11n
41
(K11n
41
)
1
Arc Sequences
4 1 7 2 10 9 4 11 1 8 6
Solving Sequence
1,4
2
5,10
6 9 7 3 11 8
c
1
c
4
c
5
c
9
c
6
c
3
c
11
c
8
c
2
, c
7
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hb
6
+ b
5
b
4
2b
3
+ b + 1, u 1, b
4
+ 2b
2
+ a 2i
I
u
2
= hu
3
+ u
2
1, b, 3u
2
+ a 5u 4i
I
u
3
= hu
35
+ 8u
34
+ ··· + 9u + 1, 5.06381 × 10
26
u
34
+ 3.18576 × 10
27
u
33
+ ··· + 7.97336 × 10
26
b 1.25386 × 10
26
,
4.65925 × 10
25
u
34
+ 3.98163 × 10
24
u
33
+ ··· + 7.97336 × 10
26
a + 6.92161 × 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
= hb
6
+ b
5
b
4
2b
3
+ b + 1, u 1, b
4
+ 2b
2
+ a 2i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
1
0
a
10
=
b
4
2b
2
+ 2
b
a
6
=
b
5
2b
3
+ 2b 1
b
2
a
9
=
b
4
2b
2
b + 2
b
a
7
=
0
b
4
a
3
=
0
1
a
11
=
b
4
2b
2
+ 2
b
4
a
8
=
0
b
4
a
8
=
0
b
4
(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 = 0.415113 0.381252I
b = 1.073950 0.558752I
1.64493 + 5.69302I 11.7058 8.3306I
u = 1.00000
a = 0.415113 + 0.381252I
b = 1.073950 + 0.558752I
1.64493 5.69302I 11.7058 + 8.3306I
u = 1.00000
a = 2.25915 1.43225I
b = 0.428243 0.664531I
3.53554 0.92430I 12.60470 5.55069I
u = 1.00000
a = 2.25915 + 1.43225I
b = 0.428243 + 0.664531I
3.53554 + 0.92430I 12.60470 + 5.55069I
u = 1.00000
a = 0.655968 + 0.098281I
b = 1.002193 0.295542I
0.245672 0.924305I 5.68949 + 0.25702I
u = 1.00000
a = 0.655968 0.098281I
b = 1.002193 + 0.295542I
0.245672 + 0.924305I 5.68949 0.25702I
3
II. I
u
2
= hu
3
+ u
2
1, b, 3u
2
+ a 5u 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
10
=
3u
2
+ 5u + 4
0
a
6
=
9u
2
+ 15u + 12
u
2
+ u 1
a
9
=
3u
2
+ 5u + 4
0
a
7
=
u
u
2
+ u 1
a
3
=
u
2
+ 1
u
2
a
11
=
3u
2
+ 6u + 4
2u
2
u + 2
a
8
=
u
2u
2
+ u 2
a
8
=
u
2u
2
+ u 2
(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 = 0.877439 0.744862I
a = 0.258045 + 0.197115I
b = 0
1.37919 2.82812I 9.0124 + 12.0277I
u = 0.877439 + 0.744862I
a = 0.258045 0.197115I
b = 0
1.37919 + 2.82812I 9.0124 12.0277I
u = 0.754878
a = 9.48391
b = 0
2.75839 102.975
5
III. I
u
3
=
hu
35
+ 8u
34
+ · · · +9u + 1, 5.06 × 10
26
u
34
+ 3.19 × 10
27
u
33
+ · · · + 7.97 × 10
26
b
1.25× 10
26
, 4.66 ×10
25
u
34
+3.98 × 10
24
u
33
+· · · +7.97 × 10
26
a + 6.92 ×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
10
=
0.0584352u
34
0.00499367u
33
+ ··· + 87.4344u 0.868092
0.635091u
34
3.99551u
33
+ ··· + 1.26311u + 0.157257
a
6
=
1.72094u
34
17.6948u
33
+ ··· + 67.6060u + 13.1818
0.291426u
34
1.97724u
33
+ ··· 9.21351u 0.566549
a
9
=
0.693526u
34
+ 3.99051u
33
+ ··· + 86.1713u 1.02535
0.635091u
34
3.99551u
33
+ ··· + 1.26311u + 0.157257
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
11
=
0.242664u
34
0.573718u
33
+ ··· 63.3575u 0.192908
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
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 = 0.777922 + 1.044169I
b = 1.93922 + 1.67518I
14.4753 2.5419I 13.83151 + 2.70823I
u = 1.75068 + 0.06096I
a = 0.777922 1.044169I
b = 1.93922 1.67518I
14.4753 + 2.5419I 13.83151 2.70823I
u = 1.72022 0.43828I
a = 0.296229 + 1.018866I
b = 0.61061 + 1.35195I
15.2163 5.6356I 10.75742 + 2.79420I
u = 1.72022 + 0.43828I
a = 0.296229 1.018866I
b = 0.61061 1.35195I
15.2163 + 5.6356I 10.75742 2.79420I
u = 1.70107 0.39786I
a = 0.17360 1.42986I
b = 1.20989 1.48650I
15.7071 13.7623I 9.89343 + 6.24021I
u = 1.70107 + 0.39786I
a = 0.17360 + 1.42986I
b = 1.20989 + 1.48650I
15.7071 + 13.7623I 9.89343 6.24021I
u = 1.68600
a = 0.808929
b = 0.920335
11.4779 0.0900530
u = 1.68299 0.18586I
a = 0.18783 + 1.58377I
b = 1.25568 + 1.90551I
9.90660 6.51942I 9.48439 + 5.38265I
u = 1.68299 + 0.18586I
a = 0.18783 1.58377I
b = 1.25568 1.90551I
9.90660 + 6.51942I 9.48439 5.38265I
u = 1.63022 0.11868I
a = 0.279124 1.207661I
b = 0.397690 0.969208I
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 = 0.279124 + 1.207661I
b = 0.397690 + 0.969208I
9.25057 + 1.88240I 8.12432 + 1.64929I
u = 0.990139 0.655507I
a = 0.328445 + 0.034132I
b = 0.537541 0.273251I
1.54213 2.47872I 0.53499 5.93000I
u = 0.990139 + 0.655507I
a = 0.328445 0.034132I
b = 0.537541 + 0.273251I
1.54213 + 2.47872I 0.53499 + 5.93000I
u = 0.686181 0.154265I
a = 0.838455 + 0.370991I
b = 0.977826 + 0.650468I
1.08296 + 5.42643I 0.21975 3.30530I
u = 0.686181 + 0.154265I
a = 0.838455 0.370991I
b = 0.977826 0.650468I
1.08296 5.42643I 0.21975 + 3.30530I
u = 0.294421 0.137620I
a = 1.30767 + 1.56416I
b = 0.805847 + 0.442462I
1.40601 1.20005I 2.74470 + 1.99044I
u = 0.294421 + 0.137620I
a = 1.30767 1.56416I
b = 0.805847 0.442462I
1.40601 + 1.20005I 2.74470 1.99044I
u = 0.0306219 0.0974691I
a = 7.02532 6.31954I
b = 0.458623 0.535266I
1.92040 + 0.80331I 4.44102 + 0.15082I
u = 0.0306219 + 0.0974691I
a = 7.02532 + 6.31954I
b = 0.458623 + 0.535266I
1.92040 0.80331I 4.44102 0.15082I
u = 0.605532 0.380104I
a = 1.43703 0.00546I
b = 0.043259 + 0.568475I
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 = 1.43703 + 0.00546I
b = 0.043259 0.568475I
1.46738 0.11420I 8.20214 + 0.34884I
u = 0.656190 1.188229I
a = 0.395829 + 0.005674I
b = 0.109496 1.311600I
7.58070 0.56154I 10.57137 + 0.23440I
u = 0.656190 + 1.188229I
a = 0.395829 0.005674I
b = 0.109496 + 1.311600I
7.58070 + 0.56154I 10.57137 0.23440I
u = 0.679243 0.583622I
a = 0.692444 + 0.020033I
b = 0.491434 1.250359I
1.63296 + 3.48211I 7.94104 7.54592I
u = 0.679243 + 0.583622I
a = 0.692444 0.020033I
b = 0.491434 + 1.250359I
1.63296 3.48211I 7.94104 + 7.54592I
u = 0.704998
a = 12.6158
b = 0.141812
2.72892 194.387
u = 0.730316 1.119096I
a = 0.586120 + 0.340107I
b = 0.70143 + 1.39478I
7.84770 + 8.00129I 9.44018 5.26349I
u = 0.730316 + 1.119096I
a = 0.586120 0.340107I
b = 0.70143 1.39478I
7.84770 8.00129I 9.44018 + 5.26349I
u = 0.779230
a = 0.816856
b = 0.118472
1.12597 9.35809
u = 0.964380 0.326022I
a = 1.34533 + 1.00277I
b = 1.165204 0.364382I
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 = 1.34533 1.00277I
b = 1.165204 + 0.364382I
4.47629 0.99972I 15.2464 + 0.4133I
u = 1.204595 0.063415I
a = 1.45117 + 1.83159I
b = 0.035022 + 0.979858I
3.08874 1.42303I 6.41632 + 5.79805I
u = 1.204595 + 0.063415I
a = 1.45117 1.83159I
b = 0.035022 0.979858I
3.08874 + 1.42303I 6.41632 5.79805I
u = 1.74717 0.03675I
a = 0.316839 1.172532I
b = 0.41033 1.45360I
10.19429 4.27290I 10.69965 + 2.77091I
u = 1.74717 + 0.03675I
a = 0.316839 + 1.172532I
b = 0.41033 + 1.45360I
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
u
5
u
4
+ 2u
3
u + 1)
(u
35
+ 4u
34
+ ··· + 1417u 1219)
c
6
(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
7
u
6
(u
3
+ u
2
+ 2u + 1)(u
35
+ 2u
34
+ ··· 320u 64)
c
8
(u 1)
3
(u
6
+ u
5
+ ··· + u + 1)(u
35
+ 5u
34
+ ··· + 67u + 1)
c
9
u
3
(u
6
u
5
+ ··· u + 1)(u
35
+ 6u
34
+ ··· + 124u 8)
c
10
(u + 1)
3
(u
6
u
5
+ ··· u + 1)(u
35
+ 5u
34
+ ··· + 67u + 1)
c
11
(u
3
+ 3u
2
+ 2u 1)(u
6
3u
5
+ 5u
4
4u
3
+ 2u
2
u + 1)
(u
35
+ 3u
34
+ ··· + 2u + 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
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
(y
35
4y
34
+ ··· + 25178641y 1485961)
c
6
(y
3
10y
2
+ 5y 1)(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
(y
35
52y
34
+ ··· + 29509y 961)
c
8
, c
10
(y 1)
3
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
(y
35
33y
34
+ ··· + 5091y 1)
c
9
y
3
(y
6
3y
5
+ ··· y + 1)(y
35
+ 18y
34
+ ··· + 7312y 64)
c
11
(y
3
5y
2
+ 10y 1)(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
(y
35
+ y
34
+ ··· + 14y 1)
12