11a
68
(K11a
68
)
1
Arc Sequences
5 1 11 2 9 10 4 3 6 7 8
Solving Sequence
1,5
2
3,9
6 10 7 4 8 11
c
1
c
2
c
5
c
9
c
6
c
4
c
8
c
11
c
3
, c
7
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
2
+ u + 1, b + u + 1, a u 1i
I
u
2
= hu
53
+ 2u
52
+ ··· 11u 1, 6.97735 × 10
34
u
52
+ 1.95213 × 10
35
u
51
+ ··· + 9.74701 × 10
34
b 4.22732 × 10
35
,
1.87093 × 10
35
u
52
3.02649 × 10
35
u
51
+ ··· + 9.74701 × 10
34
a 1.29584 × 10
34
i
There are 2 irreducible components with 55 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
2
+ u + 1, b + u + 1, a u 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u + 1
a
3
=
u
u + 1
a
9
=
u + 1
u 1
a
6
=
u + 1
1
a
10
=
0
u
a
7
=
u + 1
u 1
a
4
=
u
u + 1
a
8
=
u + 2
1
a
11
=
u 1
1
a
11
=
u 1
1
(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 = 0.500000 0.866025I
a = 0.500000 0.866025I
b = 0.500000 + 0.866025I
1.64493 + 2.02988I 3.00000 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 0.500000 0.866025I
1.64493 2.02988I 3.00000 + 3.46410I
3
II. I
u
2
=
hu
53
+2u
52
+· · ·11u 1, 6.98×10
34
u
52
+1.95×10
35
u
51
+· · ·+9.75×10
34
b
4.23× 10
35
, 1.87×10
35
u
52
3.03 ×10
35
u
51
+· · · +9.75 ×10
34
a 1.30× 10
34
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
=
1.91949u
52
+ 3.10504u
51
+ ··· 21.1378u + 0.132948
0.715845u
52
2.00280u
51
+ ··· + 36.8512u + 4.33704
a
6
=
3.40198u
52
+ 5.73946u
51
+ ··· 65.0708u 6.80774
0.805349u
52
2.72120u
51
+ ··· + 40.3132u + 4.81406
a
10
=
1.76625u
52
3.10503u
51
+ ··· + 43.5616u + 8.14328
0.476173u
52
+ 1.11611u
51
+ ··· 4.05426u 0.567895
a
7
=
3.68215u
52
+ 6.25703u
51
+ ··· 57.2633u 4.21960
1.26052u
52
3.21687u
51
+ ··· + 48.5694u + 5.61450
a
4
=
u
u
3
+ u
a
8
=
3.44825u
52
+ 5.74613u
51
+ ··· 52.4307u 3.66150
1.45100u
52
3.81914u
51
+ ··· + 44.4446u + 5.09949
a
11
=
7.05445u
52
11.8825u
51
+ ··· + 140.972u + 19.1530
2.22640u
52
+ 6.15113u
51
+ ··· 58.4460u 7.05445
a
11
=
7.05445u
52
11.8825u
51
+ ··· + 140.972u + 19.1530
2.22640u
52
+ 6.15113u
51
+ ··· 58.4460u 7.05445
(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 = 0.956160 0.388075I
a = 1.52670 0.46709I
b = 0.594603 0.819276I
8.29591 8.86132I 4.03408 + 4.58173I
u = 0.956160 + 0.388075I
a = 1.52670 + 0.46709I
b = 0.594603 + 0.819276I
8.29591 + 8.86132I 4.03408 4.58173I
u = 0.848973 0.421707I
a = 0.987184 0.112074I
b = 0.133162 + 1.035427I
0.02021 6.36572I 1.71267 + 6.01738I
u = 0.848973 + 0.421707I
a = 0.987184 + 0.112074I
b = 0.133162 1.035427I
0.02021 + 6.36572I 1.71267 6.01738I
u = 0.691596 0.477793I
a = 0.346213 + 0.744621I
b = 0.425112 1.132977I
2.06084 2.35698I 2.09384 + 2.25421I
u = 0.691596 + 0.477793I
a = 0.346213 0.744621I
b = 0.425112 + 1.132977I
2.06084 + 2.35698I 2.09384 2.25421I
u = 0.650090 1.173964I
a = 0.491602 1.145132I
b = 2.15908 + 1.58846I
10.7030 + 14.7003I 6.36734 7.95879I
u = 0.650090 + 1.173964I
a = 0.491602 + 1.145132I
b = 2.15908 1.58846I
10.7030 14.7003I 6.36734 + 7.95879I
u = 0.626693 1.121904I
a = 0.062100 + 0.799942I
b = 1.43055 1.44481I
2.08339 + 11.83766I 4.42819 9.48553I
u = 0.626693 + 1.121904I
a = 0.062100 0.799942I
b = 1.43055 + 1.44481I
2.08339 11.83766I 4.42819 + 9.48553I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.623522 0.278275I
a = 2.23107 + 0.80236I
b = 0.951810 + 0.721250I
9.21922 2.19279I 5.41459 + 1.06225I
u = 0.623522 + 0.278275I
a = 2.23107 0.80236I
b = 0.951810 0.721250I
9.21922 + 2.19279I 5.41459 1.06225I
u = 0.583629 1.059955I
a = 0.604017 0.277132I
b = 0.618931 + 1.164525I
0.34476 + 7.28943I 1.89770 7.29263I
u = 0.583629 + 1.059955I
a = 0.604017 + 0.277132I
b = 0.618931 1.164525I
0.34476 7.28943I 1.89770 + 7.29263I
u = 0.514455 1.092182I
a = 0.435572 + 1.300023I
b = 2.17222 1.29061I
11.49847 + 6.63397I 8.72640 5.80042I
u = 0.514455 + 1.092182I
a = 0.435572 1.300023I
b = 2.17222 + 1.29061I
11.49847 6.63397I 8.72640 + 5.80042I
u = 0.483358 1.013927I
a = 0.822443 0.636025I
b = 0.206175 0.361391I
2.96394 + 2.74414I 9.09376 5.57408I
u = 0.483358 + 1.013927I
a = 0.822443 + 0.636025I
b = 0.206175 + 0.361391I
2.96394 2.74414I 9.09376 + 5.57408I
u = 0.413624 1.013734I
a = 0.292864 0.843265I
b = 1.36946 + 1.22188I
3.39428 + 3.43400I 10.20122 5.71858I
u = 0.413624 + 1.013734I
a = 0.292864 + 0.843265I
b = 1.36946 1.22188I
3.39428 3.43400I 10.20122 + 5.71858I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.398755 0.662277I
a = 0.45556 1.45281I
b = 1.21265 + 0.83430I
1.68806 + 1.03417I 4.06111 + 3.56756I
u = 0.398755 + 0.662277I
a = 0.45556 + 1.45281I
b = 1.21265 0.83430I
1.68806 1.03417I 4.06111 3.56756I
u = 0.316553 1.065113I
a = 0.51542 + 1.45751I
b = 0.516053 1.072164I
12.82682 + 0.50129I 10.28284 1.75936I
u = 0.316553 + 1.065113I
a = 0.51542 1.45751I
b = 0.516053 + 1.072164I
12.82682 0.50129I 10.28284 + 1.75936I
u = 0.229892 0.048634I
a = 3.09820 + 0.49951I
b = 0.500793 0.556020I
1.33343 0.48154I 6.01943 + 1.63910I
u = 0.229892 + 0.048634I
a = 3.09820 0.49951I
b = 0.500793 + 0.556020I
1.33343 + 0.48154I 6.01943 1.63910I
u = 0.209100 0.945558I
a = 0.689574 0.132763I
b = 0.388703 0.762249I
2.15945 0.98457I 6.47095 + 3.55570I
u = 0.209100 + 0.945558I
a = 0.689574 + 0.132763I
b = 0.388703 + 0.762249I
2.15945 + 0.98457I 6.47095 3.55570I
u = 0.098247 1.348148I
a = 0.154725 1.222239I
b = 0.175223 + 1.369220I
14.5372 5.4639I 9.66226 + 4.35002I
u = 0.098247 + 1.348148I
a = 0.154725 + 1.222239I
b = 0.175223 1.369220I
14.5372 + 5.4639I 9.66226 4.35002I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.071350 1.206913I
a = 0.305057 + 0.719780I
b = 0.101426 0.458787I
5.68960 3.88038I 9.36604 + 5.67169I
u = 0.071350 + 1.206913I
a = 0.305057 0.719780I
b = 0.101426 + 0.458787I
5.68960 + 3.88038I 9.36604 5.67169I
u = 0.437475 0.833838I
a = 0.369541 0.400784I
b = 1.29501 1.44927I
0.90053 1.58968I 1.24595 + 11.67218I
u = 0.437475 + 0.833838I
a = 0.369541 + 0.400784I
b = 1.29501 + 1.44927I
0.90053 + 1.58968I 1.24595 11.67218I
u = 0.482724 0.895174I
a = 0.231297 0.607827I
b = 1.68518 + 3.18410I
1.15754 2.23299I 20.7265 16.6825I
u = 0.482724 + 0.895174I
a = 0.231297 + 0.607827I
b = 1.68518 3.18410I
1.15754 + 2.23299I 20.7265 + 16.6825I
u = 0.490506 0.975564I
a = 0.64385 + 1.33683I
b = 2.80301 3.31368I
9.14547 2.68229I 3.32958 9.15387I
u = 0.490506 + 0.975564I
a = 0.64385 1.33683I
b = 2.80301 + 3.31368I
9.14547 + 2.68229I 3.32958 + 9.15387I
u = 0.490850 0.618356I
a = 1.24224 + 1.57044I
b = 0.11410 + 1.79539I
8.08310 1.37745I 11.53885 + 3.59145I
u = 0.490850 + 0.618356I
a = 1.24224 1.57044I
b = 0.11410 1.79539I
8.08310 + 1.37745I 11.53885 3.59145I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.598899 1.150435I
a = 0.052794 + 0.538827I
b = 0.802342 1.041558I
0.87805 4.69027I 3.22278 + 10.95263I
u = 0.598899 + 1.150435I
a = 0.052794 0.538827I
b = 0.802342 + 1.041558I
0.87805 + 4.69027I 3.22278 10.95263I
u = 0.60448 1.28732I
a = 0.443233 1.040935I
b = 1.26698 + 1.56718I
8.69900 5.79684I 4.92869 + 8.02474I
u = 0.60448 + 1.28732I
a = 0.443233 + 1.040935I
b = 1.26698 1.56718I
8.69900 + 5.79684I 4.92869 8.02474I
u = 0.612381 0.946401I
a = 0.253075 0.045452I
b = 0.487867 + 0.154283I
0.54222 2.63785I 2.19810 + 0.73658I
u = 0.612381 + 0.946401I
a = 0.253075 + 0.045452I
b = 0.487867 0.154283I
0.54222 + 2.63785I 2.19810 0.73658I
u = 0.694807 0.319248I
a = 0.569358 0.175582I
b = 0.395174 + 0.336327I
1.60066 0.35812I 5.75025 1.86999I
u = 0.694807 + 0.319248I
a = 0.569358 + 0.175582I
b = 0.395174 0.336327I
1.60066 + 0.35812I 5.75025 + 1.86999I
u = 0.801200 0.670609I
a = 0.174199 + 0.487361I
b = 0.525424 0.430680I
1.29588 2.67274I 3.47265 + 8.90150I
u = 0.801200 + 0.670609I
a = 0.174199 0.487361I
b = 0.525424 + 0.430680I
1.29588 + 2.67274I 3.47265 8.90150I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.981473 0.829936I
a = 0.919093 0.809551I
b = 1.122084 + 0.216377I
5.77422 3.45795I 0.40888 + 5.77843I
u = 0.981473 + 0.829936I
a = 0.919093 + 0.809551I
b = 1.122084 0.216377I
5.77422 + 3.45795I 0.40888 5.77843I
u = 1.04242
a = 1.49658
b = 0.305785
4.89339 2.70842
10
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)(u
53
+ 2u
52
+ ··· 11u 1)
c
2
(u
2
+ u + 1)(u
53
+ 24u
52
+ ··· + 25u 1)
c
3
u
2
(u
53
+ 5u
52
+ ··· + 4u 4)
c
4
(u
2
u + 1)(u
53
+ 2u
52
+ ··· 11u 1)
c
5
, c
6
(u 1)
2
(u
53
+ 3u
52
+ ··· 2u + 1)
c
7
(u
2
u + 1)(u
53
+ 2u
52
+ ··· 127u 29)
c
8
(u
2
u + 1)(u
53
20u
51
+ ··· + 127u + 59)
c
9
, c
10
(u + 1)
2
(u
53
+ 3u
52
+ ··· 2u + 1)
c
11
(u + 1)
2
(u
53
+ 3u
52
+ ··· 2u 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
2
+ y + 1)(y
53
+ 24y
52
+ ··· + 25y 1)
c
2
(y
2
+ y + 1)(y
53
+ 12y
52
+ ··· + 425y 1)
c
3
y
2
(y
53
+ 15y
52
+ ··· + 8y 16)
c
5
, c
6
, c
9
c
10
(y 1)
2
(y
53
63y
52
+ ··· + 8y 1)
c
7
(y
2
+ y + 1)(y
53
60y
52
+ ··· + 14273y 841)
c
8
(y
2
+ y + 1)(y
53
40y
52
+ ··· 44759y 3481)
c
11
(y 1)
2
(y
53
7y
52
+ ··· + 8y 1)
12