11n
106
(K11n
106
)
1
Arc Sequences
5 1 7 8 2 11 9 5 4 6 7
Solving Sequence
1,5 2,9
8 4 10 7 3 11 6
c
1
c
8
c
4
c
9
c
7
c
3
c
11
c
6
c
2
, c
5
, c
10
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hu 1, a, b 1i
I
u
2
= hu
3
u 1, u
2
+ a + 1, u
2
+ b ui
I
u
3
= ha
4
2a
2
+ 2, u + 1, a
3
+ a
2
+ b a 1i
I
u
4
= hu
17
2u
16
+ 3u
15
+ 9u
13
14u
12
+ 24u
11
+ 2u
10
+ 4u
9
+ 21u
8
3u
7
10u
6
9u
5
34u
4
+ 8u
3
+ 11u + 5,
43789804u
16
+ 273859943u
15
+ ··· + 1476039512b + 2472690329,
1402009421u
16
3746051727u
15
+ ··· + 7380197560a + 5637736551i
There are 4 irreducible components with 25 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 1, a, b 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
a
2
=
1
1
a
9
=
0
1
a
8
=
0
1
a
4
=
0
1
a
10
=
0
1
a
7
=
0
1
a
3
=
0
1
a
11
=
1
1
a
6
=
1
0
a
6
=
1
0
(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
b = 1.00000
0 0
3
II. I
u
2
= hu
3
u 1, u
2
+ a + 1, u
2
+ b ui
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
9
=
u
2
1
u
2
+ u
a
8
=
u
2
1
u
2
+ 2u
a
4
=
u
2
+ 1
u
2
u
a
10
=
u
2
1
u
2
+ u
a
7
=
0
u
a
3
=
u
2
+ 1
u
2
a
11
=
1
u
2
a
6
=
u
1
a
6
=
u
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 = 0.662359 0.562280I
a = 0.877439 + 0.744862I
b = 0.78492 1.30714I
1.64493 6.00000
u = 0.662359 + 0.562280I
a = 0.877439 0.744862I
b = 0.78492 + 1.30714I
1.64493 6.00000
u = 1.32472
a = 0.754878
b = 0.430160
1.64493 6.00000
5
III. I
u
3
= ha
4
2a
2
+ 2, u + 1, a
3
+ a
2
+ b a 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
a
2
=
1
1
a
9
=
a
a
3
a
2
+ a + 1
a
8
=
a
a
3
a
2
+ 2a + 1
a
4
=
a
2
a
3
+ a + 1
a
10
=
a
3
a
1
a
7
=
a
3
+ a
1
a
3
=
0
1
a
11
=
a
3
a + 1
1
a
6
=
1
0
a
6
=
1
0
(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 = 1.00000
a = 1.098684 0.455090I
b = 0.455090 + 0.098684I
2.46740 + 3.66386I 4.00000 4.00000I
u = 1.00000
a = 1.098684 + 0.455090I
b = 0.455090 0.098684I
2.46740 3.66386I 4.00000 + 4.00000I
u = 1.00000
a = 1.098684 0.455090I
b = 0.45509 + 2.09868I
2.46740 3.66386I 4.00000 + 4.00000I
u = 1.00000
a = 1.098684 + 0.455090I
b = 0.45509 2.09868I
2.46740 + 3.66386I 4.00000 4.00000I
7
IV.
I
u
4
= hu
17
2u
16
+ · · · + 11u + 5, 4.38 × 10
7
u
16
+ 2.74 × 10
8
u
15
+ · · · + 1.48 ×
10
9
b +2.47 ×10
9
, 1.40× 10
9
u
16
3.75× 10
9
u
15
+· · ·+ 7.38 ×10
9
a +5.64 ×10
9
i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
9
=
0.189969u
16
+ 0.507581u
15
+ ··· + 0.732229u 0.763900
0.0296671u
16
0.185537u
15
+ ··· 0.740141u 1.67522
a
8
=
0.189969u
16
+ 0.507581u
15
+ ··· + 0.732229u 0.763900
0.0206064u
16
0.219826u
15
+ ··· 1.19437u 2.31344
a
4
=
0.0136971u
16
0.139975u
15
+ ··· 2.79220u 0.241157
0.187648u
16
0.169545u
15
+ ··· + 4.56051u + 2.34383
a
10
=
0.164938u
16
+ 0.216359u
15
+ ··· 2.34588u 2.04206
0.101612u
16
0.158274u
15
+ ··· + 1.78525u 0.404645
a
7
=
0.349694u
16
+ 0.785424u
15
+ ··· 2.79714u 1.07137
0.155247u
16
0.405587u
15
+ ··· + 1.29961u 0.779403
a
3
=
u
2
+ 1
u
2
a
11
=
0.0698451u
16
0.145243u
15
+ ··· 0.141235u + 1.26583
0.0950934u
16
0.0711159u
15
+ ··· + 2.48712u + 1.77623
a
6
=
u
u
3
+ u
a
6
=
u
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.19233 1.21311I
a = 0.453684 0.382519I
b = 0.344784 + 0.561209I
13.5757 4.4662I 2.93957 + 1.91782I
u = 1.19233 + 1.21311I
a = 0.453684 + 0.382519I
b = 0.344784 0.561209I
13.5757 + 4.4662I 2.93957 1.91782I
u = 0.944025 0.315827I
a = 0.819174 0.703142I
b = 0.30791 + 1.68630I
0.05634 3.87007I 0.55814 + 7.00568I
u = 0.944025 + 0.315827I
a = 0.819174 + 0.703142I
b = 0.30791 1.68630I
0.05634 + 3.87007I 0.55814 7.00568I
u = 0.366752
a = 1.93747
b = 1.01848
1.26530 7.99448
u = 0.228487 0.823991I
a = 0.27775 + 1.43908I
b = 0.93908 1.59767I
5.04445 4.17066I 6.60682 + 3.70952I
u = 0.228487 + 0.823991I
a = 0.27775 1.43908I
b = 0.93908 + 1.59767I
5.04445 + 4.17066I 6.60682 3.70952I
u = 0.127840 0.766680I
a = 0.913824 + 0.406300I
b = 0.274898 + 0.378400I
2.46422 0.66350I 3.92785 + 1.28554I
u = 0.127840 + 0.766680I
a = 0.913824 0.406300I
b = 0.274898 0.378400I
2.46422 + 0.66350I 3.92785 1.28554I
u = 0.336838 1.133591I
a = 0.016621 1.106933I
b = 0.46322 + 2.04947I
6.78062 + 4.50780I 6.98768 3.92800I
9
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.336838 + 1.133591I
a = 0.016621 + 1.106933I
b = 0.46322 2.04947I
6.78062 4.50780I 6.98768 + 3.92800I
u = 0.876256 0.096452I
a = 0.547958 + 0.410962I
b = 0.426652 0.425827I
1.48592 + 0.26904I 6.62187 0.62877I
u = 0.876256 + 0.096452I
a = 0.547958 0.410962I
b = 0.426652 + 0.425827I
1.48592 0.26904I 6.62187 + 0.62877I
u = 1.14210 1.37916I
a = 0.675667 0.421236I
b = 1.25795 + 1.54860I
18.0935 0.6930I 6.24024 + 0.75440I
u = 1.14210 + 1.37916I
a = 0.675667 + 0.421236I
b = 1.25795 1.54860I
18.0935 + 0.6930I 6.24024 0.75440I
u = 1.32086 1.15950I
a = 0.351916 + 0.720069I
b = 0.48688 2.87606I
17.3459 + 9.9963I 5.48062 4.80381I
u = 1.32086 + 1.15950I
a = 0.351916 0.720069I
b = 0.48688 + 2.87606I
17.3459 9.9963I 5.48062 + 4.80381I
10
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)(u + 1)
4
(u
3
u + 1)(u
17
+ 2u
16
+ ··· + 11u 5)
c
2
(u + 1)
5
(u
3
+ 2u
2
+ u + 1)(u
17
2u
16
+ ··· + 121u + 25)
c
3
(u)(u 1)
3
(u
4
+ 2u
2
+ 2)(u
17
+ 4u
16
+ ··· 7540u 3866)
c
4
, c
8
(u)(u + 1)
3
(u
4
2u
2
+ 2)(u
17
2u
16
+ ··· 4u + 2)
c
5
(u 1)
4
(u + 1)(u
3
u + 1)(u
17
+ 2u
16
+ ··· + 11u 5)
c
6
(u 1)
4
(u + 1)(u
3
u 1)(u
17
+ 2u
16
+ ··· 17u + 5)
c
7
(u)(u + 1)
3
(2 + 2u + u
2
)
2
(u
17
+ 10u
16
+ ··· + 8u + 4)
c
9
u
4
(u
4
+ 2u
2
+ 2)(u
17
+ 3u
16
+ ··· + 32u + 46)
c
10
, c
11
(u 1)(u + 1)
4
(u
3
u 1)(u
17
+ 2u
16
+ ··· 17u + 5)
11
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y 1)
5
(y
3
2y
2
+ y 1)(y
17
+ 2y
16
+ ··· + 121y 25)
c
2
(y 1)
5
(y
3
2y
2
3y 1)(y
17
+ 50y
16
+ ··· + 40441y 625)
c
3
y(y 1)
3
(y
2
+ 2y + 2)
2
(y
17
+ 70y
16
+ ··· + 33732920y 14945956)
c
4
, c
8
(y)(y 1)
3
(2 2y + y
2
)
2
(y
17
10y
16
+ ··· + 8y 4)
c
6
, c
10
, c
11
(y 1)
5
(y
3
2y
2
+ y 1)(y
17
30y
16
+ ··· + 329y 25)
c
7
(y)(y 1)
3
(y
2
+ 4)
2
(y
17
6y
16
+ ··· 96y 16)
c
9
y
4
(y
2
+ 2y + 2)
2
(y
17
+ 31y
16
+ ··· + 14640y 2116)
12