11a
67
(K11a
67
)
1
Arc Sequences
5 1 10 2 9 4 11 3 6 8 7
Solving Sequence
1,5
2
3,9
6 10 4 7 8 11
c
1
c
2
c
5
c
9
c
4
c
6
c
8
c
11
c
3
, c
7
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
66
3u
65
+ ··· 41u + 9, 9.81051 × 10
60
u
65
5.47738 × 10
62
u
64
+ ··· + 1.47186 × 10
63
b 2.87133 × 10
63
,
1.19331 × 10
63
u
65
+ 3.50804 × 10
63
u
64
+ ··· + 4.41558 × 10
63
a + 4.81025 × 10
63
i
I
u
2
= hu
2
+ u + 1, a 1, 3b 4u 2i
I
u
3
= hu
2
+ u + 1, a u, 3b + 4u + 5i
There are 3 irreducible components with 70 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
66
3u
65
+· · ·41u+9, 9.81×10
60
u
65
5.48×10
62
u
64
+· · ·+1.47×10
63
b
2.87× 10
63
, 1.19 × 10
63
u
65
+3.51 ×10
63
u
64
+· · · +4.42 ×10
63
a + 4.81× 10
63
i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
9
=
0.270250u
65
0.794469u
64
+ ··· + 4.90086u 1.08938
0.00666538u
65
+ 0.372140u
64
+ ··· 6.63626u + 1.95081
a
6
=
0.748834u
65
1.21030u
64
+ ··· 3.69300u + 1.08551
1.24191u
65
+ 3.40219u
64
+ ··· 26.3774u + 5.04613
a
10
=
0.326052u
65
0.806467u
64
+ ··· 1.31458u + 1.11497
0.171690u
65
+ 0.787810u
64
+ ··· 14.4831u + 2.93447
a
4
=
u
u
3
+ u
a
7
=
0.556836u
65
1.04706u
64
+ ··· 1.41116u + 0.0253487
0.670953u
65
+ 1.85058u
64
+ ··· 13.4642u + 2.39147
a
8
=
0.673009u
65
1.57681u
64
+ ··· + 6.69251u 1.59627
0.338424u
65
+ 1.14946u
64
+ ··· 14.5248u + 3.82271
a
11
=
0.282928u
65
+ 0.623979u
64
+ ··· + 1.73222u 0.615946
0.326521u
65
0.860783u
64
+ ··· + 9.33068u 2.10352
a
11
=
0.282928u
65
+ 0.623979u
64
+ ··· + 1.73222u 0.615946
0.326521u
65
0.860783u
64
+ ··· + 9.33068u 2.10352
(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 = 0.935300 0.273219I
a = 0.572823 + 0.487015I
b = 0.057009 0.304073I
2.35847 + 1.56855I 4.39447 3.89698I
u = 0.935300 + 0.273219I
a = 0.572823 0.487015I
b = 0.057009 + 0.304073I
2.35847 1.56855I 4.39447 + 3.89698I
u = 0.925761 0.963630I
a = 0.292558 + 0.630615I
b = 0.826100 0.753612I
3.68434 + 4.73747I 1.30727 7.90880I
u = 0.925761 + 0.963630I
a = 0.292558 0.630615I
b = 0.826100 + 0.753612I
3.68434 4.73747I 1.30727 + 7.90880I
u = 0.850565 0.604928I
a = 0.506519 0.001870I
b = 0.401943 + 0.273456I
2.64330 + 1.72779I 3.37557 1.36335I
u = 0.850565 + 0.604928I
a = 0.506519 + 0.001870I
b = 0.401943 0.273456I
2.64330 1.72779I 3.37557 + 1.36335I
u = 0.811477 0.733493I
a = 0.399576 0.654533I
b = 0.756543 + 0.252393I
0.71398 + 2.89026I 7.98157 7.76196I
u = 0.811477 + 0.733493I
a = 0.399576 + 0.654533I
b = 0.756543 0.252393I
0.71398 2.89026I 7.98157 + 7.76196I
u = 0.633036 0.187323I
a = 0.909858 + 0.122210I
b = 0.308390 0.185739I
1.191740 + 0.072864I 10.27225 + 0.64255I
u = 0.633036 + 0.187323I
a = 0.909858 0.122210I
b = 0.308390 + 0.185739I
1.191740 0.072864I 10.27225 0.64255I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.597692 1.092472I
a = 0.328387 0.083666I
b = 0.458000 0.124520I
4.56797 + 3.83881I 1.28636 2.41019I
u = 0.597692 + 1.092472I
a = 0.328387 + 0.083666I
b = 0.458000 + 0.124520I
4.56797 3.83881I 1.28636 + 2.41019I
u = 0.594015 0.896392I
a = 0.259628 0.007248I
b = 0.286059 0.035184I
0.45918 + 2.40111I 3.44791 1.32342I
u = 0.594015 + 0.896392I
a = 0.259628 + 0.007248I
b = 0.286059 + 0.035184I
0.45918 2.40111I 3.44791 + 1.32342I
u = 0.578870 1.167114I
a = 0.171817 0.795086I
b = 0.97417 + 1.37815I
1.62753 + 4.82509I 1.62654 10.22463I
u = 0.578870 + 1.167114I
a = 0.171817 + 0.795086I
b = 0.97417 1.37815I
1.62753 4.82509I 1.62654 + 10.22463I
u = 0.54028 1.34674I
a = 0.071775 + 0.761191I
b = 0.74488 1.26598I
7.07880 + 6.83377I 3.18882 8.56009I
u = 0.54028 + 1.34674I
a = 0.071775 0.761191I
b = 0.74488 + 1.26598I
7.07880 6.83377I 3.18882 + 8.56009I
u = 0.494896 0.961583I
a = 0.319647 + 0.904467I
b = 1.82671 2.45749I
1.87964 + 2.60684I 3.35251 + 10.49831I
u = 0.494896 + 0.961583I
a = 0.319647 0.904467I
b = 1.82671 + 2.45749I
1.87964 2.60684I 3.35251 10.49831I
4
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.437299 0.709945I
a = 0.516450 + 0.696973I
b = 0.377110 + 1.113225I
1.03551 + 1.38591I 5.22665 5.92733I
u = 0.437299 + 0.709945I
a = 0.516450 0.696973I
b = 0.377110 1.113225I
1.03551 1.38591I 5.22665 + 5.92733I
u = 0.381546 0.907422I
a = 0.534318 0.799412I
b = 2.87264 1.37415I
7.08795 + 1.79943I 1.64607 0.65133I
u = 0.381546 + 0.907422I
a = 0.534318 + 0.799412I
b = 2.87264 + 1.37415I
7.08795 1.79943I 1.64607 + 0.65133I
u = 0.364468 0.789088I
a = 0.497179 1.218644I
b = 1.46483 + 2.98434I
6.72143 + 1.46998I 3.43133 5.90939I
u = 0.364468 + 0.789088I
a = 0.497179 + 1.218644I
b = 1.46483 2.98434I
6.72143 1.46998I 3.43133 + 5.90939I
u = 0.159471 0.573771I
a = 0.277901 + 1.116695I
b = 0.141782 + 0.476457I
1.16566 + 1.36283I 2.49105 4.63362I
u = 0.159471 + 0.573771I
a = 0.277901 1.116695I
b = 0.141782 0.476457I
1.16566 1.36283I 2.49105 + 4.63362I
u = 0.112571 1.208744I
a = 0.339730 1.044665I
b = 0.144323 + 0.899701I
6.61424 + 3.82375I 4.50346 5.15104I
u = 0.112571 + 1.208744I
a = 0.339730 + 1.044665I
b = 0.144323 0.899701I
6.61424 3.82375I 4.50346 + 5.15104I
5
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.144965 1.388673I
a = 0.349066 + 0.881495I
b = 0.122909 0.748714I
13.4953 + 6.4032I 6.39234 4.65248I
u = 0.144965 + 1.388673I
a = 0.349066 0.881495I
b = 0.122909 + 0.748714I
13.4953 6.4032I 6.39234 + 4.65248I
u = 0.155427 1.130608I
a = 0.731166 0.132523I
b = 0.317310 0.917737I
8.91388 + 2.47472I 3.77224 2.13911I
u = 0.155427 + 1.130608I
a = 0.731166 + 0.132523I
b = 0.317310 + 0.917737I
8.91388 2.47472I 3.77224 + 2.13911I
u = 0.300452 1.063622I
a = 0.605783 + 1.176205I
b = 0.538913 0.602866I
5.41831 0.47417I 4.00028 + 1.68304I
u = 0.300452 + 1.063622I
a = 0.605783 1.176205I
b = 0.538913 + 0.602866I
5.41831 + 0.47417I 4.00028 1.68304I
u = 0.401394 0.744321I
a = 0.644399 + 1.244546I
b = 1.07882 1.26582I
0.01721 3.43805I 0.24678 3.46839I
u = 0.401394 + 0.744321I
a = 0.644399 1.244546I
b = 1.07882 + 1.26582I
0.01721 + 3.43805I 0.24678 + 3.46839I
u = 0.403574 1.119675I
a = 0.162736 1.138557I
b = 1.88880 + 1.44116I
11.48108 3.00110I 6.29523 + 3.94269I
u = 0.403574 + 1.119675I
a = 0.162736 + 1.138557I
b = 1.88880 1.44116I
11.48108 + 3.00110I 6.29523 3.94269I
6
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.417197 0.960057I
a = 1.176462 0.073344I
b = 0.092506 + 1.338630I
0.683814 0.021668I 7.29120 + 0.84008I
u = 0.417197 + 0.960057I
a = 1.176462 + 0.073344I
b = 0.092506 1.338630I
0.683814 + 0.021668I 7.29120 0.84008I
u = 0.455012 1.123044I
a = 0.761983 0.978268I
b = 0.412889 + 0.224443I
11.13053 4.68214I 6.02220 + 4.41776I
u = 0.455012 + 1.123044I
a = 0.761983 + 0.978268I
b = 0.412889 0.224443I
11.13053 + 4.68214I 6.02220 4.41776I
u = 0.537011 1.034034I
a = 0.960557 + 0.345345I
b = 0.36110 1.43359I
0.42286 5.79491I 0.74565 + 8.60428I
u = 0.537011 + 1.034034I
a = 0.960557 0.345345I
b = 0.36110 + 1.43359I
0.42286 + 5.79491I 0.74565 8.60428I
u = 0.539949 1.088433I
a = 0.005620 + 1.208237I
b = 1.78311 1.56416I
3.79681 6.60858I 1.76186 + 5.54202I
u = 0.539949 + 1.088433I
a = 0.005620 1.208237I
b = 1.78311 + 1.56416I
3.79681 + 6.60858I 1.76186 5.54202I
u = 0.587688 0.502670I
a = 0.504219 1.121993I
b = 0.533679 + 1.214779I
1.99050 + 1.28842I 6.23643 3.32220I
u = 0.587688 + 0.502670I
a = 0.504219 + 1.121993I
b = 0.533679 1.214779I
1.99050 1.28842I 6.23643 + 3.32220I
7
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.595202 1.124180I
a = 0.797953 0.339012I
b = 0.47005 + 1.39138I
6.06522 10.21103I 0.83627 + 6.94060I
u = 0.595202 + 1.124180I
a = 0.797953 + 0.339012I
b = 0.47005 1.39138I
6.06522 + 10.21103I 0.83627 6.94060I
u = 0.613804 1.137651I
a = 0.085943 1.126616I
b = 1.77542 + 1.65606I
3.35255 12.00268I 0.12656 + 9.09333I
u = 0.613804 + 1.137651I
a = 0.085943 + 1.126616I
b = 1.77542 1.65606I
3.35255 + 12.00268I 0.12656 9.09333I
u = 0.625470 0.073721I
a = 1.87319 0.82388I
b = 0.799498 + 1.060929I
8.25951 + 0.62444I 2.71689 0.04866I
u = 0.625470 + 0.073721I
a = 1.87319 + 0.82388I
b = 0.799498 1.060929I
8.25951 0.62444I 2.71689 + 0.04866I
u = 0.626248 0.346796I
a = 1.75455 + 0.14423I
b = 0.523218 + 0.944583I
1.69354 + 2.00274I 1.39705 1.11024I
u = 0.626248 + 0.346796I
a = 1.75455 0.14423I
b = 0.523218 0.944583I
1.69354 2.00274I 1.39705 + 1.11024I
u = 0.644585 1.195853I
a = 0.096287 + 1.058208I
b = 1.80039 1.72432I
9.9328 16.0998I 3.14964 + 8.72516I
u = 0.644585 + 1.195853I
a = 0.096287 1.058208I
b = 1.80039 + 1.72432I
9.9328 + 16.0998I 3.14964 8.72516I
8
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.807014 0.382013I
a = 0.316807 + 1.001679I
b = 0.322907 1.334108I
3.85129 + 4.96920I 2.15468 2.88447I
u = 0.807014 + 0.382013I
a = 0.316807 1.001679I
b = 0.322907 + 1.334108I
3.85129 4.96920I 2.15468 + 2.88447I
u = 0.854961 0.385031I
a = 1.43067 0.02071I
b = 0.433803 1.037999I
1.08969 + 6.56887I 2.69460 5.73061I
u = 0.854961 + 0.385031I
a = 1.43067 + 0.02071I
b = 0.433803 + 1.037999I
1.08969 6.56887I 2.69460 + 5.73061I
u = 0.982154 0.351456I
a = 1.305208 0.065083I
b = 0.391840 + 1.117065I
7.34078 + 10.21729I 1.02164 5.54036I
u = 0.982154 + 0.351456I
a = 1.305208 + 0.065083I
b = 0.391840 1.117065I
7.34078 10.21729I 1.02164 + 5.54036I
9
II. I
u
2
= hu
2
+ u + 1, a 1, 3b 4u 2i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u + 1
a
3
=
u
u + 1
a
9
=
1
4
3
u +
2
3
a
6
=
u
5
3
u +
4
3
a
10
=
u + 2
u 1
a
4
=
u
u + 1
a
7
=
2
3
u
1
3
u + 1
a
8
=
2
3
u +
4
3
u
a
11
=
1
3
u +
4
3
u
a
11
=
1
3
u +
4
3
u
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
10
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 1.00000
b = 1.15470I
0 5.33333
u = 0.500000 + 0.866025I
a = 1.00000
b = 1.15470I
0 5.33333
11
III. I
u
3
= hu
2
+ u + 1, a u, 3b + 4u + 5i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u + 1
a
3
=
u
u + 1
a
9
=
u
4
3
u
5
3
a
6
=
1
2
3
u
1
3
a
10
=
2u + 1
u 2
a
4
=
u
u + 1
a
7
=
1
3
u
5
3
u
a
8
=
1
3
u
1
3
u 1
a
11
=
4
3
u +
2
3
u 1
a
11
=
4
3
u +
2
3
u 1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
12
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 0.500000 0.866025I
b = 1.00000 + 1.15470I
4.05977I 0.33333 9.23760I
u = 0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 1.00000 1.15470I
4.05977I 0.33333 + 9.23760I
13
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)
2
(u
66
+ 3u
65
+ ··· + 41u + 9)
c
2
(u
2
+ u + 1)
2
(u
66
+ 31u
65
+ ··· + 677u + 81)
c
3
u
4
(u
66
+ 3u
65
+ ··· 720u + 432)
c
4
(u
2
u + 1)
2
(u
66
+ 3u
65
+ ··· + 41u + 9)
c
5
(u
2
u + 1)
2
(u
66
+ 3u
65
+ ··· + 3u + 1)
c
6
(3u
2
+ 1)(3u
2
+ 3u + 1)(9u
66
+ 6u
65
+ ··· + 14606u + 2729)
c
7
(u
2
+ u + 1)
2
(u
66
+ 3u
65
+ ··· + 3u + 1)
c
8
(3u
2
3u + 1)(3u
2
+ 3u + 1)(9u
66
+ 39u
65
+ ··· 10089u + 1177)
c
9
(u
2
+ u + 1)
2
(u
66
+ 3u
65
+ ··· + 3u + 1)
c
10
(u
2
u + 1)
2
(u
66
+ 3u
65
+ ··· + 3u + 1)
c
11
(u
2
u + 1)
2
(u
66
+ 3u
65
+ ··· + 3u + 1)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
2
+ y + 1)
2
(y
66
+ 31y
65
+ ··· + 677y + 81)
c
2
(y
2
+ y + 1)
2
(y
66
+ 11y
65
+ ··· 42151y + 6561)
c
3
y
4
(y
66
+ 25y
65
+ ··· + 1835136y + 186624)
c
5
(y
2
+ y + 1)
2
(y
66
37y
65
+ ··· 7y + 1)
c
6
(3y + 1)
2
(9y
2
3y + 1)
(81y
66
1368y
65
+ ··· + 220603054y + 7447441)
c
7
, c
11
(y
2
+ y + 1)
2
(y
66
+ 63y
65
+ ··· 7y + 1)
c
8
(9y
2
3y + 1)
2
(81y
66
+ 2655y
65
+ ··· 3529607y + 1385329)
c
9
(y
2
+ y + 1)
2
(y
66
37y
65
+ ··· 7y + 1)
c
10
(y
2
+ y + 1)
2
(y
66
+ 63y
65
+ ··· 7y + 1)
15