11a
345
(K11a
345
)
1
Arc Sequences
9 7 1 8 11 10 2 4 3 6 5
Solving Sequence
5,8
4
1,9
2 3 7 11 6 10
c
4
c
8
c
1
c
3
c
7
c
11
c
5
c
10
c
2
, c
6
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
10
+ 7u
8
+ 17u
6
+ 17u
4
+ u
3
+ 7u
2
+ 2u + 1, u
9
6u
7
12u
5
+ u
4
9u
3
+ 2u
2
+ b 2u,
u
9
u
8
+ 7u
7
6u
6
+ 17u
5
12u
4
+ 17u
3
9u
2
+ a + 6u 2i
I
u
2
= hu
55
+ u
54
+ ··· 5u + 7, 2.36023 × 10
42
u
54
3.88651 × 10
42
u
53
+ ··· + 1.28841 × 10
43
b 5.67990 × 10
43
,
8.53343 × 10
42
u
54
4.03677 × 10
43
u
53
+ ··· + 9.01884 × 10
43
a 1.11633 × 10
45
i
There are 2 irreducible components with 65 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
10
+ 7u
8
+ 17u
6
+ 17u
4
+ u
3
+ 7u
2
+ 2u + 1, u
9
6u
7
12u
5
+
u
4
9u
3
+ 2u
2
+ b 2u, u
9
u
8
+ · · · + a 2i
(i) Arc colorings
a
5
=
0
u
a
8
=
u
9
+ u
8
7u
7
+ 6u
6
17u
5
+ 12u
4
17u
3
+ 9u
2
6u + 2
u
9
+ 6u
7
+ 12u
5
u
4
+ 9u
3
2u
2
+ 2u
a
4
=
u
9
7u
7
u
6
16u
5
5u
4
13u
3
8u
2
3u 5
u
6
+ 4u
4
+ 4u
2
+ 1
a
1
=
1
0
a
9
=
2u
9
+ 13u
7
+ 29u
5
+ 26u
3
+ 3u
2
+ 8u + 5
u
4
3u
2
1
a
2
=
u
9
7u
7
u
6
16u
5
5u
4
13u
3
8u
2
3u 4
u
8
+ 6u
6
+ 11u
4
+ 6u
2
+ 1
a
3
=
u
9
7u
7
16u
5
u
4
13u
3
4u
2
3u 4
u
6
+ 4u
4
+ 4u
2
+ 1
a
7
=
u
3
2u
u
5
+ 3u
3
+ u
a
11
=
1
u
2
a
6
=
u
u
3
+ u
a
10
=
u
2
+ 1
u
4
2u
2
a
10
=
u
2
+ 1
u
4
2u
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 = 0.211333 0.326245I
a = 1.60956 + 3.00167I
b = 0.204272 0.894627I
1.26483 3.13412I 9.57651 + 7.99526I
u = 0.211333 + 0.326245I
a = 1.60956 3.00167I
b = 0.204272 + 0.894627I
1.26483 + 3.13412I 9.57651 7.99526I
u = 0.177185 1.148898I
a = 0.882229 0.191506I
b = 0.653016 0.262322I
1.42305 + 1.66512I 4.81318 3.74793I
u = 0.177185 + 1.148898I
a = 0.882229 + 0.191506I
b = 0.653016 + 0.262322I
1.42305 1.66512I 4.81318 + 3.74793I
u = 0.06987 1.53463I
a = 0.827339 0.616572I
b = 0.668870 + 1.154944I
5.27080 4.15690I 7.93433 + 8.62435I
u = 0.06987 + 1.53463I
a = 0.827339 + 0.616572I
b = 0.668870 1.154944I
5.27080 + 4.15690I 7.93433 8.62435I
u = 0.07477 1.69713I
a = 0.354669 + 0.137758I
b = 0.12814 1.55099I
12.44746 + 3.08863I 2.40432 + 0.06420I
u = 0.07477 + 1.69713I
a = 0.354669 0.137758I
b = 0.12814 + 1.55099I
12.44746 3.08863I 2.40432 0.06420I
u = 0.383617 0.756267I
a = 0.690218 0.458086I
b = 0.18344 + 1.96392I
3.50766 + 1.39846I 4.77165 3.39480I
u = 0.383617 + 0.756267I
a = 0.690218 + 0.458086I
b = 0.18344 1.96392I
3.50766 1.39846I 4.77165 + 3.39480I
3
II. I
u
2
=
hu
55
+u
54
+· · ·5u +7, 2.36×10
42
u
54
3.89× 10
42
u
53
+· · ·+1.29 ×10
43
b
5.68 × 10
43
, 8.53 × 10
42
u
54
4.04 × 10
43
u
53
+ · · · + 9.02 × 10
43
a 1.12 × 10
45
i
(i) Arc colorings
a
5
=
0
u
a
8
=
0.0946178u
54
+ 0.447593u
53
+ ··· 6.65513u + 12.3777
0.183190u
54
+ 0.301652u
53
+ ··· + 1.05917u + 4.40847
a
4
=
0.182511u
54
+ 0.526097u
53
+ ··· 15.9244u + 6.58342
0.399356u
54
0.198299u
53
+ ··· + 9.59641u 8.03634
a
1
=
1
0
a
9
=
0.566872u
54
+ 0.0415916u
53
+ ··· 0.0444348u + 14.9662
0.253953u
54
+ 0.228244u
53
+ ··· + 3.45810u 0.919730
a
2
=
0.372891u
54
0.404659u
53
+ ··· + 17.1159u 8.33712
0.190401u
54
+ 0.443357u
53
+ ··· 6.55735u + 8.36128
a
3
=
0.216844u
54
+ 0.327798u
53
+ ··· 6.32800u 1.45292
0.399356u
54
0.198299u
53
+ ··· + 9.59641u 8.03634
a
7
=
u
3
2u
u
5
+ 3u
3
+ u
a
11
=
1
u
2
a
6
=
u
u
3
+ u
a
10
=
u
2
+ 1
u
4
2u
2
a
10
=
u
2
+ 1
u
4
2u
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 = 0.864118 0.012965I
a = 0.59833 1.32242I
b = 0.20052 + 1.50584I
0.07105 + 6.42965I 5.72472 6.18635I
u = 0.864118 + 0.012965I
a = 0.59833 + 1.32242I
b = 0.20052 1.50584I
0.07105 6.42965I 5.72472 + 6.18635I
u = 0.610435 1.043968I
a = 0.876594 + 0.114471I
b = 0.321004 0.820079I
3.21288 + 1.41327I 0.79892 2.47902I
u = 0.610435 + 1.043968I
a = 0.876594 0.114471I
b = 0.321004 + 0.820079I
3.21288 1.41327I 0.79892 + 2.47902I
u = 0.588670 0.914922I
a = 0.640704 1.085408I
b = 0.76787 + 1.74002I
2.82511 11.22664I 3.78897 + 8.41121I
u = 0.588670 + 0.914922I
a = 0.640704 + 1.085408I
b = 0.76787 1.74002I
2.82511 + 11.22664I 3.78897 8.41121I
u = 0.472889 0.479834I
a = 0.83011 + 1.49859I
b = 0.236312 1.103502I
1.01300 1.69462I 7.06539 + 2.31882I
u = 0.472889 + 0.479834I
a = 0.83011 1.49859I
b = 0.236312 + 1.103502I
1.01300 + 1.69462I 7.06539 2.31882I
u = 0.456547 0.410612I
a = 1.20776 + 1.51344I
b = 0.128001 0.912756I
1.14656 1.61384I 9.05144 + 4.20142I
u = 0.456547 + 0.410612I
a = 1.20776 1.51344I
b = 0.128001 + 0.912756I
1.14656 + 1.61384I 9.05144 4.20142I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.346289
a = 0.931204
b = 0.365005
0.742548 13.8049
u = 0.17113 1.68532I
a = 0.699430 + 0.494208I
b = 1.24530 1.85586I
11.7421 14.2160I 2.13501 + 7.06878I
u = 0.17113 + 1.68532I
a = 0.699430 0.494208I
b = 1.24530 + 1.85586I
11.7421 + 14.2160I 2.13501 7.06878I
u = 0.151090 0.780801I
a = 0.150117 + 0.759458I
b = 0.371738 0.034513I
1.58788 1.71112I 2.31712 + 4.70415I
u = 0.151090 + 0.780801I
a = 0.150117 0.759458I
b = 0.371738 + 0.034513I
1.58788 + 1.71112I 2.31712 4.70415I
u = 0.140808 0.966052I
a = 0.868671 0.450454I
b = 0.663742 + 0.830897I
3.17927 3.14967I 1.09390 + 6.92052I
u = 0.140808 + 0.966052I
a = 0.868671 + 0.450454I
b = 0.663742 0.830897I
3.17927 + 3.14967I 1.09390 6.92052I
u = 0.12278 1.74080I
a = 0.437520 + 0.224363I
b = 0.233480 0.146536I
13.12826 1.45980I 0.69339 1.95269I
u = 0.12278 + 1.74080I
a = 0.437520 0.224363I
b = 0.233480 + 0.146536I
13.12826 + 1.45980I 0.69339 + 1.95269I
u = 0.11048 1.52706I
a = 0.729203 0.564094I
b = 0.678357 + 1.206568I
5.67173 3.68365I 1.03535 0.96847I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.11048 + 1.52706I
a = 0.729203 + 0.564094I
b = 0.678357 1.206568I
5.67173 + 3.68365I 1.03535 + 0.96847I
u = 0.04652 1.56492I
a = 0.937953 0.234960I
b = 1.083142 + 0.379926I
5.43122 2.80843I 6.31485 + 1.24846I
u = 0.04652 + 1.56492I
a = 0.937953 + 0.234960I
b = 1.083142 0.379926I
5.43122 + 2.80843I 6.31485 1.24846I
u = 0.04042 1.69986I
a = 0.441337 + 0.214004I
b = 0.96011 1.21387I
12.61374 3.89122I 0.20719 + 8.96757I
u = 0.04042 + 1.69986I
a = 0.441337 0.214004I
b = 0.96011 + 1.21387I
12.61374 + 3.89122I 0.20719 8.96757I
u = 0.02833 1.64604I
a = 0.033197 0.553614I
b = 0.381820 + 0.488632I
10.07854 2.31202I 1.07799 + 2.78280I
u = 0.02833 + 1.64604I
a = 0.033197 + 0.553614I
b = 0.381820 0.488632I
10.07854 + 2.31202I 1.07799 2.78280I
u = 0.01279 1.63475I
a = 0.568018 + 0.241148I
b = 1.50764 2.44427I
12.00641 + 0.63513I 0.597872 + 0.536133I
u = 0.01279 + 1.63475I
a = 0.568018 0.241148I
b = 1.50764 + 2.44427I
12.00641 0.63513I 0.597872 0.536133I
u = 0.02860 1.62809I
a = 0.693590 + 0.625630I
b = 0.40686 1.42621I
7.90755 + 3.97377I 3.08656 3.64965I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.02860 + 1.62809I
a = 0.693590 0.625630I
b = 0.40686 + 1.42621I
7.90755 3.97377I 3.08656 + 3.64965I
u = 0.073778 0.672365I
a = 1.123499 0.267831I
b = 0.74025 + 2.06124I
3.83511 + 0.35945I 0.43782 + 1.63118I
u = 0.073778 + 0.672365I
a = 1.123499 + 0.267831I
b = 0.74025 2.06124I
3.83511 0.35945I 0.43782 1.63118I
u = 0.09200 1.63412I
a = 0.729011 0.353886I
b = 1.65022 + 1.91119I
6.56631 + 7.31158I 3.30832 6.27982I
u = 0.09200 + 1.63412I
a = 0.729011 + 0.353886I
b = 1.65022 1.91119I
6.56631 7.31158I 3.30832 + 6.27982I
u = 0.117271 0.670556I
a = 1.40648 1.46549I
b = 0.145074 + 1.047413I
0.18164 + 3.45397I 3.45985 6.03391I
u = 0.117271 + 0.670556I
a = 1.40648 + 1.46549I
b = 0.145074 1.047413I
0.18164 3.45397I 3.45985 + 6.03391I
u = 0.118294 0.339429I
a = 3.56060 + 0.27683I
b = 0.049659 + 0.362752I
1.10271 2.46020I 5.86646 2.22286I
u = 0.118294 + 0.339429I
a = 3.56060 0.27683I
b = 0.049659 0.362752I
1.10271 + 2.46020I 5.86646 + 2.22286I
u = 0.12025 1.66551I
a = 0.295988 + 0.720116I
b = 0.288497 0.639755I
14.5361 + 7.4191I 0.09894 4.83420I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.12025 + 1.66551I
a = 0.295988 0.720116I
b = 0.288497 + 0.639755I
14.5361 7.4191I 0.09894 + 4.83420I
u = 0.178670 1.238418I
a = 0.690553 0.041565I
b = 0.764420 + 1.048999I
1.097691 0.148579I 6.50751 0.03024I
u = 0.178670 + 1.238418I
a = 0.690553 + 0.041565I
b = 0.764420 1.048999I
1.097691 + 0.148579I 6.50751 + 0.03024I
u = 0.19570 1.65977I
a = 0.507646 + 0.346994I
b = 1.73777 1.44055I
12.54411 + 5.30776I 0.93873 5.45617I
u = 0.19570 + 1.65977I
a = 0.507646 0.346994I
b = 1.73777 + 1.44055I
12.54411 5.30776I 0.93873 + 5.45617I
u = 0.371626 0.726720I
a = 1.02392 + 1.11200I
b = 0.94383 1.70269I
1.61640 + 5.63449I 6.38690 8.36375I
u = 0.371626 + 0.726720I
a = 1.02392 1.11200I
b = 0.94383 + 1.70269I
1.61640 5.63449I 6.38690 + 8.36375I
u = 0.420813 0.866276I
a = 0.745448 0.862678I
b = 0.198503 + 0.106722I
5.77965 + 5.30825I 0.39212 6.19930I
u = 0.420813 + 0.866276I
a = 0.745448 + 0.862678I
b = 0.198503 0.106722I
5.77965 5.30825I 0.39212 + 6.19930I
u = 0.518495 0.178316I
a = 1.16097 + 1.62640I
b = 0.12502 1.51983I
3.26165 2.54833I 12.38074 + 1.07618I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.518495 + 0.178316I
a = 1.16097 1.62640I
b = 0.12502 + 1.51983I
3.26165 + 2.54833I 12.38074 1.07618I
u = 0.583790 0.094289I
a = 1.220270 0.019989I
b = 0.102394 + 0.614835I
2.94002 + 1.90835I 3.01939 3.01750I
u = 0.583790 + 0.094289I
a = 1.220270 + 0.019989I
b = 0.102394 0.614835I
2.94002 1.90835I 3.01939 + 3.01750I
u = 0.645292 0.720286I
a = 0.235484 0.717538I
b = 0.89975 + 1.46055I
4.35568 + 2.03916I 0.91927 6.28505I
u = 0.645292 + 0.720286I
a = 0.235484 + 0.717538I
b = 0.89975 1.46055I
4.35568 2.03916I 0.91927 + 6.28505I
10
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
10
2u
9
+ u
8
u
6
u
5
+ 3u
4
+ u
3
u
2
u + 1)
(u
55
+ 3u
54
+ ··· + 3u
2
+ 1)
c
2
(u
10
+ 5u
8
+ u
7
+ 10u
6
+ 3u
5
+ 9u
4
+ 2u
3
+ 4u
2
+ u + 1)
(u
55
+ u
54
+ ··· 8u 88)
c
3
(u
10
+ 3u
7
+ 3u
6
2u
5
3u
4
+ u
3
+ 4u
2
+ 3u + 1)
(u
55
+ 9u
54
+ ··· 76u 7)
c
4
(u
10
u
9
+ 4u
8
2u
7
+ 9u
6
3u
5
+ 10u
4
u
3
+ 5u
2
+ 1)
(u
55
+ 2u
54
+ ··· 21u 19)
c
5
, c
6
(u
10
+ 7u
8
+ ··· 2u + 1)(u
55
+ u
54
+ ··· 5u + 7)
c
7
(u
10
+ 5u
8
u
7
+ 10u
6
3u
5
+ 9u
4
2u
3
+ 4u
2
u + 1)
(u
55
+ u
54
+ ··· 8u 88)
c
8
(u
10
+ u
9
+ 4u
8
+ 2u
7
+ 9u
6
+ 3u
5
+ 10u
4
+ u
3
+ 5u
2
+ 1)
(u
55
+ 2u
54
+ ··· 21u 19)
c
9
(u
10
+ u
9
u
8
u
7
+ 3u
6
+ u
5
u
4
+ u
2
+ 2u + 1)
(u
55
+ 12u
53
+ ··· 2271u 6677)
c
10
, c
11
(u
10
+ 7u
8
+ ··· + 2u + 1)(u
55
+ u
54
+ ··· 5u + 7)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
(y
10
2y
9
y
8
+ 9y
6
11y
5
+ 15y
4
11y
3
+ 9y
2
3y + 1)
(y
55
3y
54
+ ··· 6y 1)
c
2
, c
7
(y
10
+ 10y
9
+ ··· + 7y + 1)(y
55
+ 45y
54
+ ··· 134048y 7744)
c
3
(y
10
+ 6y
8
15y
7
+ 29y
6
26y
5
+ 19y
4
7y
3
+ 4y
2
y + 1)
(y
55
+ 3y
54
+ ··· 160y 49)
c
4
, c
8
(y
10
+ 7y
9
+ ··· + 10y + 1)(y
55
+ 30y
54
+ ··· 927y 361)
c
5
, c
6
, c
10
c
11
(y
10
+ 14y
9
+ ··· + 10y + 1)(y
55
+ 69y
54
+ ··· 843y 49)
c
9
(y
10
3y
9
+ 9y
8
11y
7
+ 15y
6
11y
5
+ 9y
4
y
2
2y + 1)
(y
55
+ 24y
54
+ ··· 671436323y 44582329)
12