11a
293
(K11a
293
)
1
Arc Sequences
9 5 1 8 2 10 11 3 6 7 4
Solving Sequence
6,9
10 7
2,11
1 5 3 8 4
c
9
c
6
c
10
c
1
c
5
c
2
c
8
c
4
c
3
, c
7
, c
11
Representation Ideals
I =
6
\
i=1
I
u
i
I
u
1
= ha
2
+ a 1, u 1, 2b ai
I
u
2
= hu
2
+ 1, a + u, b + u + 1i
I
u
3
= hu
2
+ u + 2, b 1, 2a u 1i
I
u
4
= hu
4
u
3
+ u
2
2u 1, b 1, u
3
2u
2
+ a + 2u 3i
I
u
5
= hu
14
u
13
+ 7u
12
11u
11
+ 24u
10
42u
9
+ 55u
8
79u
7
+ 82u
6
81u
5
+ 65u
4
42u
3
+ 19u
2
4u 1,
7u
13
+ 14u
12
+ ··· + 32a + 71, 19u
13
+ 54u
12
+ ··· + 64b + 3i
I
u
6
= hu
24
+ 4u
23
+ ··· + 4u + 1, 30156352u
23
+ 117877310u
22
+ ··· + 26009126b + 19822861,
27550425u
23
+ 103986434u
22
+ ··· + 26009126a + 104398322i
There are 6 irreducible components with 48 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
= ha
2
+ a 1, u 1, 2b ai
(i) Arc colorings
a
6
=
0
1
a
9
=
a
1
2
a
a
10
=
a
1
2
a
a
7
=
a + 1
1
2
a +
1
2
a
2
=
1
0
a
11
=
a + 1
1
2
a
1
2
a
1
=
1
2
a +
3
2
1
4
a +
1
4
a
5
=
1
1
a
3
=
2
1
a
8
=
a
1
2
a
a
4
=
1
2
a +
1
2
1
4
a +
3
4
a
4
=
1
2
a +
1
2
1
4
a +
3
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 = 1.61803
b = 0.809017
7.23771 3.81763
u = 1.00000
a = 0.618034
b = 0.309017
0.657974 4.56763
3
II. I
u
2
= hu
2
+ 1, a + u, b + u + 1i
(i) Arc colorings
a
6
=
0
u
a
9
=
u
u 1
a
10
=
u
2u 1
a
7
=
u
u 1
a
2
=
1
0
a
11
=
0
u
a
1
=
u
2u
a
5
=
u
u
a
3
=
0
1
a
8
=
u
2u 1
a
4
=
1
3
a
4
=
1
3
(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 = 1.00000I
a = 1.00000I
b = 1.00000 + 1.00000I
4.93480 12.0000
u = 1.00000I
a = 1.00000I
b = 1.00000 1.00000I
4.93480 12.0000
5
III. I
u
3
= hu
2
+ u + 2, b 1, 2a u 1i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
2
u +
1
2
1
a
10
=
1
2
u +
1
2
u + 1
a
7
=
1
2
u
1
2
1
a
2
=
1
0
a
11
=
0
u
a
1
=
1
2
u +
3
2
1
a
5
=
u
u
a
3
=
u 1
u 2
a
8
=
1
2
u
1
2
u 1
a
4
=
1
2
u
1
2
1
a
4
=
1
2
u
1
2
1
(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 = 0.50000 1.32288I
a = 0.250000 0.661438I
b = 1.00000
8.22467 14.0000
u = 0.50000 + 1.32288I
a = 0.250000 + 0.661438I
b = 1.00000
8.22467 14.0000
7
IV. I
u
4
= hu
4
u
3
+ u
2
2u 1, b 1, u
3
2u
2
+ a + 2u 3i
(i) Arc colorings
a
6
=
0
u
a
9
=
u
3
+ 2u
2
2u + 3
1
a
10
=
u
3
+ 2u
2
2u + 3
u
a
7
=
2u
3
2u
2
+ 3u 5
1
a
2
=
1
0
a
11
=
2u
3
3u
2
+ 2u 5
u
3
u
2
1
a
1
=
u
3
+ 2u
2
2u + 4
1
a
5
=
u
u
a
3
=
u
2
+ 1
u
2
a
8
=
u
3
+ 2u
2
3u + 2
u
2
2u
a
4
=
u
3
+ 2u
2
2u + 2
u
2
u
a
4
=
u
3
+ 2u
2
2u + 2
u
2
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 = 0.385795
a = 4.12668
b = 1.00000
8.22467 14.0000
u = 0.061094 1.309640I
a = 0.614787 + 0.707750I
b = 1.00000
8.22467 14.0000
u = 0.061094 + 1.309640I
a = 0.614787 0.707750I
b = 1.00000
8.22467 14.0000
u = 1.50798
a = 1.10289
b = 1.00000
8.22467 14.0000
9
V. I
u
5
= hu
14
u
13
+ · · · 4u 1, 7u
13
+ 14u
12
+ · · · + 32a +
71, 19u
13
+ 54u
12
+ · · · + 64b + 3i
(i) Arc colorings
a
6
=
0
u
a
9
=
0.218750u
13
0.437500u
12
+ ··· + 9.34375u 2.21875
0.296875u
13
0.843750u
12
+ ··· + 1.60938u 0.0468750
a
10
=
0.218750u
13
0.437500u
12
+ ··· + 9.34375u 2.21875
0.0781250u
13
0.406250u
12
+ ··· + 2.26563u + 0.171875
a
7
=
0.531250u
13
0.562500u
12
+ ··· + 5.40625u 1.03125
0.484375u
13
0.468750u
12
+ ··· + 1.54688u + 0.0156250
a
2
=
1
0
a
11
=
0.0312500u
13
0.0625000u
12
+ ··· + 4.90625u 2.03125
0.0156250u
13
0.0312500u
12
+ ··· + 0.953125u 0.0156250
a
1
=
0.0156250u
13
+ 0.0312500u
12
+ ··· 3.95313u + 2.01563
0.00781250u
13
+ 0.0156250u
12
+ ··· 0.976563u + 0.00781250
a
5
=
u
u
a
3
=
u
2
+ 1
u
2
a
8
=
0.468750u
13
+ 0.0625000u
12
+ ··· + 6.59375u 2.46875
0.421875u
13
+ 0.406250u
12
+ ··· 0.765625u 0.671875
a
4
=
0.0156250u
13
+ 0.0312500u
12
+ ··· 1.95313u + 1.01563
0.00781250u
13
+ 0.0156250u
12
+ ··· + 0.0234375u + 0.00781250
a
4
=
0.0156250u
13
+ 0.0312500u
12
+ ··· 1.95313u + 1.01563
0.00781250u
13
+ 0.0156250u
12
+ ··· + 0.0234375u + 0.00781250
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
10
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.52839 1.47236I
a = 0.274970 + 1.069607I
b = 1.74897 + 0.88809I
18.7924 13.4596I 11.49654 + 6.20510I
u = 0.52839 + 1.47236I
a = 0.274970 1.069607I
b = 1.74897 0.88809I
18.7924 + 13.4596I 11.49654 6.20510I
u = 0.41976 1.38103I
a = 0.172129 0.788550I
b = 1.44582 0.72488I
9.30919 10.50754I 10.66498 + 7.31370I
u = 0.41976 + 1.38103I
a = 0.172129 + 0.788550I
b = 1.44582 + 0.72488I
9.30919 + 10.50754I 10.66498 7.31370I
u = 0.258574 1.300317I
a = 0.660273 + 0.223942I
b = 1.131492 + 0.435989I
5.81499 5.60499I 9.56216 + 5.32481I
u = 0.258574 + 1.300317I
a = 0.660273 0.223942I
b = 1.131492 0.435989I
5.81499 + 5.60499I 9.56216 5.32481I
u = 0.133734
a = 3.93745
b = 0.422086
0.784313 13.0946
u = 0.150905 1.395376I
a = 0.008238 1.261701I
b = 1.029022 0.477997I
17.7001 + 3.9528I 13.79713 2.45311I
u = 0.150905 + 1.395376I
a = 0.008238 + 1.261701I
b = 1.029022 + 0.477997I
17.7001 3.9528I 13.79713 + 2.45311I
u = 0.364310 0.747897I
a = 0.74609 + 1.38706I
b = 0.162438 + 0.623510I
3.65387 + 1.40001I 5.70050 4.92983I
11
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.364310 + 0.747897I
a = 0.74609 1.38706I
b = 0.162438 0.623510I
3.65387 1.40001I 5.70050 + 4.92983I
u = 0.644907 0.288035I
a = 0.198268 0.487587I
b = 0.042110 0.515580I
1.20190 + 0.85736I 4.43900 4.77044I
u = 0.644907 + 0.288035I
a = 0.198268 + 0.487587I
b = 0.042110 + 0.515580I
1.20190 0.85736I 4.43900 + 4.77044I
u = 1.22694
a = 0.601015
b = 0.559695
0.355949 22.0908
12
VI.
I
u
6
= hu
24
+ 4u
23
+ · · · + 4u + 1, 3.02 × 10
7
u
23
+ 1.18 × 10
8
u
22
+ · · · + 2.60 ×
10
7
b +1.98 ×10
7
, 2.76× 10
7
u
23
+1.04× 10
8
u
22
+· · ·+ 2.60 ×10
7
a +1.04 ×10
8
i
(i) Arc colorings
a
6
=
0
u
a
9
=
1.05926u
23
3.99807u
22
+ ··· 14.5362u 4.01391
1.15945u
23
4.53215u
22
+ ··· 1.20981u 0.762150
a
10
=
1.05926u
23
3.99807u
22
+ ··· 14.5362u 4.01391
0.980122u
23
4.01235u
22
+ ··· 1.31321u 0.523185
a
7
=
0.000114306u
23
+ 0.116188u
22
+ ··· 6.10770u 1.82570
0.825385u
23
3.01695u
22
+ ··· 0.219096u 0.0234969
a
2
=
1
0
a
11
=
1.66766u
23
5.93828u
22
+ ··· 14.0868u 3.39050
0.654187u
23
2.44023u
22
+ ··· 0.209915u 0.555847
a
1
=
0.139227u
23
0.353565u
22
+ ··· 13.7036u 0.125222
2.05934u
23
+ 8.44844u
22
+ ··· + 16.0316u + 3.90693
a
5
=
u
u
a
3
=
u
2
+ 1
u
2
a
8
=
0.349995u
23
1.45967u
22
+ ··· 13.6700u 3.10578
1.27750u
23
5.02370u
22
+ ··· 0.919981u 0.482866
a
4
=
0.654302u
23
2.55642u
22
+ ··· + 8.89778u + 4.26985
0.516544u
23
+ 1.79023u
22
+ ··· 2.86287u 2.70477
a
4
=
0.654302u
23
2.55642u
22
+ ··· + 8.89778u + 4.26985
0.516544u
23
+ 1.79023u
22
+ ··· 2.86287u 2.70477
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
13
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
6
1(vol +
1CS) Cusp shape
u = 1.265993 0.145887I
a = 1.309027 + 0.221065I
b = 1.196622 + 0.373126I
13.6389 7.2036I 10.08749 + 4.71657I
u = 1.265993 + 0.145887I
a = 1.309027 0.221065I
b = 1.196622 0.373126I
13.6389 + 7.2036I 10.08749 4.71657I
u = 0.991263 0.128941I
a = 0.854960 0.459140I
b = 0.908844 0.555382I
4.52195 5.52285I 8.56374 + 6.48307I
u = 0.991263 + 0.128941I
a = 0.854960 + 0.459140I
b = 0.908844 + 0.555382I
4.52195 + 5.52285I 8.56374 6.48307I
u = 0.67504 1.53852I
a = 0.409206 + 0.932641I
b = 1.064025 + 0.463555I
17.8795 13.6671
u = 0.67504 + 1.53852I
a = 0.409206 0.932641I
b = 1.064025 0.463555I
17.8795 13.6671
u = 0.580478 0.088669I
a = 0.011635 + 0.951222I
b = 0.651494 + 0.761585I
1.52068 2.46907I 2.47747 + 3.95252I
u = 0.580478 + 0.088669I
a = 0.011635 0.951222I
b = 0.651494 0.761585I
1.52068 + 2.46907I 2.47747 3.95252I
u = 0.110618 1.018189I
a = 0.041056 + 0.377904I
b = 1.94289 1.68580I
3.95056 11.0692
u = 0.110618 + 1.018189I
a = 0.041056 0.377904I
b = 1.94289 + 1.68580I
3.95056 11.0692
14
Solution to I
u
6
1(vol +
1CS) Cusp shape
u = 0.099763 1.246625I
a = 0.058167 1.324987I
b = 1.28010 1.70804I
11.96397 1.70959I 7.87181 + 0.16720I
u = 0.099763 + 1.246625I
a = 0.058167 + 1.324987I
b = 1.28010 + 1.70804I
11.96397 + 1.70959I 7.87181 0.16720I
u = 0.002396 1.116615I
a = 0.269066 + 0.669543I
b = 1.21547 + 0.75603I
3.49764 0.49850I 6.63137 + 1.38008I
u = 0.002396 + 1.116615I
a = 0.269066 0.669543I
b = 1.21547 0.75603I
3.49764 + 0.49850I 6.63137 1.38008I
u = 0.009071 0.303466I
a = 0.92092 + 2.48902I
b = 0.813521 0.946836I
3.49764 + 0.49850I 6.63137 1.38008I
u = 0.009071 + 0.303466I
a = 0.92092 2.48902I
b = 0.813521 + 0.946836I
3.49764 0.49850I 6.63137 + 1.38008I
u = 0.255684 1.161479I
a = 0.440245 + 0.163717I
b = 0.917583 + 0.121966I
1.52068 + 2.46907I 2.47747 3.95252I
u = 0.255684 + 1.161479I
a = 0.440245 0.163717I
b = 0.917583 0.121966I
1.52068 2.46907I 2.47747 + 3.95252I
u = 0.42275 1.37969I
a = 0.044442 0.670790I
b = 1.124532 0.590415I
4.52195 + 5.52285I 8.56374 6.48307I
u = 0.42275 + 1.37969I
a = 0.044442 + 0.670790I
b = 1.124532 + 0.590415I
4.52195 5.52285I 8.56374 + 6.48307I
15
Solution to I
u
6
1(vol +
1CS) Cusp shape
u = 0.498557 0.415422I
a = 1.90343 1.70573I
b = 1.56144 + 0.76644I
11.96397 + 1.70959I 7.87181 0.16720I
u = 0.498557 + 0.415422I
a = 1.90343 + 1.70573I
b = 1.56144 0.76644I
11.96397 1.70959I 7.87181 + 0.16720I
u = 0.53950 1.55444I
a = 0.285623 + 0.987736I
b = 1.33643 + 0.88146I
13.6389 + 7.2036I 10.08749 4.71657I
u = 0.53950 + 1.55444I
a = 0.285623 0.987736I
b = 1.33643 0.88146I
13.6389 7.2036I 10.08749 + 4.71657I
16
VII. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
6
(u
2
+ 2u + 2)(4u
2
+ 2u 1)(4u
14
+ 22u
13
+ ··· + 12u + 2)
(u
24
+ 7u
23
+ ··· + 5492u + 2488)
c
2
, c
11
(u 1)
2
(u
2
+ 1)(u
2
+ u + 2)(u
4
u
3
+ u
2
2u 1)
(u
14
u
13
+ ··· 4u 1)(u
24
+ 4u
23
+ ··· + 4u + 1)
c
3
, c
5
(u + 1)
2
(u
2
+ 1)(u
2
+ u + 2)(u
4
u
3
+ u
2
2u 1)
(u
14
u
13
+ ··· 4u 1)(u
24
+ 4u
23
+ ··· + 4u + 1)
c
4
(u 1)
6
(u
2
2u + 2)(4u
2
2u 1)(4u
14
+ 22u
13
+ ··· + 12u + 2)
(u
24
+ 7u
23
+ ··· + 5492u + 2488)
c
6
, c
7
(u 1)
2
(u + 1)
2
(u
2
u 1)(u
4
+ 2u
3
u
2
5u 4)
(1 + 4u + 5u
2
+ 3u
3
7u
4
19u
5
u
6
+ 27u
7
+ 10u
8
13u
9
6u
10
+ 2u
11
+ u
12
)
2
(u
14
8u
12
+ ··· + 3u 1)
c
8
u
2
(u 1)
2
(u
2
+ 1)(u
4
2u
3
+ u
2
+ 5u + 2)
(u
12
2u
10
+ u
9
+ 4u
8
u
7
3u
6
+ 3u
5
+ 3u
4
u
3
u
2
+ 2u + 1)
2
(u
14
+ 5u
13
+ ··· 44u + 16)
c
9
, c
10
(u 1)
4
(u
2
+ u 1)(u
4
+ 2u
3
u
2
5u 4)
(1 + 4u + 5u
2
+ 3u
3
7u
4
19u
5
u
6
+ 27u
7
+ 10u
8
13u
9
6u
10
+ 2u
11
+ u
12
)
2
(u
14
8u
12
+ ··· + 3u 1)
17
VIII. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
6
(y
2
+ 4)(16y
2
12y + 1)(16y
14
172y
13
+ ··· 148y
2
+ 4)
(y
24
15y
23
+ ··· 39497040y + 6190144)
c
2
, c
3
, c
5
c
11
(y 1)
2
(y + 1)
2
(y
2
+ 3y + 4)(y
4
+ y
3
5y
2
6y + 1)
(y
14
+ 13y
13
+ ··· 54y + 1)(y
24
+ 16y
23
+ ··· + 20y + 1)
c
6
, c
7
, c
9
c
10
(y 1)
4
(y
2
3y + 1)(y
4
6y
3
+ 13y
2
17y + 16)
(1 6y 13y
2
+ 71y
3
43y
4
317y
5
+ 891y
6
1161y
7
+ 876y
8
399y
9
+ 108y
10
16y
11
+ y
12
)
2
(y
14
16y
13
+ ··· 7y + 1)
c
8
y
2
(y 1)
2
(y + 1)
2
(y
4
2y
3
+ 25y
2
21y + 4)
(1 6y + 11y
2
25y
3
+ 33y
4
45y
5
+ 47y
6
45y
7
+ 36y
8
23y
9
+ 12y
10
4y
11
+ y
12
)
2
(y
14
y
13
+ ··· 1584y + 256)
18