732 + 0.046184I
a = 0.549190 0.690086I
b = 0.17320 1.89525I
0.78716 2.48122I 3.04347 + 4.76589I
u = 1.187512 0.743870I
a = 0.072042 + 0.928580I
b = 0.22698 + 2.87369I
12.1487 12.9478I 2.37539 + 7.27389I
u = 1.187512 + 0.743870I
a = 0.072042 0.928580I
b = 0.22698 2.87369I
12.1487 + 12.9478I 2.37539 7.27389I
u = 1.142271 0.782721I
a = 0.834159 0.049636I
b = 0.657108 0.344323I
13.2084 6.7061I 0.79872 + 2.70879I
u = 1.142271 + 0.782721I
a = 0.834159 + 0.049636I
b = 0.657108 + 0.344323I
13.2084 + 6.7061I 0.79872 2.70879I
u = 1.040167 0.307743I
a = 0.115187 0.816672I
b = 0.43757 2.33356I
3.30905 1.49655I 13.54204 0.04800I
u = 1.040167 + 0.307743I
a = 0.115187 + 0.816672I
b = 0.43757 + 2.33356I
3.30905 + 1.49655I 13.54204 + 0.04800I
u = 0.994742 0.649050I
a = 0.362490 0.927648I
b = 0.75685 2.71673I
5.28483 6.46491I 5.51136 + 5.66813I
u = 0.994742 + 0.649050I
a = 0.362490 + 0.927648I
b = 0.75685 + 2.71673I
5.28483 + 6.46491I 5.51136 5.66813I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.915192 0.660915I
a = 0.051046 + 1.058876I
b = 0.56397 + 2.04326I
1.87890 5.45220I 5.89309 + 4.91113I
u = 0.915192 + 0.660915I
a = 0.051046 1.058876I
b = 0.56397 2.04326I
1.87890 + 5.45220I 5.89309 4.91113I
u = 0.868984 0.770535I
a = 0.485413 0.538975I
b = 0.563760 0.225232I
9.56075 2.89885I 0.23004 + 2.78105I
u = 0.868984 + 0.770535I
a = 0.485413 + 0.538975I
b = 0.563760 + 0.225232I
9.56075 + 2.89885I 0.23004 2.78105I
u = 0.796516 0.671250I
a = 1.069165 0.122272I
b = 0.124624 + 0.407415I
2.24775 + 0.28638I 4.98856 0.17511I
u = 0.796516 + 0.671250I
a = 1.069165 + 0.122272I
b = 0.124624 0.407415I
2.24775 0.28638I 4.98856 + 0.17511I
u = 0.742479
a = 0.407884
b = 0.294991
0.995192 10.6148
u = 0.694965 0.663592I
a = 1.093173 + 0.559949I
b = 0.90248 + 1.51031I
6.21731 + 1.33164I 3.44169 0.67479I
u = 0.694965 + 0.663592I
a = 1.093173 0.559949I
b = 0.90248 1.51031I
6.21731 1.33164I 3.44169 + 0.67479I
u = 0.588843 1.039071I
a = 0.207523 + 0.983424I
b = 0.323618 + 0.824642I
14.9288 + 0.1276I 0.71810 + 1.65207I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.588843 + 1.039071I
a = 0.207523 0.983424I
b = 0.323618 0.824642I
14.9288 0.1276I 0.71810 1.65207I
u = 0.500593 1.055939I
a = 1.071280 + 0.262615I
b = 0.847372 + 0.071564I
14.2846 + 6.4583I 0.08655 3.23273I
u = 0.500593 + 1.055939I
a = 1.071280 0.262615I
b = 0.847372 0.071564I
14.2846 6.4583I 0.08655 + 3.23273I
u = 0.094889 0.410145I
a = 1.83128 + 0.14328I
b = 0.154457 0.321024I
0.430811 1.192017I 5.27697 + 5.64355I
u = 0.094889 + 0.410145I
a = 1.83128 0.14328I
b = 0.154457 + 0.321024I
0.430811 + 1.192017I 5.27697 5.64355I
u = 0.333842 0.234428I
a = 0.05196 2.42052I
b = 0.933026 + 0.364690I
4.91236 + 2.29181I 0.89336 4.09144I
u = 0.333842 + 0.234428I
a = 0.05196 + 2.42052I
b = 0.933026 0.364690I
4.91236 2.29181I 0.89336 + 4.09144I
u = 0.569429 0.827480I
a = 1.190610 0.109807I
b = 0.476385 0.339369I
5.31668 4.26099I 1.24439 + 3.98671I
u = 0.569429 + 0.827480I
a = 1.190610 + 0.109807I
b = 0.476385 + 0.339369I
5.31668 + 4.26099I 1.24439 3.98671I
u = 0.668649 0.765905I
a = 0.111980 1.122998I
b = 0.312009 1.130877I
5.57757 + 1.59310I 0.70414 1.88658I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.668649 + 0.765905I
a = 0.111980 + 1.122998I
b = 0.312009 + 1.130877I
5.57757 1.59310I 0.70414 + 1.88658I
u = 0.782063 0.490678I
a = 0.496006 + 0.447495I
b = 0.214606 + 0.360884I
1.25627 + 2.00304I 0.90089 4.84629I
u = 0.782063 + 0.490678I
a = 0.496006 0.447495I
b = 0.214606 0.360884I
1.25627 2.00304I 0.90089 + 4.84629I
u = 0.787608 0.218573I
a = 0.857516 0.995741I
b = 0.20979 1.61031I
1.01308 1.54097I 3.12750 1.49729I
u = 0.787608 + 0.218573I
a = 0.857516 + 0.995741I
b = 0.20979 + 1.61031I
1.01308 + 1.54097I 3.12750 + 1.49729I
u = 0.988691 0.467315I
a = 0.236593 + 0.943426I
b = 0.23771 + 2.54690I
2.33846 + 4.64978I 9.14624 8.10002I
u = 0.988691 + 0.467315I
a = 0.236593 0.943426I
b = 0.23771 2.54690I
2.33846 4.64978I 9.14624 + 8.10002I
u = 0.999315 0.709768I
a = 0.907408 + 0.099888I
b = 0.509084 0.027139I
4.60680 + 4.00969I 2.37481 3.81201I
u = 0.999315 + 0.709768I
a = 0.907408 0.099888I
b = 0.509084 + 0.027139I
4.60680 4.00969I 2.37481 + 3.81201I
u = 1.070764 0.695416I
a = 0.071680 0.979523I
b = 0.18302 2.59107I
3.82469 + 9.97213I 4.07711 8.66746I
10
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.070764 + 0.695416I
a = 0.071680 + 0.979523I
b = 0.18302 + 2.59107I
3.82469 9.97213I 4.07711 + 8.66746I
u = 1.090284 0.213034I
a = 0.189732 + 0.629553I
b = 1.52500 + 2.02902I
2.28059 0.33440I 8.55924 0.81533I
u = 1.090284 + 0.213034I
a = 0.189732 0.629553I
b = 1.52500 2.02902I
2.28059 + 0.33440I 8.55924 + 0.81533I
u = 1.43222 0.04392I
a = 0.554003 0.599044I
b = 0.45629 2.11601I
6.93943 2.89736I 2.44194 + 2.79046I
u = 1.43222 + 0.04392I
a = 0.554003 + 0.599044I
b = 0.45629 + 2.11601I
6.93943 + 2.89736I 2.44194 2.79046I
11
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
2
(u + 1)
4
(u
45
+ 3u
44
+ ··· 10u + 3)
c
2
(u + 1)
6
(u
45
+ 19u
44
+ ··· + 58u + 9)
c
3
, c
4
, c
7
c
8
u
2
(u
2
+ 2)
2
(u
45
+ u
44
+ ··· 28u
2
4)
c
5
(u 1)
4
(u + 1)
2
(u
45
+ 3u
44
+ ··· 10u + 3)
c
6
(u
2
u + 1)
2
(u
2
+ u + 1)(u
45
+ 2u
44
+ ··· + 3u 3)
c
9
(u
2
u + 1)
3
(u
45
+ 14u
44
+ ··· + 21u 9)
c
10
(u
2
u + 1)(u
2
+ u + 1)
2
(u
45
+ 2u
44
+ ··· + 3u 3)
c
11
(u
2
+ u + 1)
3
(u
45
+ 14u
44
+ ··· + 21u 9)
12
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y 1)
6
(y
45
19y
44
+ ··· + 58y 9)
c
2
(y 1)
6
(y
45
+ 21y
44
+ ··· 1838y 81)
c
3
, c
4
, c
7
c
8
y
2
(y + 2)
4
(y
45
+ 55y
44
+ ··· 224y 16)
c
6
, c
10
(y
2
+ y + 1)
3
(y
45
+ 14y
44
+ ··· + 21y 9)
c
9
, c
11
(y
2
+ y + 1)
3
(y
45
+ 38y
44
+ ··· + 10737y 81)
13