11n
23
(K11n
23
)
1
Arc Sequences
4 1 7 2 8 10 4 11 6 1 9
Solving Sequence
9,11 1,4
2 5 8 6 7 3 10
c
11
c
1
c
4
c
8
c
5
c
7
c
3
c
10
c
2
, c
6
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
2
u 1, b 1, a u + 1i
I
u
2
= hu
6
+ 4u
4
+ u
3
+ 4u
2
+ 1, u
5
+ u
4
2u
3
+ u
2
+ 3b + u + 2, u
5
+ u
4
5u
3
+ 4u
2
+ 3a 5u + 5i
I
u
3
= hu
22
2u
21
+ ··· + 112u + 31,
7.21440 × 10
40
u
21
+ 2.06732 × 10
41
u
20
+ ··· + 1.03131 × 10
43
b 7.73692 × 10
42
,
8.57722 × 10
42
u
21
2.50879 × 10
43
u
20
+ ··· + 3.19707 × 10
44
a 4.85712 × 10
44
i
There are 3 irreducible components with 30 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
9
=
0
u
a
11
=
u 1
1
a
1
=
u 1
u + 1
a
4
=
1
0
a
2
=
2u 2
u + 1
a
5
=
2u 3
u + 2
a
8
=
u + 1
u 1
a
6
=
0
u 1
a
7
=
0
u 1
a
3
=
1
u 2
a
10
=
0
u
a
10
=
0
u
(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 = 0.618034
a = 1.61803
b = 1.00000
7.23771 1.00000
u = 1.61803
a = 0.618034
b = 1.00000
0.657974 1.00000
3
II. I
u
2
= hu
6
+ 4u
4
+ u
3
+ 4u
2
+ 1, u
5
+ u
4
2u
3
+ u
2
+ 3b + u + 2, u
5
+
u
4
5u
3
+ 4u
2
+ 3a 5u + 5i
(i) Arc colorings
a
9
=
0
u
a
11
=
1
3
u
5
1
3
u
4
+ ··· +
5
3
u
5
3
1
3
u
5
1
3
u
4
+ ···
1
3
u
2
3
a
1
=
1
3
u
5
1
3
u
4
+ ··· +
5
3
u
5
3
1
a
4
=
1
0
a
2
=
1
3
u
5
1
3
u
4
+ ··· +
5
3
u
2
3
1
a
5
=
1
3
u
5
+
1
3
u
4
+ ···
5
3
u +
5
3
1
a
8
=
2
3
u
5
1
3
u
4
+ ···
4
3
u
5
3
0
a
6
=
4
3
u
5
2
3
u
4
+ ···
11
3
u
1
3
1
a
7
=
2
3
u
5
1
3
u
4
+ ···
4
3
u
5
3
0
a
3
=
1
0
a
10
=
5
3
u
5
2
3
u
4
+ ··· +
13
3
u
4
3
2
3
u
5
2
3
u
4
+ ··· +
4
3
u
4
3
a
10
=
5
3
u
5
2
3
u
4
+ ··· +
13
3
u
4
3
2
3
u
5
2
3
u
4
+ ··· +
4
3
u
4
3
(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 = 0.341164 0.940004I
a = 0.158836 1.200143I
b = 0.108378 + 0.818891I
1.64493 5.69302I 8.89162 + 7.09196I
u = 0.341164 + 0.940004I
a = 0.158836 + 1.200143I
b = 0.108378 0.818891I
1.64493 + 5.69302I 8.89162 7.09196I
u = 0.084211 0.566250I
a = 1.258209 0.569162I
b = 0.659772 + 0.298454I
3.53554 + 0.92430I 10.88169 1.11590I
u = 0.084211 + 0.566250I
a = 1.258209 + 0.569162I
b = 0.659772 0.298454I
3.53554 0.92430I 10.88169 + 1.11590I
u = 0.25695 1.72779I
a = 0.082955 + 0.592379I
b = 0.23185 1.65564I
0.245672 + 0.924305I 6.22669 + 0.83820I
u = 0.25695 + 1.72779I
a = 0.082955 0.592379I
b = 0.23185 + 1.65564I
0.245672 0.924305I 6.22669 0.83820I
5
III. I
u
3
=
hu
22
2u
21
+· · ·+112u+31, 7.21×10
40
u
21
+2.07×10
41
u
20
+· · ·+1.03×10
43
b
7.74 × 10
42
, 8.58 × 10
42
u
21
2.51 × 10
43
u
20
+ · · · + 3.20 × 10
44
a 4.86 × 10
44
i
(i) Arc colorings
a
9
=
0
u
a
11
=
0.0268284u
21
+ 0.0784715u
20
+ ··· + 1.45270u + 1.51924
0.00699535u
21
0.0200455u
20
+ ··· 0.954967u + 0.750201
a
1
=
0.0268284u
21
+ 0.0784715u
20
+ ··· + 1.45270u + 1.51924
0.0177629u
21
0.0462456u
20
+ ··· + 0.992606u + 1.51946
a
4
=
1
0
a
2
=
0.0445912u
21
+ 0.124717u
20
+ ··· + 0.460090u 0.000218937
0.0177629u
21
0.0462456u
20
+ ··· + 0.992606u + 1.51946
a
5
=
0.102508u
21
+ 0.260561u
20
+ ··· 11.5117u 3.91868
0.0327906u
21
0.0843196u
20
+ ··· + 2.88618u + 2.26626
a
8
=
0.0361949u
21
0.0822923u
20
+ ··· + 1.47809u + 6.12161
0.0277521u
21
+ 0.0642706u
20
+ ··· 4.16882u 1.85424
a
6
=
0.0100498u
21
+ 0.00871961u
20
+ ··· 11.0552u 1.49420
0.0177629u
21
+ 0.0462456u
20
+ ··· 0.992606u 1.51946
a
7
=
0.00844279u
21
0.0180216u
20
+ ··· 2.69073u + 4.26737
0.0277521u
21
+ 0.0642706u
20
+ ··· 4.16882u 1.85424
a
3
=
0.0830533u
21
0.192993u
20
+ ··· + 16.1097u + 7.57835
0.0539918u
21
+ 0.135230u
20
+ ··· 6.39021u 3.13813
a
10
=
0.0849522u
21
0.195970u
20
+ ··· + 21.0393u + 4.71313
0.0313914u
21
+ 0.0795010u
20
+ ··· 2.93678u 1.48386
a
10
=
0.0849522u
21
0.195970u
20
+ ··· + 21.0393u + 4.71313
0.0313914u
21
+ 0.0795010u
20
+ ··· 2.93678u 1.48386
(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 = 1.23634 1.25211I
a = 0.332965 + 0.126245I
b = 1.40111 0.75463I
0.990371 + 0.924237I 8.66470 0.43219I
u = 1.23634 + 1.25211I
a = 0.332965 0.126245I
b = 1.40111 + 0.75463I
0.990371 0.924237I 8.66470 + 0.43219I
u = 0.55625 1.78181I
a = 0.666785 0.085232I
b = 0.27191 + 1.80152I
18.8826 3.9450I 10.61812 + 1.00166I
u = 0.55625 + 1.78181I
a = 0.666785 + 0.085232I
b = 0.27191 1.80152I
18.8826 + 3.9450I 10.61812 1.00166I
u = 0.551322
a = 0.242776
b = 0.391099
0.845350 11.9656
u = 0.489488
a = 3.04886
b = 1.18058
10.9429 8.28675
u = 0.307182 0.702948I
a = 1.145983 0.369509I
b = 0.122623 + 0.098224I
2.04648 0.07308I 6.61841 0.32192I
u = 0.307182 + 0.702948I
a = 1.145983 + 0.369509I
b = 0.122623 0.098224I
2.04648 + 0.07308I 6.61841 + 0.32192I
u = 0.140598 0.270065I
a = 3.34058 2.11350I
b = 0.431084 + 0.709735I
0.69505 + 5.13446I 2.61215 4.09914I
u = 0.140598 + 0.270065I
a = 3.34058 + 2.11350I
b = 0.431084 0.709735I
0.69505 5.13446I 2.61215 + 4.09914I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.04554 2.54475I
a = 0.295335 0.488495I
b = 0.11518 + 2.75329I
16.6374 + 10.8083I 8.69774 4.99684I
u = 0.04554 + 2.54475I
a = 0.295335 + 0.488495I
b = 0.11518 2.75329I
16.6374 10.8083I 8.69774 + 4.99684I
u = 0.33395 2.00823I
a = 0.479936 0.409661I
b = 0.113345 + 1.400723I
6.06672 5.11915I 9.50515 + 3.92885I
u = 0.33395 + 2.00823I
a = 0.479936 + 0.409661I
b = 0.113345 1.400723I
6.06672 + 5.11915I 9.50515 3.92885I
u = 0.346023 1.045974I
a = 0.996654 0.470899I
b = 0.79848 + 1.81489I
6.52288 1.21996I 10.21847 + 1.61822I
u = 0.346023 + 1.045974I
a = 0.996654 + 0.470899I
b = 0.79848 1.81489I
6.52288 + 1.21996I 10.21847 1.61822I
u = 0.54416 2.47711I
a = 0.392068 + 0.280485I
b = 0.40292 2.59906I
12.75242 3.06700I 8.27093 + 2.19380I
u = 0.54416 + 2.47711I
a = 0.392068 0.280485I
b = 0.40292 + 2.59906I
12.75242 + 3.06700I 8.27093 2.19380I
u = 0.596875 0.272500I
a = 1.46986 + 0.26415I
b = 0.606013 0.510709I
1.72824 0.76607I 3.12936 + 1.22783I
u = 0.596875 + 0.272500I
a = 1.46986 0.26415I
b = 0.606013 + 0.510709I
1.72824 + 0.76607I 3.12936 1.22783I
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.700392
a = 1.18728
b = 0.991565
7.60774 21.1716
u = 3.08804
a = 0.346088
b = 2.44319
0.373053 36.4235
8
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
6
(u
2
+ u 1)(u
22
+ 8u
21
+ ··· + 10u + 1)
c
2
(u + 1)
6
(u
2
+ 3u + 1)(u
22
+ 36u
21
+ ··· + 6u + 1)
c
3
, c
7
u
6
(u
2
+ u 1)(u
22
+ 2u
21
+ ··· + 128u
2
+ 64)
c
4
(u + 1)
6
(u
2
+ u 1)(u
22
+ 8u
21
+ ··· + 10u + 1)
c
5
(u
2
+ 3u + 1)(u
6
3u
5
+ 5u
4
4u
3
+ 2u
2
u + 1)
(u
22
+ 3u
21
+ ··· + u + 1)
c
6
u
2
(u
6
+ u
5
+ ··· + u + 1)(u
22
+ 2u
21
+ ··· + 28u + 4)
c
8
(u 1)
2
(u
6
u
5
+ ··· u + 1)(u
22
+ 4u
21
+ ··· + 8u + 1)
c
9
u
2
(u
6
u
5
+ ··· u + 1)(u
22
+ 2u
21
+ ··· + 28u + 4)
c
10
(u + 1)
2
(u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)
(u
22
+ 8u
21
+ ··· + 64u + 1)
c
11
(u 1)
2
(u
6
+ u
5
+ ··· + u + 1)(u
22
+ 4u
21
+ ··· + 8u + 1)
9
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
6
(y
2
3y + 1)(y
22
36y
21
+ ··· 6y + 1)
c
2
(y 1)
6
(y
2
7y + 1)(y
22
92y
21
+ ··· + 3898y + 1)
c
3
, c
7
y
6
(y
2
3y + 1)(y
22
42y
21
+ ··· + 16384y + 4096)
c
5
(y
2
7y + 1)(y
6
+ y
5
+ ··· + 3y + 1)(y
22
49y
21
+ ··· 17y + 1)
c
6
, c
9
y
2
(y
6
3y
5
+ ··· y + 1)(y
22
18y
21
+ ··· 264y + 16)
c
8
, c
11
(y 1)
2
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
(y
22
8y
21
+ ··· 64y + 1)
c
10
(y 1)
2
(y
6
+ y
5
+ ··· + 3y + 1)(y
22
+ 16y
21
+ ··· 3112y + 1)
10