10
57
(K10a
6
)
1
Arc Sequences
3 9 1 7 4 10 5 2 6 8
Solving Sequence
2,8
9 3
1,5
7 4 6 10
c
8
c
2
c
1
c
7
c
4
c
5
c
10
c
3
, c
6
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
3
u
2
+ 2u 1, u
2
+ a u + 2, u
2
+ b + u 1i
I
u
2
= hu
42
4u
41
+ ··· 27870u 3617,
8.16418 × 10
180
u
41
3.75194 × 10
181
u
40
+ ··· + 7.71641 × 10
185
b 8.55025 × 10
185
,
8.88604 × 10
185
u
41
3.70890 × 10
186
u
40
+ ··· + 2.79102 × 10
189
a 2.06417 × 10
190
i
There are 2 irreducible components with 45 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
3
u
2
+ 2u 1, u
2
+ a u + 2, u
2
+ b + u 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
u
2
+ u 2
u
2
u + 1
a
9
=
1
u
2
u + 1
a
3
=
u
2
u + 2
u
a
1
=
0
u
2
a
5
=
0
u
a
7
=
u
2
+ u 2
u
2
+ 1
a
4
=
u
2
u + 2
u
2
+ u 1
a
6
=
u
2
+ u 2
u
2
+ 1
a
10
=
1
u
2
u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
3u + 3
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.215080 1.307141I
a = 0.122561 0.744862I
b = 0.877439 + 0.744862I
4.66906 2.82812I 0.69240 + 3.35914I
u = 0.215080 + 1.307141I
a = 0.122561 + 0.744862I
b = 0.877439 0.744862I
4.66906 + 2.82812I 0.69240 3.35914I
u = 0.569840
a = 1.75488
b = 0.754878
0.531480 1.61520
3
II. I
u
2
= hu
42
4u
41
+ · · · 27870u 3617, 8.16 × 10
180
u
41
3.75 ×
10
181
u
40
+ · · · + 7.72 × 10
185
b 8.55 × 10
185
, 8.89 × 10
185
u
41
3.71 ×
10
186
u
40
+ · · · + 2.79 × 10
189
a 2.06 × 10
190
i
(i) Arc colorings
a
2
=
1
0
a
8
=
0.000318379u
41
+ 0.00132887u
40
+ ··· + 8.00693u + 7.39573
0.0000105803u
41
+ 0.0000486228u
40
+ ··· + 2.41991u + 1.10806
a
9
=
0.000328959u
41
+ 0.00137749u
40
+ ··· + 10.4268u + 8.50379
0.0000105803u
41
+ 0.0000486228u
40
+ ··· + 2.41991u + 1.10806
a
3
=
0.000172078u
41
0.000720713u
40
+ ··· 14.6535u 0.0141903
6.40387 × 10
7
u
41
7.67170 × 10
6
u
40
+ ··· 1.42594u 0.282905
a
1
=
0.0000919194u
41
0.000411850u
40
+ ··· 4.27659u 2.53900
3.55866 × 10
6
u
41
+ 0.0000148673u
40
+ ··· 2.44205u 0.583284
a
5
=
0
u
a
7
=
0.000318379u
41
+ 0.00132887u
40
+ ··· + 8.00693u + 7.39573
0.0000183744u
41
+ 0.0000741340u
40
+ ··· + 2.02882u + 0.907853
a
4
=
0.000147967u
41
0.000634862u
40
+ ··· 19.9277u 2.66820
0.0000150554u
41
0.0000622188u
40
+ ··· 2.41937u 0.469219
a
6
=
0.000158493u
41
0.000669516u
40
+ ··· + 2.09100u 5.33279
3.87193 × 10
6
u
41
0.0000150203u
40
+ ··· + 0.925577u 0.454487
a
10
=
0.000141310u
41
+ 0.000587034u
40
+ ··· + 9.77229u + 2.46660
0.0000150554u
41
+ 0.0000622188u
40
+ ··· + 2.41937u + 0.469219
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6.51148 ×10
6
u
41
0.0000520169u
40
+ ···0.984999u + 5.49711
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.54139 2.28043I
a = 0.321367 + 0.153645I
b = 0.991138 0.067760I
1.67988 2.03798I 8.18964 + 3.67578I
u = 1.54139 + 2.28043I
a = 0.321367 0.153645I
b = 0.991138 + 0.067760I
1.67988 + 2.03798I 8.18964 3.67578I
u = 1.173455 0.376345I
a = 0.676218 + 0.632570I
b = 0.703889 + 0.756112I
3.91253 + 1.78828I 0.036224 1.373729I
u = 1.173455 + 0.376345I
a = 0.676218 0.632570I
b = 0.703889 0.756112I
3.91253 1.78828I 0.036224 + 1.373729I
u = 1.08775 2.33696I
a = 0.409600 0.273004I
b = 1.083348 + 0.141922I
4.86295 7.53350I 9.04295 + 6.51119I
u = 1.08775 + 2.33696I
a = 0.409600 + 0.273004I
b = 1.083348 0.141922I
4.86295 + 7.53350I 9.04295 6.51119I
u = 1.044483 0.234103I
a = 1.167396 + 0.320333I
b = 0.670918 + 0.832205I
1.77790 7.76497I 1.88925 + 4.74518I
u = 1.044483 + 0.234103I
a = 1.167396 0.320333I
b = 0.670918 0.832205I
1.77790 + 7.76497I 1.88925 4.74518I
u = 1.00012 1.83275I
a = 0.472585 0.020384I
b = 1.074440 0.080759I
6.58974 + 1.93798I 11.95326 1.38361I
u = 1.00012 + 1.83275I
a = 0.472585 + 0.020384I
b = 1.074440 + 0.080759I
6.58974 1.93798I 11.95326 + 1.38361I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.389947
a = 0.453072
b = 0.647067
0.883120 11.7260
u = 0.16996 1.42099I
a = 0.014812 + 0.901777I
b = 0.962070 0.695356I
3.92956 4.75718I 2.72048 + 5.86296I
u = 0.16996 + 1.42099I
a = 0.014812 0.901777I
b = 0.962070 + 0.695356I
3.92956 + 4.75718I 2.72048 5.86296I
u = 0.1241943 0.0494556I
a = 2.05982 6.25586I
b = 0.368496 0.622797I
2.03700 0.16365I 5.74023 + 0.29295I
u = 0.1241943 + 0.0494556I
a = 2.05982 + 6.25586I
b = 0.368496 + 0.622797I
2.03700 + 0.16365I 5.74023 0.29295I
u = 0.009130 1.034780I
a = 0.325349 0.901866I
b = 0.923145 + 0.781924I
4.47229 1.63203I 2.91298 2.62995I
u = 0.009130 + 1.034780I
a = 0.325349 + 0.901866I
b = 0.923145 0.781924I
4.47229 + 1.63203I 2.91298 + 2.62995I
u = 0.197795 0.302435I
a = 2.47511 + 0.44500I
b = 0.645452 + 0.781684I
0.56632 2.39851I 5.00404 + 0.87866I
u = 0.197795 + 0.302435I
a = 2.47511 0.44500I
b = 0.645452 0.781684I
0.56632 + 2.39851I 5.00404 0.87866I
u = 0.314282 0.746075I
a = 1.16727 0.88863I
b = 0.209332 + 0.676070I
0.63189 + 5.08816I 2.51962 5.57765I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.314282 + 0.746075I
a = 1.16727 + 0.88863I
b = 0.209332 0.676070I
0.63189 5.08816I 2.51962 + 5.57765I
u = 0.43454 1.56838I
a = 0.077767 + 0.767805I
b = 0.745202 0.733734I
4.59267 0.70618I 0.622977 0.556758I
u = 0.43454 + 1.56838I
a = 0.077767 0.767805I
b = 0.745202 + 0.733734I
4.59267 + 0.70618I 0.622977 + 0.556758I
u = 0.50922 1.38468I
a = 0.014651 0.707609I
b = 0.836375 + 0.809644I
4.73966 4.32552I 1.66531 + 7.57694I
u = 0.50922 + 1.38468I
a = 0.014651 + 0.707609I
b = 0.836375 0.809644I
4.73966 + 4.32552I 1.66531 7.57694I
u = 0.602587 0.511748I
a = 1.114755 + 0.543683I
b = 0.145920 0.358325I
1.72875 + 0.76607I 3.12845 1.30178I
u = 0.602587 + 0.511748I
a = 1.114755 0.543683I
b = 0.145920 + 0.358325I
1.72875 0.76607I 3.12845 + 1.30178I
u = 0.60381 3.06472I
a = 0.269477 0.325201I
b = 1.028179 + 0.723271I
0.68940 + 13.58856I 3.64913 9.29837I
u = 0.60381 + 3.06472I
a = 0.269477 + 0.325201I
b = 1.028179 0.723271I
0.68940 13.58856I 3.64913 + 9.29837I
u = 0.61307 1.96776I
a = 0.457502 0.202967I
b = 0.999660 + 0.570752I
3.66366 + 4.35155I 8.59858 5.33139I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.61307 + 1.96776I
a = 0.457502 + 0.202967I
b = 0.999660 0.570752I
3.66366 4.35155I 8.59858 + 5.33139I
u = 0.802948
a = 1.53006
b = 0.938084
0.325164 11.1789
u = 0.812549 0.830092I
a = 0.976659 0.183444I
b = 0.988336 0.481239I
2.85726 + 1.06689I 7.69538 0.36183I
u = 0.812549 + 0.830092I
a = 0.976659 + 0.183444I
b = 0.988336 + 0.481239I
2.85726 1.06689I 7.69538 + 0.36183I
u = 0.827693 0.324495I
a = 0.554501 0.355300I
b = 0.794934 0.673703I
2.06220 + 2.20756I 3.08817 4.39193I
u = 0.827693 + 0.324495I
a = 0.554501 + 0.355300I
b = 0.794934 + 0.673703I
2.06220 2.20756I 3.08817 + 4.39193I
u = 0.91194 1.24381I
a = 0.474052 + 0.070417I
b = 0.932953 + 0.658227I
1.62453 2.94974I 4.00088 + 1.92478I
u = 0.91194 + 1.24381I
a = 0.474052 0.070417I
b = 0.932953 0.658227I
1.62453 + 2.94974I 4.00088 1.92478I
u = 0.97503 1.16280I
a = 0.608831 + 0.191393I
b = 1.020287 0.695366I
1.68665 7.98804I 6.75545 + 5.63639I
u = 0.97503 + 1.16280I
a = 0.608831 0.191393I
b = 1.020287 + 0.695366I
1.68665 + 7.98804I 6.75545 5.63639I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.14146 3.00590I
a = 0.268135 + 0.235188I
b = 0.988556 0.699620I
3.05223 + 7.32917I 2.09146 6.67478I
u = 1.14146 + 3.00590I
a = 0.268135 0.235188I
b = 0.988556 + 0.699620I
3.05223 7.32917I 2.09146 + 6.67478I
8
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
3
+ u
2
+ 2u + 1)(u
42
+ 14u
41
+ ··· 2u + 1)
c
2
(u
3
+ u
2
1)(u
42
+ 2u
41
+ ··· 2u 1)
c
3
(u
3
u
2
+ 2u 1)(u
42
+ 14u
41
+ ··· 2u + 1)
c
4
(u 1)
3
(u
42
+ 4u
41
+ ··· 7u 1)
c
5
(u + 1)
3
(u
42
+ 20u
41
+ ··· + 39u + 1)
c
6
u
3
(u
42
+ u
41
+ ··· + 28u + 8)
c
7
(u + 1)
3
(u
42
+ 4u
41
+ ··· 7u 1)
c
8
(u
3
u
2
+ 1)(u
42
+ 2u
41
+ ··· 2u 1)
c
9
u
3
(u
42
+ u
41
+ ··· + 28u + 8)
c
10
(u
3
u
2
+ 2u 1)(u
42
+ 2u
41
+ ··· 168u 49)
9
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
(y
3
+ 3y
2
+ 2y 1)(y
42
+ 30y
41
+ ··· + 2y + 1)
c
2
, c
8
(y
3
y
2
+ 2y 1)(y
42
14y
41
+ ··· + 2y + 1)
c
4
(y 1)
3
(y
42
20y
41
+ ··· 39y + 1)
c
5
(y 1)
3
(y
42
+ 8y
41
+ ··· 999y + 1)
c
6
y
3
(y
42
21y
41
+ ··· 784y + 64)
c
7
(y 1)
3
(y
42
20y
41
+ ··· 39y + 1)
c
9
y
3
(y
42
21y
41
+ ··· 784y + 64)
c
10
(y
3
+ 3y
2
+ 2y 1)(y
42
6y
41
+ ··· 7154y + 2401)
10