11a
163
(K11a
163
)
1
Arc Sequences
6 1 11 8 2 10 4 5 3 7 9
Solving Sequence
2,5
6 1
3,9
10 8 4 7 11
c
5
c
1
c
2
c
9
c
8
c
4
c
7
c
11
c
3
, c
6
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
13
u
12
+ 4u
11
3u
10
+ 8u
9
6u
8
+ 8u
7
7u
6
+ 3u
5
6u
4
u
3
3u
2
u 1,
2u
11
+ u
10
6u
9
+ 3u
8
11u
7
+ 7u
6
7u
5
+ 10u
4
+ 8u
2
+ a + 3u + 2,
u
11
+ u
10
4u
9
+ 4u
8
8u
7
+ 7u
6
7u
5
+ 8u
4
2u
3
+ 3u
2
+ b + u + 1i
I
u
2
= hu
72
+ 17u
70
+ ··· u + 1, 5.69300 × 10
65
u
71
+ 3.66420 × 10
64
u
70
+ ··· + 5.98806 × 10
65
a 1.18567 × 10
66
,
7.77862 × 10
65
u
71
6.44760 × 10
65
u
70
+ ··· + 5.98806 × 10
65
b 1.86215 × 10
65
i
There are 2 irreducible components with 85 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
13
u
12
+· · ·u 1, 2u
11
+u
10
+· · ·+a + 2, u
11
+u
10
+· · ·+b + 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
+ u
2
+ 1
u
4
a
9
=
2u
11
u
10
+ 6u
9
3u
8
+ 11u
7
7u
6
+ 7u
5
10u
4
8u
2
3u 2
u
11
u
10
+ 4u
9
4u
8
+ 8u
7
7u
6
+ 7u
5
8u
4
+ 2u
3
3u
2
u 1
a
10
=
2u
11
u
10
+ 6u
9
3u
8
+ 11u
7
7u
6
+ 7u
5
10u
4
7u
2
3u 1
u
11
+ 3u
9
u
8
+ 6u
7
3u
6
+ 4u
5
7u
4
4u
2
2u 1
a
8
=
2u
11
u
10
+ 6u
9
3u
8
+ 11u
7
7u
6
+ 7u
5
10u
4
8u
2
3u 2
u
12
+ 3u
11
+ ··· 3u 3
a
4
=
u
12
+ 3u
10
+ 6u
8
u
7
+ 4u
6
4u
5
6u
3
4u
2
4u 1
2u
11
+ u
10
+ ··· + 2u + 1
a
7
=
u
12
2u
11
+ ··· + 6u + 3
2u
12
4u
11
+ ··· + 2u + 2
a
11
=
2u
12
+ u
11
+ ··· + 4u + 1
u
12
2u
11
+ 3u
10
6u
9
+ 6u
8
11u
7
+ 5u
6
8u
5
+ 6u
4
+ 5u
2
+ 3u + 3
a
11
=
2u
12
+ u
11
+ ··· + 4u + 1
u
12
2u
11
+ 3u
10
6u
9
+ 6u
8
11u
7
+ 5u
6
8u
5
+ 6u
4
+ 5u
2
+ 3u + 3
(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 = 0.564862 1.080816I
a = 0.449076 + 0.192847I
b = 0.013032 0.333605I
1.43963 + 4.27361I 3.28652 1.94242I
u = 0.564862 + 1.080816I
a = 0.449076 0.192847I
b = 0.013032 + 0.333605I
1.43963 4.27361I 3.28652 + 1.94242I
u = 0.443976 0.410014I
a = 0.700932 0.990323I
b = 1.011605 0.353847I
0.639447 + 0.198383I 2.33125 + 0.81736I
u = 0.443976 + 0.410014I
a = 0.700932 + 0.990323I
b = 1.011605 + 0.353847I
0.639447 0.198383I 2.33125 0.81736I
u = 0.349870 0.909420I
a = 0.571675 + 0.302017I
b = 1.04357 + 1.88317I
3.76256 + 1.44897I 0.77634 5.07895I
u = 0.349870 + 0.909420I
a = 0.571675 0.302017I
b = 1.04357 1.88317I
3.76256 1.44897I 0.77634 + 5.07895I
u = 0.272707 0.834669I
a = 0.73959 + 1.61876I
b = 0.569619 0.253022I
9.45221 + 1.59908I 10.36413 + 2.46917I
u = 0.272707 + 0.834669I
a = 0.73959 1.61876I
b = 0.569619 + 0.253022I
9.45221 1.59908I 10.36413 2.46917I
u = 0.345453 1.027117I
a = 0.36668 1.41355I
b = 0.97263 2.34146I
10.26050 4.15031I 8.69293 + 2.72489I
u = 0.345453 + 1.027117I
a = 0.36668 + 1.41355I
b = 0.97263 + 2.34146I
10.26050 + 4.15031I 8.69293 2.72489I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.700273 1.221277I
a = 0.370003 + 0.867283I
b = 1.21901 + 1.03656I
4.69184 6.21694I 5.54626 + 10.85275I
u = 0.700273 + 1.221277I
a = 0.370003 0.867283I
b = 1.21901 1.03656I
4.69184 + 6.21694I 5.54626 10.85275I
u = 1.08055
a = 1.13288
b = 0.263539
1.28306 7.99487
4
II. I
u
2
=
hu
72
+17u
70
+· · · u + 1, 5.69× 10
65
u
71
+3.66 × 10
64
u
70
+· · · +5.99 × 10
65
a
1.19 × 10
66
, 7.78 × 10
65
u
71
6.45 × 10
65
u
70
+ · · · + 5.99 × 10
65
b 1.86 × 10
65
i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
+ u
2
+ 1
u
4
a
9
=
0.950725u
71
0.0611917u
70
+ ··· 8.38931u + 1.98005
1.29902u
71
+ 1.07674u
70
+ ··· 0.0872015u + 0.310978
a
10
=
1.47703u
71
0.254667u
70
+ ··· 5.67977u + 1.66319
1.15579u
71
+ 0.480884u
70
+ ··· + 0.329368u + 0.0969772
a
8
=
0.950725u
71
0.0611917u
70
+ ··· 8.38931u + 1.98005
1.07463u
71
+ 0.758478u
70
+ ··· 0.976735u + 0.249786
a
4
=
2.05357u
71
0.873645u
70
+ ··· 10.0426u + 2.19385
2.74506u
71
1.05708u
70
+ ··· + 2.80468u 2.35209
a
7
=
1.23094u
71
+ 0.614017u
70
+ ··· + 5.66095u 2.52917
1.42755u
71
0.813110u
70
+ ··· 0.382667u + 0.518603
a
11
=
1.60445u
71
1.06712u
70
+ ··· + 2.27844u + 2.50716
0.223371u
71
0.0291713u
70
+ ··· + 2.30028u 1.75192
a
11
=
1.60445u
71
1.06712u
70
+ ··· + 2.27844u + 2.50716
0.223371u
71
0.0291713u
70
+ ··· + 2.30028u 1.75192
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
5
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.950154 0.340310I
a = 1.53648 + 0.25711I
b = 0.472314 0.868510I
7.89489 10.18184I 4.39970 + 5.17085I
u = 0.950154 + 0.340310I
a = 1.53648 0.25711I
b = 0.472314 + 0.868510I
7.89489 + 10.18184I 4.39970 5.17085I
u = 0.794162 0.453233I
a = 1.44902 0.64499I
b = 0.529750 + 1.073076I
3.22516 3.95142I 2.96229 + 3.51289I
u = 0.794162 + 0.453233I
a = 1.44902 + 0.64499I
b = 0.529750 1.073076I
3.22516 + 3.95142I 2.96229 3.51289I
u = 0.775035 0.705599I
a = 0.304839 0.533982I
b = 0.478468 0.691938I
0.45501 + 2.74412I 2.62061 6.58454I
u = 0.775035 + 0.705599I
a = 0.304839 + 0.533982I
b = 0.478468 + 0.691938I
0.45501 2.74412I 2.62061 + 6.58454I
u = 0.681999 0.181568I
a = 0.057013 + 0.232464I
b = 0.535948 + 0.164385I
1.48009 + 0.06732I 8.50693 + 1.19475I
u = 0.681999 + 0.181568I
a = 0.057013 0.232464I
b = 0.535948 0.164385I
1.48009 0.06732I 8.50693 1.19475I
u = 0.630641 1.185246I
a = 0.360506 1.147988I
b = 1.99091 1.74440I
10.4703 + 15.9193I 6.85189 8.36957I
u = 0.630641 + 1.185246I
a = 0.360506 + 1.147988I
b = 1.99091 + 1.74440I
10.4703 15.9193I 6.85189 + 8.36957I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.614768 0.453497I
a = 0.046552 1.369771I
b = 0.621560 1.116640I
1.24059 0.70092I 3.96458 + 1.25636I
u = 0.614768 + 0.453497I
a = 0.046552 + 1.369771I
b = 0.621560 + 1.116640I
1.24059 + 0.70092I 3.96458 1.25636I
u = 0.613054 1.092111I
a = 0.451264 + 1.098316I
b = 2.45792 + 1.51317I
5.13451 + 9.23572I 5.81827 8.30669I
u = 0.613054 + 1.092111I
a = 0.451264 1.098316I
b = 2.45792 1.51317I
5.13451 9.23572I 5.81827 + 8.30669I
u = 0.600309 1.184395I
a = 0.028265 + 0.223843I
b = 0.495089 + 0.645877I
1.42281 + 5.04884I 2.82719 12.32546I
u = 0.600309 + 1.184395I
a = 0.028265 0.223843I
b = 0.495089 0.645877I
1.42281 5.04884I 2.82719 + 12.32546I
u = 0.550811 1.056744I
a = 1.004973 + 0.081788I
b = 0.054236 0.976249I
2.99967 + 5.33296I 7.64505 6.23927I
u = 0.550811 + 1.056744I
a = 1.004973 0.081788I
b = 0.054236 + 0.976249I
2.99967 5.33296I 7.64505 + 6.23927I
u = 0.541899 0.922731I
a = 0.420513 0.436000I
b = 0.584653 0.865697I
0.03676 + 2.24980I 0.26399 2.57020I
u = 0.541899 + 0.922731I
a = 0.420513 + 0.436000I
b = 0.584653 + 0.865697I
0.03676 2.24980I 0.26399 + 2.57020I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.474907 0.952227I
a = 0.707063 0.433856I
b = 0.121999 + 0.813962I
0.15843 + 2.64508I 2.08016 2.19137I
u = 0.474907 + 0.952227I
a = 0.707063 + 0.433856I
b = 0.121999 0.813962I
0.15843 2.64508I 2.08016 + 2.19137I
u = 0.470200 1.042547I
a = 0.529327 1.173864I
b = 2.74644 2.23625I
10.28458 + 1.13066I 9.68211 1.28236I
u = 0.470200 + 1.042547I
a = 0.529327 + 1.173864I
b = 2.74644 + 2.23625I
10.28458 1.13066I 9.68211 + 1.28236I
u = 0.442925 0.726814I
a = 0.304097 + 1.118310I
b = 1.45654 + 1.18670I
0.609165 + 1.192599I 2.86974 6.15662I
u = 0.442925 + 0.726814I
a = 0.304097 1.118310I
b = 1.45654 1.18670I
0.609165 1.192599I 2.86974 + 6.15662I
u = 0.426028 1.029616I
a = 0.499707 1.317761I
b = 0.03958 2.51058I
10.60602 + 5.33661I 10.20513 8.68617I
u = 0.426028 + 1.029616I
a = 0.499707 + 1.317761I
b = 0.03958 + 2.51058I
10.60602 5.33661I 10.20513 + 8.68617I
u = 0.314455 0.987227I
a = 0.872079 + 0.521784I
b = 0.86687 + 1.57007I
4.69737 + 1.04071I 11.41192 0.81876I
u = 0.314455 + 0.987227I
a = 0.872079 0.521784I
b = 0.86687 1.57007I
4.69737 1.04071I 11.41192 + 0.81876I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.293343 0.413513I
a = 1.44996 + 2.46945I
b = 0.13705 1.40851I
8.45650 + 2.58799I 6.38429 2.67811I
u = 0.293343 + 0.413513I
a = 1.44996 2.46945I
b = 0.13705 + 1.40851I
8.45650 2.58799I 6.38429 + 2.67811I
u = 0.157141 1.357404I
a = 0.270798 1.110863I
b = 0.045316 1.262510I
13.8147 6.4968I 9.98960 + 4.59431I
u = 0.157141 + 1.357404I
a = 0.270798 + 1.110863I
b = 0.045316 + 1.262510I
13.8147 + 6.4968I 9.98960 4.59431I
u = 0.079295 0.696769I
a = 0.41355 + 2.22957I
b = 0.849433 + 0.925172I
8.96742 2.37559I 5.52309 + 3.97031I
u = 0.079295 + 0.696769I
a = 0.41355 2.22957I
b = 0.849433 0.925172I
8.96742 + 2.37559I 5.52309 3.97031I
u = 0.005532 1.088180I
a = 0.037898 + 1.412052I
b = 0.482869 + 1.170887I
8.67044 2.20266I 8.24814 + 3.06701I
u = 0.005532 + 1.088180I
a = 0.037898 1.412052I
b = 0.482869 1.170887I
8.67044 + 2.20266I 8.24814 3.06701I
u = 0.086155 1.193606I
a = 0.499146 + 0.511741I
b = 0.167882 + 0.028889I
6.57891 + 3.67515I 8.82815 5.11413I
u = 0.086155 + 1.193606I
a = 0.499146 0.511741I
b = 0.167882 0.028889I
6.57891 3.67515I 8.82815 + 5.11413I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.208920 0.825464I
a = 0.577853 0.573348I
b = 0.436247 0.199872I
1.46471 + 0.85730I 6.08976 4.46736I
u = 0.208920 + 0.825464I
a = 0.577853 + 0.573348I
b = 0.436247 + 0.199872I
1.46471 0.85730I 6.08976 + 4.46736I
u = 0.335948 0.212346I
a = 3.52584 + 1.53941I
b = 0.814425 1.015502I
8.60610 + 2.68144I 5.49009 1.95050I
u = 0.335948 + 0.212346I
a = 3.52584 1.53941I
b = 0.814425 + 1.015502I
8.60610 2.68144I 5.49009 + 1.95050I
u = 0.341711 0.959879I
a = 0.131891 + 0.182609I
b = 1.59766 + 1.61716I
4.43635 1.24068I 13.95090 + 0.78074I
u = 0.341711 + 0.959879I
a = 0.131891 0.182609I
b = 1.59766 1.61716I
4.43635 + 1.24068I 13.95090 0.78074I
u = 0.421636 1.054606I
a = 0.75840 1.30202I
b = 0.637238 0.802524I
11.08039 0.40194I 9.60031 + 1.91699I
u = 0.421636 + 1.054606I
a = 0.75840 + 1.30202I
b = 0.637238 + 0.802524I
11.08039 + 0.40194I 9.60031 1.91699I
u = 0.462289 1.065052I
a = 0.233516 1.377762I
b = 2.27698 1.43491I
10.78700 6.40801I 9.24549 + 6.85551I
u = 0.462289 + 1.065052I
a = 0.233516 + 1.377762I
b = 2.27698 + 1.43491I
10.78700 + 6.40801I 9.24549 6.85551I
10
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.482865 0.381048I
a = 1.158218 0.230495I
b = 0.361676 0.060781I
1.50033 + 0.49934I 4.97310 1.55605I
u = 0.482865 + 0.381048I
a = 1.158218 + 0.230495I
b = 0.361676 + 0.060781I
1.50033 0.49934I 4.97310 + 1.55605I
u = 0.533946 1.013709I
a = 0.193018 + 0.551546I
b = 0.164536 + 1.052487I
3.06053 4.69031I 8.68269 + 6.75230I
u = 0.533946 + 1.013709I
a = 0.193018 0.551546I
b = 0.164536 1.052487I
3.06053 + 4.69031I 8.68269 6.75230I
u = 0.575499 1.031323I
a = 0.294285 0.524607I
b = 1.23740 1.33152I
0.91167 6.46667I 0.27924 + 8.67512I
u = 0.575499 + 1.031323I
a = 0.294285 + 0.524607I
b = 1.23740 + 1.33152I
0.91167 + 6.46667I 0.27924 8.67512I
u = 0.59705 1.31039I
a = 0.388554 0.940654I
b = 1.11302 1.53684I
7.22620 5.83293I 6.15082 + 5.68855I
u = 0.59705 + 1.31039I
a = 0.388554 + 0.940654I
b = 1.11302 + 1.53684I
7.22620 + 5.83293I 6.15082 5.68855I
u = 0.609332 1.155207I
a = 0.327103 + 1.005163I
b = 1.35350 + 1.27843I
4.80243 5.41014I 7.26346 + 0.72558I
u = 0.609332 + 1.155207I
a = 0.327103 1.005163I
b = 1.35350 1.27843I
4.80243 + 5.41014I 7.26346 0.72558I
11
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.614621 1.121983I
a = 0.366269 + 0.769880I
b = 1.04552 + 1.57884I
3.19452 11.57247I 4.62517 + 9.19158I
u = 0.614621 + 1.121983I
a = 0.366269 0.769880I
b = 1.04552 1.57884I
3.19452 + 11.57247I 4.62517 9.19158I
u = 0.651335 0.521113I
a = 0.651925 0.264644I
b = 0.296598 1.181027I
2.41607 + 1.66211I 3.86075 3.01699I
u = 0.651335 + 0.521113I
a = 0.651925 + 0.264644I
b = 0.296598 + 1.181027I
2.41607 1.66211I 3.86075 + 3.01699I
u = 0.834732 0.404612I
a = 0.915701 + 0.531066I
b = 0.002299 + 1.151678I
1.04725 + 6.18742I 1.76655 5.50914I
u = 0.834732 + 0.404612I
a = 0.915701 0.531066I
b = 0.002299 1.151678I
1.04725 6.18742I 1.76655 + 5.50914I
u = 0.838953 0.988789I
a = 0.585981 + 0.864643I
b = 1.078270 + 0.536662I
4.06336 5.47241I 0.38615 + 5.26925I
u = 0.838953 + 0.988789I
a = 0.585981 0.864643I
b = 1.078270 0.536662I
4.06336 + 5.47241I 0.38615 5.26925I
u = 0.840708 0.557697I
a = 1.122850 0.716168I
b = 0.978982 + 0.131303I
2.78766 0.58394I 2.05211 + 2.44149I
u = 0.840708 + 0.557697I
a = 1.122850 + 0.716168I
b = 0.978982 0.131303I
2.78766 + 0.58394I 2.05211 2.44149I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.880421
a = 1.27930
b = 0.248247
1.79480 8.89994
u = 1.05938
a = 1.33817
b = 0.192189
3.29623 2.05434
12
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
13
u
12
+ ··· u 1)(u
72
+ 17u
70
+ ··· + u + 1)
c
2
(u
13
+ 7u
12
+ ··· 5u 1)(u
72
+ 34u
71
+ ··· + 11u + 1)
c
3
(u
13
+ u
12
u
11
5u
10
2u
9
+ u
8
+ u
7
u
6
+ 3u
5
+ 3u
4
+ u
3
u
2
u 1)
(u
72
+ 6u
71
+ ··· + 50529u + 18761)
c
4
(u
13
+ u
12
+ ··· u 1)(u
72
+ 2u
71
+ ··· + 17u 1)
c
5
(u
13
+ u
12
+ ··· u + 1)(u
72
+ 17u
70
+ ··· + u + 1)
c
6
(u
13
+ 2u
12
+ ··· 7u
2
+ 1)(u
72
+ u
71
+ ··· 704u + 121)
c
7
(u
13
u
12
+ ··· u + 1)(u
72
+ 2u
71
+ ··· + 17u 1)
c
8
(u
13
u
12
+ ··· u + 1)(u
72
+ 2u
71
+ ··· + 17u 1)
c
9
(u
13
u
12
+ u
11
+ u
10
3u
9
+ 3u
8
+ u
7
+ u
6
u
5
2u
4
+ 5u
3
u
2
u + 1)
(u
72
+ 2u
71
+ ··· + 25u 1)
c
10
(u
13
2u
12
+ ··· + 7u
2
1)(u
72
+ u
71
+ ··· 704u + 121)
c
11
(u
13
+ u
12
+ ··· + 3u 1)(u
72
+ 12u
71
+ ··· 916u 88)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
13
+ 7y
12
+ ··· 5y 1)(y
72
+ 34y
71
+ ··· + 11y + 1)
c
2
(y
13
+ 3y
12
+ ··· 13y 1)(y
72
+ 14y
71
+ ··· + 87y + 1)
c
3
(y
13
3y
12
+ ··· y 1)
(y
72
+ 28y
71
+ ··· + 9764167099y + 351975121)
c
4
(y
13
15y
12
+ ··· + 3y 1)(y
72
76y
71
+ ··· 13y + 1)
c
6
(y
13
14y
12
+ ··· + 14y 1)(y
72
59y
71
+ ··· 471900y + 14641)
c
7
(y
13
15y
12
+ ··· + 3y 1)(y
72
76y
71
+ ··· 13y + 1)
c
8
(y
13
15y
12
+ ··· + 3y 1)(y
72
76y
71
+ ··· 13y + 1)
c
9
(y
13
+ y
12
+ ··· + 3y 1)(y
72
+ 4y
71
+ ··· 33y + 1)
c
10
(y
13
14y
12
+ ··· + 14y 1)(y
72
59y
71
+ ··· 471900y + 14641)
c
11
(y
13
+ y
12
+ ··· + 3y 1)(y
72
+ 8y
71
+ ··· + 92336y + 7744)
14