11n
124
(K11n
124
)
1
Arc Sequences
7 1 11 8 9 2 5 3 7 8 9
Solving Sequence
7,9 2,10
1 3 6 5 8 11 4
c
9
c
1
c
2
c
6
c
5
c
8
c
11
c
3
c
4
, c
7
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
10
+ u
9
+ 3u
8
+ 2u
7
+ 5u
6
+ 3u
5
+ 6u
4
+ 2u
3
+ 4u
2
+ u + 1, u
8
+ u
6
u
5
+ 2u
4
u
3
+ u
2
+ b 2u + 1,
u
9
u
8
2u
7
u
6
3u
5
2u
4
3u
3
+ a u 1i
I
u
2
= hu
39
+ 12u
37
+ ··· + 14u + 7, 3.71252 × 10
29
u
38
1.89092 × 10
29
u
37
+ ··· + 1.67323 × 10
29
b 6.57733 × 10
29
,
1.82393 × 10
30
u
38
+ 7.78408 × 10
29
u
37
+ ··· + 1.17126 × 10
30
a + 1.39048 × 10
30
i
There are 2 irreducible components with 49 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
+u
9
+· · ·+u+1, u
8
+u
6
u
5
+2u
4
u
3
+u
2
+b2u+1, u
9
u
8
+· · ·+a1i
(i) Arc colorings
a
7
=
0
u
a
9
=
u
9
+ u
8
+ 2u
7
+ u
6
+ 3u
5
+ 2u
4
+ 3u
3
+ u + 1
u
8
u
6
+ u
5
2u
4
+ u
3
u
2
+ 2u 1
a
2
=
1
0
a
10
=
u
9
+ u
8
+ 2u
7
+ u
6
+ 3u
5
+ 2u
4
+ 3u
3
+ u + 1
u
9
2u
8
2u
7
2u
6
2u
5
4u
4
2u
3
u
2
+ u 1
a
1
=
1
u
2
a
3
=
u
2
+ 1
u
4
a
6
=
u
u
a
5
=
2u
9
2u
8
5u
7
3u
6
7u
5
4u
4
8u
3
2u
2
4u 1
u
9
+ 2u
8
+ 3u
7
+ 4u
6
+ 4u
5
+ 5u
4
+ 5u
3
+ 5u
2
+ 2u + 2
a
8
=
2u
9
+ 2u
8
+ 5u
7
+ 3u
6
+ 7u
5
+ 4u
4
+ 7u
3
+ u
2
+ 3u + 1
u
9
2u
8
3u
7
3u
6
3u
5
4u
4
3u
3
2u
2
+ u 1
a
11
=
2u
8
+ u
7
+ 4u
6
+ u
5
+ 6u
4
+ u
3
+ 6u
2
u + 4
2u
9
+ 4u
7
+ 7u
5
+ 8u
3
u
2
+ 5u
a
4
=
3u
9
3u
8
8u
7
6u
6
13u
5
9u
4
14u
3
5u
2
7u 3
u
8
+ 3u
6
+ u
5
+ 5u
4
+ u
3
+ 5u
2
u + 3
a
4
=
3u
9
3u
8
8u
7
6u
6
13u
5
9u
4
14u
3
5u
2
7u 3
u
8
+ 3u
6
+ u
5
+ 5u
4
+ u
3
+ 5u
2
u + 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 = 0.879557 0.842212I
a = 0.319829 0.445214I
b = 0.496572 + 0.841721I
1.28605 + 3.24415I 6.98007 6.78944I
u = 0.879557 + 0.842212I
a = 0.319829 + 0.445214I
b = 0.496572 0.841721I
1.28605 3.24415I 6.98007 + 6.78944I
u = 0.372175 1.177672I
a = 0.299227 1.055981I
b = 0.21685 + 2.20703I
4.06424 + 5.03997I 7.43062 5.98899I
u = 0.372175 + 1.177672I
a = 0.299227 + 1.055981I
b = 0.21685 2.20703I
4.06424 5.03997I 7.43062 + 5.98899I
u = 0.220878 0.599013I
a = 1.263493 0.484575I
b = 1.07007 1.03582I
1.69097 2.38428I 3.68022 + 2.86338I
u = 0.220878 + 0.599013I
a = 1.263493 + 0.484575I
b = 1.07007 + 1.03582I
1.69097 + 2.38428I 3.68022 2.86338I
u = 0.283646 0.889181I
a = 0.85633 + 1.49046I
b = 0.10259 1.96070I
1.77560 1.23703I 1.12210 + 5.29149I
u = 0.283646 + 0.889181I
a = 0.85633 1.49046I
b = 0.10259 + 1.96070I
1.77560 + 1.23703I 1.12210 5.29149I
u = 0.688964 0.877700I
a = 0.359578 + 0.617446I
b = 0.812246 0.334913I
4.33849 2.65528I 5.99132 + 3.67032I
u = 0.688964 + 0.877700I
a = 0.359578 0.617446I
b = 0.812246 + 0.334913I
4.33849 + 2.65528I 5.99132 3.67032I
3
II. I
u
2
=
hu
39
+12u
37
+· · ·+14u+7, 3.71×10
29
u
38
1.89×10
29
u
37
+· · ·+1.67×10
29
b
6.58× 10
29
, 1.82× 10
30
u
38
+7.78 ×10
29
u
37
+· · · +1.17 ×10
30
a + 1.39× 10
30
i
(i) Arc colorings
a
7
=
0
u
a
9
=
1.55724u
38
0.664589u
37
+ ··· + 7.55322u 1.18716
2.21877u
38
+ 1.13010u
37
+ ··· 8.05855u + 3.93091
a
2
=
1
0
a
10
=
1.55724u
38
0.664589u
37
+ ··· + 7.55322u 1.18716
3.36931u
38
+ 1.93095u
37
+ ··· 9.65496u + 8.58303
a
1
=
1
u
2
a
3
=
u
2
+ 1
u
4
a
6
=
u
u
a
5
=
0.737177u
38
+ 0.407586u
37
+ ··· + 13.5161u + 4.88745
2.96524u
38
0.543373u
37
+ ··· 39.7761u 10.1662
a
8
=
1.34253u
38
0.801447u
37
+ ··· + 4.38094u 2.85956
3.03311u
38
+ 1.84827u
37
+ ··· 6.65770u + 8.18752
a
11
=
0.596770u
38
0.670897u
37
+ ··· 3.69102u 2.60571
1.64150u
38
+ 2.89153u
37
+ ··· + 21.0263u + 18.8892
a
4
=
0.639576u
38
+ 0.925361u
37
+ ··· + 6.56459u + 5.74668
3.13251u
38
2.07609u
37
+ ··· + 7.20606u 8.19814
a
4
=
0.639576u
38
+ 0.925361u
37
+ ··· + 6.56459u + 5.74668
3.13251u
38
2.07609u
37
+ ··· + 7.20606u 8.19814
(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 = 1.067817 0.145596I
a = 1.383671 0.195517I
b = 0.233676 0.045350I
5.61750 7.80414I 3.64994 + 4.73695I
u = 1.067817 + 0.145596I
a = 1.383671 + 0.195517I
b = 0.233676 + 0.045350I
5.61750 + 7.80414I 3.64994 4.73695I
u = 0.661560 0.856108I
a = 0.105096 + 0.363168I
b = 0.724386 + 0.130762I
5.00877 + 2.57683I 6.58497 2.64677I
u = 0.661560 + 0.856108I
a = 0.105096 0.363168I
b = 0.724386 0.130762I
5.00877 2.57683I 6.58497 + 2.64677I
u = 0.660029
a = 1.75790
b = 0.191844
1.67672 6.74229
u = 0.58423 1.30883I
a = 0.088865 + 1.209821I
b = 0.03733 2.67531I
9.2430 + 13.6991I 5.63720 7.27460I
u = 0.58423 + 1.30883I
a = 0.088865 1.209821I
b = 0.03733 + 2.67531I
9.2430 13.6991I 5.63720 + 7.27460I
u = 0.461639 1.163798I
a = 0.093896 1.347319I
b = 0.29393 + 2.37615I
4.83553 + 4.15063I 11.12509 1.85834I
u = 0.461639 + 1.163798I
a = 0.093896 + 1.347319I
b = 0.29393 2.37615I
4.83553 4.15063I 11.12509 + 1.85834I
u = 0.410850 0.313768I
a = 1.30683 + 1.45682I
b = 0.952365 0.587886I
0.53003 2.77733I 1.12295 + 3.77344I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.410850 + 0.313768I
a = 1.30683 1.45682I
b = 0.952365 + 0.587886I
0.53003 + 2.77733I 1.12295 3.77344I
u = 0.39136 1.40665I
a = 0.309024 + 0.965300I
b = 0.58142 2.29137I
10.72517 2.62440I 7.66574 + 2.07464I
u = 0.39136 + 1.40665I
a = 0.309024 0.965300I
b = 0.58142 + 2.29137I
10.72517 + 2.62440I 7.66574 2.07464I
u = 0.365080 1.061567I
a = 1.16765 0.98033I
b = 0.933649 + 1.054949I
2.63598 + 6.13265I 3.84561 9.40841I
u = 0.365080 + 1.061567I
a = 1.16765 + 0.98033I
b = 0.933649 1.054949I
2.63598 6.13265I 3.84561 + 9.40841I
u = 0.358275 0.727858I
a = 1.31473 0.56113I
b = 0.470884 0.319022I
2.12535 1.41634I 6.42200 1.56979I
u = 0.358275 + 0.727858I
a = 1.31473 + 0.56113I
b = 0.470884 + 0.319022I
2.12535 + 1.41634I 6.42200 + 1.56979I
u = 0.145422 0.960485I
a = 0.312403 1.125533I
b = 0.15416 + 2.89600I
2.60397 + 3.58012I 5.99936 4.69227I
u = 0.145422 + 0.960485I
a = 0.312403 + 1.125533I
b = 0.15416 2.89600I
2.60397 3.58012I 5.99936 + 4.69227I
u = 0.075811 0.941393I
a = 0.57379 + 1.58798I
b = 0.40235 2.05076I
1.322387 + 0.422831I 4.94003 + 1.41438I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.075811 + 0.941393I
a = 0.57379 1.58798I
b = 0.40235 + 2.05076I
1.322387 0.422831I 4.94003 1.41438I
u = 0.142576 1.181589I
a = 0.256537 0.091644I
b = 1.313089 0.223447I
4.72844 2.99253I 9.56242 + 1.88756I
u = 0.142576 + 1.181589I
a = 0.256537 + 0.091644I
b = 1.313089 + 0.223447I
4.72844 + 2.99253I 9.56242 1.88756I
u = 0.244496 0.924229I
a = 0.771798 0.143039I
b = 0.288487 0.166887I
0.62997 1.59285I 3.23591 + 4.35949I
u = 0.244496 + 0.924229I
a = 0.771798 + 0.143039I
b = 0.288487 + 0.166887I
0.62997 + 1.59285I 3.23591 4.35949I
u = 0.399747 1.326196I
a = 0.135365 1.304921I
b = 0.13768 + 2.63610I
10.43667 5.23929I 7.47031 + 3.45363I
u = 0.399747 + 1.326196I
a = 0.135365 + 1.304921I
b = 0.13768 2.63610I
10.43667 + 5.23929I 7.47031 3.45363I
u = 0.421432 1.340551I
a = 0.081054 + 0.871471I
b = 0.04004 2.43201I
3.60682 6.09591I 4.01419 + 11.09242I
u = 0.421432 + 1.340551I
a = 0.081054 0.871471I
b = 0.04004 + 2.43201I
3.60682 + 6.09591I 4.01419 11.09242I
u = 0.518178 0.693040I
a = 0.925341 + 0.451847I
b = 0.068871 1.190035I
0.23938 2.14007I 2.40854 + 4.03571I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.518178 + 0.693040I
a = 0.925341 0.451847I
b = 0.068871 + 1.190035I
0.23938 + 2.14007I 2.40854 4.03571I
u = 0.577975 1.276162I
a = 0.280369 0.960950I
b = 0.83703 + 2.11813I
9.10381 5.07821I 6.63468 + 3.17849I
u = 0.577975 + 1.276162I
a = 0.280369 + 0.960950I
b = 0.83703 2.11813I
9.10381 + 5.07821I 6.63468 3.17849I
u = 0.715910 0.207951I
a = 0.864160 + 0.809698I
b = 0.283616 0.541442I
1.08867 1.79567I 3.61269 + 3.73862I
u = 0.715910 + 0.207951I
a = 0.864160 0.809698I
b = 0.283616 + 0.541442I
1.08867 + 1.79567I 3.61269 3.73862I
u = 0.861374 0.909425I
a = 0.420286 + 0.450008I
b = 0.142444 0.650507I
0.74799 3.17123I 8.85439 + 3.48619I
u = 0.861374 + 0.909425I
a = 0.420286 0.450008I
b = 0.142444 + 0.650507I
0.74799 + 3.17123I 8.85439 3.48619I
u = 0.970374 0.165160I
a = 1.38216 0.38547I
b = 0.088430 0.255820I
5.72155 0.54332I 4.48404 + 0.44801I
u = 0.970374 + 0.165160I
a = 1.38216 + 0.38547I
b = 0.088430 + 0.255820I
5.72155 + 0.54332I 4.48404 0.44801I
8
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
10
+ u
9
+ 3u
8
+ 2u
7
+ 5u
6
+ 3u
5
+ 6u
4
+ 2u
3
+ 4u
2
+ u + 1)
(u
39
+ 12u
37
+ ··· + 14u + 7)
c
2
(u
10
+ 5u
9
+ ··· + 7u + 1)(u
39
+ 24u
38
+ ··· 280u 49)
c
3
(u
10
+ 4u
8
+ 2u
7
+ 6u
6
+ 3u
5
+ 8u
4
+ u
3
+ 5u
2
+ 1)
(u
39
+ 3u
38
+ ··· + 9u + 1)
c
4
(u
10
2u
9
+ 5u
8
6u
7
+ 4u
6
u
5
3u
4
+ 3u
3
u
2
+ 1)
(u
39
+ 3u
38
+ ··· 9u 1)
c
5
(u
10
u
8
+ 3u
7
3u
6
u
5
+ 4u
4
6u
3
+ 5u
2
2u + 1)
(u
39
+ u
38
+ ··· 1365u + 253)
c
6
(u
10
u
9
+ 3u
8
2u
7
+ 5u
6
3u
5
+ 6u
4
2u
3
+ 4u
2
u + 1)
(u
39
+ 12u
37
+ ··· + 14u + 7)
c
7
(u
10
+ 2u
9
+ 5u
8
+ 6u
7
+ 4u
6
+ u
5
3u
4
3u
3
u
2
+ 1)
(u
39
+ 3u
38
+ ··· 9u 1)
c
8
(u
10
+ 5u
8
+ u
7
+ 8u
6
+ 3u
5
+ 6u
4
+ 2u
3
+ 4u
2
+ 1)
(u
39
+ u
38
+ ··· 11u + 1)
c
9
(u
10
4u
8
3u
7
+ 7u
6
+ 8u
5
2u
4
7u
3
2u
2
+ 2u + 1)
(u
39
+ u
38
+ ··· 265u 47)
c
10
(u
10
2u
9
+ 4u
8
6u
7
+ 6u
6
6u
5
+ 6u
4
u
3
2u
2
+ 1)
(u
39
+ 3u
38
+ ··· 19u + 1)
c
11
(u
10
+ 8u
9
+ ··· + 5u + 1)(u
39
+ u
38
+ ··· + 90u 209)
9
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y
10
+ 5y
9
+ ··· + 7y + 1)(y
39
+ 24y
38
+ ··· 280y 49)
c
2
(y
10
+ 5y
9
+ 11y
8
+ 28y
7
+ 69y
6
+ 87y
5
+ 50y
4
+ 32y
3
+ 36y
2
y + 1)
(y
39
12y
38
+ ··· 29400y 2401)
c
3
(y
10
+ 8y
9
+ ··· + 10y + 1)(y
39
+ 35y
38
+ ··· 63y 1)
c
4
(y
10
+ 6y
9
+ 9y
8
6y
7
16y
6
+ 3y
5
+ 17y
4
+ 5y
3
5y
2
2y + 1)
(y
39
+ y
38
+ ··· + 37y 1)
c
5
(y
10
2y
9
5y
8
+ 5y
7
+ 17y
6
+ 3y
5
16y
4
6y
3
+ 9y
2
+ 6y + 1)
(y
39
43y
38
+ ··· 1224386y 64009)
c
7
(y
10
+ 6y
9
+ 9y
8
6y
7
16y
6
+ 3y
5
+ 17y
4
+ 5y
3
5y
2
2y + 1)
(y
39
+ y
38
+ ··· + 37y 1)
c
8
(y
10
+ 10y
9
+ ··· + 8y + 1)(y
39
+ 9y
38
+ ··· + 31y 1)
c
9
(y
10
8y
9
+ ··· 8y + 1)(y
39
37y
38
+ ··· + 14483y 2209)
c
10
(y
10
+ 4y
9
+ 4y
8
+ 4y
6
+ 10y
5
+ 8y
4
13y
3
+ 16y
2
4y + 1)
(y
39
+ 39y
38
+ ··· 61y 1)
c
11
(y
10
2y
9
+ ··· + 15y + 1)(y
39
23y
38
+ ··· + 796448y 43681)
10