11a
25
(K11a
25
)
1
Arc Sequences
4 1 7 2 8 3 10 11 5 6 9
Solving Sequence
9,11 1,4
2 5 10 8 6 7 3
c
11
c
1
c
4
c
9
c
8
c
5
c
7
c
3
c
2
, c
6
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
5
+ a
4
2a
3
a
2
+ a 1, u + 1, b + a + 1i
I
u
2
= hu
82
+ 4u
81
+ ··· + 185419986536u + 19815397013,
4.15682 × 10
728
u
81
+ 1.28769 × 10
729
u
80
+ ··· + 7.19287 × 10
733
b + 9.13398 × 10
738
,
1.54962 × 10
739
u
81
4.78468 × 10
739
u
80
+ ··· + 1.42529 × 10
744
a 3.36544 × 10
749
i
There are 2 irreducible components with 87 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
5
+ a
4
2a
3
a
2
+ a 1, u + 1, b + a + 1i
(i) Arc colorings
a
9
=
0
1
a
11
=
a
a 1
a
1
=
a
1
a
4
=
1
0
a
2
=
a + 1
1
a
5
=
a
1
a
10
=
a
2
a 1
a
8
=
a
2
a
2
+ a 1
a
6
=
a
4
+ a
3
+ a
2
2a + 1
0
a
7
=
a
4
+ a
3
+ a
2
2a + 1
0
a
3
=
1
0
a
3
=
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 = 1.41878 0.21917I
b = 0.418784 + 0.219165I
7.51750 4.40083I 3.44484 + 1.78781I
u = 1.00000
a = 1.41878 + 0.21917I
b = 0.418784 0.219165I
7.51750 + 4.40083I 3.44484 1.78781I
u = 1.00000
a = 0.309916 0.549911I
b = 1.30992 + 0.54991I
1.97403 + 1.53058I 3.57269 4.45807I
u = 1.00000
a = 0.309916 + 0.549911I
b = 1.30992 0.54991I
1.97403 1.53058I 3.57269 + 4.45807I
u = 1.00000
a = 1.21774
b = 2.21774
4.04602 15.9649
3
II. I
u
2
= hu
82
+ 4u
81
+ · · · + 1.85 × 10
11
u + 1.98 × 10
10
, 4.16 × 10
728
u
81
+
1.29 × 10
729
u
80
+ · · · + 7.19 × 10
733
b + 9.13 × 10
738
, 1.55 × 10
739
u
81
4.78 × 10
739
u
80
+ · · · + 1.43 × 10
744
a 3.37 × 10
749
i
(i) Arc colorings
a
9
=
0
u
a
11
=
0.0000108723u
81
+ 0.0000335698u
80
+ ··· + 1.95075 × 10
6
u + 236122.
5.77909 × 10
6
u
81
0.0000179023u
80
+ ··· 1.04734 × 10
6
u 126987.
a
1
=
0.0000108723u
81
+ 0.0000335698u
80
+ ··· + 1.95075 × 10
6
u + 236122.
3.27096 × 10
6
u
81
+ 0.0000100402u
80
+ ··· + 576450.u + 69567.5
a
4
=
1
0
a
2
=
7.60130 × 10
6
u
81
+ 0.0000235296u
80
+ ··· + 1.37430 × 10
6
u + 166555.
3.27096 × 10
6
u
81
+ 0.0000100402u
80
+ ··· + 576450.u + 69567.5
a
5
=
4.93050 × 10
6
u
81
0.0000153257u
80
+ ··· 901934.u 109481.
1.73370 × 10
6
u
81
5.24860 × 10
6
u
80
+ ··· 292664.u 35064.3
a
10
=
0.0000100114u
81
0.0000312198u
80
+ ··· 1.84888 × 10
6
u 224766.
0.0000109828u
81
0.0000338977u
80
+ ··· 1.96783 × 10
6
u 238131.
a
8
=
5.86214 × 10
6
u
81
+ 0.0000182733u
80
+ ··· + 1.08149 × 10
6
u + 131451.
1.51696 × 10
6
u
81
+ 4.54396 × 10
6
u
80
+ ··· + 247690.u + 29503.4
a
6
=
9.53794 × 10
6
u
81
+ 0.0000298729u
80
+ ··· + 1.78419 × 10
6
u + 217341.
8.63834 × 10
6
u
81
+ 0.0000267944u
80
+ ··· + 1.56960 × 10
6
u + 190319.
a
7
=
6.49077 × 10
6
u
81
0.0000199113u
80
+ ··· 1.14092 × 10
6
u 137639.
1.99027 × 10
6
u
81
5.98664 × 10
6
u
80
+ ··· 329058.u 39295.2
a
3
=
2.84484 × 10
7
u
81
7.39387 × 10
7
u
80
+ ··· 26056.8u 2654.30
3.30357 × 10
6
u
81
+ 0.0000103159u
80
+ ··· + 612901.u + 74572.0
a
3
=
2.84484 × 10
7
u
81
7.39387 × 10
7
u
80
+ ··· 26056.8u 2654.30
3.30357 × 10
6
u
81
+ 0.0000103159u
80
+ ··· + 612901.u + 74572.0
(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 = 3.54700 0.90383I
a = 0.255168 0.021069I
b = 3.95942 1.55029I
2.85067 0.96287I 12.24182 4.62200I
u = 3.54700 + 0.90383I
a = 0.255168 + 0.021069I
b = 3.95942 + 1.55029I
2.85067 + 0.96287I 12.24182 + 4.62200I
u = 3.44051 1.04039I
a = 0.290324 0.117842I
b = 2.87049 1.35355I
2.13971 + 6.01246I 8.23433 + 2.70516I
u = 3.44051 + 1.04039I
a = 0.290324 + 0.117842I
b = 2.87049 + 1.35355I
2.13971 6.01246I 8.23433 2.70516I
u = 2.71509
a = 0.403585
b = 4.04235
3.80714 30.5601
u = 1.92035 1.53508I
a = 0.268269 0.619370I
b = 2.87562 1.29334I
4.37377 + 8.54127I 9.27955 9.08846I
u = 1.92035 + 1.53508I
a = 0.268269 + 0.619370I
b = 2.87562 + 1.29334I
4.37377 8.54127I 9.27955 + 9.08846I
u = 1.70464 2.12904I
a = 0.166463 + 0.506562I
b = 2.29565 + 2.59894I
5.89382 + 11.02702I 6.79585 7.24164I
u = 1.70464 + 2.12904I
a = 0.166463 0.506562I
b = 2.29565 2.59894I
5.89382 11.02702I 6.79585 + 7.24164I
u = 1.59343 0.25776I
a = 0.110774 0.784797I
b = 0.78226 + 1.44506I
0.45408 + 11.38862I 3.22823 9.94815I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.59343 + 0.25776I
a = 0.110774 + 0.784797I
b = 0.78226 1.44506I
0.45408 11.38862I 3.22823 + 9.94815I
u = 1.147889 0.077523I
a = 0.157509 0.021928I
b = 1.316179 + 0.200491I
2.36954 + 0.63816I 5.46759 + 1.52326I
u = 1.147889 + 0.077523I
a = 0.157509 + 0.021928I
b = 1.316179 0.200491I
2.36954 0.63816I 5.46759 1.52326I
u = 1.115548 0.311169I
a = 1.001354 + 0.782258I
b = 1.96069 + 0.74365I
1.22744 + 12.16919I 2.52898 7.09852I
u = 1.115548 + 0.311169I
a = 1.001354 0.782258I
b = 1.96069 0.74365I
1.22744 12.16919I 2.52898 + 7.09852I
u = 1.103101 0.720481I
a = 0.309619 + 0.494805I
b = 0.114702 0.447528I
0.28541 1.54658I 0.57343 + 2.06881I
u = 1.103101 + 0.720481I
a = 0.309619 0.494805I
b = 0.114702 + 0.447528I
0.28541 + 1.54658I 0.57343 2.06881I
u = 1.03599 1.62898I
a = 0.145489 0.388533I
b = 0.57618 1.68729I
0.603497 0.216471I 2.65199 + 5.07001I
u = 1.03599 + 1.62898I
a = 0.145489 + 0.388533I
b = 0.57618 + 1.68729I
0.603497 + 0.216471I 2.65199 5.07001I
u = 0.974987 0.176971I
a = 0.066056 0.996039I
b = 1.50144 + 0.72596I
2.09067 + 2.95787I 3.92137 7.14092I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.974987 + 0.176971I
a = 0.066056 + 0.996039I
b = 1.50144 0.72596I
2.09067 2.95787I 3.92137 + 7.14092I
u = 0.931481 0.513910I
a = 0.725808 0.590391I
b = 0.598083 + 0.333211I
5.22827 + 3.35824I 3.04121 4.47637I
u = 0.931481 + 0.513910I
a = 0.725808 + 0.590391I
b = 0.598083 0.333211I
5.22827 3.35824I 3.04121 + 4.47637I
u = 0.919895
a = 1.45865
b = 1.03727
2.55123 4.11145
u = 0.897223 0.019803I
a = 1.51384 0.49006I
b = 0.135479 + 0.710516I
6.58899 8.11200I 6.71308 + 6.24405I
u = 0.897223 + 0.019803I
a = 1.51384 + 0.49006I
b = 0.135479 0.710516I
6.58899 + 8.11200I 6.71308 6.24405I
u = 0.891519 0.143867I
a = 1.215022 + 0.221669I
b = 0.442857 + 0.386483I
5.52870 + 0.86398I 20.6612 3.3440I
u = 0.891519 + 0.143867I
a = 1.215022 0.221669I
b = 0.442857 0.386483I
5.52870 0.86398I 20.6612 + 3.3440I
u = 0.819968 0.385481I
a = 1.37520 0.95056I
b = 1.098339 0.864418I
2.65606 0.12519I 6.98986 + 0.10633I
u = 0.819968 + 0.385481I
a = 1.37520 + 0.95056I
b = 1.098339 + 0.864418I
2.65606 + 0.12519I 6.98986 0.10633I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.81139 1.20292I
a = 0.378605 + 0.658497I
b = 0.18807 + 1.89415I
4.70519 + 2.39134I 16.5054 10.8772I
u = 0.81139 + 1.20292I
a = 0.378605 0.658497I
b = 0.18807 1.89415I
4.70519 2.39134I 16.5054 + 10.8772I
u = 0.706492 0.827720I
a = 1.29279 0.92164I
b = 0.26140 2.17909I
5.00012 + 4.29669I 10.41022 3.63392I
u = 0.706492 + 0.827720I
a = 1.29279 + 0.92164I
b = 0.26140 + 2.17909I
5.00012 4.29669I 10.41022 + 3.63392I
u = 0.611677 0.487979I
a = 0.623340 + 1.162822I
b = 1.50789 + 1.07819I
3.48456 + 2.48996I 0.90365 4.62403I
u = 0.611677 + 0.487979I
a = 0.623340 1.162822I
b = 1.50789 1.07819I
3.48456 2.48996I 0.90365 + 4.62403I
u = 0.546463 0.738573I
a = 0.204781 1.258090I
b = 0.449202 0.604555I
0.34685 + 2.85468I 2.55427 7.34325I
u = 0.546463 + 0.738573I
a = 0.204781 + 1.258090I
b = 0.449202 + 0.604555I
0.34685 2.85468I 2.55427 + 7.34325I
u = 0.416921 0.218817I
a = 1.78811 + 1.44846I
b = 0.686377 + 1.137109I
0.01560 + 3.91593I 3.20651 9.86113I
u = 0.416921 + 0.218817I
a = 1.78811 1.44846I
b = 0.686377 1.137109I
0.01560 3.91593I 3.20651 + 9.86113I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.321741 0.743270I
a = 0.202136 + 0.014721I
b = 0.876829 0.486347I
1.87549 1.38403I 4.93593 + 0.98614I
u = 0.321741 + 0.743270I
a = 0.202136 0.014721I
b = 0.876829 + 0.486347I
1.87549 + 1.38403I 4.93593 0.98614I
u = 0.283276 0.661835I
a = 0.558480 0.265028I
b = 0.814930 0.481448I
0.34729 + 6.79546I 2.04718 3.52041I
u = 0.283276 + 0.661835I
a = 0.558480 + 0.265028I
b = 0.814930 + 0.481448I
0.34729 6.79546I 2.04718 + 3.52041I
u = 0.114047 0.756375I
a = 0.539794 + 0.019112I
b = 0.088440 0.337986I
0.041508 1.376627I 0.16544 + 4.98668I
u = 0.114047 + 0.756375I
a = 0.539794 0.019112I
b = 0.088440 + 0.337986I
0.041508 + 1.376627I 0.16544 4.98668I
u = 0.156436 0.954361I
a = 0.339005 + 1.083918I
b = 0.59639 + 1.86743I
2.85407 + 3.43010I 8.3578 12.3401I
u = 0.156436 + 0.954361I
a = 0.339005 1.083918I
b = 0.59639 1.86743I
2.85407 3.43010I 8.3578 + 12.3401I
u = 0.620902 1.101458I
a = 0.885434 0.801956I
b = 0.78279 2.18549I
5.91812 + 0.62603I 12.66614 3.47864I
u = 0.620902 + 1.101458I
a = 0.885434 + 0.801956I
b = 0.78279 + 2.18549I
5.91812 0.62603I 12.66614 + 3.47864I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.635267 0.108056I
a = 0.259293 0.445131I
b = 0.374737 0.582371I
2.44851 1.57419I 1.58009 0.44639I
u = 0.635267 + 0.108056I
a = 0.259293 + 0.445131I
b = 0.374737 + 0.582371I
2.44851 + 1.57419I 1.58009 + 0.44639I
u = 0.81475 1.58964I
a = 0.320915 0.488166I
b = 1.02561 1.07702I
0.006489 + 1.319229I 7.9750 + 14.9866I
u = 0.81475 + 1.58964I
a = 0.320915 + 0.488166I
b = 1.02561 + 1.07702I
0.006489 1.319229I 7.9750 14.9866I
u = 0.831414 0.429688I
a = 0.90160 1.32654I
b = 1.66101 0.68827I
1.85423 3.78257I 5.86247 + 6.02702I
u = 0.831414 + 0.429688I
a = 0.90160 + 1.32654I
b = 1.66101 + 0.68827I
1.85423 + 3.78257I 5.86247 6.02702I
u = 0.84727 1.31140I
a = 0.103051 + 0.761261I
b = 0.68952 + 1.78277I
2.97750 8.50086I 0.82594 + 8.56912I
u = 0.84727 + 1.31140I
a = 0.103051 0.761261I
b = 0.68952 1.78277I
2.97750 + 8.50086I 0.82594 8.56912I
u = 0.885525 0.229111I
a = 0.242789 1.314391I
b = 1.73413 + 0.40200I
1.65757 + 0.93442I 5.02283 0.17696I
u = 0.885525 + 0.229111I
a = 0.242789 + 1.314391I
b = 1.73413 0.40200I
1.65757 0.93442I 5.02283 + 0.17696I
10
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.950405 0.049959I
a = 1.55228 0.25821I
b = 0.317000 + 0.645244I
8.00907 + 3.75643I 8.56422 + 3.52534I
u = 0.950405 + 0.049959I
a = 1.55228 + 0.25821I
b = 0.317000 0.645244I
8.00907 3.75643I 8.56422 3.52534I
u = 1.025775 0.058583I
a = 1.374674 + 0.200792I
b = 0.435523 + 0.177315I
7.77694 4.95365I 9.5185 + 11.5933I
u = 1.025775 + 0.058583I
a = 1.374674 0.200792I
b = 0.435523 0.177315I
7.77694 + 4.95365I 9.5185 11.5933I
u = 1.139716 0.446175I
a = 0.537983 + 0.058718I
b = 1.56701 0.08835I
0.96667 4.63901I 3.87963 + 8.57640I
u = 1.139716 + 0.446175I
a = 0.537983 0.058718I
b = 1.56701 + 0.08835I
0.96667 + 4.63901I 3.87963 8.57640I
u = 1.188111 0.468209I
a = 0.818849 + 0.687165I
b = 2.11415 + 0.83095I
3.71598 6.06156I 4.81312 + 3.61533I
u = 1.188111 + 0.468209I
a = 0.818849 0.687165I
b = 2.11415 0.83095I
3.71598 + 6.06156I 4.81312 3.61533I
u = 1.279540 0.303089I
a = 0.175697 + 0.877632I
b = 0.089632 0.656215I
2.93914 + 6.11947I 0.48338 6.25574I
u = 1.279540 + 0.303089I
a = 0.175697 0.877632I
b = 0.089632 + 0.656215I
2.93914 6.11947I 0.48338 + 6.25574I
11
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.43318 1.21313I
a = 0.393297 + 0.516767I
b = 0.78876 + 1.66228I
3.13741 9.59200I 7.14919 + 9.30743I
u = 1.43318 + 1.21313I
a = 0.393297 0.516767I
b = 0.78876 1.66228I
3.13741 + 9.59200I 7.14919 9.30743I
u = 1.48638 0.57080I
a = 0.138595 + 0.564697I
b = 0.348376 0.150644I
4.80113 1.92406I 2.63229 + 2.70065I
u = 1.48638 + 0.57080I
a = 0.138595 0.564697I
b = 0.348376 + 0.150644I
4.80113 + 1.92406I 2.63229 2.70065I
u = 1.73512 0.58134I
a = 0.160559 0.574170I
b = 0.31539 + 1.38178I
1.46560 5.42569I 4.53155 + 7.12185I
u = 1.73512 + 0.58134I
a = 0.160559 + 0.574170I
b = 0.31539 1.38178I
1.46560 + 5.42569I 4.53155 7.12185I
u = 1.96066 2.24482I
a = 0.160658 0.480708I
b = 2.94046 2.37920I
5.73183 3.04738I 12.16397 + 4.67101I
u = 1.96066 + 2.24482I
a = 0.160658 + 0.480708I
b = 2.94046 + 2.37920I
5.73183 + 3.04738I 12.16397 4.67101I
u = 1.99486 1.78114I
a = 0.250760 + 0.512649I
b = 2.63470 + 1.97142I
4.0922 17.7863I 5.47301 + 10.24572I
u = 1.99486 + 1.78114I
a = 0.250760 0.512649I
b = 2.63470 1.97142I
4.0922 + 17.7863I 5.47301 10.24572I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 5.76782 1.88095I
a = 0.162166 0.057730I
b = 5.02878 2.12159I
3.74856 1.03244I 28.3012 20.0875I
u = 5.76782 + 1.88095I
a = 0.162166 + 0.057730I
b = 5.02878 + 2.12159I
3.74856 + 1.03244I 28.3012 + 20.0875I
12
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
5
(u
82
+ 6u
81
+ ··· 8u 1)
c
2
(u + 1)
5
(u
82
+ 42u
81
+ ··· + 32u + 1)
c
3
, c
6
u
5
(u
82
+ u
81
+ ··· 160u + 32)
c
4
(u + 1)
5
(u
82
+ 6u
81
+ ··· 8u 1)
c
5
(u
5
3u
4
+ ··· u + 1)(u
82
+ 6u
81
+ ··· 2u 1)
c
7
(u
5
u
4
+ ··· + u 1)(u
82
+ 14u
81
+ ··· + 2u + 1)
c
8
(u + 1)(u
5
+ u
4
+ ··· + u 1)(u
81
+ u
80
+ ··· 15u + 1)
c
9
(u
5
u
4
+ ··· + u + 1)(u
82
+ 2u
81
+ ··· 20520u 1647)
c
10
(u
5
+ u
4
+ ··· + u + 1)(u
82
+ 2u
81
+ ··· + 2362u 484)
c
11
(u
5
u
4
+ ··· + u + 1)(u
82
+ 2u
81
+ ··· 14u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
5
(y
82
42y
81
+ ··· 32y + 1)
c
2
(y 1)
5
(y
82
+ 2y
81
+ ··· 484y + 1)
c
3
, c
6
y
5
(y
82
33y
81
+ ··· 19968y + 1024)
c
5
(y
5
y
4
+ ··· + 3y 1)(y
82
14y
81
+ ··· 6y + 1)
c
7
(y
5
+ 3y
4
+ ··· y 1)(y
82
+ 6y
81
+ ··· + 14y + 1)
c
8
(y
5
5y
4
+ ··· y 1)(y
82
58y
81
+ ··· + 14y + 1)
c
9
(y
5
5y
4
+ 8y
3
3y
2
y 1)
(y
82
+ 50y
81
+ ··· 261288342y + 2712609)
c
10
(y
5
+ 3y
4
+ 4y
3
+ y
2
y 1)
(y
82
+ 90y
81
+ ··· + 10862436y + 234256)
c
11
(y
5
5y
4
+ ··· y 1)(y
82
58y
81
+ ··· + 14y + 1)
14