11n
141
(K11n
141
)
1
Arc Sequences
8 1 9 8 11 10 2 4 1 6 5
Solving Sequence
1,8
2
3,5
4 7 11 6 10 9
c
1
c
2
c
4
c
7
c
11
c
5
c
10
c
9
c
3
, c
6
, c
8
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
4
+ 3u
2
+ 1, u
2
+ a 2, u
3
+ b 2ui
I
u
2
= ha
6
+ 3a
4
7a
3
a + 5, 11a
5
28a
4
56a
3
28a
2
+ 95a + 59u + 49,
20a
5
+ 8a
4
+ 75a
3
110a
2
+ 59b + 15a 73i
I
u
3
= hu
8
+ 3u
7
+ 9u
6
+ 16u
5
+ 23u
4
+ 24u
3
+ 18u
2
+ 7u + 2, u
4
u
3
3u
2
+ b 2u 1,
u
7
+ 3u
6
+ 9u
5
+ 16u
4
+ 21u
3
+ 22u
2
+ 2a + 12u + 3i
There are 3 irreducible components with 18 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
4
+ 3u
2
+ 1, u
2
+ a 2, u
3
+ b 2ui
(i) Arc colorings
a
1
=
1
0
a
8
=
u
2
+ 2
u
3
+ 2u
a
2
=
u
3
3u + 1
1
a
3
=
u
3
3u
1
a
5
=
0
u
a
4
=
u
3
3u
u + 1
a
7
=
u
3
2u
0
a
11
=
1
u
2
a
6
=
u
u
3
+ u
a
10
=
u
2
+ 1
u
2
+ 1
a
9
=
0
u
2
+ 1
a
9
=
0
u
2
+ 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 = 0.618034I
a = 1.61803
b = 1.00000I
2.30291 0
u = 0.618034I
a = 1.61803
b = 1.00000I
2.30291 0
u = 1.61803I
a = 0.618034
b = 1.00000I
5.59278 0
u = 1.61803I
a = 0.618034
b = 1.00000I
5.59278 0
3
II. I
u
2
=
ha
6
+3a
4
7a
3
a+5, 11a
5
+59u+· · ·+95a+49, 20a
5
+59b+· · ·+15a73i
(i) Arc colorings
a
1
=
1
0
a
8
=
a
0.338983a
5
0.135593a
4
+ ··· 0.254237a + 1.23729
a
2
=
0.135593a
5
+ 0.254237a
4
+ ··· 0.898305a 0.694915
0.474576a
5
+ 0.389831a
4
+ ··· 1.64407a + 0.0677966
a
3
=
0.338983a
5
0.135593a
4
+ ··· + 0.745763a 0.762712
0.474576a
5
+ 0.389831a
4
+ ··· 1.64407a + 0.0677966
a
5
=
0
0.186441a
5
+ 0.474576a
4
+ ··· 1.61017a 0.830508
a
4
=
0.389831a
5
0.355932a
4
+ ··· + 0.457627a + 2.37288
1
a
7
=
0.0508475a
5
0.220339a
4
+ ··· + 0.711864a + 1.13559
0.322034a
5
0.728814a
4
+ ··· + 2.50847a + 1.52542
a
11
=
1
0.0508475a
5
0.220339a
4
+ ··· + 0.711864a + 2.13559
a
6
=
0.186441a
5
0.474576a
4
+ ··· + 1.61017a + 0.830508
0.135593a
5
0.254237a
4
+ ··· + 0.898305a 0.305085
a
10
=
0.0508475a
5
+ 0.220339a
4
+ ··· 0.711864a 1.13559
0.135593a
5
+ 0.254237a
4
+ ··· 0.898305a + 0.305085
a
9
=
0.0847458a
5
0.0338983a
4
+ ··· + 0.186441a 1.44068
0.135593a
5
+ 0.254237a
4
+ ··· 0.898305a + 0.305085
a
9
=
0.0847458a
5
0.0338983a
4
+ ··· + 0.186441a 1.44068
0.135593a
5
+ 0.254237a
4
+ ··· 0.898305a + 0.305085
(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 = 0.569840
a = 0.66236 2.09661I
b = 0.377439 + 1.194730I
4.40332 5.01951
u = 0.569840
a = 0.66236 + 2.09661I
b = 0.377439 1.194730I
4.40332 5.01951
u = 0.215080 + 1.307141I
a = 0.479348 0.738390I
b = 0.01536 + 1.53025I
0.26574 2.82812I 1.50976 + 2.97945I
u = 0.215080 1.307141I
a = 0.479348 + 0.738390I
b = 0.01536 1.53025I
0.26574 + 2.82812I 1.50976 2.97945I
u = 0.215080 1.307141I
a = 1.141707 0.176111I
b = 0.862082 + 0.785389I
0.26574 + 2.82812I 1.50976 2.97945I
u = 0.215080 + 1.307141I
a = 1.141707 + 0.176111I
b = 0.862082 0.785389I
0.26574 2.82812I 1.50976 + 2.97945I
5
III.
I
u
3
= hu
8
+3u
7
+· · ·+7u+2, u
4
u
3
3u
2
+b2u1, u
7
+3u
6
+· · ·+2a+3i
(i) Arc colorings
a
1
=
1
0
a
8
=
1
2
u
7
3
2
u
6
+ ··· 6u
3
2
u
4
+ u
3
+ 3u
2
+ 2u + 1
a
2
=
1
2
u
7
1
2
u
6
+ ··· u +
1
2
u
7
+ 2u
6
+ 6u
5
+ 8u
4
+ 10u
3
+ 8u
2
+ 3u + 1
a
3
=
3
2
u
7
5
2
u
6
+ ··· 4u
1
2
u
7
+ 2u
6
+ 6u
5
+ 8u
4
+ 10u
3
+ 8u
2
+ 3u + 1
a
5
=
0
u
a
4
=
1
2
u
7
1
2
u
6
+ ··· u
1
2
u
7
+ 2u
6
+ 6u
5
+ 8u
4
+ 10u
3
+ 8u
2
+ 4u + 1
a
7
=
u
3
2u
u
5
+ 3u
3
+ u
a
11
=
1
u
2
a
6
=
u
u
3
+ u
a
10
=
u
2
+ 1
u
4
2u
2
a
9
=
u
4
+ 3u
2
+ 1
u
4
2u
2
a
9
=
u
4
+ 3u
2
+ 1
u
4
2u
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 = 0.927504 0.597003I
a = 0.84716 1.52084I
b = 0.12220 + 1.91634I
16.1792 3.0379I 2.50917 + 2.22003I
u = 0.927504 + 0.597003I
a = 0.84716 + 1.52084I
b = 0.12220 1.91634I
16.1792 + 3.0379I 2.50917 2.22003I
u = 0.34865 1.60107I
a = 1.021154 + 0.429864I
b = 0.33222 1.78481I
9.04281 7.80349I 0.02756 + 3.21559I
u = 0.34865 + 1.60107I
a = 1.021154 0.429864I
b = 0.33222 + 1.78481I
9.04281 + 7.80349I 0.02756 3.21559I
u = 0.218002 0.455338I
a = 0.456034 + 0.648640I
b = 0.195934 0.349055I
0.133570 0.902562I 2.84755 + 7.78366I
u = 0.218002 + 0.455338I
a = 0.456034 0.648640I
b = 0.195934 + 0.349055I
0.133570 + 0.902562I 2.84755 7.78366I
u = 0.00585 1.54991I
a = 0.380027 0.263352I
b = 0.405950 + 0.590547I
6.99421 1.46497I 5.63406 + 4.72165I
u = 0.00585 + 1.54991I
a = 0.380027 + 0.263352I
b = 0.405950 0.590547I
6.99421 + 1.46497I 5.63406 4.72165I
7
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
3
, c
4
c
7
, c
8
(u
2
+ 1)
2
(u
6
+ u
5
+ 4u
4
+ 4u
3
+ 6u
2
+ 4u + 5)
(u
8
+ 7u
6
u
5
+ 14u
4
4u
3
+ 5u
2
+ u + 1)
c
2
(u + 1)
4
(u
6
+ 7u
5
+ 20u
4
+ 34u
3
+ 44u
2
+ 44u + 25)
(u
8
+ 14u
7
+ 77u
6
+ 205u
5
+ 260u
4
+ 140u
3
+ 61u
2
+ 9u + 1)
c
5
, c
6
, c
10
c
11
(u
3
u
2
+ 2u 1)
2
(u
4
+ 3u
2
+ 1)
(u
8
+ 3u
7
+ 9u
6
+ 16u
5
+ 23u
4
+ 24u
3
+ 18u
2
+ 7u + 2)
c
9
(u
2
u 1)
2
(u
3
+ u
2
1)
2
(u
8
u
7
+ 21u
6
24u
5
+ 109u
4
142u
3
10u
2
+ 23u + 24)
8
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
4
c
7
, c
8
(y + 1)
4
(y
6
+ 7y
5
+ 20y
4
+ 34y
3
+ 44y
2
+ 44y + 25)
(y
8
+ 14y
7
+ 77y
6
+ 205y
5
+ 260y
4
+ 140y
3
+ 61y
2
+ 9y + 1)
c
2
(y 1)
4
(y
6
9y
5
+ 12y
4
+ 38y
3
56y
2
+ 264y + 625)
(y
8
42y
7
+ ··· + 41y + 1)
c
5
, c
6
, c
10
c
11
(y
2
+ 3y + 1)
2
(y
3
+ 3y
2
+ 2y 1)
2
(y
8
+ 9y
7
+ 31y
6
+ 50y
5
+ 47y
4
+ 64y
3
+ 80y
2
+ 23y + 4)
c
9
(y
2
3y + 1)
2
(1 + 2y y
2
+ y
3
)
2
(y
8
+ 41y
7
+ ··· 1009y + 576)
9