11n
47
(K11n
47
)
1
Arc Sequences
4 1 7 2 11 10 4 6 1 8 9
Solving Sequence
1,4
2
5,9
10 11 6 7 3 8
c
1
c
4
c
9
c
11
c
5
c
6
c
3
c
8
c
2
, c
7
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
3
+ u
2
1, b + 1, 2u
2
+ a + 4u + 4i
I
u
2
= hb
6
b
5
b
4
+ 2b
3
b + 1, a 1, u 1i
I
u
3
= hu
35
+ 8u
34
+ ··· + 9u + 1, 1.64269 × 10
25
u
34
1.05850 × 10
26
u
33
+ ··· + 2.49168 × 10
25
a 1.38382 × 10
26
,
7.57271 × 10
25
u
34
+ 6.48827 × 10
26
u
33
+ ··· + 7.97336 × 10
26
b 7.26255 × 10
26
i
There are 3 irreducible components with 44 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
3
+ u
2
1, b + 1, 2u
2
+ a + 4u + 4i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
u
u
2
+ u 1
a
9
=
2u
2
4u 4
1
a
10
=
2u
2
4u 3
1
a
11
=
2u
2
4u 3
1
a
6
=
7u
2
13u 9
u 2
a
7
=
u
u
2
+ u 1
a
3
=
u
2
+ 1
u
2
a
8
=
u
2u
2
+ u 2
a
8
=
u
2u
2
+ u 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 = 0.877439 0.744862I
a = 0.920404 + 0.365165I
b = 1.00000
1.37919 2.82812I 9.0124 + 12.0277I
u = 0.877439 + 0.744862I
a = 0.920404 0.365165I
b = 1.00000
1.37919 + 2.82812I 9.0124 12.0277I
u = 0.754878
a = 8.15919
b = 1.00000
2.75839 102.975
3
II. I
u
2
= hb
6
b
5
b
4
+ 2b
3
b + 1, a 1, u 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
1
0
a
9
=
1
b
a
10
=
b + 1
b
a
11
=
b + 1
b
2
a
6
=
b
3
+ b
2
1
b
4
a
7
=
0
b
5
b
4
2b
3
+ b
2
+ b 1
a
3
=
0
1
a
8
=
0
b
5
b
4
2b
3
+ b
2
+ b 1
a
8
=
0
b
5
b
4
2b
3
+ b
2
+ b 1
(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 = 1.00000
b = 1.002193 0.295542I
3.53554 0.92430I 12.60470 5.55069I
u = 1.00000
a = 1.00000
b = 1.002193 + 0.295542I
3.53554 + 0.92430I 12.60470 + 5.55069I
u = 1.00000
a = 1.00000
b = 0.428243 0.664531I
0.245672 0.924305I 5.68949 + 0.25702I
u = 1.00000
a = 1.00000
b = 0.428243 + 0.664531I
0.245672 + 0.924305I 5.68949 0.25702I
u = 1.00000
a = 1.00000
b = 1.073950 0.558752I
1.64493 + 5.69302I 11.7058 8.3306I
u = 1.00000
a = 1.00000
b = 1.073950 + 0.558752I
1.64493 5.69302I 11.7058 + 8.3306I
5
III. I
u
3
=
hu
35
+8u
34
+· · ·+9u+1, 1.64×10
25
u
34
1.06×10
26
u
33
+· · ·+2.49×10
25
a
1.38 × 10
26
, 7.57 × 10
25
u
34
+ 6.49 × 10
26
u
33
+ · · · + 7.97 × 10
26
b 7.26 × 10
26
i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
u
u
3
+ u
a
9
=
0.659269u
34
+ 4.24813u
33
+ ··· + 120.228u + 5.55377
0.0949751u
34
0.813743u
33
+ ··· + 1.09979u + 0.910851
a
10
=
0.754244u
34
+ 5.06188u
33
+ ··· + 119.128u + 4.64292
0.0949751u
34
0.813743u
33
+ ··· + 1.09979u + 0.910851
a
11
=
0.894280u
34
6.14049u
33
+ ··· 123.987u 2.97552
0.0130827u
34
0.0126909u
33
+ ··· 5.06131u 1.20991
a
6
=
1.77774u
34
+ 17.8440u
33
+ ··· + 23.8681u 10.0326
0.414794u
34
1.96166u
33
+ ··· + 6.00083u + 0.386377
a
7
=
0.135945u
34
+ 1.32830u
33
+ ··· + 28.8699u 0.0316372
0.201606u
34
+ 1.17050u
33
+ ··· + 2.73205u + 0.337551
a
3
=
u
2
+ 1
u
2
a
8
=
0.135945u
34
+ 1.32830u
33
+ ··· + 28.8699u 0.0316372
0.239495u
34
1.52839u
33
+ ··· + 0.429444u + 0.0968115
a
8
=
0.135945u
34
+ 1.32830u
33
+ ··· + 28.8699u 0.0316372
0.239495u
34
1.52839u
33
+ ··· + 0.429444u + 0.0968115
(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 = 1.75068 0.06096I
a = 1.46330 0.02059I
b = 1.82396 + 0. 77537I
14.4753 2.5419I 13.83151 + 2.70823I
u = 1.75068 + 0 .06096I
a = 1.46330 + 0.02059I
b = 1.82396 0. 77537I
14.4753 + 2.5419I 13.83151 2.70823I
u = 1.72022 0.43828I
a = 1.34465 + 0.58112I
b = 1.49438 0.29415I
15.2163 5.6356I 10.75742 + 2.79420I
u = 1.72022 + 0 .43828I
a = 1.34465 0.58112I
b = 1.49438 + 0.29415I
15.2163 + 5.6356I 10.75742 2.79420I
u = 1.70107 0.39786I
a = 1.50913 + 0.63392I
b = 1.58042 0.58476I
15.7071 13.7623I 9.89343 + 6.24021I
u = 1.70107 + 0 .39786I
a = 1.50913 0.63392I
b = 1.58042 + 0.58476I
15.7071 + 13 .7623I 9.89343 6.24021I
u = 1.68600
a = 2.34569
b = 1.28757
11.4779 0.0900530
u = 1.68299 0.18586I
a = 0.383087 + 0 .343481I
b = 0.25892 + 1.52764I
9.90660 6.51942I 9.48439 + 5.38265I
u = 1.68299 + 0 .18586I
a = 0.383087 0.343481I
b = 0.25892 1.52764I
9.90660 + 6 .51942I 9.48439 5.38265I
u = 1.63022 0.11868I
a = 0.069536 0.327596I
b = 0.340175 + 0.686101I
9.25057 1.88240I 8.12432 1.64929I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.63022 + 0 .11868I
a = 0.069536 + 0 .327596I
b = 0.340175 0.686101I
9.25057 + 1.88240I 8.12432 + 1 .64929I
u = 0.990139 0.655507I
a = 0.886995 + 0.313990I
b = 0.934664 + 0 .185167I
1.54213 2.47872I 0.53499 5.93000I
u = 0.990139 + 0 .655507I
a = 0.886995 0.313990I
b = 0.934664 0.185167I
1.54213 + 2.47872I 0.53499 + 5 .93000I
u = 0.686181 0.154265I
a = 1.246624 + 0.427424I
b = 1.126278 + 0 .497250I
1.08296 + 5.42643I 0.21975 3.30530I
u = 0.686181 + 0 .154265I
a = 1.246624 0.427424I
b = 1.126278 0.497250I
1.08296 5.42643I 0.21975 + 3 .30530I
u = 0.294421 0.137620I
a = 0.647696 + 1.089553I
b = 0.225869 + 0 .594231I
1.40601 1.20005I 2.74470 + 1.99044I
u = 0.294421 + 0 .137620I
a = 0.647696 1.089553I
b = 0.225869 0.594231I
1.40601 + 1.20005I 2.74470 1.99044I
u = 0.0306219 0.0974691I
a = 1.93881 9.66636I
b = 1.038378 0 .224787I
1.92040 + 0.80331I 4.44102 + 0 .15082I
u = 0.0306219 + 0 .0974691I
a = 1.93881 + 9.66636I
b = 1.038378 + 0 .224787I
1.92040 0.80331I 4.44102 0.15082I
u = 0.605532 0.380104I
a = 1.67784 + 0.73020I
b = 0.361624 + 0.080090I
1.46738 + 0.11420I 8.20214 0.34884I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.605532 + 0.380104I
a = 1.67784 0.73020I
b = 0.361624 0 .080090I
1.46738 0.11420I 8.20214 + 0 .34884I
u = 0.656190 1.188229I
a = 0.419118 0.434391I
b = 1.47252 0.05220I
7.58070 0.56154I 10.57137 + 0.23440I
u = 0.656190 + 1.188229I
a = 0.419118 + 0.434391I
b = 1.47252 + 0.05220I
7.58070 + 0 .56154I 10.57137 0.23440I
u = 0.679243 0.583622I
a = 0.155632 + 1.137359I
b = 0.397949 0 .909235I
1.63296 + 3 .48211I 7.94104 7.54592I
u = 0.679243 + 0.583622I
a = 0.155632 1.137359I
b = 0.397949 + 0 .909235I
1.63296 3.48211I 7.94104 + 7 .54592I
u = 0.704998
a = 11.0900
b = 1.00991
2.72892 194.387
u = 0.730316 1.119096I
a = 0.576099 0.700870I
b = 1.51689 + 0.33803I
7.84770 + 8 .00129I 9.44018 5.26349I
u = 0.730316 + 1.119096I
a = 0.576099 + 0.700870I
b = 1.51689 0.33803I
7.84770 8.00129I 9.44018 + 5 .26349I
u = 0.779230
a = 1.08295
b = 0.0140385
1.12597 9.35809
u = 0.964380 0.326022I
a = 0.35491 + 1.77356I
b = 1.401851 0 .174541I
4.47629 + 0 .99972I 15.2464 0.4133I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.964380 + 0.326022I
a = 0.35491 1.77356I
b = 1.401851 + 0 .174541I
4.47629 0.99972I 15.2464 + 0 .4133I
u = 1.204595 0.063415I
a = 0.412107 + 0.706810I
b = 0.628022 + 0 .554154I
3.08874 1.42303I 6.41632 + 5 .79805I
u = 1.204595 + 0.063415I
a = 0.412107 0.706810I
b = 0.628022 0 .554154I
3.08874 + 1.42303I 6.41632 5.79805I
u = 1.74717 0 .03675I
a = 1.63248 0.11344I
b = 1.54159 0.19584I
10.19429 4.27290I 10.69965 + 2.77091I
u = 1.74717 + 0 .03675I
a = 1.63248 + 0.11344I
b = 1.54159 + 0 .19584I
10.19429 + 4.27290I 10.69965 2.77091I
10
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
6
(u
3
+ u
2
1)(u
35
+ 8u
34
+ ··· + 9u + 1)
c
2
(u + 1)
6
(u
3
+ u
2
+ 2u + 1)(u
35
+ 42u
34
+ ··· 129u + 1)
c
3
u
6
(u
3
u
2
+ 2u 1)(u
35
+ 2u
34
+ ··· 320u 64)
c
4
(u + 1)
6
(u
3
u
2
+ 1)(u
35
+ 8u
34
+ ··· + 9u + 1)
c
5
(u
3
2u
2
3u 1)(u
6
3u
5
+ 5u
4
4u
3
+ 2u
2
u + 1)
(u
35
+ 8u
34
+ ··· + 73u 31)
c
6
(u
3
2u
2
3u 1)(u
6
u
5
u
4
+ 2u
3
u + 1)
(u
35
+ 4u
34
+ ··· + 1417u 1219)
c
7
u
6
(u
3
+ u
2
+ 2u + 1)(u
35
+ 2u
34
+ ··· 320u 64)
c
8
(u
3
3u
2
+ 2u + 1)(u
6
3u
5
+ 5u
4
4u
3
+ 2u
2
u + 1)
(u
35
+ 3u
34
+ ··· + 2u + 1)
c
9
(u 1)
3
(u
6
+ u
5
+ ··· + u + 1)(u
35
+ 5u
34
+ ··· + 67u + 1)
c
10
u
3
(u
6
+ u
5
+ ··· + u + 1)(u
35
+ 6u
34
+ ··· + 124u 8)
c
11
(u + 1)
3
(u
6
u
5
+ ··· u + 1)(u
35
+ 5u
34
+ ··· + 67u + 1)
11
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
6
(y
3
y
2
+ 2y 1)(y
35
42y
34
+ ··· 129y 1)
c
2
(y 1)
6
(y
3
+ 3y
2
+ 2y 1)(y
35
90y
34
+ ··· + 6323y 1)
c
3
, c
7
y
6
(y
3
+ 3y
2
+ 2y 1)(y
35
36y
34
+ ··· 20480y 4096)
c
5
(y
3
10y
2
+ 5y 1)(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
(y
35
52y
34
+ ··· + 29509y 961)
c
6
(y
3
10y
2
+ 5y 1)(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
(y
35
4y
34
+ ··· + 25178641y 1485961)
c
8
(y
3
5y
2
+ 10y 1)(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
(y
35
+ y
34
+ ··· + 14y 1)
c
9
, c
11
(y 1)
3
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
(y
35
33y
34
+ ··· + 5091y 1)
c
10
y
3
(y
6
3y
5
+ ··· y + 1)(y
35
+ 18y
34
+ ··· + 7312y 64)
12