11n
56
(K11n
56
)
1
Arc Sequences
4 1 8 2 9 10 4 1 11 7 6
Solving Sequence
6,10
7 11
1,4
2 3 9 5 8
c
6
c
10
c
11
c
1
c
2
c
9
c
5
c
8
c
3
, c
4
, c
7
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
6
3u
5
+ 5u
4
6u
3
+ 5u
2
2u + 1, u
2
+ b u + 1, u
5
3u
4
+ 4u
3
4u
2
+ a + 3ui
I
u
2
= hu
23
2u
22
+ ··· + 367u 49,
5.68751 × 10
50
u
22
1.18514 × 10
51
u
21
+ ··· + 5.75345 × 10
52
b + 1.35846 × 10
53
,
3.01665 × 10
52
u
22
+ 4.90207 × 10
52
u
21
+ ··· + 2.81919 × 10
54
a + 7.70665 × 10
54
i
There are 2 irreducible components with 29 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
6
3u
5
+5u
4
6u
3
+5u
2
2u+1, u
2
+bu+1, u
5
3u
4
+4u
3
4u
2
+a+3ui
(i) Arc colorings
a
6
=
0
u
a
10
=
u
5
+ 3u
4
4u
3
+ 4u
2
3u
u
2
+ u 1
a
7
=
u
5
2u
4
+ 2u
3
u
2
+ 1
0
a
11
=
u
5
+ 3u
4
5u
3
+ 6u
2
5u + 1
u
2
+ u 1
a
1
=
u
5
+ 3u
4
5u
3
+ 6u
2
5u + 1
1
a
4
=
1
0
a
2
=
u
5
+ 3u
4
5u
3
+ 6u
2
5u + 2
1
a
3
=
1
0
a
9
=
u
5
+ 4u
4
7u
3
+ 8u
2
6u + 2
u
5
+ 2u
4
2u
3
+ u
2
1
a
5
=
u
5
3u
4
+ 5u
3
6u
2
+ 5u 1
1
a
8
=
u
5
2u
4
+ 2u
3
u
2
+ 1
0
a
8
=
u
5
2u
4
+ 2u
3
u
2
+ 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 = 0.082955 0.592379I
a = 1.000936 + 0.863088I
b = 0.573013 0.494098I
0.245672 + 0.924305I 1.96974 0.88960I
u = 0.082955 + 0.592379I
a = 1.000936 0.863088I
b = 0.573013 + 0.494098I
0.245672 0.924305I 1.96974 + 0.88960I
u = 0.158836 1.200143I
a = 0.573950 + 0.818891I
b = 0.573950 0.818891I
1.64493 + 5.69302I 5.23279 6.15196I
u = 0.158836 + 1.200143I
a = 0.573950 0.818891I
b = 0.573950 + 0.818891I
1.64493 5.69302I 5.23279 + 6.15196I
u = 1.258209 0.569162I
a = 0.573013 0.494098I
b = 1.000936 + 0.863088I
3.53554 0.92430I 6.79748 + 1.68215I
u = 1.258209 + 0.569162I
a = 0.573013 + 0.494098I
b = 1.000936 0.863088I
3.53554 + 0.92430I 6.79748 1.68215I
3
II. I
u
2
=
hu
23
2u
22
+· · ·+367u49, 5.69×10
50
u
22
1.19×10
51
u
21
+· · ·+5.75×10
52
b+
1.36× 10
53
, 3.02× 10
52
u
22
+4.90 ×10
52
u
21
+· · · +2.82 ×10
54
a + 7.71× 10
54
i
(i) Arc colorings
a
6
=
0
u
a
10
=
0.0107004u
22
0.0173882u
21
+ ··· 1.05033u 2.73364
0.00988539u
22
+ 0.0205987u
21
+ ··· + 14.3739u 2.36113
a
7
=
0.0328763u
22
0.0594760u
21
+ ··· 26.0161u + 1.49329
0.00781349u
22
+ 0.0173350u
21
+ ··· + 13.0699u 2.37221
a
11
=
0.0351693u
22
+ 0.0665411u
21
+ ··· + 38.9804u 7.11048
0.00460608u
22
+ 0.00773529u
21
+ ··· + 4.24228u 0.128227
a
1
=
0.0351693u
22
+ 0.0665411u
21
+ ··· + 38.9804u 7.11048
0.00598730u
22
+ 0.00966183u
21
+ ··· + 4.57192u + 0.0578458
a
4
=
1
0
a
2
=
0.0291820u
22
+ 0.0568793u
21
+ ··· + 34.4084u 7.16833
0.00598730u
22
+ 0.00966183u
21
+ ··· + 4.57192u + 0.0578458
a
3
=
0.0412034u
22
+ 0.0761334u
21
+ ··· + 40.9608u 6.70310
0.00330400u
22
+ 0.00397147u
21
+ ··· + 0.335622u + 0.840868
a
9
=
0.0136810u
22
0.0201398u
21
+ ··· 0.801159u 1.71464
0.00759533u
22
+ 0.0156870u
21
+ ··· + 10.3240u 1.69836
a
5
=
0.00592089u
22
0.00459915u
21
+ ··· + 4.67825u 2.12275
0.00710141u
22
+ 0.0129475u
21
+ ··· + 7.13226u 0.882426
a
8
=
0.0250628u
22
0.0421410u
21
+ ··· 12.9462u 0.878920
0.00781349u
22
+ 0.0173350u
21
+ ··· + 13.0699u 2.37221
a
8
=
0.0250628u
22
0.0421410u
21
+ ··· 12.9462u 0.878920
0.00781349u
22
+ 0.0173350u
21
+ ··· + 13.0699u 2.37221
(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 = 1.07712 1.98419I
a = 0.152782 + 0.539582I
b = 0.18589 3.42522I
17.5522 1.4869I 8.02521 0.25180I
u = 1.07712 + 1.98419I
a = 0.152782 0.539582I
b = 0.18589 + 3.42522I
17.5522 + 1.4869I 8.02521 + 0.25180I
u = 0.450330 0.386164I
a = 1.192137 0.349994I
b = 0.274299 0.316242I
1.43616 + 0.72615I 6.25783 0.91942I
u = 0.450330 + 0.386164I
a = 1.192137 + 0.349994I
b = 0.274299 + 0.316242I
1.43616 0.72615I 6.25783 + 0.91942I
u = 0.323737 0.843029I
a = 0.126553 + 1.328597I
b = 0.341364 1.077299I
0.53628 + 5.30661I 1.77241 5.11876I
u = 0.323737 + 0.843029I
a = 0.126553 1.328597I
b = 0.341364 + 1.077299I
0.53628 5.30661I 1.77241 + 5.11876I
u = 0.29675 1.85371I
a = 0.155497 0.643499I
b = 1.20424 + 2.27978I
5.12386 + 5.67209I 6.64054 5.01271I
u = 0.29675 + 1.85371I
a = 0.155497 + 0.643499I
b = 1.20424 2.27978I
5.12386 5.67209I 6.64054 + 5.01271I
u = 0.17909 3.14085I
a = 0.201654 0.365084I
b = 1.46869 + 3.76733I
15.5409 10.3372I 6.00224 + 5.46879I
u = 0.17909 + 3.14085I
a = 0.201654 + 0.365084I
b = 1.46869 3.76733I
15.5409 + 10.3372I 6.00224 5.46879I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.09815 2.48508I
a = 0.339801 0.116043I
b = 0.209654 + 0.712700I
12.76380 5.09874I 3.17808 + 1.98307I
u = 0.09815 + 2.48508I
a = 0.339801 + 0.116043I
b = 0.209654 0.712700I
12.76380 + 5.09874I 3.17808 1.98307I
u = 0.158914 0.203116I
a = 2.75483 1.26024I
b = 0.093987 0.797110I
0.87588 + 1.51254I 2.24997 5.09221I
u = 0.158914 + 0.203116I
a = 2.75483 + 1.26024I
b = 0.093987 + 0.797110I
0.87588 1.51254I 2.24997 + 5.09221I
u = 0.263822 0.275329I
a = 1.16937 2.61099I
b = 0.424385 + 0.547005I
1.85876 1.44380I 2.27537 + 0.68239I
u = 0.263822 + 0.275329I
a = 1.16937 + 2.61099I
b = 0.424385 0.547005I
1.85876 + 1.44380I 2.27537 0.68239I
u = 0.316473 1.333068I
a = 0.438410 0.189559I
b = 0.516877 0.245886I
2.25261 + 1.36983I 3.69794 1.43293I
u = 0.316473 + 1.333068I
a = 0.438410 + 0.189559I
b = 0.516877 + 0.245886I
2.25261 1.36983I 3.69794 + 1.43293I
u = 0.843379 0.494457I
a = 0.340280 1.032145I
b = 1.43017 0.55226I
9.64155 2.65369I 2.71409 + 2.86915I
u = 0.843379 + 0.494457I
a = 0.340280 + 1.032145I
b = 1.43017 + 0.55226I
9.64155 + 2.65369I 2.71409 2.86915I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.03478 2.11364I
a = 0.054363 + 0.513460I
b = 0.65023 2.97975I
5.57676 2.33070I 7.43736 + 2.84176I
u = 1.03478 + 2.11364I
a = 0.054363 0.513460I
b = 0.65023 + 2.97975I
5.57676 + 2.33070I 7.43736 2.84176I
u = 1.61562
a = 0.444987
b = 1.23934
2.53646 1.61888
7
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
6
(u
23
+ 7u
22
+ ··· 7u 1)
c
2
(u + 1)
6
(u
23
+ 35u
22
+ ··· + 11u + 1)
c
3
, c
7
u
6
(u
23
+ u
22
+ ··· + 128u 64)
c
4
(u + 1)
6
(u
23
+ 7u
22
+ ··· 7u 1)
c
5
(u
6
u
5
+ ··· u + 1)(u
23
+ 2u
22
+ ··· 108u + 36)
c
6
(u
6
+ u
5
+ ··· + u + 1)(u
23
+ 2u
22
+ ··· 2u 1)
c
8
(u
6
u
5
+ ··· u + 1)(u
23
+ 24u
21
+ ··· + 2u 1)
c
9
(u
6
3u
5
+ ··· u + 1)(u
23
+ 12u
22
+ ··· + 2u + 1)
c
10
(u
6
u
5
+ ··· u + 1)(u
23
+ 2u
22
+ ··· 2u 1)
c
11
(u
6
3u
5
+ ··· u + 1)(u
23
+ 6u
22
+ ··· 18u 7)
8
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
6
(y
23
35y
22
+ ··· + 11y 1)
c
2
(y 1)
6
(y
23
87y
22
+ ··· + 15y 1)
c
3
, c
7
y
6
(y
23
+ 39y
22
+ ··· + 28672y 4096)
c
5
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
(y
23
+ 12y
22
+ ··· + 10296y 1296)
c
6
, c
10
(y
6
3y
5
+ ··· y + 1)(y
23
12y
22
+ ··· + 2y 1)
c
8
(y
6
3y
5
+ ··· y + 1)(y
23
+ 48y
22
+ ··· + 2y 1)
c
9
(y
6
+ y
5
+ ··· + 3y + 1)(y
23
+ 24y
21
+ ··· + 6y 1)
c
11
(y
6
+ y
5
+ ··· + 3y + 1)(y
23
+ 12y
22
+ ··· + 282y 49)
9