11n
151
(K11n
151
)
1
Arc Sequences
4 8 1 2 9 10 3 11 1 8 6
Solving Sequence
8,10
11
3,9
2 7 6 1 5 4
c
10
c
8
c
2
c
7
c
6
c
11
c
5
c
4
c
1
, c
3
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
\
I
v
1
I
u
1
= hu
5
u
4
+ 2u
3
u
2
+ u 1, u
4
2u
2
+ b 2, u
4
u
3
+ 2u
2
+ a u + 1i
I
u
2
= hu
10
u
9
10u
8
+ 11u
7
+ 17u
6
5u
5
+ 12u
4
144u
3
+ 248u
2
160u + 32,
1615u
9
5704u
8
+ ··· + 102868b 69876,
101143u
9
+ 66205u
8
+ ··· + 822944a + 7931776i
I
u
3
= hu
11
u
10
4u
9
+ 5u
8
+ 12u
7
10u
6
+ 24u
5
7u
4
+ 9u
3
3u
2
+ 2u 1,
39502u
10
38316u
9
+ ··· + 40941a + 87943, 54952u
10
+ 52350u
9
+ ··· + 40941b 162724i
I
v
1
= hb
5
+ 5b
4
+ 6b
3
+ 3b
2
+ b + 1, 2b
4
14b
3
30b
2
21b + 5v 4, ai
There are 4 irreducible components with 31 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
5
u
4
+ 2u
3
u
2
+ u 1, u
4
2u
2
+ b 2, u
4
u
3
+ 2u
2
+ a u + 1i
(i) Arc colorings
a
8
=
0
u
a
10
=
u
4
+ u
3
2u
2
+ u 1
u
4
+ 2u
2
+ 2
a
11
=
u
4
+ u
3
2u
2
+ u 1
u
4
+ 2u
2
u + 2
a
3
=
1
0
a
9
=
u
4
+ u
3
2u
2
+ u 1
u
4
+ 2u
2
+ 2
a
2
=
1
u
2
a
7
=
u
u
a
6
=
u
3
2u 1
2u
4
+ 3u
2
+ 3u + 2
a
1
=
u
3
u
3
u
a
5
=
u
u
a
4
=
u
3
u
4
u
3
+ u
2
+ 1
a
4
=
u
3
u
4
u
3
+ 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 = 0.339110 0.822375I
a = 0.428550 1.039275I
b = 0.881366 + 0.489365I
1.97403 + 1.53058I 3.57269 4.45807I
u = 0.339110 + 0.822375I
a = 0.428550 + 1.039275I
b = 0.881366 0.489365I
1.97403 1.53058I 3.57269 + 4.45807I
u = 0.455697 1.200152I
a = 0.276511 0.728237I
b = 0.142272 + 0.509071I
7.51750 4.40083I 3.44484 + 1.78781I
u = 0.455697 + 1.200152I
a = 0.276511 + 0.728237I
b = 0.142272 0.509071I
7.51750 + 4.40083I 3.44484 1.78781I
u = 0.766826
a = 1.30408
b = 3.52181
4.04602 15.9649
3
II. I
u
2
= hu
10
u
9
+ · · · 160u + 32, 1615u
9
5704u
8
+ · · · + 102868b
69876, 1.01 × 10
5
u
9
+ 6.62 × 10
4
u
8
+ · · · + 8.23 × 10
5
a + 7.93 × 10
6
i
(i) Arc colorings
a
8
=
0
u
a
10
=
0.122904u
9
0.0804490u
8
+ ··· + 22.6749u 9.63829
0.0156997u
9
+ 0.0554497u
8
+ ··· 0.929930u + 0.679278
a
11
=
0.122904u
9
0.0804490u
8
+ ··· + 22.6749u 9.63829
0.0156997u
9
0.0554497u
8
+ ··· + 1.92993u 0.679278
a
3
=
1
0
a
9
=
0.187002u
9
0.0779980u
8
+ ··· + 33.1162u 13.5285
0.0762798u
9
+ 0.113104u
8
+ ··· + 5.29834u 1.06478
a
2
=
1
u
2
a
7
=
u
u
a
6
=
0.142121u
9
+ 0.0802023u
8
+ ··· 26.9580u + 11.1052
0.0879161u
9
+ 0.0336135u
8
+ ··· + 10.5503u 2.80838
a
1
=
0.0212274u
9
0.0369272u
8
+ ··· + 5.01316u 2.46646
0.0544873u
9
0.0428510u
8
+ ··· + 9.24319u 2.46603
a
5
=
0.0757148u
9
+ 0.0797782u
8
+ ··· 14.2563u + 4.93250
0.193491u
9
+ 0.258001u
8
+ ··· + 12.3162u 2.59606
a
4
=
0.0212274u
9
+ 0.0369272u
8
+ ··· 5.01316u + 2.46646
0.125637u
9
+ 0.0740755u
8
+ ··· + 12.4344u 2.96843
a
4
=
0.0212274u
9
+ 0.0369272u
8
+ ··· 5.01316u + 2.46646
0.125637u
9
+ 0.0740755u
8
+ ··· + 12.4344u 2.96843
(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 = 2.55655 0.21740I
a = 0.319009 + 0.542444I
b = 0.05818 1.69128I
9.31336 + 3.33174I 3.28666 2.53508I
u = 2.55655 + 0.21740I
a = 0.319009 0.542444I
b = 0.05818 + 1.69128I
9.31336 3.33174I 3.28666 + 2.53508I
u = 0.65997 1.49495I
a = 0.360854 + 0.242256I
b = 0.233677 0.885557I
0.17487 + 2.21397I 2.88087 4.04855I
u = 0.65997 + 1.49495I
a = 0.360854 0.242256I
b = 0.233677 + 0.885557I
0.17487 2.21397I 2.88087 + 4.04855I
u = 0.361047
a = 3.61721
b = 0.416284
2.52712 3.66494
u = 0.824006 0.612321I
a = 0.955128 + 0.641215I
b = 0.233677 0.885557I
0.17487 + 2.21397I 2.88087 4.04855I
u = 0.824006 + 0.612321I
a = 0.955128 0.641215I
b = 0.233677 + 0.885557I
0.17487 2.21397I 2.88087 + 4.04855I
u = 0.959943
a = 0.138228
b = 0.416284
2.52712 3.66494
u = 2.23202 0.03584I
a = 0.402776 + 0.684883I
b = 0.05818 1.69128I
9.31336 + 3.33174I 3.28666 2.53508I
u = 2.23202 + 0.03584I
a = 0.402776 0.684883I
b = 0.05818 + 1.69128I
9.31336 3.33174I 3.28666 + 2.53508I
5
III. I
u
3
= hu
11
u
10
+ · · · + 2u 1, 39502u
10
38316u
9
+ · · · + 40941a +
87943, 54952u
10
+ 52350u
9
+ · · · + 40941b 162724i
(i) Arc colorings
a
8
=
0
u
a
10
=
0.964852u
10
+ 0.935883u
9
+ ··· + 1.95716u 2.14804
1.34222u
10
1.27867u
9
+ ··· + 0.504336u + 3.97460
a
11
=
0.964852u
10
+ 0.935883u
9
+ ··· + 1.95716u 2.14804
1.58570u
10
1.47402u
9
+ ··· 0.402579u + 3.94563
a
3
=
1
0
a
9
=
0.620845u
10
+ 0.538140u
9
+ ··· + 0.445421u 1.79759
1.09875u
10
1.08332u
9
+ ··· + 1.41125u + 4.00357
a
2
=
1
u
2
a
7
=
u
u
a
6
=
0.548961u
10
+ 0.491097u
9
+ ··· 2.31211u 2.28868
1.46384u
10
0.459515u
9
+ ··· + 5.64102u + 4.43665
a
1
=
0.350456u
10
0.00644830u
9
+ ··· + 0.128771u 0.810825
0.0289685u
10
0.272441u
9
+ ··· + 0.218339u + 0.964852
a
5
=
0.379424u
10
+ 0.278889u
9
+ ··· 0.347109u 0.154027
0.130822u
10
0.243277u
9
+ ··· + 0.0399844u + 0.864317
a
4
=
0.350456u
10
+ 0.00644830u
9
+ ··· 0.128771u + 0.810825
0.0827044u
10
0.0116509u
9
+ ··· + 0.555898u + 0.620845
a
4
=
0.350456u
10
+ 0.00644830u
9
+ ··· 0.128771u + 0.810825
0.0827044u
10
0.0116509u
9
+ ··· + 0.555898u + 0.620845
(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 = 1.84060 0.85046I
a = 0.390361 + 0.701026I
b = 0.10260 1.75202I
7.84139 + 5.06071I 4.48302 2.40182I
u = 1.84060 + 0.85046I
a = 0.390361 0.701026I
b = 0.10260 + 1.75202I
7.84139 5.06071I 4.48302 + 2.40182I
u = 0.313592 0.489723I
a = 0.319343 0.436527I
b = 0.322788 0.550650I
0.003691 + 1.266704I 0.27668 5.30833I
u = 0.313592 + 0.489723I
a = 0.319343 + 0.436527I
b = 0.322788 + 0.550650I
0.003691 1.266704I 0.27668 + 5.30833I
u = 0.168753 0.621565I
a = 0.641624 1.164851I
b = 1.76349 + 0.21001I
2.76312 + 1.08944I 13.75530 + 1.30535I
u = 0.168753 + 0.621565I
a = 0.641624 + 1.164851I
b = 1.76349 0.21001I
2.76312 1.08944I 13.75530 1.30535I
u = 0.299337 0.876682I
a = 0.298312 + 1.339046I
b = 0.009586 0.293616I
8.16883 4.71969I 15.8344 + 7.6612I
u = 0.299337 + 0.876682I
a = 0.298312 1.339046I
b = 0.009586 + 0.293616I
8.16883 + 4.71969I 15.8344 7.6612I
u = 0.410996
a = 2.61719
b = 6.69648
3.78211 32.5964
u = 1.98061 0.99673I
a = 0.361613 + 0.691909I
b = 0.38195 1.94651I
7.4453 12.4339I 4.94880 + 5.95992I
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.98061 + 0.99673I
a = 0.361613 0.691909I
b = 0.38195 + 1.94651I
7.4453 + 12.4339I 4.94880 5.95992I
7
IV. I
v
1
= hb
5
+ 5b
4
+ 6b
3
+ 3b
2
+ b + 1, 2b
4
14b
3
30b
2
21b + 5v 4, ai
(i) Arc colorings
a
8
=
2
5
b
4
+
14
5
b
3
+ ··· +
21
5
b +
4
5
0
a
10
=
0
b
a
11
=
4
5
b
4
+
23
5
b
3
+ ··· +
32
5
b +
8
5
b
a
3
=
1
0
a
9
=
3
5
b
4
16
5
b
3
+ ···
24
5
b
11
5
2
5
b
4
9
5
b
3
+ ···
6
5
b
4
5
a
2
=
1
0
a
7
=
2
5
b
4
+
14
5
b
3
+ ··· +
21
5
b +
4
5
0
a
6
=
2
5
b
4
+
14
5
b
3
+ ··· +
21
5
b +
4
5
2
5
b
4
+
9
5
b
3
+ 2b
2
+
6
5
b +
4
5
a
1
=
4
5
b
4
+
18
5
b
3
+ 3b
2
+
2
5
b
7
5
1
a
5
=
4
5
b
4
18
5
b
3
+ ···
2
5
b +
7
5
1
a
4
=
4
5
b
4
18
5
b
3
+ ···
2
5
b +
12
5
1
a
4
=
4
5
b
4
18
5
b
3
+ ···
2
5
b +
12
5
1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 0.345770
a = 0
b = 3.52181
4.04602 15.9649
v = 0.003977 + 0.626138I
a = 0
b = 0.881366 0.489365I
1.97403 + 1.53058I 3.57269 4.45807I
v = 0.003977 0.626138I
a = 0
b = 0.881366 + 0.489365I
1.97403 1.53058I 3.57269 + 4.45807I
v = 0.32314 2.69669I
a = 0
b = 0.142272 0.509071I
7.51750 4.40083I 3.44484 + 1.78781I
v = 0.32314 + 2.69669I
a = 0
b = 0.142272 + 0.509071I
7.51750 + 4.40083I 3.44484 1.78781I
9
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
8
(u 1)
5
(u
5
+ u
4
2u
3
u
2
+ u 1)
(u
10
+ 2u
9
+ 5u
8
+ u
7
+ 3u
6
8u
5
2u
4
19u
3
6u
2
8u 1)
(u
11
+ 5u
10
+ ··· + 2u 1)
c
2
, c
9
u
5
(u
5
u
4
+ ··· + u 1)(u
10
+ u
9
+ ··· + 160u + 32)
(u
11
+ u
10
+ ··· + 2u + 1)
c
3
, c
4
, c
10
(u + 1)
5
(u
5
u
4
2u
3
+ u
2
+ u + 1)
(u
10
+ 2u
9
+ 5u
8
+ u
7
+ 3u
6
8u
5
2u
4
19u
3
6u
2
8u 1)
(u
11
+ 5u
10
+ ··· + 2u 1)
c
5
(u
5
u
4
u
3
+ 4u
2
3u + 1)(u
5
u
4
+ 2u
3
u
2
+ u 1)
(u
10
10u
8
+ 43u
6
17u
5
35u
4
46u
3
+ 64u
2
+ 38u 29)
(u
11
+ u
10
3u
8
+ 12u
7
10u
6
6u
5
+ 33u
4
31u
3
+ 33u
2
10u + 11)
c
6
(u
5
u
4
u
3
+ 4u
2
3u + 1)(u
5
+ u
4
2u
3
u
2
+ u 1)
(u
10
+ 2u
9
+ ··· 100u 43)(u
11
+ u
10
+ ··· 33u
2
27)
c
7
u
5
(u
5
+ u
4
+ ··· + u + 1)(u
10
+ u
9
+ ··· + 160u + 32)
(u
11
+ u
10
+ ··· + 2u + 1)
c
11
(u
5
3u
4
+ 4u
3
u
2
u + 1)(u
5
+ u
4
u
2
+ u + 1)
2
(u
5
+ 3u
4
+ 4u
3
+ u
2
u 1)
(u
11
+ u
10
u
8
+ 8u
7
+ 12u
6
+ 8u
5
3u
4
+ 3u
3
+ 3u
2
1)
10
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
4
c
8
, c
10
(y 1)
5
(y
5
5y
4
+ ··· y 1)(y
10
+ 6y
9
+ ··· 52y + 1)
(y
11
9y
10
+ ··· 10y 1)
c
2
, c
7
, c
9
y
5
(y
5
+ 3y
4
+ ··· y 1)(y
10
21y
9
+ ··· 9728y + 1024)
(y
11
9y
10
+ ··· 2y 1)
c
5
(y
5
3y
4
+ 3y
3
8y
2
+ y 1)(y
5
+ 3y
4
+ 4y
3
+ y
2
y 1)
(y
10
20y
9
+ ··· 5156y + 841)(y
11
y
10
+ ··· 626y 121)
c
6
(y
5
5y
4
+ 8y
3
3y
2
y 1)(y
5
3y
4
+ 3y
3
8y
2
+ y 1)
(y
10
+ 20y
9
+ ··· 13440y + 1849)
(y
11
+ 15y
10
+ ··· 1782y 729)
c
11
(y
5
y
4
+ 4y
3
3y
2
+ 3y 1)
2
(y
5
y
4
+ 8y
3
3y
2
+ 3y 1)
2
(y
11
y
10
+ ··· + 6y 1)
11