11a
11
(K11a
11
)
1
Arc Sequences
5 1 8 2 3 11 9 4 7 6 10
Solving Sequence
4,9
8 3 7
10,11
1 2 6 5
c
8
c
3
c
7
c
9
c
11
c
2
c
6
c
5
c
1
, c
4
, c
10
Representation Ideals
I = I
u
1
\
I
v
1
I
u
1
= hu
58
+ u
57
+ ··· 4u + 4, 3.09096 × 10
31
u
57
6.61165 × 10
31
u
56
+ ··· + 4.71586 × 10
31
b 1.87421 × 10
32
,
6.39352 × 10
31
u
57
+ 8.81969 × 10
31
u
56
+ ··· + 2.35793 × 10
31
a 3.61858 × 10
32
i
I
v
1
= h−b + v 1, b
2
+ b + 1, ai
There are 2 irreducible components with 60 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
58
+u
57
+· · ·4u +4, 3.09×10
31
u
57
6.61× 10
31
u
56
+· · ·+4.72 ×10
31
b
1.87 × 10
32
, 6.39 × 10
31
u
57
+ 8.82 × 10
31
u
56
+ · · · + 2.36 × 10
31
a 3.62 × 10
32
i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
8
=
u
u
a
3
=
u
2
+ 1
u
2
a
7
=
u
u
3
+ u
a
10
=
u
3
u
5
u
3
+ u
a
11
=
2.71150u
57
3.74044u
56
+ ··· + 2.09467u + 15.3464
0.655440u
57
+ 1.40200u
56
+ ··· 2.24430u + 3.97426
a
1
=
2.70789u
57
4.52920u
56
+ ··· + 6.30614u + 15.3067
0.656171u
57
+ 1.40311u
56
+ ··· + 0.932406u 1.22587
a
2
=
2.94156u
57
+ 5.06615u
56
+ ··· 14.3092u 16.1539
1.30694u
57
+ 2.20766u
56
+ ··· 3.11530u 4.20133
a
6
=
2.36713u
57
+ 4.08110u
56
+ ··· 1.82738u 9.24737
0.340761u
57
0.448100u
56
+ ··· + 4.47876u + 6.05934
a
5
=
2.83660u
57
+ 4.52459u
56
+ ··· 1.25804u 7.48548
0.527460u
57
1.40772u
56
+ ··· + 4.11569u + 9.04710
a
5
=
2.83660u
57
+ 4.52459u
56
+ ··· 1.25804u 7.48548
0.527460u
57
1.40772u
56
+ ··· + 4.11569u + 9.04710
(ii) Obstruction class = 1
(iii) Cusp Shapes = 1.35369u
57
+ 2.48411u
56
+ ··· 5.92434u 3.41763
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.048561 0.067957I
a = 0.559839 0.037746I
b = 0.685431 0.626108I
5.05702 + 2.63543I 8.96800 3.93457I
u = 1.048561 + 0.067957I
a = 0.559839 + 0.037746I
b = 0.685431 + 0.626108I
5.05702 2.63543I 8.96800 + 3.93457I
u = 1.038689 0.260534I
a = 0.382607 + 0.196103I
b = 0.476634 0.716148I
4.36361 4.61978I 7.52611 + 4.69531I
u = 1.038689 + 0.260534I
a = 0.382607 0.196103I
b = 0.476634 + 0.716148I
4.36361 + 4.61978I 7.52611 4.69531I
u = 1.025183 0.826072I
a = 1.56413 + 1.82853I
b = 3.60552 0.60183I
5.5002 14.7239I 1.22769 + 9.89191I
u = 1.025183 + 0.826072I
a = 1.56413 1.82853I
b = 3.60552 + 0.60183I
5.5002 + 14.7239I 1.22769 9.89191I
u = 0.990315 0.389853I
a = 0.911414 + 0.955546I
b = 0.229881 + 0.238137I
0.45426 4.86179I 0.56072 + 6.44358I
u = 0.990315 + 0.389853I
a = 0.911414 0.955546I
b = 0.229881 0.238137I
0.45426 + 4.86179I 0.56072 6.44358I
u = 0.988463 0.757489I
a = 1.39536 + 1.73246I
b = 2.91390 0.15577I
2.01751 7.25755I 2.01563 + 5.53872I
u = 0.988463 + 0.757489I
a = 1.39536 1.73246I
b = 2.91390 + 0.15577I
2.01751 + 7.25755I 2.01563 5.53872I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.944641 0.777587I
a = 0.202921 + 0.404867I
b = 0.0521931 + 0.0917099I
4.07167 4.42773I 1.08988 + 2.76799I
u = 0.944641 + 0.777587I
a = 0.202921 0.404867I
b = 0.0521931 0.0917099I
4.07167 + 4.42773I 1.08988 2.76799I
u = 0.898879 0.794879I
a = 2.27583 1.17813I
b = 2.55245 + 1.82789I
6.42683 5.27485I 2.53353 + 4.61239I
u = 0.898879 + 0.794879I
a = 2.27583 + 1.17813I
b = 2.55245 1.82789I
6.42683 + 5.27485I 2.53353 4.61239I
u = 0.881786 0.800253I
a = 1.22945 + 1.96308I
b = 3.32632 + 0.91591I
6.48025 0.70893I 2.72501 + 1.07342I
u = 0.881786 + 0.800253I
a = 1.22945 1.96308I
b = 3.32632 0.91591I
6.48025 + 0.70893I 2.72501 1.07342I
u = 0.826470 0.811218I
a = 0.254122 + 0.271069I
b = 0.379361 + 0.111056I
4.43701 1.53415I 1.72366 + 2.51421I
u = 0.826470 + 0.811218I
a = 0.254122 0.271069I
b = 0.379361 0.111056I
4.43701 + 1.53415I 1.72366 2.51421I
u = 0.792798 0.939540I
a = 1.87151 1.33967I
b = 3.41570 + 0.60471I
6.24695 + 8.22888I 2.36913 5.79102I
u = 0.792798 + 0.939540I
a = 1.87151 + 1.33967I
b = 3.41570 0.60471I
6.24695 8.22888I 2.36913 + 5.79102I
4
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.746191 0.812452I
a = 1.94463 1.01558I
b = 2.38492 + 0.69473I
2.74272 + 1.35672I 0.686842 0.658171I
u = 0.746191 + 0.812452I
a = 1.94463 + 1.01558I
b = 2.38492 0.69473I
2.74272 1.35672I 0.686842 + 0.658171I
u = 0.732349 0.313831I
a = 0.221327 + 1.099831I
b = 0.266285 + 1.344798I
1.56865 3.63401I 0.50067 + 8.81328I
u = 0.732349 + 0.313831I
a = 0.221327 1.099831I
b = 0.266285 1.344798I
1.56865 + 3.63401I 0.50067 8.81328I
u = 0.536435 0.432411I
a = 2.17911 + 0.03542I
b = 0.543168 + 0.254424I
2.16202 + 0.77457I 3.14237 + 2.08066I
u = 0.536435 + 0.432411I
a = 2.17911 0.03542I
b = 0.543168 0.254424I
2.16202 0.77457I 3.14237 2.08066I
u = 0.327071 0.663667I
a = 1.233135 0.055842I
b = 0.770460 0.508189I
1.77616 + 0.94992I 4.44755 1.71410I
u = 0.327071 + 0.663667I
a = 1.233135 + 0.055842I
b = 0.770460 + 0.508189I
1.77616 0.94992I 4.44755 + 1.71410I
u = 0.036590 0.771690I
a = 0.350529 0.136230I
b = 0.093556 1.021657I
0.81275 + 1.40752I 3.05190 0.70072I
u = 0.036590 + 0.771690I
a = 0.350529 + 0.136230I
b = 0.093556 + 1.021657I
0.81275 1.40752I 3.05190 + 0.70072I
5
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.278867 0.850545I
a = 0.874500 0.473068I
b = 0.92049 1.15445I
0.11517 4.84774I 0.96980 + 7.14409I
u = 0.278867 + 0.850545I
a = 0.874500 + 0.473068I
b = 0.92049 + 1.15445I
0.11517 + 4.84774I 0.96980 7.14409I
u = 0.291033 0.534414I
a = 0.0153111 0.0176376I
b = 0.339612 0.487861I
0.32061 + 1.54716I 2.16073 4.65280I
u = 0.291033 + 0.534414I
a = 0.0153111 + 0.0176376I
b = 0.339612 + 0.487861I
0.32061 1.54716I 2.16073 + 4.65280I
u = 0.649706 0.200927I
a = 0.157184 + 0.869311I
b = 0.95807 + 1.28027I
1.09083 1.19632I 2.67942 2.86295I
u = 0.649706 + 0.200927I
a = 0.157184 0.869311I
b = 0.95807 1.28027I
1.09083 + 1.19632I 2.67942 + 2.86295I
u = 0.715542 0.182951I
a = 2.75787 0.05269I
b = 0.138663 + 0.400271I
0.90066 + 2.88860I 3.50834 6.47238I
u = 0.715542 + 0.182951I
a = 2.75787 + 0.05269I
b = 0.138663 0.400271I
0.90066 2.88860I 3.50834 + 6.47238I
u = 0.768453 0.836694I
a = 0.280171 + 0.206896I
b = 0.535760 + 0.093490I
2.81999 3.64889I 0.72605 + 2.41695I
u = 0.768453 + 0.836694I
a = 0.280171 0.206896I
b = 0.535760 0.093490I
2.81999 + 3.64889I 0.72605 2.41695I
6
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.813150 0.906222I
a = 1.95775 1.29698I
b = 3.23685 + 0.88444I
8.06195 2.97069I 5.15213 + 1.06762I
u = 0.813150 + 0.906222I
a = 1.95775 + 1.29698I
b = 3.23685 0.88444I
8.06195 + 2.97069I 5.15213 1.06762I
u = 0.870741 0.827532I
a = 2.17780 1.20916I
b = 2.77149 + 1.54849I
8.16003 + 0.02626I 5.24638 + 0.26541I
u = 0.870741 + 0.827532I
a = 2.17780 + 1.20916I
b = 2.77149 1.54849I
8.16003 0.02626I 5.24638 0.26541I
u = 0.903498 0.662172I
a = 0.082491 + 0.331374I
b = 0.161407 0.181206I
1.07636 + 2.53447I 5.45479 2.49966I
u = 0.903498 + 0.662172I
a = 0.082491 0.331374I
b = 0.161407 + 0.181206I
1.07636 2.53447I 5.45479 + 2.49966I
u = 0.913527 0.181757I
a = 0.301584 + 0.019035I
b = 0.554517 0.572023I
1.79712 + 0.58239I 4.29390 0.53701I
u = 0.913527 + 0.181757I
a = 0.301584 0.019035I
b = 0.554517 + 0.572023I
1.79712 0.58239I 4.29390 + 0.53701I
u = 0.920258 0.810339I
a = 1.32251 + 1.93351I
b = 3.45813 + 0.53171I
8.00524 + 6.08837I 4.76268 5.76942I
u = 0.920258 + 0.810339I
a = 1.32251 1.93351I
b = 3.45813 0.53171I
8.00524 6.08837I 4.76268 + 5.76942I
7
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.987501 0.771501I
a = 0.186336 + 0.456524I
b = 0.0758896 + 0.0706432I
2.14956 + 9.66192I 1.84579 7.09878I
u = 0.987501 + 0.771501I
a = 0.186336 0.456524I
b = 0.0758896 0.0706432I
2.14956 9.66192I 1.84579 + 7.09878I
u = 0.998202 0.822274I
a = 1.50191 + 1.85613I
b = 3.58439 0.30006I
7.47129 + 9.35808I 4.04876 5.69041I
u = 0.998202 + 0.822274I
a = 1.50191 1.85613I
b = 3.58439 + 0.30006I
7.47129 9.35808I 4.04876 + 5.69041I
u = 1.053535 0.258866I
a = 0.886093 + 0.610626I
b = 0.361952 + 0.187903I
4.33981 + 1.99854I 7.41077 3.19574I
u = 1.053535 + 0.258866I
a = 0.886093 0.610626I
b = 0.361952 0.187903I
4.33981 1.99854I 7.41077 + 3.19574I
u = 1.077230 0.420247I
a = 1.13071 + 0.90891I
b = 0.225654 0.241080I
2.89029 + 9.39325I 4.11066 9.87037I
u = 1.077230 + 0.420247I
a = 1.13071 0.90891I
b = 0.225654 + 0.241080I
2.89029 9.39325I 4.11066 + 9.87037I
8
II. I
v
1
= h−b + v 1, b
2
+ b + 1, ai
(i) Arc colorings
a
4
=
1
0
a
9
=
b + 1
0
a
8
=
b + 1
0
a
3
=
1
0
a
7
=
b + 1
0
a
10
=
b + 1
0
a
11
=
0
b
a
1
=
b 1
b
a
2
=
0
b + 1
a
6
=
b + 1
b
a
5
=
1
b
a
5
=
1
b
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4b + 1
9
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 0.500000 0.866025I
a = 0
b = 0.500000 0.866025I
1.64493 2.02988I 3.00000 + 3.46410I
v = 0.500000 + 0.866025I
a = 0
b = 0.500000 + 0.866025I
1.64493 + 2.02988I 3.00000 3.46410I
10
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)(u
58
+ 2u
57
+ ··· + 3u + 1)
c
2
(u
2
u + 1)(u
58
+ 26u
57
+ ··· + 5u + 1)
c
3
, c
8
u
2
(u
58
+ u
57
+ ··· 4u + 4)
c
4
(u
2
u + 1)(u
58
+ 2u
57
+ ··· + 3u + 1)
c
5
(u
2
u + 1)(u
58
+ 2u
57
+ ··· + 5u + 1)
c
6
, c
10
(u 1)
2
(u
58
+ 3u
57
+ ··· + 2u + 1)
c
7
, c
9
u
2
(u
58
+ 15u
57
+ ··· + 168u + 16)
c
11
(u + 1)
2
(u
58
+ 33u
57
+ ··· 2u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
2
+ y + 1)(y
58
+ 26y
57
+ ··· + 5y + 1)
c
2
(y
2
+ y + 1)(y
58
+ 14y
57
+ ··· + 29y + 1)
c
3
, c
8
y
2
(y
58
15y
57
+ ··· 168y + 16)
c
5
(y
2
+ y + 1)(y
58
+ 2y
57
+ ··· + 53y + 1)
c
6
, c
10
(y 1)
2
(y
58
33y
57
+ ··· + 2y + 1)
c
7
, c
9
y
2
(y
58
+ 53y
57
+ ··· + 2784y + 256)
c
11
(y 1)
2
(y
58
13y
57
+ ··· + 42y + 1)
12