11a
82
(K11a
82
)
1
Arc Sequences
5 1 9 11 2 3 10 4 7 8 6
Solving Sequence
2,6
5 1 3 11
4,8
10 7 9
c
5
c
1
c
2
c
11
c
4
c
10
c
7
c
9
c
3
, c
6
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
6
+ u
5
u
4
2u
3
+ u + 1, u
5
u
3
u
2
+ b, u
5
+ u
4
u
3
2u
2
+ a + 1i
I
u
2
= hu
53
+ 2u
52
+ ··· + u + 1, u
51
+ u
50
+ ··· 3u
2
+ b, u
51
+ u
50
+ ··· + a + 1i
There are 2 irreducible components with 59 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
6
+ u
5
u
4
2u
3
+ u + 1, u
5
u
3
u
2
+ b, u
5
+ u
4
u
3
2u
2
+ a + 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
5
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
u
2
+ 1
u
4
a
11
=
u
2
+ 1
u
4
a
4
=
u
4
u
2
+ 1
u
4
a
8
=
u
5
u
4
+ u
3
+ 2u
2
1
u
5
+ u
3
+ u
2
a
10
=
u
5
u
4
+ u
3
+ u
2
u
5
u
4
+ u
3
+ u
2
a
7
=
u
2
1
u
4
a
9
=
u
5
u
4
+ u
3
+ 2u
2
1
u
5
+ u
3
+ u
2
a
9
=
u
5
u
4
+ u
3
+ 2u
2
1
u
5
+ u
3
+ u
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 = 1.073950 0.558752I
a = 0.732786 + 0.381252I
b = 1.30674 + 1.20014I
1.64493 5.69302I 5.23279 + 4.86918I
u = 1.073950 + 0.558752I
a = 0.732786 0.381252I
b = 1.30674 1.20014I
1.64493 + 5.69302I 5.23279 4.86918I
u = 0.428243 0.664531I
a = 0.685196 + 1.063260I
b = 0.315740 + 0.200172I
3.53554 + 0.92430I 9.40317 0.69886I
u = 0.428243 + 0.664531I
a = 0.685196 1.063260I
b = 0.315740 0.200172I
3.53554 0.92430I 9.40317 + 0.69886I
u = 1.002193 0.295542I
a = 0.917982 + 0.270708I
b = 1.49099 0.22339I
0.245672 + 0.924305I 0.635956 + 0.093695I
u = 1.002193 + 0.295542I
a = 0.917982 0.270708I
b = 1.49099 + 0.22339I
0.245672 0.924305I 0.635956 0.093695I
3
II.
I
u
2
= hu
53
+ 2u
52
+ · · · + u + 1, u
51
+ u
50
+ · · · 3u
2
+ b, u
51
+ u
50
+ · · · + a + 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
5
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
u
2
+ 1
u
4
a
11
=
u
2
+ 1
u
4
a
4
=
u
7
2u
5
+ 2u
3
u
9
u
7
+ u
5
+ u
a
8
=
u
51
u
50
+ ··· + 2u 1
u
51
u
50
+ ··· 2u
3
+ 3u
2
a
10
=
u
51
u
50
+ ··· + 2u 1
u
52
2u
51
+ ··· u 1
a
7
=
u
9
2u
7
+ 3u
5
2u
3
+ u
u
9
u
7
+ u
5
+ u
a
9
=
u
51
u
50
+ ··· + 3u 2
2u
52
3u
51
+ ··· u 2
a
9
=
u
51
u
50
+ ··· + 3u 2
2u
52
3u
51
+ ··· u 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 = 1.125346 0.320255I
a = 1.064506 + 0.530920I
b = 1.78896 + 0.12727I
1.92863 + 0.10640I 2.34145 + 0.07111I
u = 1.125346 + 0.320255I
a = 1.064506 0.530920I
b = 1.78896 0.12727I
1.92863 0.10640I 2.34145 0.07111I
u = 1.107261 0.619290I
a = 2.07065 1.65840I
b = 4.37365 2.66482I
11.3318 13.8883I 6.08047 + 8.01135I
u = 1.107261 + 0.619290I
a = 2.07065 + 1.65840I
b = 4.37365 + 2.66482I
11.3318 + 13.8883I 6.08047 8.01135I
u = 1.087344 0.612889I
a = 0.876764 + 0.989537I
b = 1.38334 + 1.45236I
4.52609 9.53770I 4.27083 + 7.79920I
u = 1.087344 + 0.612889I
a = 0.876764 0.989537I
b = 1.38334 1.45236I
4.52609 + 9.53770I 4.27083 7.79920I
u = 1.06609
a = 1.56401
b = 4.35073
3.31477 2.11819
u = 1.062541 0.617182I
a = 0.107607 + 0.183921I
b = 0.777476 + 0.837410I
4.96232 3.90423I 5.42928 + 2.26868I
u = 1.062541 + 0.617182I
a = 0.107607 0.183921I
b = 0.777476 0.837410I
4.96232 + 3.90423I 5.42928 2.26868I
u = 1.041328 0.643287I
a = 0.54618 1.95501I
b = 1.95250 3.74455I
12.49885 0.07857I 7.52232 + 1.52860I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.041328 + 0.643287I
a = 0.54618 + 1.95501I
b = 1.95250 + 3.74455I
12.49885 + 0.07857I 7.52232 1.52860I
u = 1.038314 0.379912I
a = 0.760266 0.233442I
b = 0.708899 0.688815I
2.65480 1.42970I 5.16065 + 0.45006I
u = 1.038314 + 0.379912I
a = 0.760266 + 0.233442I
b = 0.708899 + 0.688815I
2.65480 + 1.42970I 5.16065 0.45006I
u = 1.015082 0.482408I
a = 1.46062 + 0.43285I
b = 1.87831 + 1.80401I
0.80550 3.99450I 2.19146 + 3.84882I
u = 1.015082 + 0.482408I
a = 1.46062 0.43285I
b = 1.87831 1.80401I
0.80550 + 3.99450I 2.19146 3.84882I
u = 0.974772 0.241666I
a = 0.274312 0.520281I
b = 0.858608 0.806299I
1.72966 0.52144I 5.77451 + 0.49909I
u = 0.974772 + 0.241666I
a = 0.274312 + 0.520281I
b = 0.858608 + 0.806299I
1.72966 + 0.52144I 5.77451 0.49909I
u = 0.547871 0.781328I
a = 2.45359 1.28278I
b = 0.090999 + 0.627833I
13.9720 5.2869I 9.56069 + 3.45269I
u = 0.547871 + 0.781328I
a = 2.45359 + 1.28278I
b = 0.090999 0.627833I
13.9720 + 5.2869I 9.56069 3.45269I
u = 0.500888 0.763334I
a = 0.404675 + 0.479078I
b = 0.368325 0.387535I
6.63504 1.32263I 7.88405 + 2.69846I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.500888 + 0.763334I
a = 0.404675 0.479078I
b = 0.368325 + 0.387535I
6.63504 + 1.32263I 7.88405 2.69846I
u = 0.458619 0.779636I
a = 1.01323 + 1.05264I
b = 0.417733 + 0.253698I
6.39750 + 4.28616I 7.23871 3.29000I
u = 0.458619 + 0.779636I
a = 1.01323 1.05264I
b = 0.417733 0.253698I
6.39750 4.28616I 7.23871 + 3.29000I
u = 0.436206 0.813289I
a = 1.64223 2.94790I
b = 0.485668 + 0.215668I
13.3377 + 8.5290I 8.89957 3.73071I
u = 0.436206 + 0.813289I
a = 1.64223 + 2.94790I
b = 0.485668 0.215668I
13.3377 8.5290I 8.89957 + 3.73071I
u = 0.435442 0.365869I
a = 1.15070 + 1.40335I
b = 0.976752 + 0.598553I
2.39706 + 0.15379I 3.69002 + 1.57866I
u = 0.435442 + 0.365869I
a = 1.15070 1.40335I
b = 0.976752 0.598553I
2.39706 0.15379I 3.69002 1.57866I
u = 0.193784 0.702298I
a = 0.05344 + 1.41842I
b = 0.737914 + 0.078271I
5.78851 3.34050I 7.33924 + 3.06497I
u = 0.193784 + 0.702298I
a = 0.05344 1.41842I
b = 0.737914 0.078271I
5.78851 + 3.34050I 7.33924 3.06497I
u = 0.204119 0.487719I
a = 0.666298 1.042686I
b = 0.087258 0.441591I
0.239475 1.389702I 2.08792 + 5.18976I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.204119 + 0.487719I
a = 0.666298 + 1.042686I
b = 0.087258 + 0.441591I
0.239475 + 1.389702I 2.08792 5.18976I
u = 0.447001 0.697020I
a = 0.981890 + 0.771451I
b = 0.1143164 0.0807952I
2.42469 1.08462I 0.919907 + 0.841939I
u = 0.447001 + 0.697020I
a = 0.981890 0.771451I
b = 0.1143164 + 0.0807952I
2.42469 + 1.08462I 0.919907 0.841939I
u = 0.480967 0.776724I
a = 2.77352 2.55490I
b = 0.480129 + 0.581903I
8.70701 1.51183I 8.31702 + 0.27451I
u = 0.480967 + 0.776724I
a = 2.77352 + 2.55490I
b = 0.480129 0.581903I
8.70701 + 1.51183I 8.31702 0.27451I
u = 0.721542 0.368379I
a = 0.006166 + 0.837730I
b = 0.684153 + 0.192507I
1.12660 + 1.52721I 6.78225 4.43587I
u = 0.721542 + 0.368379I
a = 0.006166 0.837730I
b = 0.684153 0.192507I
1.12660 1.52721I 6.78225 + 4.43587I
u = 0.799486 0.620081I
a = 1.09585 + 1.07705I
b = 0.45724 + 1.42320I
8.47576 + 2.42942I 9.05009 3.27749I
u = 0.799486 + 0.620081I
a = 1.09585 1.07705I
b = 0.45724 1.42320I
8.47576 2.42942I 9.05009 + 3.27749I
u = 0.972428 0.431066I
a = 1.260667 0.148749I
b = 1.83249 1.04935I
0.32639 + 1.89843I 4.06618 4.82062I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.972428 + 0.431066I
a = 1.260667 + 0.148749I
b = 1.83249 + 1.04935I
0.32639 1.89843I 4.06618 + 4.82062I
u = 1.063424 0.475788I
a = 0.979310 + 0.402712I
b = 1.158098 + 0.628193I
1.98458 + 5.32256I 2.05214 8.22615I
u = 1.063424 + 0.475788I
a = 0.979310 0.402712I
b = 1.158098 0.628193I
1.98458 5.32256I 2.05214 + 8.22615I
u = 1.068649 0.060095I
a = 0.649567 0.634275I
b = 1.294521 0.489632I
1.15819 2.52423I 1.16527 + 3.38233I
u = 1.068649 + 0.060095I
a = 0.649567 + 0.634275I
b = 1.294521 + 0.489632I
1.15819 + 2.52423I 1.16527 3.38233I
u = 1.071556 0.575084I
a = 0.497701 + 0.875049I
b = 1.06541 + 1.57159I
0.58719 + 5.99085I 2.37433 5.87203I
u = 1.071556 + 0.575084I
a = 0.497701 0.875049I
b = 1.06541 1.57159I
0.58719 5.99085I 2.37433 + 5.87203I
u = 1.076356 0.618201I
a = 1.54725 2.47130I
b = 3.81869 4.24242I
6.93298 + 6.77646I 5.61640 4.97069I
u = 1.076356 + 0.618201I
a = 1.54725 + 2.47130I
b = 3.81869 + 4.24242I
6.93298 6.77646I 5.61640 + 4.97069I
u = 1.124888 0.501254I
a = 0.933579 0.122384I
b = 1.74457 + 0.77799I
3.12807 + 7.84635I 3.71152 7.15898I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.124888 + 0.501254I
a = 0.933579 + 0.122384I
b = 1.74457 0.77799I
3.12807 7.84635I 3.71152 + 7.15898I
u = 1.139854 0.088394I
a = 1.74758 + 0.04344I
b = 4.11285 0.40579I
7.92687 6.35005I 3.13792 + 3.33110I
u = 1.139854 + 0.088394I
a = 1.74758 0.04344I
b = 4.11285 + 0.40579I
7.92687 + 6.35005I 3.13792 3.33110I
10
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
6
+ u
5
+ ··· + u + 1)(u
53
+ 2u
52
+ ··· + u + 1)
c
2
(u
6
+ 3u
5
+ ··· + u + 1)(u
53
+ 24u
52
+ ··· + 5u + 1)
c
3
, c
8
u
6
(u
53
+ u
52
+ ··· 64u + 64)
c
4
, c
6
(u
6
+ u
5
+ ··· + u + 1)(u
53
+ 2u
52
+ ··· 144u 36)
c
5
(u
6
u
5
+ ··· u + 1)(u
53
+ 2u
52
+ ··· + u + 1)
c
7
(u + 1)
6
(u
53
+ 7u
52
+ ··· 6u 1)
c
9
, c
10
(u 1)
6
(u
53
+ 7u
52
+ ··· 6u 1)
c
11
(u
6
+ 3u
5
+ ··· + u + 1)(u
53
+ 6u
52
+ ··· 5u 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
6
3y
5
+ ··· y + 1)(y
53
24y
52
+ ··· + 5y 1)
c
2
(y
6
+ y
5
+ ··· + 3y + 1)(y
53
+ 12y
52
+ ··· 27y 1)
c
3
, c
8
y
6
(y
53
+ 39y
52
+ ··· + 8192y 4096)
c
4
, c
6
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
(y
53
48y
52
+ ··· + 10728y 1296)
c
7
, c
9
, c
10
(y 1)
6
(y
53
55y
52
+ ··· + 14y 1)
c
11
(y
6
+ y
5
+ ··· + 3y + 1)(y
53
+ 54y
51
+ ··· + 45y 1)
12