10
80
(K10a
8
)
1
Arc Sequences
9 6 10 7 3 8 5 1 2 4
Solving Sequence
2,6 3,10
4 5 9 1 8 7
c
2
c
3
c
5
c
9
c
1
c
8
c
7
c
4
, c
6
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
\
I
v
1
I
u
1
= hu
3
u
2
+ 2u 1, b + 1, 2u
2
+ a u + 3i
I
u
2
= hu
40
2u
39
+ ··· + 4u 4, 1.26420 × 10
36
u
39
+ 2.07440 × 10
36
u
38
+ ··· + 2.35525 × 10
36
b 3.32893 × 10
36
,
2.30624 × 10
36
u
39
4.87025 × 10
36
u
38
+ ··· + 4.71051 × 10
36
a + 2.43924 × 10
37
i
I
v
1
= h−b + v + 2, b
2
b 1, ai
There are 3 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, b + 1, 2u
2
+ a u + 3i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
10
=
2u
2
+ u 3
1
a
4
=
1
u
2
a
5
=
u
u
2
u + 1
a
9
=
2u
2
+ u 4
1
a
1
=
2u
2
+ u 3
1
a
8
=
1
0
a
7
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = u 8
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.215080 1.307141I
a = 0.539798 0.182582I
b = 1.00000
1.37919 + 2.82812I 7.78492 1.30714I
u = 0.215080 + 1.307141I
a = 0.539798 + 0.182582I
b = 1.00000
1.37919 2.82812I 7.78492 + 1.30714I
u = 0.569840
a = 3.07960
b = 1.00000
2.75839 7.43016
3
II. I
u
2
=
hu
40
2u
39
+· · ·+4u4, 1.26×10
36
u
39
+2.07×10
36
u
38
+· · ·+2.36×10
36
b
3.33 × 10
36
, 2.31 × 10
36
u
39
4.87 × 10
36
u
38
+ · · · + 4.71 × 10
36
a + 2.44 × 10
37
i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
10
=
0.489595u
39
+ 1.03391u
38
+ ··· 5.49347u 5.17829
0.536759u
39
0.880756u
38
+ ··· 3.25662u + 1.41341
a
4
=
0.473705u
39
+ 0.973998u
38
+ ··· 0.321153u 2.70973
0.581299u
39
0.726984u
38
+ ··· 4.02214u + 1.17964
a
5
=
u
u
3
+ u
a
9
=
0.0471637u
39
+ 0.153157u
38
+ ··· 8.75009u 3.76489
0.536759u
39
0.880756u
38
+ ··· 3.25662u + 1.41341
a
1
=
0.597452u
39
+ 0.737046u
38
+ ··· 3.78265u 2.75349
0.659942u
39
+ 0.591523u
38
+ ··· + 3.19008u 0.0976319
a
8
=
0.628885u
39
+ 1.24099u
38
+ ··· + 1.69981u 3.78302
0.155180u
39
+ 0.266989u
38
+ ··· + 2.02096u 1.07329
a
7
=
0.961214u
39
+ 1.72033u
38
+ ··· + 3.18201u 4.78644
0.486102u
39
+ 0.886499u
38
+ ··· + 2.91511u 2.81798
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2.96903u
39
+ 5.08961u
38
+ ··· 14.8512u 32.4878
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.105286 0.361243I
a = 0.192027 0.003144I
b = 1.42528 0.13732I
6.42531 + 1.37910I 14.4871 0.1126I
u = 1.105286 + 0.361243I
a = 0.192027 + 0.003144I
b = 1.42528 + 0.13732I
6.42531 1.37910I 14.4871 + 0.1126I
u = 0.712026 0.592514I
a = 1.35270 + 1.43314I
b = 1.310003 + 0.051508I
4.81110 + 1.15004I 14.8249 0.1630I
u = 0.712026 + 0.592514I
a = 1.35270 1.43314I
b = 1.310003 0.051508I
4.81110 1.15004I 14.8249 + 0.1630I
u = 0.682230 1.218514I
a = 0.29633 1.41563I
b = 1.43155 + 0.31022I
3.73330 7.68923I 11.60024 + 4.76581I
u = 0.682230 + 1.218514I
a = 0.29633 + 1.41563I
b = 1.43155 0.31022I
3.73330 + 7.68923I 11.60024 4.76581I
u = 0.661490 0.051964I
a = 0.750383 + 0.021748I
b = 0.074897 0.213008I
0.945608 + 0.085520I 9.49008 + 0.83288I
u = 0.661490 + 0.051964I
a = 0.750383 0.021748I
b = 0.074897 + 0.213008I
0.945608 0.085520I 9.49008 0.83288I
u = 0.620815 1.023939I
a = 0.554031 + 1.141011I
b = 1.45816 0.17967I
3.50534 6.28261I 13.0953 + 5.4809I
u = 0.620815 + 1.023939I
a = 0.554031 1.141011I
b = 1.45816 + 0.17967I
3.50534 + 6.28261I 13.0953 5.4809I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.477122 1.075674I
a = 0.086144 + 1.347713I
b = 0.271495 0.796081I
1.73108 3.69196I 7.37427 + 4.06105I
u = 0.477122 + 1.075674I
a = 0.086144 1.347713I
b = 0.271495 + 0.796081I
1.73108 + 3.69196I 7.37427 4.06105I
u = 0.418716
a = 0.995068
b = 0.409236
0.821503 11.8785
u = 0.361248 1.173912I
a = 0.366981 0.724767I
b = 0.377778 + 0.443628I
2.47440 3.90124I 5.43445 + 4.68146I
u = 0.361248 + 1.173912I
a = 0.366981 + 0.724767I
b = 0.377778 0.443628I
2.47440 + 3.90124I 5.43445 4.68146I
u = 0.14806 1.51121I
a = 0.874630 0.251624I
b = 1.250124 + 0.055687I
0.49614 3.22180I 15.2960 + 4.0561I
u = 0.14806 + 1.51121I
a = 0.874630 + 0.251624I
b = 1.250124 0.055687I
0.49614 + 3.22180I 15.2960 4.0561I
u = 0.143356 0.859264I
a = 0.527613 1.068941I
b = 1.003622 + 0.442463I
0.414732 + 0.767581I 9.73697 1.10255I
u = 0.143356 + 0.859264I
a = 0.527613 + 1.068941I
b = 1.003622 0.442463I
0.414732 0.767581I 9.73697 + 1.10255I
u = 0.072074 1.152144I
a = 0.368515 + 0.957274I
b = 0.160673 0.580088I
3.38024 1.21441I 4.33120 + 2.38202I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.072074 + 1.152144I
a = 0.368515 0.957274I
b = 0.160673 + 0.580088I
3.38024 + 1.21441I 4.33120 2.38202I
u = 0.380635
a = 8.84579
b = 0.932243
2.98695 59.3923
u = 0.416919 0.891552I
a = 0.35784 1.46982I
b = 1.306179 + 0.207684I
1.17381 + 1.71654I 9.22754 1.14237I
u = 0.416919 + 0.891552I
a = 0.35784 + 1.46982I
b = 1.306179 0.207684I
1.17381 1.71654I 9.22754 + 1.14237I
u = 0.470686 0.578246I
a = 0.191234 0.000903I
b = 1.63869 + 0.09271I
10.88698 + 0.63545I 16.1019 7.3224I
u = 0.470686 + 0.578246I
a = 0.191234 + 0.000903I
b = 1.63869 0.09271I
10.88698 0.63545I 16.1019 + 7.3224I
u = 0.473036 0.681631I
a = 0.26300 2.22678I
b = 0.477456 + 0.529520I
3.04518 + 0.83928I 13.6876 5.4055I
u = 0.473036 + 0.681631I
a = 0.26300 + 2.22678I
b = 0.477456 0.529520I
3.04518 0.83928I 13.6876 + 5.4055I
u = 0.522827 1.072493I
a = 0.85009 + 1.78859I
b = 1.46212 0.19943I
9.24274 + 3.54815I 15.7354 3.2017I
u = 0.522827 + 1.072493I
a = 0.85009 1.78859I
b = 1.46212 + 0.19943I
9.24274 3.54815I 15.7354 + 3.2017I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.523308 0.966470I
a = 0.455655 + 0.673858I
b = 0.858789 0.701991I
2.11016 + 3.32020I 13.06049 3.76837I
u = 0.523308 + 0.966470I
a = 0.455655 0.673858I
b = 0.858789 + 0.701991I
2.11016 3.32020I 13.06049 + 3.76837I
u = 0.657661 1.113155I
a = 0.283876 1.253827I
b = 0.374538 + 0.908825I
0.64027 + 8.82354I 11.18744 7.65851I
u = 0.657661 + 1.113155I
a = 0.283876 + 1.253827I
b = 0.374538 0.908825I
0.64027 8.82354I 11.18744 + 7.65851I
u = 0.79345 1.18299I
a = 0.03462 + 1.49209I
b = 1.48955 0.35340I
6.6238 + 13.3940I 14.1442 7.8976I
u = 0.79345 + 1.18299I
a = 0.03462 1.49209I
b = 1.48955 + 0.35340I
6.6238 13.3940I 14.1442 + 7.8976I
u = 0.876256 0.500142I
a = 0.579698 + 0.340133I
b = 0.388591 0.672832I
2.52882 3.14028I 13.2871 + 4.9220I
u = 0.876256 + 0.500142I
a = 0.579698 0.340133I
b = 0.388591 + 0.672832I
2.52882 + 3.14028I 13.2871 4.9220I
u = 1.124457 0.612008I
a = 0.196520 + 0.001971I
b = 1.45872 + 0.24956I
8.48566 6.50843I 15.7623 + 4.5910I
u = 1.124457 + 0.612008I
a = 0.196520 0.001971I
b = 1.45872 0.24956I
8.48566 + 6.50843I 15.7623 4.5910I
8
III. I
v
1
= h−b + v + 2, b
2
b 1, ai
(i) Arc colorings
a
2
=
1
0
a
6
=
b 2
0
a
3
=
1
0
a
10
=
0
b
a
4
=
1
b + 1
a
5
=
b 2
0
a
9
=
b
b
a
1
=
b
b 1
a
8
=
1
b 1
a
7
=
b 3
b 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 11
9
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 2.61803
a = 0
b = 0.618034
2.63189 11.0000
v = 0.381966
a = 0
b = 1.61803
10.5276 11.0000
10
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u + 1)
3
(u
2
u 1)(u
40
+ 5u
39
+ ··· 6u + 1)
c
2
u
2
(u
3
u
2
+ 2u 1)(u
40
+ 2u
39
+ ··· 4u 4)
c
3
u
3
(u
2
u 1)(u
40
+ 2u
39
+ ··· 28u 8)
c
4
(u 1)
2
(u
3
+ u
2
1)(u
40
+ 4u
39
+ ··· 2u + 1)
c
5
u
2
(u
3
+ u
2
+ 2u + 1)(u
40
+ 2u
39
+ ··· 4u 4)
c
6
(u 1)
2
(u
3
u
2
+ 2u 1)(u
40
+ 20u
39
+ ··· + 38u + 1)
c
7
(u + 1)
2
(u
3
u
2
+ 1)(u
40
+ 4u
39
+ ··· 2u + 1)
c
8
, c
9
(u 1)
3
(u
2
+ u 1)(u
40
+ 5u
39
+ ··· 6u + 1)
c
10
u
3
(u
2
+ u 1)(u
40
+ 2u
39
+ ··· 28u 8)
11
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
8
, c
9
(y 1)
3
(y
2
3y + 1)(y
40
39y
39
+ ··· + 24y + 1)
c
2
, c
5
y
2
(y
3
+ 3y
2
+ 2y 1)(y
40
+ 18y
39
+ ··· 104y + 16)
c
3
, c
10
y
3
(y
2
3y + 1)(y
40
24y
39
+ ··· 1360y + 64)
c
4
, c
7
(y 1)
2
(y
3
y
2
+ 2y 1)(y
40
20y
39
+ ··· 38y + 1)
c
6
(y 1)
2
(y
3
+ 3y
2
+ 2y 1)(y
40
+ 4y
39
+ ··· 918y + 1)
12