10
93
(K10a
101
)
1
Arc Sequences
9 6 10 8 3 2 4 1 7 5
Solving Sequence
7,9 2,10
1 6 3 4 5 8
c
9
c
1
c
6
c
2
c
3
c
5
c
8
c
4
, c
7
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
2
u + 1, b 1, 3a + 2u 1i
I
u
2
= hu
35
2u
34
+ ··· 2u + 1, 1.90874 × 10
16
u
34
3.64908 × 10
16
u
33
+ ··· + 7.16599 × 10
16
b 1.17575 × 10
17
,
3.71507 × 10
17
u
34
5.94517 × 10
17
u
33
+ ··· + 2.14980 × 10
17
a 4.36258 × 10
17
i
There are 2 irreducible components with 37 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 1, 3a + 2u 1i
(i) Arc colorings
a
7
=
0
u
a
9
=
2
3
u +
1
3
1
a
2
=
1
0
a
10
=
2
3
u +
1
3
1
3
u +
2
3
a
1
=
2
3
u +
4
3
1
a
6
=
u
u
a
3
=
u
u 1
a
4
=
u
u 1
a
5
=
2u 1
u 1
a
8
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes =
4
3
u + 5
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 0.577350I
b = 1.00000
1.64493 + 2.02988I 5.66667 1.15470I
u = 0.500000 + 0.866025I
a = 0.577350I
b = 1.00000
1.64493 2.02988I 5.66667 + 1.15470I
3
II. I
u
2
=
hu
35
2u
34
+ · · · 2u + 1, 1.91 × 10
16
u
34
3.65 × 10
16
u
33
+ · · · + 7.17 × 10
16
b
1.18 × 10
17
, 3.72 × 10
17
u
34
5.95 × 10
17
u
33
+ · · · + 2.15 × 10
17
a 4.36 × 10
17
i
(i) Arc colorings
a
7
=
0
u
a
9
=
1.72810u
34
+ 2.76545u
33
+ ··· 13.4887u + 2.02930
0.266360u
34
+ 0.509222u
33
+ ··· 0.453751u + 1.64074
a
2
=
1
0
a
10
=
1.72810u
34
+ 2.76545u
33
+ ··· 13.4887u + 2.02930
0.162937u
34
+ 0.546731u
33
+ ··· 0.800358u + 0.949992
a
1
=
1.67732u
34
+ 2.78475u
33
+ ··· 13.2887u + 3.25498
0.188347u
34
+ 0.378143u
33
+ ··· 0.398180u + 1.39270
a
6
=
u
u
a
3
=
u
2
+ 1
u
2
a
4
=
0.219453u
34
+ 0.0692611u
33
+ ··· 1.00475u 1.94792
0.0524374u
34
+ 0.0392743u
33
+ ··· 0.383921u 0.300462
a
5
=
u
3
+ 2u
u
3
+ u
a
8
=
0.338982u
34
+ 0.333102u
33
+ ··· 0.677979u 1.04295
0.171966u
34
+ 0.303115u
33
+ ··· 0.0571521u + 0.604507
(ii) Obstruction class = 1
(iii) Cusp Shapes =
43902992914495201
128987874023997969
u
34
+
378946062727999073
214979790039996615
u
33
+ ··· +
1287205910561212151
214979790039996615
u +
3437895448731260974
644939370119989845
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.890046 0.661300I
a = 0.525336 + 0.747490I
b = 1.231702 + 0.126872I
2.19099 3.04973I 6.83792 + 5.73006I
u = 0.890046 + 0.661300I
a = 0.525336 0.747490I
b = 1.231702 0.126872I
2.19099 + 3.04973I 6.83792 5.73006I
u = 0.518141 0.274512I
a = 0.193186 0.806817I
b = 0.114792 0.464614I
1.063792 0.837639I 5.45708 + 2.88305I
u = 0.518141 + 0.274512I
a = 0.193186 + 0.806817I
b = 0.114792 + 0.464614I
1.063792 + 0.837639I 5.45708 2.88305I
u = 0.368379
a = 6.68898
b = 1.15662
3.85534 11.8895
u = 0.319146 0.974832I
a = 0.502502 + 0.310871I
b = 0.550694 + 0.058879I
0.84892 2.27938I 3.03865 + 4.27236I
u = 0.319146 + 0.974832I
a = 0.502502 0.310871I
b = 0.550694 0.058879I
0.84892 + 2.27938I 3.03865 4.27236I
u = 0.27157 1.59599I
a = 0.362471 + 0.901697I
b = 1.39753 + 0.33365I
9.65156 7.25912I 5.71385 + 4.77180I
u = 0.27157 + 1.59599I
a = 0.362471 0.901697I
b = 1.39753 0.33365I
9.65156 + 7.25912I 5.71385 4.77180I
u = 0.177701 0.568169I
a = 0.79433 1.23827I
b = 1.40644 0.46218I
5.61974 1.75521I 9.80898 + 3.99717I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.177701 + 0.568169I
a = 0.79433 + 1.23827I
b = 1.40644 + 0.46218I
5.61974 + 1.75521I 9.80898 3.99717I
u = 0.13283 1.44979I
a = 0.008629 0.654957I
b = 0.183478 0.844596I
4.57837 3.04741I 1.15855 + 3.13738I
u = 0.13283 + 1.44979I
a = 0.008629 + 0.654957I
b = 0.183478 + 0.844596I
4.57837 + 3.04741I 1.15855 3.13738I
u = 0.04304 1.53065I
a = 0.253812 0.565522I
b = 1.81945 0.72263I
12.61949 2.50960I 11.52662 + 2.71271I
u = 0.04304 + 1.53065I
a = 0.253812 + 0.565522I
b = 1.81945 + 0.72263I
12.61949 + 2.50960I 11.52662 2.71271I
u = 0.03075 1.47995I
a = 1.003811 + 0.893249I
b = 1.239832 + 0.289302I
7.85287 + 1.23959I 3.27188 0.82535I
u = 0.03075 + 1.47995I
a = 1.003811 0.893249I
b = 1.239832 0.289302I
7.85287 1.23959I 3.27188 + 0.82535I
u = 0.07117 1.42730I
a = 0.85426 + 1.31677I
b = 0.520982 + 0.140994I
7.48516 + 0.26471I 8.26333 + 1.55245I
u = 0.07117 + 1.42730I
a = 0.85426 1.31677I
b = 0.520982 0.140994I
7.48516 0.26471I 8.26333 1.55245I
u = 0.13948 1.49938I
a = 0.270236 + 0.740224I
b = 0.35483 + 1.42531I
8.34974 + 6.42549I 7.15577 5.95651I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.13948 + 1.49938I
a = 0.270236 0.740224I
b = 0.35483 1.42531I
8.34974 6.42549I 7.15577 + 5.95651I
u = 0.171298 0.343458I
a = 0.20317 + 2.57033I
b = 0.987729 + 0.158150I
1.76589 + 0.63046I 5.20787 + 1.46477I
u = 0.171298 + 0.343458I
a = 0.20317 2.57033I
b = 0.987729 0.158150I
1.76589 0.63046I 5.20787 1.46477I
u = 0.26639 1.57422I
a = 0.381354 1.144451I
b = 1.54844 0.51246I
14.3602 + 13.0165I 7.50230 6.40526I
u = 0.26639 + 1.57422I
a = 0.381354 + 1.144451I
b = 1.54844 + 0.51246I
14.3602 13.0165I 7.50230 + 6.40526I
u = 0.31084 1.58997I
a = 0.106390 0.739193I
b = 1.43361 0.04259I
13.76818 + 0.95076I 9.63320 0.41706I
u = 0.31084 + 1.58997I
a = 0.106390 + 0.739193I
b = 1.43361 + 0.04259I
13.76818 0.95076I 9.63320 + 0.41706I
u = 0.379190 0.370732I
a = 2.06545 0.12179I
b = 0.232736 0.423076I
1.88873 1.15770I 2.26234 1.26872I
u = 0.379190 + 0.370732I
a = 2.06545 + 0.12179I
b = 0.232736 + 0.423076I
1.88873 + 1.15770I 2.26234 + 1.26872I
u = 0.485797 0.446415I
a = 0.10410 + 1.60626I
b = 0.226754 + 1.002745I
1.92901 + 4.20671I 2.61467 7.67969I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.485797 + 0.446415I
a = 0.10410 1.60626I
b = 0.226754 1.002745I
1.92901 4.20671I 2.61467 + 7.67969I
u = 0.797949 0.618523I
a = 0.63930 1.30182I
b = 1.41929 0.39124I
7.15893 + 9.08856I 5.28395 6.85993I
u = 0.797949 + 0.618523I
a = 0.63930 + 1.30182I
b = 1.41929 + 0.39124I
7.15893 9.08856I 5.28395 + 6.85993I
u = 0.883803 0.527645I
a = 0.988164 0.381138I
b = 1.354487 + 0.193009I
6.81152 3.53470I 6.64372 + 2.46356I
u = 0.883803 + 0.527645I
a = 0.988164 + 0.381138I
b = 1.354487 0.193009I
6.81152 + 3.53470I 6.64372 2.46356I
8
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
2
(u
35
+ 3u
34
+ ··· + 5u 9)
c
2
(u
2
u + 1)(u
35
+ 2u
34
+ ··· 2u 1)
c
3
u
2
(u
35
+ 3u
34
+ ··· 60u + 36)
c
4
(u
2
+ u + 1)(u
35
+ 2u
34
+ ··· + 2u + 1)
c
5
, c
6
(u
2
+ u + 1)(u
35
+ 2u
34
+ ··· 2u 1)
c
7
(u
2
u + 1)(u
35
+ 2u
34
+ ··· + 2u + 1)
c
8
(u + 1)
2
(u
35
+ 3u
34
+ ··· + 5u 9)
c
9
(3u
2
3u + 1)(3u
35
+ 2u
34
+ ··· + 2024u + 529)
c
10
(3u
2
+ 1)(3u
35
+ 13u
34
+ ··· 793u + 173)
9
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
8
(y 1)
2
(y
35
31y
34
+ ··· + 439y 81)
c
2
, c
5
, c
6
(y
2
+ y + 1)(y
35
+ 36y
34
+ ··· 8y 1)
c
3
y
2
(y
35
+ 15y
34
+ ··· 8136y 1296)
c
4
, c
7
(y
2
+ y + 1)(y
35
24y
34
+ ··· 8y 1)
c
9
(9y
2
3y + 1)(9y
35
280y
34
+ ··· + 2073680y 279841)
c
10
(3y + 1)
2
(9y
35
+ 137y
34
+ ··· + 24387y 29929)
10