11a
130
(K11a
130
)
1
Arc Sequences
6 1 11 10 2 9 3 5 7 4 8
Solving Sequence
1,6
2
3,9
7 10 5 4 8 11
c
1
c
2
c
6
c
9
c
5
c
4
c
8
c
11
c
3
, c
7
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu + 1, a + 1, 3b 4i
I
u
2
= hu
62
+ 2u
61
+ ··· + 2u + 1, 2.85458 × 10
50
u
61
+ 4.03585 × 10
49
u
60
+ ··· + 8.51424 × 10
50
b + 2.78875 × 10
49
,
2.98376 × 10
50
u
61
+ 6.37708 × 10
50
u
60
+ ··· + 2.83808 × 10
50
a + 6.40961 × 10
50
i
There are 2 irreducible components with 63 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 + 1, a + 1, 3b 4i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
1
a
2
=
1
1
a
3
=
2
1
a
9
=
1
4
3
a
7
=
1
1
3
a
10
=
0
1
a
5
=
1
2
a
4
=
1
1
a
8
=
1
3
0
a
11
=
1
0
a
11
=
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 = 1.00000
b = 1.33333
3.28987 19.1111
3
II. I
u
2
=
hu
62
+ 2u
61
+ · · · +2u + 1, 2.85 × 10
50
u
61
+ 4.04 × 10
49
u
60
+ · · · + 8.51 × 10
50
b +
2.79 × 10
49
, 2.98 × 10
50
u
61
+ 6.38 × 10
50
u
60
+ · · · + 2.84 × 10
50
a + 6.41 × 10
50
i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
9
=
1.05133u
61
2.24697u
60
+ ··· + 3.69845u 2.25843
0.335271u
61
0.0474012u
60
+ ··· + 0.636087u 0.0327539
a
7
=
1.02842u
61
2.17650u
60
+ ··· + 5.74168u 2.19565
0.243266u
61
+ 0.159474u
60
+ ··· + 1.94873u + 0.101743
a
10
=
0.0640050u
61
0.204956u
60
+ ··· 4.13365u 0.176772
0.167886u
61
0.370564u
60
+ ··· 1.53024u 0.235976
a
5
=
u
u
3
+ u
a
4
=
0.378896u
61
+ 1.00408u
60
+ ··· + 0.00761576u + 0.969819
0.246292u
61
0.348316u
60
+ ··· + 0.212027u 0.378896
a
8
=
1.10694u
61
1.73632u
60
+ ··· + 3.37517u 1.75994
0.0562646u
61
+ 0.150179u
60
+ ··· + 2.14749u + 0.0906228
a
11
=
0.235976u
61
+ 0.304067u
60
+ ··· + 0.223734u 1.05829
0.0769461u
61
0.0781366u
60
+ ··· 0.0487618u + 0.0640050
a
11
=
0.235976u
61
+ 0.304067u
60
+ ··· + 0.223734u 1.05829
0.0769461u
61
0.0781366u
60
+ ··· 0.0487618u + 0.0640050
(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 = 1.16418
a = 0.794736
b = 1.51241
3.07751 20.4252
u = 0.994996 0.610567I
a = 0.737295 0.649733I
b = 1.65200 0.67537I
4.56334 0.20821I 10.45314 + 4.67004I
u = 0.994996 + 0.610567I
a = 0.737295 + 0.649733I
b = 1.65200 + 0.67537I
4.56334 + 0.20821I 10.45314 4.67004I
u = 0.902984 0.556883I
a = 0.783507 1.123853I
b = 1.70739 0.58303I
1.20729 10.07416I 5.18115 + 4.93729I
u = 0.902984 + 0.556883I
a = 0.783507 + 1.123853I
b = 1.70739 + 0.58303I
1.20729 + 10.07416I 5.18115 4.93729I
u = 0.742285 1.100859I
a = 0.720083 0.587607I
b = 2.18544 0.82844I
3.00308 + 6.49550I 6.56838 5.96882I
u = 0.742285 + 1.100859I
a = 0.720083 + 0.587607I
b = 2.18544 + 0.82844I
3.00308 6.49550I 6.56838 + 5.96882I
u = 0.701257 1.097869I
a = 0.987932 0.610384I
b = 2.65408 0.93933I
2.8670 + 15.9906I 2.98705 8.93821I
u = 0.701257 + 1.097869I
a = 0.987932 + 0.610384I
b = 2.65408 + 0.93933I
2.8670 15.9906I 2.98705 + 8.93821I
u = 0.698170 0.530206I
a = 0.849543 + 1.031523I
b = 0.196075 0.905744I
5.19208 4.01888I 2.73793 + 2.84274I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.698170 + 0.530206I
a = 0.849543 1.031523I
b = 0.196075 + 0.905744I
5.19208 + 4.01888I 2.73793 2.84274I
u = 0.659387 0.732568I
a = 0.27677 + 1.57956I
b = 1.186756 0.055930I
0.24430 1.55160I 7.25020 + 1.87836I
u = 0.659387 + 0.732568I
a = 0.27677 1.57956I
b = 1.186756 + 0.055930I
0.24430 + 1.55160I 7.25020 1.87836I
u = 0.656882 0.844482I
a = 1.07234 + 1.24211I
b = 2.27766 + 0.68264I
1.05860 + 2.54967I 8.07115 3.38859I
u = 0.656882 + 0.844482I
a = 1.07234 1.24211I
b = 2.27766 0.68264I
1.05860 2.54967I 8.07115 + 3.38859I
u = 0.647942 0.935076I
a = 1.348177 + 0.370353I
b = 2.01490 + 0.89553I
0.85413 + 6.64291I 5.52706 8.01438I
u = 0.647942 + 0.935076I
a = 1.348177 0.370353I
b = 2.01490 0.89553I
0.85413 6.64291I 5.52706 + 8.01438I
u = 0.629062 1.035700I
a = 0.863029 0.706501I
b = 0.267757 + 1.129682I
6.64439 + 9.14751I 0.63477 7.47677I
u = 0.629062 + 1.035700I
a = 0.863029 + 0.706501I
b = 0.267757 1.129682I
6.64439 9.14751I 0.63477 + 7.47677I
u = 0.576585 0.858853I
a = 0.685269 + 0.875028I
b = 3.10260 + 1.51311I
2.17222 + 2.29029I 6.76562 + 2.81680I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.576585 + 0.858853I
a = 0.685269 0.875028I
b = 3.10260 1.51311I
2.17222 2.29029I 6.76562 2.81680I
u = 0.545219 1.002890I
a = 0.362764 0.205266I
b = 0.251171 0.154629I
0.34287 + 3.16265I 3.47830 1.04615I
u = 0.545219 + 1.002890I
a = 0.362764 + 0.205266I
b = 0.251171 + 0.154629I
0.34287 3.16265I 3.47830 + 1.04615I
u = 0.453175 0.667312I
a = 0.093495 + 0.563326I
b = 0.587804 0.249337I
0.771077 + 1.084407I 6.64507 5.83005I
u = 0.453175 + 0.667312I
a = 0.093495 0.563326I
b = 0.587804 + 0.249337I
0.771077 1.084407I 6.64507 + 5.83005I
u = 0.227124
a = 4.88292
b = 0.214799
2.13260 1.50213
u = 0.150735 1.228251I
a = 0.428838 0.698808I
b = 0.672836 0.257976I
2.11951 + 4.18883I 1.15636 7.37516I
u = 0.150735 + 1.228251I
a = 0.428838 + 0.698808I
b = 0.672836 + 0.257976I
2.11951 4.18883I 1.15636 + 7.37516I
u = 0.125644 0.540647I
a = 0.671685 + 0.939334I
b = 0.176229 0.671009I
0.631390 + 1.069145I 6.33384 6.12642I
u = 0.125644 + 0.540647I
a = 0.671685 0.939334I
b = 0.176229 + 0.671009I
0.631390 1.069145I 6.33384 + 6.12642I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.055192 1.081899I
a = 0.908543 0.369254I
b = 1.072680 0.583147I
10.23622 2.66149I 4.14851 + 1.11224I
u = 0.055192 + 1.081899I
a = 0.908543 + 0.369254I
b = 1.072680 + 0.583147I
10.23622 + 2.66149I 4.14851 1.11224I
u = 0.081888 1.222834I
a = 0.560235 0.878664I
b = 0.768435 0.052862I
8.08648 8.33462I 1.00778 + 6.22474I
u = 0.081888 + 1.222834I
a = 0.560235 + 0.878664I
b = 0.768435 + 0.052862I
8.08648 + 8.33462I 1.00778 6.22474I
u = 0.147039 1.062938I
a = 0.606301 0.316931I
b = 0.796135 0.542972I
3.46490 + 0.19843I 2.36990 1.02428I
u = 0.147039 + 1.062938I
a = 0.606301 + 0.316931I
b = 0.796135 + 0.542972I
3.46490 0.19843I 2.36990 + 1.02428I
u = 0.169706 0.789053I
a = 0.801761 + 0.622556I
b = 0.084834 1.408499I
4.73418 2.70185I 0.85446 + 2.99856I
u = 0.169706 + 0.789053I
a = 0.801761 0.622556I
b = 0.084834 + 1.408499I
4.73418 + 2.70185I 0.85446 2.99856I
u = 0.362305 0.172735I
a = 2.98011 + 0.56930I
b = 0.354031 0.102042I
2.08593 1.24865I 4.04804 0.25829I
u = 0.362305 + 0.172735I
a = 2.98011 0.56930I
b = 0.354031 + 0.102042I
2.08593 + 1.24865I 4.04804 + 0.25829I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.505068 0.904960I
a = 0.428066 + 0.836649I
b = 3.90112 7.11244I
3.33744 1.94948I 26.1819 + 39.3616I
u = 0.505068 + 0.904960I
a = 0.428066 0.836649I
b = 3.90112 + 7.11244I
3.33744 + 1.94948I 26.1819 39.3616I
u = 0.545167 1.169648I
a = 0.027408 0.667074I
b = 0.814817 0.011236I
4.97349 0.07836I 2.07213 3.23818I
u = 0.545167 + 1.169648I
a = 0.027408 + 0.667074I
b = 0.814817 + 0.011236I
4.97349 + 0.07836I 2.07213 + 3.23818I
u = 0.581310 0.693806I
a = 0.256001 0.225470I
b = 0.948976 0.648855I
4.86855 2.39184I 1.23135 + 2.56041I
u = 0.581310 + 0.693806I
a = 0.256001 + 0.225470I
b = 0.948976 + 0.648855I
4.86855 + 2.39184I 1.23135 2.56041I
u = 0.605270 0.552472I
a = 0.453300 + 0.920362I
b = 0.129776 0.529358I
0.78887 + 1.60795I 6.50613 4.85102I
u = 0.605270 + 0.552472I
a = 0.453300 0.920362I
b = 0.129776 + 0.529358I
0.78887 1.60795I 6.50613 + 4.85102I
u = 0.606375 1.017388I
a = 0.764145 0.432270I
b = 0.490376 + 0.690143I
0.54466 6.46296I 3.86014 + 8.94603I
u = 0.606375 + 1.017388I
a = 0.764145 + 0.432270I
b = 0.490376 0.690143I
0.54466 + 6.46296I 3.86014 8.94603I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.633930 0.901953I
a = 1.141884 + 0.688505I
b = 2.31553 + 0.71170I
3.74985 4.26844I 11.92903 + 7.28144I
u = 0.633930 + 0.901953I
a = 1.141884 0.688505I
b = 2.31553 0.71170I
3.74985 + 4.26844I 11.92903 7.28144I
u = 0.634236 0.783021I
a = 0.59363 + 1.35933I
b = 1.53204 + 0.60523I
4.11698 0.69808I 13.39590 0.06993I
u = 0.634236 + 0.783021I
a = 0.59363 1.35933I
b = 1.53204 0.60523I
4.11698 + 0.69808I 13.39590 + 0.06993I
u = 0.714389 1.100810I
a = 0.871910 0.628101I
b = 2.46774 0.84422I
3.57519 11.87876I 6.63470 + 8.97401I
u = 0.714389 + 1.100810I
a = 0.871910 + 0.628101I
b = 2.46774 + 0.84422I
3.57519 + 11.87876I 6.63470 8.97401I
u = 0.776428 0.978919I
a = 0.529613 0.348603I
b = 1.72977 1.12815I
5.19332 2.98425I 2.12446 + 6.24469I
u = 0.776428 + 0.978919I
a = 0.529613 + 0.348603I
b = 1.72977 + 1.12815I
5.19332 + 2.98425I 2.12446 6.24469I
u = 0.929142 0.570567I
a = 0.807883 0.928250I
b = 1.67922 0.63563I
5.21542 + 5.84748I 9.04556 5.18141I
u = 0.929142 + 0.570567I
a = 0.807883 + 0.928250I
b = 1.67922 + 0.63563I
5.21542 5.84748I 9.04556 + 5.18141I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.942919 0.364850I
a = 1.107068 0.055587I
b = 1.29930 0.65086I
2.26534 5.42283I 4.72502 + 7.90618I
u = 0.942919 + 0.364850I
a = 1.107068 + 0.055587I
b = 1.29930 + 0.65086I
2.26534 + 5.42283I 4.72502 7.90618I
10
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u + 1)(u
62
+ 2u
61
+ ··· + 2u + 1)
c
2
(u 1)(u
62
+ 22u
61
+ ··· 8u + 1)
c
3
, c
4
(u 1)(u
62
+ 2u
61
+ ··· 2u 1)
c
5
(u 1)(u
62
+ 2u
61
+ ··· + 2u + 1)
c
6
(u + 1)(u
62
+ 2u
61
+ ··· 4u 9)
c
7
(3u 1)(3u
62
+ 52u
61
+ ··· 308u + 49)
c
8
(3u 2)(3u
62
+ 43u
61
+ ··· + 708u 62)
c
9
(u 1)(u
62
+ 2u
61
+ ··· 4u 9)
c
10
(u + 1)(u
62
+ 2u
61
+ ··· 2u 1)
c
11
(u)(u
62
+ 5u
61
+ ··· + 36u + 18)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y 1)(y
62
+ 22y
61
+ ··· 8y + 1)
c
2
(y 1)(y
62
+ 30y
61
+ ··· 352y + 1)
c
3
, c
4
, c
10
(y 1)(y
62
+ 58y
61
+ ··· 8y + 1)
c
6
, c
9
(y 1)(y
62
38y
61
+ ··· + 524y + 81)
c
7
(9y 1)(9y
62
1042y
61
+ ··· 72128y + 2401)
c
8
(9y 4)(9y
62
1081y
61
+ ··· + 151348y + 3844)
c
11
(y)(y
62
9y
61
+ ··· 180y + 324)
12