10
82
(K10a
83
)
1
Arc Sequences
7 8 10 1 2 9 5 6 3 4
Solving Sequence
1,7 2,5
8 4 10 3 9 6
c
1
c
7
c
4
c
10
c
3
c
9
c
6
c
2
, c
5
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu 1, a, b + 1i
I
u
2
= hu
32
2u
31
+ ··· + u + 1,
1796669401u
31
+ 1802524431u
30
+ ··· + 15215838414a + 29808471797,
9543021035u
31
15574254375u
30
+ ··· + 7607919207b 15574499611i
There are 2 irreducible components with 33 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 1, a, b + 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
1
a
2
=
1
1
a
5
=
0
1
a
8
=
0
1
a
4
=
1
1
a
10
=
0
1
a
3
=
1
0
a
9
=
1
1
a
6
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 0
b = 1.00000
0 0
3
II.
I
u
2
= hu
32
2u
31
+ · · · + u + 1, 1.80 × 10
9
u
31
+ 1.80 × 10
9
u
30
+ · · · + 1.52 ×
10
10
a+2.98×10
10
, 9.54×10
9
u
31
1.56×10
10
u
30
+· · ·+7.61×10
9
b1.56×10
10
i
(i) Arc colorings
a
1
=
1
0
a
7
=
0.118079u
31
0.118464u
30
+ ··· + 0.412685u 1.95904
1.25435u
31
+ 2.04711u
30
+ ··· + 8.05367u + 2 .04714
a
2
=
0.999382u
31
+ 1.00005u
30
+ ··· + 3 .56630u + 0.283430
1.42658u
31
2.23424u
30
+ ··· 1.94251 u 2.22835
a
5
=
0
u
a
8
=
0.118079u
31
0.118464u
30
+ ··· + 0 .412685u 1.95904
1.21967u
31
+ 1.92903u
30
+ ··· + 7.81790u + 1 .92945
a
4
=
u
u
a
10
=
u
2
+ 1
u
2
a
3
=
u
3
+ 2u
u
3
+ u
a
9
=
u
4
3u
2
+ 1
u
4
2u
2
a
6
=
u
4
+ 3u
2
1
1.40694u
31
+ 2.22362u
30
+ ··· + 8 .44715u + 2.22354
(ii) Obstruction class = 1
(iii) Cusp Shapes =
18210579048
2535973069
u
31
+
32359838926
2535973069
u
30
+ ···+
94883406442
2535973069
u +
41946667180
2535973069
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.67671 0.06666I
a = 0.536466 0.098385I
b = 0.654826 + 0.660177I
10.51010 0.53898I 8.46673 2.10572I
u = 1.67671 + 0.06666I
a = 0.536466 + 0.098385I
b = 0.654826 0.660177I
10.51010 + 0.53898I 8.46673 + 2.10572I
u = 1.61612 0.17777I
a = 0.632286 + 0.201988I
b = 0.306209 1.201451I
8.99388 4.78654I 6.99769 + 4.71106I
u = 1.61612 + 0.17777I
a = 0.632286 0.201988I
b = 0.306209 + 1.201451I
8.99388 + 4.78654I 6.99769 4.71106I
u = 1.60015 0.02565I
a = 0.221331 0.375842I
b = 0.70363 + 2.68902I
7.09081 0.79638I 8.17185 8.18456I
u = 1.60015 + 0.02565I
a = 0.221331 + 0.375842I
b = 0.70363 2.68902I
7.09081 + 0.79638I 8.17185 + 8.18456I
u = 0.820983 0.567595I
a = 1.381692 0.080236I
b = 0.98755 + 1.10218I
0.60537 9.61260I 2.87987 + 8.20248I
u = 0.820983 + 0.567595I
a = 1.381692 + 0.080236I
b = 0.98755 1.10218I
0.60537 + 9.61260I 2.87987 8.20248I
u = 0.795955 0.349102I
a = 1.58365 + 0.35537I
b = 0.685271 0.866968I
2.75563 4.13382I 6.93448 + 6.73749I
u = 0.795955 + 0.349102I
a = 1.58365 0.35537I
b = 0.685271 + 0.866968I
2.75563 + 4.13382I 6.93448 6.73749I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.560858 0.310184I
a = 0.04973 + 2.07642I
b = 0.448395 0.126836I
1.86601 2.61443I 0.82365 + 8.13996I
u = 0.560858 + 0.310184I
a = 0.04973 2.07642I
b = 0.448395 + 0.126836I
1.86601 + 2.61443I 0.82365 8.13996I
u = 0.269938 0.288721I
a = 2.38342 + 0.84703I
b = 1.008544 + 0.055119I
2.64104 + 0.25879I 3.85203 + 2.96045I
u = 0.269938 + 0.288721I
a = 2.38342 0.84703I
b = 1.008544 0.055119I
2.64104 0.25879I 3.85203 2.96045I
u = 0.102445 0.771273I
a = 0.732522 + 1.203864I
b = 0.760666 + 0.764334I
2.78249 + 5.16401I 0.17525 5.43243I
u = 0.102445 + 0.771273I
a = 0.732522 1.203864I
b = 0.760666 0.764334I
2.78249 5.16401I 0.17525 + 5.43243I
u = 0.086458 0.449548I
a = 1.40221 1.08937I
b = 0.327058 0.712943I
0.227616 + 1.394369I 2.60146 4.04487I
u = 0.086458 + 0.449548I
a = 1.40221 + 1.08937I
b = 0.327058 + 0.712943I
0.227616 1.394369I 2.60146 + 4.04487I
u = 0.598750 0.114970I
a = 0.106728 0.622053I
b = 0.40414 + 2.50188I
0.576409 + 0.313871I 8.1378 + 17.1065I
u = 0.598750 + 0.114970I
a = 0.106728 + 0.622053I
b = 0.40414 2.50188I
0.576409 0.313871I 8.1378 17.1065I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.643643 0.579820I
a = 0.735849 0.491532I
b = 0.069756 0.917675I
1.27204 + 1.92248I 7.80216 5.91516I
u = 0.643643 + 0.579820I
a = 0.735849 + 0.491532I
b = 0.069756 + 0.917675I
1.27204 1.92248I 7.80216 + 5.91516I
u = 0.788048
a = 0.423528
b = 0.523396
1.36694 7.37901
u = 1.076155 0.444148I
a = 0.444604 + 0.479356I
b = 0.282418 + 0.474577I
0.800175 0.941991I 6.40540 + 5.25085I
u = 1.076155 + 0.444148I
a = 0.444604 0.479356I
b = 0.282418 0.474577I
0.800175 + 0.941991I 6.40540 5.25085I
u = 1.55208
a = 1.13958
b = 1.33607
3.73390 0.293662
u = 1.57850 0.06009I
a = 0.318476 + 1.091374I
b = 0.083790 0.485002I
5.46664 + 3.81790I 2.30098 5.52630I
u = 1.57850 + 0.06009I
a = 0.318476 1.091374I
b = 0.083790 + 0.485002I
5.46664 3.81790I 2.30098 + 5.52630I
u = 1.63927 0.09770I
a = 0.955274 + 0.519291I
b = 0.95326 1.04581I
11.15787 + 5.83644I 7.71930 4.74826I
u = 1.63927 + 0.09770I
a = 0.955274 0.519291I
b = 0.95326 + 1.04581I
11.15787 5.83644I 7.71930 + 4.74826I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.65031 0.16673I
a = 0.836945 0.484675I
b = 1.12916 + 1.41089I
7.8166 + 12.4315I 5.16615 6.75497I
u = 1.65031 + 0.16673I
a = 0.836945 + 0.484675I
b = 1.12916 1.41089I
7.8166 12.4315I 5.16615 + 6.75497I
7
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u + 1)(u
32
+ 2u
31
+ ··· + 12u + 8)
c
2
(u + 1)(u
32
+ 11u
30
+ ··· 13u 1)
c
3
, c
4
(u + 1)(u
32
+ 2u
31
+ ··· u + 1)
c
5
(u + 1)(u
32
+ 2u
31
+ ··· u 1)
c
6
(u + 1)(u
32
+ 2u
31
+ ··· 13u 1)
c
7
(u)(u
32
+ 5u
31
+ ··· + 6u + 2)
c
8
(u 1)(u
32
+ 2u
31
+ ··· 13u 1)
c
9
, c
10
(u 1)(u
32
+ 2u
31
+ ··· u + 1)
8
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
(y 1)(y
32
+ 30y
31
+ ··· + 240y + 64)
c
2
(y 1)(y
32
+ 22y
31
+ ··· 121y + 1)
c
3
, c
4
, c
9
c
10
(y 1)(y
32
38y
31
+ ··· 5 y + 1)
c
5
(y 1)(y
32
6y
31
+ ··· 5y + 1)
c
6
, c
8
(y 1)(y
32
18y
31
+ ··· 81y + 1)
c
7
(y)(y
32
9y
31
+ ··· 32y + 4)
9