11a
88
(K11a
88
)
1
Arc Sequences
6 1 10 9 2 3 11 4 5 7 8
Solving Sequence
2,5
6 1 3
7,10
9 4 8 11
c
5
c
1
c
2
c
6
c
9
c
4
c
8
c
11
c
3
, c
7
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= ha
4
+ 2a
2
+ 4, a
2
+ 2u, a
3
+ a
2
+ 2b + 2i
I
u
2
= hu
2
+ u + 1, a, b + u + 1i
I
u
3
= hu
56
+ 2u
55
+ ··· + 2u + 1, 1.70370 × 10
18
u
55
+ 8.32383 × 10
18
u
54
+ ··· + 7.25836 × 10
18
b + 1.23640 × 10
19
,
6.54872 × 10
18
u
55
9.28010 × 10
18
u
54
+ ··· + 7.25836 × 10
18
a 1.54729 × 10
19
i
There are 3 irreducible components with 62 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
4
+ 2a
2
+ 4, a
2
+ 2u, a
3
+ a
2
+ 2b + 2i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
1
2
a
2
a
6
=
1
2
a
2
1
2
a
2
a
1
=
1
2
a
2
1
2
a
2
1
a
3
=
0
1
2
a
2
a
7
=
1
2
a
2
1
2
a
2
1
a
10
=
a
1
2
a
3
1
2
a
2
1
a
9
=
a
a
3
1
2
a
2
a 1
a
4
=
a
2
2
1
2
a
2
+ a 2
a
8
=
a
1
2
a
3
+
1
2
a
2
+ 1
a
11
=
1
2
a
2
+ a
1
2
a
3
a
2
2
a
11
=
1
2
a
2
+ a
1
2
a
3
a
2
2
(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 = 0.500000 0.866025I
a = 0.70711 1.22474I
b = 1.91421 0.86603I
3.28987 2.02988I 2.00000 + 3.46410I
u = 0.500000 + 0.866025I
a = 0.70711 + 1.22474I
b = 1.91421 + 0.86603I
3.28987 + 2.02988I 2.00000 3.46410I
u = 0.500000 + 0.866025I
a = 0.70711 1.22474I
b = 0.914214 + 0.866025I
3.28987 + 2.02988I 2.00000 3.46410I
u = 0.500000 0.866025I
a = 0.70711 + 1.22474I
b = 0.914214 0.866025I
3.28987 2.02988I 2.00000 + 3.46410I
3
II. I
u
2
= hu
2
+ u + 1, a, b + u + 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
u
u 1
a
3
=
0
u
a
7
=
u
u + 1
a
10
=
0
u 1
a
9
=
0
u 1
a
4
=
0
u
a
8
=
0
u 1
a
11
=
u
2u 2
a
11
=
u
2u 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 = 0.500000 0.866025I
a = 0
b = 0.500000 + 0.866025I
1.64493 + 2.02988I 3.46410I
u = 0.500000 + 0.866025I
a = 0
b = 0.500000 0.866025I
1.64493 2.02988I 3.46410I
5
III. I
u
3
=
hu
56
+ 2u
55
+ · · · +2u + 1, 1.70 × 10
18
u
55
+ 8.32 × 10
18
u
54
+ · · · + 7.26 × 10
18
b +
1.24× 10
19
, 6.55× 10
18
u
55
9.28 × 10
18
u
54
+· · · +7.26 × 10
18
a 1.55 ×10
19
i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
+ u
2
+ 1
u
4
a
7
=
u
7
2u
5
2u
3
u
7
u
5
+ u
a
10
=
0.902232u
55
+ 1.27854u
54
+ ··· + 4.24456u + 2.13174
0.234722u
55
1.14679u
54
+ ··· 1.65066u 1.70341
a
9
=
0.902232u
55
+ 1.27854u
54
+ ··· + 4.24456u + 2.13174
0.346913u
55
+ 0.241661u
54
+ ··· 1.50104u 1.17748
a
4
=
1.11145u
55
+ 2.57511u
54
+ ··· + 7.20845u + 3.40735
0.284126u
55
+ 1.99077u
54
+ ··· + 0.887335u 0.0147560
a
8
=
0.899349u
55
1.34939u
54
+ ··· 4.69836u 1.34103
0.234930u
55
+ 0.402020u
54
+ ··· + 0.821736u + 1.10278
a
11
=
0.917813u
55
+ 1.35526u
54
+ ··· + 4.63727u + 2.04654
0.495942u
55
+ 0.276494u
54
+ ··· 1.46763u 1.33623
a
11
=
0.917813u
55
+ 1.35526u
54
+ ··· + 4.63727u + 2.04654
0.495942u
55
+ 0.276494u
54
+ ··· 1.46763u 1.33623
(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 = 0.853869 0.316506I
a = 1.32152 0.87262I
b = 0.575151 + 0.969236I
0.70521 8.81361I 1.89048 + 4.78995I
u = 0.853869 + 0.316506I
a = 1.32152 + 0.87262I
b = 0.575151 0.969236I
0.70521 + 8.81361I 1.89048 4.78995I
u = 0.764797 0.121262I
a = 1.216229 0.332850I
b = 0.447879 + 0.167499I
3.86656 0.41270I 0.197348 0.929430I
u = 0.764797 + 0.121262I
a = 1.216229 + 0.332850I
b = 0.447879 0.167499I
3.86656 + 0.41270I 0.197348 + 0.929430I
u = 0.743330 0.310940I
a = 1.44862 + 0.94062I
b = 0.257124 1.050967I
4.80088 4.79415I 6.07434 + 4.15713I
u = 0.743330 + 0.310940I
a = 1.44862 0.94062I
b = 0.257124 + 1.050967I
4.80088 + 4.79415I 6.07434 4.15713I
u = 0.690346 0.801960I
a = 0.473254 0.324975I
b = 0.440782 0.200836I
2.82580 + 2.62743I 4.75305 4.09062I
u = 0.690346 + 0.801960I
a = 0.473254 + 0.324975I
b = 0.440782 + 0.200836I
2.82580 2.62743I 4.75305 + 4.09062I
u = 0.589077 1.163721I
a = 0.731129 + 0.860048I
b = 2.84613 + 1.09153I
3.2473 + 14.1444I 0.80890 8.28814I
u = 0.589077 + 1.163721I
a = 0.731129 0.860048I
b = 2.84613 1.09153I
3.2473 14.1444I 0.80890 + 8.28814I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.558020 0.203304I
a = 1.98677 0.95500I
b = 0.347195 + 0.673637I
2.93700 0.37083I 3.52953 0.41815I
u = 0.558020 + 0.203304I
a = 1.98677 + 0.95500I
b = 0.347195 0.673637I
2.93700 + 0.37083I 3.52953 + 0.41815I
u = 0.553003 1.128775I
a = 0.691704 0.881544I
b = 2.87173 1.49649I
2.40647 + 9.70272I 2.45583 7.81180I
u = 0.553003 + 1.128775I
a = 0.691704 + 0.881544I
b = 2.87173 + 1.49649I
2.40647 9.70272I 2.45583 + 7.81180I
u = 0.501395 1.168449I
a = 0.120582 0.789177I
b = 0.498958 0.834592I
6.88232 + 5.05308I 3.62442 2.62661I
u = 0.501395 + 1.168449I
a = 0.120582 + 0.789177I
b = 0.498958 + 0.834592I
6.88232 5.05308I 3.62442 + 2.62661I
u = 0.491299 1.104419I
a = 0.615204 + 0.887091I
b = 2.37361 + 2.06503I
0.50012 + 4.58916I 0.35782 3.91492I
u = 0.491299 + 1.104419I
a = 0.615204 0.887091I
b = 2.37361 2.06503I
0.50012 4.58916I 0.35782 + 3.91492I
u = 0.451244 0.989156I
a = 0.103814 + 0.501109I
b = 0.088530 + 0.628783I
0.57122 + 2.77824I 2.95589 4.20688I
u = 0.451244 + 0.989156I
a = 0.103814 0.501109I
b = 0.088530 0.628783I
0.57122 2.77824I 2.95589 + 4.20688I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.442137 0.560010I
a = 0.718092 0.087794I
b = 0.463075 + 0.039809I
0.730759 + 1.016186I 5.20362 4.92767I
u = 0.442137 + 0.560010I
a = 0.718092 + 0.087794I
b = 0.463075 0.039809I
0.730759 1.016186I 5.20362 + 4.92767I
u = 0.374837 1.075047I
a = 0.221979 + 0.933850I
b = 0.27062 + 1.80833I
0.27332 + 2.70620I 1.11123 4.07607I
u = 0.374837 + 1.075047I
a = 0.221979 0.933850I
b = 0.27062 1.80833I
0.27332 2.70620I 1.11123 + 4.07607I
u = 0.364658 1.197693I
a = 0.594584 0.688989I
b = 1.62715 1.04454I
7.81847 + 3.46546I 4.57962 3.86575I
u = 0.364658 + 1.197693I
a = 0.594584 + 0.688989I
b = 1.62715 + 1.04454I
7.81847 3.46546I 4.57962 + 3.86575I
u = 0.268445 1.111906I
a = 0.046114 1.133396I
b = 1.16400 1.38912I
0.50238 1.98879I 0.34285 + 2.44288I
u = 0.268445 + 1.111906I
a = 0.046114 + 1.133396I
b = 1.16400 + 1.38912I
0.50238 + 1.98879I 0.34285 2.44288I
u = 0.218016 1.217336I
a = 0.099701 + 1.133251I
b = 1.11666 + 1.00556I
5.76926 5.58643I 3.62087 + 2.94823I
u = 0.218016 + 1.217336I
a = 0.099701 1.133251I
b = 1.11666 1.00556I
5.76926 + 5.58643I 3.62087 2.94823I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.278365 0.729460I
a = 0.348509 + 0.488079I
b = 0.889962 0.872101I
1.99309 1.19360I 4.63700 2.51647I
u = 0.278365 + 0.729460I
a = 0.348509 0.488079I
b = 0.889962 + 0.872101I
1.99309 + 1.19360I 4.63700 + 2.51647I
u = 0.287537 1.214310I
a = 0.634828 0.281975I
b = 1.82155 0.03654I
10.53926 + 1.20311I 7.50236 0.61522I
u = 0.287537 + 1.214310I
a = 0.634828 + 0.281975I
b = 1.82155 + 0.03654I
10.53926 1.20311I 7.50236 + 0.61522I
u = 0.302548 0.409621I
a = 1.79650 2.21555I
b = 1.63181 + 0.07015I
3.41711 0.18501I 2.17012 1.34687I
u = 0.302548 + 0.409621I
a = 1.79650 + 2.21555I
b = 1.63181 0.07015I
3.41711 + 0.18501I 2.17012 + 1.34687I
u = 0.385295 1.082824I
a = 0.542329 + 0.197476I
b = 2.01943 0.14091I
3.45328 1.33657I 5.74001 + 0.61084I
u = 0.385295 + 1.082824I
a = 0.542329 0.197476I
b = 2.01943 + 0.14091I
3.45328 + 1.33657I 5.74001 0.61084I
u = 0.459731 1.077506I
a = 0.479912 + 1.234967I
b = 1.262228 + 0.173914I
1.33205 3.50361I 1.93330 + 4.45658I
u = 0.459731 + 1.077506I
a = 0.479912 1.234967I
b = 1.262228 0.173914I
1.33205 + 3.50361I 1.93330 4.45658I
10
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.502022 1.112164I
a = 0.599403 0.091064I
b = 2.13211 0.08952I
2.59517 6.04162I 2.64180 + 7.41449I
u = 0.502022 + 1.112164I
a = 0.599403 + 0.091064I
b = 2.13211 + 0.08952I
2.59517 + 6.04162I 2.64180 7.41449I
u = 0.554081 1.174851I
a = 0.678185 + 0.077988I
b = 2.05190 + 0.16862I
8.70986 9.86105I 5.02329 + 7.09179I
u = 0.554081 + 1.174851I
a = 0.678185 0.077988I
b = 2.05190 0.16862I
8.70986 + 9.86105I 5.02329 7.09179I
u = 0.569045 0.945562I
a = 0.594181 1.135017I
b = 1.290763 + 0.015699I
5.37458 1.80246I 7.34576 + 2.56666I
u = 0.569045 + 0.945562I
a = 0.594181 + 1.135017I
b = 1.290763 0.015699I
5.37458 + 1.80246I 7.34576 2.56666I
u = 0.604383 0.247576I
a = 0.065405 0.955343I
b = 0.027501 + 0.540064I
0.16557 + 1.66732I 1.05437 4.59670I
u = 0.604383 + 0.247576I
a = 0.065405 + 0.955343I
b = 0.027501 0.540064I
0.16557 1.66732I 1.05437 + 4.59670I
u = 0.631832 0.630832I
a = 1.21993 + 0.99632I
b = 1.59065 0.12118I
6.29507 2.91322I 8.51117 + 4.03861I
u = 0.631832 + 0.630832I
a = 1.21993 0.99632I
b = 1.59065 + 0.12118I
6.29507 + 2.91322I 8.51117 4.03861I
11
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.698151 0.927977I
a = 0.517170 + 0.985261I
b = 1.159655 0.176921I
0.926044 + 0.238949I 0.328621 0.369179I
u = 0.698151 + 0.927977I
a = 0.517170 0.985261I
b = 1.159655 + 0.176921I
0.926044 0.238949I 0.328621 + 0.369179I
u = 0.761612 0.706746I
a = 1.094808 0.694458I
b = 1.62773 + 0.21030I
1.58146 5.73177I 1.83404 + 5.79402I
u = 0.761612 + 0.706746I
a = 1.094808 + 0.694458I
b = 1.62773 0.21030I
1.58146 + 5.73177I 1.83404 5.79402I
u = 0.829870 0.241336I
a = 0.010188 + 0.960627I
b = 0.112725 0.446821I
5.93188 + 4.75475I 2.47991 3.65893I
u = 0.829870 + 0.241336I
a = 0.010188 0.960627I
b = 0.112725 + 0.446821I
5.93188 4.75475I 2.47991 + 3.65893I
12
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
u + 1)
2
(u
2
+ u + 1)(u
56
+ 2u
55
+ ··· + 2u + 1)
c
2
(u
2
+ u + 1)
3
(u
56
+ 28u
55
+ ··· + 4u + 1)
c
3
u
2
(u
2
2)
2
(u
56
+ 3u
55
+ ··· 276u + 172)
c
4
, c
8
, c
9
u
2
(u
2
2)
2
(u
56
+ u
55
+ ··· + 12u + 4)
c
5
(u
2
+ u + 1)
3
(u
56
+ 2u
55
+ ··· + 2u + 1)
c
6
(u
2
u + 1)
2
(u
2
+ u + 1)(u
56
+ 2u
55
+ ··· + 1554u + 481)
c
7
(u 1)
2
(u + 1)
4
(u
56
+ 3u
55
+ ··· 31u + 7)
c
10
(u 1)
6
(u
56
+ 3u
55
+ ··· 31u + 7)
c
11
(u 1)
4
(u + 1)
2
(u
56
+ 3u
55
+ ··· 31u + 7)
13
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
2
+ y + 1)
3
(y
56
+ 28y
55
+ ··· + 4y + 1)
c
2
(y
2
+ y + 1)
3
(y
56
+ 4y
55
+ ··· + 28y + 1)
c
3
y
2
(y 2)
4
(y
56
+ 9y
55
+ ··· + 99952y + 29584)
c
4
, c
8
, c
9
y
2
(y 2)
4
(y
56
51y
55
+ ··· + 48y + 16)
c
6
(y
2
+ y + 1)
3
(y
56
20y
55
+ ··· 456284y + 231361)
c
7
, c
10
, c
11
(y 1)
6
(y
56
53y
55
+ ··· + 747y + 49)
14