11n
59
(K11n
59
)
1
Arc Sequences
4 1 7 2 11 10 4 5 6 7 9
Solving Sequence
6,10
7
4,11
3 5 9 1 2 8
c
6
c
10
c
3
c
5
c
9
c
11
c
2
c
8
c
1
, c
4
, c
7
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
5
3u
4
+ 5u
3
8u
2
+ 5u 1, 2u
4
5u
3
+ 8u
2
+ b 13u + 5, 4u
4
+ 10u
3
15u
2
+ a + 25u 8i
I
u
2
= hu
31
17u
29
+ ··· 8920u + 1225,
4.03455 × 10
118
u
30
5.87288 × 10
117
u
29
+ ··· + 1.99242 × 10
122
b + 3.52460 × 10
122
,
3.07475 × 10
121
u
30
8.30865 × 10
118
u
29
+ ··· + 4.88142 × 10
124
a + 3.57044 × 10
125
i
There are 2 irreducible components with 36 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
5
3u
4
+ 5u
3
8u
2
+ 5u 1, 2u
4
5u
3
+ 8u
2
+ b 13u +
5, 4u
4
+ 10u
3
15u
2
+ a + 25u 8i
(i) Arc colorings
a
6
=
0
u
a
10
=
4u
4
10u
3
+ 15u
2
25u + 8
2u
4
+ 5u
3
8u
2
+ 13u 5
a
7
=
3u
4
8u
3
+ 12u
2
19u + 7
0
a
4
=
1
0
a
11
=
4u
4
10u
3
+ 15u
2
24u + 7
2u
4
+ 5u
3
8u
2
+ 13u 5
a
3
=
1
0
a
5
=
5u
4
+ 13u
3
20u
2
+ 32u 11
1
a
9
=
4u
4
10u
3
+ 15u
2
25u + 8
3u
4
+ 8u
3
12u
2
+ 19u 7
a
1
=
5u
4
13u
3
+ 20u
2
32u + 11
1
a
2
=
5u
4
13u
3
+ 20u
2
32u + 12
1
a
8
=
3u
4
8u
3
+ 12u
2
19u + 7
0
a
8
=
3u
4
8u
3
+ 12u
2
19u + 7
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 = 0.09043 1.60288I
a = 0.186078 + 0.874646I
b = 0.232705 1.093812I
7.51750 + 4.40083I 16.4309 2.8075I
u = 0.09043 + 1.60288I
a = 0.186078 0.874646I
b = 0.232705 + 1.093812I
7.51750 4.40083I 16.4309 + 2.8075I
u = 0.442672 0.068387I
a = 0.871221 + 1.107662I
b = 0.438694 0.557752I
1.97403 + 1.53058I 11.97286 4.76366I
u = 0.442672 + 0.068387I
a = 0.871221 1.107662I
b = 0.438694 + 0.557752I
1.97403 1.53058I 11.97286 + 4.76366I
u = 1.93379
a = 0.629714
b = 1.58802
4.04602 9.19250
3
II. I
u
2
= hu
31
17u
29
+ · · · 8920u + 1225, 4.03 × 10
118
u
30
5.87 ×
10
117
u
29
+ · · · + 1.99 × 10
122
b + 3.52 × 10
122
, 3.07 × 10
121
u
30
8.31 ×
10
118
u
29
+ · · · + 4.88 × 10
124
a + 3.57 × 10
125
i
(i) Arc colorings
a
6
=
0
u
a
10
=
0.000629888u
30
+ 1.70210 × 10
6
u
29
+ ··· 5.08802u 7.31435
0.000202495u
30
+ 0.0000294762u
29
+ ··· + 2.41576u 1.76901
a
7
=
0.000405357u
30
0.0000405879u
29
+ ··· 9.97864u 6.92023
0.000228923u
30
+ 0.0000277311u
29
+ ··· + 2.80009u 1.90833
a
4
=
1
0
a
11
=
0.000540847u
30
+ 0.000122062u
29
+ ··· + 11.8218u 1.37500
0.0000435013u
30
2.32728 × 10
6
u
29
+ ··· + 0.349086u + 0.287320
a
3
=
0.00183186u
30
+ 0.000261258u
29
+ ··· + 17.3633u 13.6680
0.000273379u
30
+ 0.0000615471u
29
+ ··· + 3.84164u 1.82706
a
5
=
0.000661367u
30
+ 0.000130819u
29
+ ··· + 12.1726u 4.12144
0.0000133647u
30
+ 8.54868 × 10
6
u
29
+ ··· + 1.29474u 0.0111420
a
9
=
0.000629888u
30
+ 1.70210 × 10
6
u
29
+ ··· 5.08802u 7.31435
0.000221399u
30
+ 0.0000319772u
29
+ ··· + 3.17219u 1.76692
a
1
=
0.00164513u
30
+ 0.000251744u
29
+ ··· + 16.8822u 11.8716
0.000201047u
30
+ 0.0000492486u
29
+ ··· + 3.17910u 1.40610
a
2
=
0.00144409u
30
+ 0.000202495u
29
+ ··· + 13.7031u 10.4655
0.000201047u
30
+ 0.0000492486u
29
+ ··· + 3.17910u 1.40610
a
8
=
0.000176434u
30
0.0000683189u
29
+ ··· 12.7787u 5.01190
0.000228923u
30
+ 0.0000277311u
29
+ ··· + 2.80009u 1.90833
a
8
=
0.000176434u
30
0.0000683189u
29
+ ··· 12.7787u 5.01190
0.000228923u
30
+ 0.0000277311u
29
+ ··· + 2.80009u 1.90833
(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 = 3.29257 0.33043I
a = 0.393778 + 0.086396I
b = 3.07499 0.29210I
5.25829 + 1.15681I 12.57201 1.48822I
u = 3.29257 + 0.33043I
a = 0.393778 0.086396I
b = 3.07499 + 0.29210I
5.25829 1.15681I 12.57201 + 1.48822I
u = 2.14173 1.56439I
a = 0.489337 0.208824I
b = 2.08726 + 1.89767I
2.95558 4.85038I 7.33363 + 2.61184I
u = 2.14173 + 1.56439I
a = 0.489337 + 0.208824I
b = 2.08726 1.89767I
2.95558 + 4.85038I 7.33363 2.61184I
u = 1.20809 2.65506I
a = 0.193201 0.458837I
b = 0.94814 + 2.80645I
1.95006 2.95334I 10.65956 + 2.73175I
u = 1.20809 + 2.65506I
a = 0.193201 + 0.458837I
b = 0.94814 2.80645I
1.95006 + 2.95334I 10.65956 2.73175I
u = 0.719965 0.236177I
a = 0.039767 + 0.660162I
b = 0.828634 0.542777I
1.33374 + 0.83076I 4.07951 + 0.75098I
u = 0.719965 + 0.236177I
a = 0.039767 0.660162I
b = 0.828634 + 0.542777I
1.33374 0.83076I 4.07951 0.75098I
u = 0.417803
a = 0.804754
b = 0.599238
0.889878 11.7875
u = 0.39056 1.87293I
a = 0.264328 + 0.711767I
b = 0.53081 1.67027I
6.67621 4.65354I 6.91647 + 4.60285I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.39056 + 1.87293I
a = 0.264328 0.711767I
b = 0.53081 + 1.67027I
6.67621 + 4.65354I 6.91647 4.60285I
u = 0.372585 0.938778I
a = 0.783024 0.102771I
b = 0.831227 + 0.684303I
7.92090 0.90622I 3.00591 + 1.13607I
u = 0.372585 + 0.938778I
a = 0.783024 + 0.102771I
b = 0.831227 0.684303I
7.92090 + 0.90622I 3.00591 1.13607I
u = 0.278908 0.464935I
a = 1.251292 0.049124I
b = 0.053667 0.506448I
0.87857 1.75044I 3.85072 + 4.54648I
u = 0.278908 + 0.464935I
a = 1.251292 + 0.049124I
b = 0.053667 + 0.506448I
0.87857 + 1.75044I 3.85072 4.54648I
u = 0.049649 1.223526I
a = 0.342264 + 0.660443I
b = 0.207529 + 0.589199I
5.21409 3.77786I 6.77733 + 1.30179I
u = 0.049649 + 1.223526I
a = 0.342264 0.660443I
b = 0.207529 0.589199I
5.21409 + 3.77786I 6.77733 1.30179I
u = 0.131445 1.325086I
a = 0.054312 1.025475I
b = 0.178276 + 0.303063I
6.05722 3.45238I 10.37888 + 2.19312I
u = 0.131445 + 1.325086I
a = 0.054312 + 1.025475I
b = 0.178276 0.303063I
6.05722 + 3.45238I 10.37888 2.19312I
u = 0.177342 0.039655I
a = 6.54625 0.75948I
b = 0.653858 0.480672I
1.84615 + 0.55694I 6.11373 + 0.01533I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.177342 + 0.039655I
a = 6.54625 + 0.75948I
b = 0.653858 + 0.480672I
1.84615 0.55694I 6.11373 0.01533I
u = 0.223825 0.806345I
a = 0.088157 + 1.206832I
b = 0.643081 + 0.693417I
5.55958 3.20800I 6.18542 + 3.44031I
u = 0.223825 + 0.806345I
a = 0.088157 1.206832I
b = 0.643081 0.693417I
5.55958 + 3.20800I 6.18542 3.44031I
u = 0.866515 0.185129I
a = 0.022555 0.740543I
b = 0.643200 + 0.045254I
1.00966 2.22196I 3.40092 + 5.38737I
u = 0.866515 + 0.185129I
a = 0.022555 + 0.740543I
b = 0.643200 0.045254I
1.00966 + 2.22196I 3.40092 5.38737I
u = 0.959132 0.550961I
a = 0.547750 0.487439I
b = 1.08147 + 0.98278I
7.07782 + 7.98216I 4.46644 5.97470I
u = 0.959132 + 0.550961I
a = 0.547750 + 0.487439I
b = 1.08147 0.98278I
7.07782 7.98216I 4.46644 + 5.97470I
u = 2.69287 0.67346I
a = 0.451409 + 0.207599I
b = 2.64106 0.89209I
3.77819 5.48065I 9.45984 + 5.94075I
u = 2.69287 + 0.67346I
a = 0.451409 0.207599I
b = 2.64106 + 0.89209I
3.77819 + 5.48065I 9.45984 5.94075I
u = 3.51253 1.80577I
a = 0.353212 0.093181I
b = 3.26405 + 1.70145I
1.77560 + 11.89502I 8.90588 6.87931I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 3.51253 + 1.80577I
a = 0.353212 + 0.093181I
b = 3.26405 1.70145I
1.77560 11.89502I 8.90588 + 6.87931I
7
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
5
(u
31
+ 6u
30
+ ··· 4u 1)
c
2
(u + 1)
5
(u
31
+ 8u
30
+ ··· + 12u + 1)
c
3
, c
7
u
5
(u
31
+ u
30
+ ··· + 64u + 32)
c
4
(u + 1)
5
(u
31
+ 6u
30
+ ··· 4u 1)
c
5
(u
5
3u
4
+ ··· u + 1)(u
31
+ 6u
30
+ ··· 18u 5)
c
6
(u
5
+ u
4
+ ··· + u 1)(u
31
+ 2u
30
+ ··· + 2u + 1)
c
8
(u
5
+ u
4
+ ··· + u + 1)(u
31
+ 2u
30
+ ··· + 2u 1)
c
9
, c
10
(u
5
u
4
+ ··· + u + 1)(u
31
+ 2u
30
+ ··· + 2u + 1)
c
11
(u
5
+ u
4
+ ··· + u + 1)(u
31
+ 8u
30
+ ··· + 30u + 7)
8
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
5
(y
31
8y
30
+ ··· + 12y 1)
c
2
(y 1)
5
(y
31
+ 36y
30
+ ··· + 48y 1)
c
3
, c
7
y
5
(y
31
+ 33y
30
+ ··· 10752y 1024)
c
5
(y
5
y
4
+ ··· + 3y 1)(y
31
+ 4y
30
+ ··· + 174y 25)
c
6
, c
9
, c
10
(y
5
5y
4
+ ··· y 1)(y
31
28y
30
+ ··· + 2y 1)
c
8
(y
5
+ 3y
4
+ ··· y 1)(y
31
36y
30
+ ··· + 2y 1)
c
11
(y
5
+ 3y
4
+ ··· y 1)(y
31
+ 32y
29
+ ··· + 1054y 49)
9