11a
15
(K11a
15
)
1
Arc Sequences
4 1 7 2 10 3 5 11 6 8 9
Solving Sequence
9,11 1,4
2 5 8 7 3 10 6
c
11
c
1
c
4
c
8
c
7
c
3
c
10
c
5
c
2
, c
6
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
6
+ 4u
4
u
3
+ 4u
2
+ 1, u
5
+ 4u
3
u
2
+ a + 4u, u
5
+ u
4
+ 5u
3
+ u
2
+ 3b + 2u 1i
I
u
2
= hu
5
u
4
+ u
3
2u
2
+ u 1, u
4
+ b + u + 1, u
4
u
3
+ a 2u + 1i
I
u
3
= hu
64
+ 72u
62
+ ··· + 2354384u + 9965467,
1.13032 × 10
457
u
63
2.39144 × 10
456
u
62
+ ··· + 1.05851 × 10
464
b 2.16710 × 10
464
,
4.91646 × 10
464
u
63
2.16918 × 10
463
u
62
+ ··· + 1.05486 × 10
471
a + 5.61219 × 10
471
i
There are 3 irreducible components with 75 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
+4u
4
u
3
+4u
2
+1, u
5
+4u
3
u
2
+a+4u, u
5
+u
4
+5u
3
+u
2
+3b +2u1i
(i) Arc colorings
a
9
=
0
u
a
11
=
u
5
4u
3
+ u
2
4u
1
3
u
5
1
3
u
4
+ ···
2
3
u +
1
3
a
1
=
u
5
4u
3
+ u
2
4u
1
3
u
5
1
3
u
4
+ ···
5
3
u +
1
3
a
4
=
1
0
a
2
=
2
3
u
5
+
1
3
u
4
+ ···
7
3
u
1
3
1
3
u
5
1
3
u
4
+ ···
5
3
u +
1
3
a
5
=
0
1
3
u
5
1
3
u
4
+ ···
5
3
u
2
3
a
8
=
u
5
+ 4u
3
u
2
+ 4u
1
3
u
5
+
1
3
u
4
+ ··· +
5
3
u
1
3
a
7
=
u
5
+ 4u
3
u
2
+ 4u
2
3
u
5
2
3
u
4
+ ···
1
3
u
4
3
a
3
=
4
3
u
5
+
2
3
u
4
+ ···
11
3
u +
4
3
2
3
u
5
+
1
3
u
4
+ ···
4
3
u +
5
3
a
10
=
0
u
a
6
=
0
1
3
u
5
1
3
u
4
+ ···
5
3
u
2
3
a
6
=
0
1
3
u
5
1
3
u
4
+ ···
5
3
u
2
3
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
5
+ u
4
+ 12u
3
+ 4u + 8
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.25695 1.72779I
a = 0.084211 + 0.566250I
b = 1.25915 1.43225I
0.245672 + 0.924305I 1.66012 2.42665I
u = 0.25695 + 1.72779I
a = 0.084211 0.566250I
b = 1.25915 + 1.43225I
0.245672 0.924305I 1.66012 + 2.42665I
u = 0.084211 0.566250I
a = 0.25695 + 1.72779I
b = 0.344032 + 0.098281I
3.53554 + 0.92430I 8.55174 0.47256I
u = 0.084211 + 0.566250I
a = 0.25695 1.72779I
b = 0.344032 0.098281I
3.53554 0.92430I 8.55174 + 0.47256I
u = 0.341164 0.940004I
a = 0.341164 + 0.940004I
b = 1.41511 0.38125I
1.64493 5.69302I 3.10838 + 3.92918I
u = 0.341164 + 0.940004I
a = 0.341164 0.940004I
b = 1.41511 + 0.38125I
1.64493 + 5.69302I 3.10838 3.92918I
3
II. I
u
2
= hu
5
u
4
+ u
3
2u
2
+ u 1, u
4
+ b + u + 1, u
4
u
3
+ a 2u + 1i
(i) Arc colorings
a
9
=
0
u
a
11
=
u
4
+ u
3
+ 2u 1
u
4
u 1
a
1
=
u
4
+ u
3
+ 2u 1
1
a
4
=
1
0
a
2
=
u
4
+ u
3
+ 2u
1
a
5
=
u
4
u
3
2u + 1
1
a
8
=
2u
4
+ 2u
3
u
2
+ 3u 2
u
4
u 1
a
7
=
u
4
+ u + 1
0
a
3
=
1
0
a
10
=
u
4
2u
3
+ u
2
2u + 2
u
a
6
=
u
4
+ u + 1
0
a
6
=
u
4
+ u + 1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5u
4
5u
3
7u + 2
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.428550 1.039275I
a = 0.557328 + 0.068387I
b = 0.561306 0.557752I
1.31583 + 1.53058I 1.49901 3.45976I
u = 0.428550 + 1.039275I
a = 0.557328 0.068387I
b = 0.561306 + 0.557752I
1.31583 1.53058I 1.49901 + 3.45976I
u = 0.276511 0.728237I
a = 0.90957 1.60288I
b = 1.23271 + 1.09381I
4.22763 4.40083I 2.37737 + 5.82971I
u = 0.276511 + 0.728237I
a = 0.90957 + 1.60288I
b = 1.23271 1.09381I
4.22763 + 4.40083I 2.37737 5.82971I
u = 1.30408
a = 0.933791
b = 0.588022
0.756147 3.75673
5
III. I
u
3
= hu
64
+ 72u
62
+ · · · + 2354384u + 9965467, 1.13 × 10
457
u
63
2.39 × 10
456
u
62
+ · · · + 1.06 × 10
464
b 2.17 × 10
464
, 4.92 × 10
464
u
63
2.17 ×
10
463
u
62
+ · · · + 1.05 × 10
471
a + 5.61 × 10
471
i
(i) Arc colorings
a
9
=
0
u
a
11
=
4.66077 × 10
7
u
63
+ 2.05637 × 10
8
u
62
+ ··· 12.1318u 5.32032
1.06783 × 10
7
u
63
+ 2.25924 × 10
8
u
62
+ ··· + 2.56477u + 2.04730
a
1
=
4.66077 × 10
7
u
63
+ 2.05637 × 10
8
u
62
+ ··· 12.1318u 5.32032
3.22541 × 10
9
u
63
+ 5.97189 × 10
8
u
62
+ ··· 2.03149u + 2.25223
a
4
=
1
0
a
2
=
4.69303 × 10
7
u
63
3.91551 × 10
8
u
62
+ ··· 10.1003u 7.57255
3.22541 × 10
9
u
63
+ 5.97189 × 10
8
u
62
+ ··· 2.03149u + 2.25223
a
5
=
1.00082 × 10
7
u
63
1.25777 × 10
7
u
62
+ ··· + 9.94880u 4.36809
1.53948 × 10
7
u
63
+ 5.73531 × 10
8
u
62
+ ··· 5.47859u 0.345441
a
8
=
1.22922 × 10
7
u
63
+ 1.72268 × 10
7
u
62
+ ··· 7.34948u + 3.73366
3.56082 × 10
8
u
63
2.74784 × 10
8
u
62
+ ··· + 7.40609u 2.25354
a
7
=
5.68155 × 10
7
u
63
4.09160 × 10
8
u
62
+ ··· + 12.0387u + 4.96050
1.90838 × 10
7
u
63
5.38065 × 10
9
u
62
+ ··· 1.93397u 3.13899
a
3
=
4.88417 × 10
7
u
63
+ 2.26685 × 10
8
u
62
+ ··· 18.3640u 4.53122
1.42161 × 10
7
u
63
+ 7.30628 × 10
8
u
62
+ ··· 0.651923u + 5.31198
a
10
=
3.05495 × 10
7
u
63
+ 6.50508 × 10
9
u
62
+ ··· 17.2002u 1.09885
2.96212 × 10
7
u
63
+ 2.75031 × 10
8
u
62
+ ··· + 6.21278u + 5.34770
a
6
=
8.76625 × 10
9
u
63
+ 5.34853 × 10
8
u
62
+ ··· 1.44733u + 5.45490
1.15727 × 10
7
u
63
1.81818 × 10
7
u
62
+ ··· + 8.71537u 4.41162
a
6
=
8.76625 × 10
9
u
63
+ 5.34853 × 10
8
u
62
+ ··· 1.44733u + 5.45490
1.15727 × 10
7
u
63
1.81818 × 10
7
u
62
+ ··· + 8.71537u 4.41162
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5.38821 ×10
7
u
63
3.39826 ×10
7
u
62
+ ···+ 24.1595u 7.51078
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.37521 1.96734I
a = 0.439577 0.467038I
b = 0.97280 + 2.63985I
8.25062 + 3.46517I 6.83292 0.71122I
u = 1.37521 + 1.96734I
a = 0.439577 + 0.467038I
b = 0.97280 2.63985I
8.25062 3.46517I 6.83292 + 0.71122I
u = 1.29960 2.23169I
a = 0.309273 + 0.476440I
b = 1.23227 2.46123I
8.42241 + 2.26761I 8.35385 0.91173I
u = 1.29960 + 2.23169I
a = 0.309273 0.476440I
b = 1.23227 + 2.46123I
8.42241 2.26761I 8.35385 + 0.91173I
u = 0.959664 0.800360I
a = 0.006888 0.221630I
b = 0.619093 0.442660I
1.81999 1.29187I 4.94153 + 1.99475I
u = 0.959664 + 0.800360I
a = 0.006888 + 0.221630I
b = 0.619093 + 0.442660I
1.81999 + 1.29187I 4.94153 1.99475I
u = 0.90814 3.07854I
a = 0.092654 + 0.447992I
b = 0.59380 3.70814I
6.60913 8.11292I 5.77074 + 5.50142I
u = 0.90814 + 3.07854I
a = 0.092654 0.447992I
b = 0.59380 + 3.70814I
6.60913 + 8.11292I 5.77074 5.50142I
u = 0.72661 3.24113I
a = 0.018052 0.463022I
b = 1.03260 + 4.08229I
7.38800 8.47452I 5.37555 + 5.77471I
u = 0.72661 + 3.24113I
a = 0.018052 + 0.463022I
b = 1.03260 4.08229I
7.38800 + 8.47452I 5.37555 5.77471I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.690032 1.058279I
a = 0.636043 0.818342I
b = 1.88818 + 0.98624I
2.70253 + 6.62281I 6.09992 7.93225I
u = 0.690032 + 1.058279I
a = 0.636043 + 0.818342I
b = 1.88818 0.98624I
2.70253 6.62281I 6.09992 + 7.93225I
u = 0.684341 0.057830I
a = 1.71795 + 0.56608I
b = 0.044711 0.410335I
2.82835 + 0.98333I 5.99923 3.36293I
u = 0.684341 + 0.057830I
a = 1.71795 0.56608I
b = 0.044711 + 0.410335I
2.82835 0.98333I 5.99923 + 3.36293I
u = 0.645381 0.527409I
a = 0.222390 0.263157I
b = 0.942122 + 0.013411I
2.28356 + 0.63564I 4.73296 + 1.47525I
u = 0.645381 + 0.527409I
a = 0.222390 + 0.263157I
b = 0.942122 0.013411I
2.28356 0.63564I 4.73296 1.47525I
u = 0.63283 1.62355I
a = 0.380663 + 0.418349I
b = 0.579544 + 0.518787I
3.96365 10.71218I 3.94181 + 8.25311I
u = 0.63283 + 1.62355I
a = 0.380663 0.418349I
b = 0.579544 0.518787I
3.96365 + 10.71218I 3.94181 8.25311I
u = 0.386490 0.929599I
a = 0.467768 1.325038I
b = 0.601503 + 0.586624I
5.65313 2.65309I 8.42225 + 3.34216I
u = 0.386490 + 0.929599I
a = 0.467768 + 1.325038I
b = 0.601503 0.586624I
5.65313 + 2.65309I 8.42225 3.34216I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.325454 0.787043I
a = 0.514804 + 1.078465I
b = 0.262136 + 0.582690I
6.56660 0.51971I 8.32944 + 2.07551I
u = 0.325454 + 0.787043I
a = 0.514804 1.078465I
b = 0.262136 0.582690I
6.56660 + 0.51971I 8.32944 2.07551I
u = 0.310155 0.426887I
a = 1.92747 1.85234I
b = 1.13883 + 0.89562I
4.72329 + 3.85111I 9.95444 + 1.38614I
u = 0.310155 + 0.426887I
a = 1.92747 + 1.85234I
b = 1.13883 0.89562I
4.72329 3.85111I 9.95444 1.38614I
u = 0.281994 1.320672I
a = 0.580135 + 0.239324I
b = 0.482182 0.896901I
0.29295 + 2.27028I 4.94099 4.77864I
u = 0.281994 + 1.320672I
a = 0.580135 0.239324I
b = 0.482182 + 0.896901I
0.29295 2.27028I 4.94099 + 4.77864I
u = 0.233608 0.718094I
a = 0.600430 + 1.012055I
b = 1.39033 + 0.25985I
1.392233 0.239734I 4.64422 + 0.33938I
u = 0.233608 + 0.718094I
a = 0.600430 1.012055I
b = 1.39033 0.25985I
1.392233 + 0.239734I 4.64422 0.33938I
u = 0.176547 0.453896I
a = 1.99624 + 0.30987I
b = 0.768160 + 0.532756I
5.97499 + 4.90532I 6.97895 4.00354I
u = 0.176547 + 0.453896I
a = 1.99624 0.30987I
b = 0.768160 0.532756I
5.97499 4.90532I 6.97895 + 4.00354I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.110371 0.780690I
a = 0.04965 1.93648I
b = 0.98736 + 1.25714I
6.16621 5.77451I 6.31555 + 4.15792I
u = 0.110371 + 0.780690I
a = 0.04965 + 1.93648I
b = 0.98736 1.25714I
6.16621 + 5.77451I 6.31555 4.15792I
u = 0.081150 0.937518I
a = 0.755778 + 0.111841I
b = 0.153847 0.624008I
0.262566 0.498963I 2.74689 0.51399I
u = 0.081150 + 0.937518I
a = 0.755778 0.111841I
b = 0.153847 + 0.624008I
0.262566 + 0.498963I 2.74689 + 0.51399I
u = 0.156217 0.409870I
a = 1.042712 + 0.283173I
b = 0.109056 0.424133I
0.309124 1.130769I 3.85993 + 6.08046I
u = 0.156217 + 0.409870I
a = 1.042712 0.283173I
b = 0.109056 + 0.424133I
0.309124 + 1.130769I 3.85993 6.08046I
u = 0.15636 3.14204I
a = 0.080517 0.487211I
b = 0.09925 + 4.04081I
10.2926 + 15.1288I 6.13774 8.45245I
u = 0.15636 + 3.14204I
a = 0.080517 + 0.487211I
b = 0.09925 4.04081I
10.2926 15.1288I 6.13774 + 8.45245I
u = 0.24539 3.05391I
a = 0.008633 + 0.317355I
b = 0.37068 2.79945I
0.047902 + 0.498911I 2.1353 + 14.8724I
u = 0.24539 + 3.05391I
a = 0.008633 0.317355I
b = 0.37068 + 2.79945I
0.047902 0.498911I 2.1353 14.8724I
10
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.324283 0.573602I
a = 0.505179 0.339599I
b = 0.444666 0.477700I
2.38826 1.27360I 1.11043 + 1.24138I
u = 0.324283 + 0.573602I
a = 0.505179 + 0.339599I
b = 0.444666 + 0.477700I
2.38826 + 1.27360I 1.11043 1.24138I
u = 0.41841 1.74450I
a = 0.354205 + 0.534621I
b = 0.65708 1.48322I
2.50515 3.47504I 6.09515 + 3.70919I
u = 0.41841 + 1.74450I
a = 0.354205 0.534621I
b = 0.65708 + 1.48322I
2.50515 + 3.47504I 6.09515 3.70919I
u = 0.422151 0.651411I
a = 0.626864 0.469117I
b = 0.553044 + 0.271752I
1.01229 4.28688I 0.54974 + 6.54536I
u = 0.422151 + 0.651411I
a = 0.626864 + 0.469117I
b = 0.553044 0.271752I
1.01229 + 4.28688I 0.54974 6.54536I
u = 0.427115 0.149322I
a = 1.06655 + 2.06565I
b = 1.51241 + 0.88157I
4.97472 5.25194I 6.17348 + 3.10447I
u = 0.427115 + 0.149322I
a = 1.06655 2.06565I
b = 1.51241 0.88157I
4.97472 + 5.25194I 6.17348 3.10447I
u = 0.549212 1.117410I
a = 0.097379 0.390315I
b = 0.507667 0.354118I
0.55603 + 6.40064I 1.64899 6.18880I
u = 0.549212 + 1.117410I
a = 0.097379 + 0.390315I
b = 0.507667 + 0.354118I
0.55603 6.40064I 1.64899 + 6.18880I
11
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.64565 2.35467I
a = 0.090341 0.647042I
b = 0.84748 + 2.66906I
14.1398 + 3.9767I 9.96670 2.84794I
u = 0.64565 + 2.35467I
a = 0.090341 + 0.647042I
b = 0.84748 2.66906I
14.1398 3.9767I 9.96670 + 2.84794I
u = 0.681809
a = 1.68327
b = 0.916013
2.19568 3.71440
u = 0.69391 1.47737I
a = 0.371327 0.760024I
b = 2.11715 + 1.67337I
2.11698 1.35062I 4.57653 + 0.61915I
u = 0.69391 + 1.47737I
a = 0.371327 + 0.760024I
b = 2.11715 1.67337I
2.11698 + 1.35062I 4.57653 0.61915I
u = 0.83189 1.96075I
a = 0.359354 0.658465I
b = 0.02210 + 2.47788I
12.97911 + 2.20987I 8.69224 2.15711I
u = 0.83189 + 1.96075I
a = 0.359354 + 0.658465I
b = 0.02210 2.47788I
12.97911 2.20987I 8.69224 + 2.15711I
u = 1.02127 1.25913I
a = 0.188641 + 0.526097I
b = 0.543312 + 0.753907I
0.93426 + 4.62344I 2.97321 6.17205I
u = 1.02127 + 1.25913I
a = 0.188641 0.526097I
b = 0.543312 0.753907I
0.93426 4.62344I 2.97321 + 6.17205I
u = 1.08501
a = 1.26203
b = 0.215201
1.81885 6.28221
12
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.17532 1.58842I
a = 0.585835 0.533283I
b = 0.89551 + 1.82490I
12.5775 9.1285I 8.90487 + 4.40512I
u = 1.17532 + 1.58842I
a = 0.585835 + 0.533283I
b = 0.89551 1.82490I
12.5775 + 9.1285I 8.90487 4.40512I
u = 1.71468 3.11971I
a = 0.177532 + 0.352173I
b = 1.71088 3.73834I
3.26372 + 2.22000I 4.54254 2.98814I
u = 1.71468 + 3.11971I
a = 0.177532 0.352173I
b = 1.71088 + 3.73834I
3.26372 2.22000I 4.54254 + 2.98814I
13
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
5
(u
6
+ u
5
+ ··· + u + 1)(u
64
+ 7u
63
+ ··· 13u 1)
c
2
(u + 1)
5
(u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)
(u
64
+ 29u
63
+ ··· + 37u + 1)
c
3
u
5
(u
6
u
5
+ ··· u + 1)(u
64
+ 2u
63
+ ··· + 128u + 32)
c
4
(u + 1)
5
(u
6
u
5
+ ··· u + 1)(u
64
+ 7u
63
+ ··· 13u 1)
c
5
u
6
(u
5
+ u
4
+ ··· + u + 1)(u
64
+ 2u
63
+ ··· + 128u 64)
c
6
u
5
(u
6
+ u
5
+ ··· + u + 1)(u
64
+ 2u
63
+ ··· + 128u + 32)
c
7
(u
5
3u
4
+ 4u
3
u
2
u + 1)(u
6
3u
5
+ 5u
4
4u
3
+ 2u
2
u + 1)
(u
64
+ 3u
63
+ ··· 2001u 1609)
c
8
(u + 1)
6
(u
5
u
4
+ ··· + u + 1)(u
64
+ 8u
63
+ ··· + 4u + 1)
c
9
u
6
(u
5
u
4
+ ··· + u 1)(u
64
+ 2u
63
+ ··· + 128u 64)
c
10
, c
11
(u 1)
6
(u
5
+ u
4
+ ··· + u 1)(u
64
+ 8u
63
+ ··· + 4u + 1)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
5
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
(y
64
29y
63
+ ··· 37y + 1)
c
2
(y 1)
5
(y
6
+ y
5
+ ··· + 3y + 1)(y
64
+ 19y
63
+ ··· 3653y + 1)
c
3
, c
6
y
5
(y
6
3y
5
+ 5y
4
4y
3
+ 2y
2
y + 1)
(y
64
36y
63
+ ··· 20992y + 1024)
c
5
, c
9
y
6
(y
5
+ 3y
4
+ ··· y 1)(y
64
+ 42y
63
+ ··· + 24576y + 4096)
c
7
(y
5
y
4
+ 8y
3
3y
2
+ 3y 1)(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
(y
64
37y
63
+ ··· + 70611765y + 2588881)
c
8
, c
10
, c
11
(y 1)
6
(y
5
5y
4
+ ··· y 1)(y
64
64y
63
+ ··· + 20y + 1)
15