10
90
(K10a
92
)
1
Arc Sequences
5 7 10 8 2 9 1 6 4 3
Solving Sequence
1,7 3,8
2 10 4 5 9 6
c
7
c
2
c
10
c
3
c
4
c
9
c
6
c
1
, c
5
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
2
u + 1, b, 3a u 1i
I
u
2
= hu
40
+ 2u
39
+ ··· + 2u + 1, 4.21804 × 10
19
u
39
+ 2.81836 × 10
20
u
38
+ ··· + 2.27356 × 10
21
b + 1.86252 × 10
20
,
6.87748 × 10
21
u
39
+ 8.80415 × 10
21
u
38
+ ··· + 6.82068 × 10
21
a 4.02376 × 10
21
i
There are 2 irreducible components with 42 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, 3a u 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
1
3
u +
1
3
0
a
3
=
0
u
a
8
=
1
3
u +
1
3
0
a
2
=
1
3
u +
1
3
u
a
10
=
1
u + 1
a
4
=
u
u 1
a
5
=
u +
1
3
u 1
a
9
=
u
u + 2
a
6
=
2
3
u +
1
3
u 2
(ii) Obstruction class = 1
(iii) Cusp Shapes =
20
3
u 9
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 0.288675I
b = 0
1.64493 + 2.02988I 5.66667 5.77350I
u = 0.500000 + 0.866025I
a = 0.500000 + 0.288675I
b = 0
1.64493 2.02988I 5.66667 + 5.77350I
3
II. I
u
2
=
hu
40
+ 2u
39
+ · · · +2u + 1, 4.22 × 10
19
u
39
+ 2.82 × 10
20
u
38
+ · · · + 2.27 × 10
21
b +
1.86 × 10
20
, 6.88 × 10
21
u
39
+ 8.80 × 10
21
u
38
+ · · · + 6.82 × 10
21
a 4.02 × 10
21
i
(i) Arc colorings
a
1
=
1
0
a
7
=
1.00833u
39
1.29080u
38
+ ··· + 9.74194u + 0.589936
0.0185526u
39
0.123963u
38
+ ··· 0.0962885u 0.0819209
a
3
=
0
u
a
8
=
1.02688u
39
1.41477u
38
+ ··· + 9.64565u + 0.508015
0.0185526u
39
0.123963u
38
+ ··· 0.0962885u 0.0819209
a
2
=
1.76783u
39
4.42874u
38
+ ··· 6.96745u + 1.61338
0.259095u
39
+ 0.292216u
38
+ ··· 1.34286u + 0.0682056
a
10
=
1
u
2
a
4
=
u
u
3
+ u
a
5
=
0.856015u
39
2.32606u
38
+ ··· 11.6614u + 2.20906
0.111452u
39
0.303733u
38
+ ··· 1.75504u 0.659792
a
9
=
u
2
+ 1
u
4
2u
2
a
6
=
1.06067u
39
1.36384u
38
+ ··· + 9.36195u 0.661431
u
4
2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes =
23709189092098357085641
20462029573507327146753
u
39
+
5967042122571150693039
2273558841500814127417
u
38
+ ··· +
151009571647058399841037
6820676524502442382251
u +
36425712582025581848210
20462029573507327146753
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.812016 0.457171I
a = 0.51150 1.48693I
b = 1.149904 + 0.802594I
0.61792 9.83239I 1.42359 + 7.89553I
u = 0.812016 + 0.457171I
a = 0.51150 + 1.48693I
b = 1.149904 0.802594I
0.61792 + 9.83239I 1.42359 7.89553I
u = 0.718542 0.684654I
a = 0.914818 + 0.108019I
b = 0.954641 + 0.530178I
1.29563 + 4.66233I 0.05723 4.37430I
u = 0.718542 + 0.684654I
a = 0.914818 0.108019I
b = 0.954641 0.530178I
1.29563 4.66233I 0.05723 + 4.37430I
u = 0.635260 0.284826I
a = 1.54927 0.54512I
b = 0.865119 0.230334I
3.12136 + 0.50572I 2.23334 + 2.05026I
u = 0.635260 + 0.284826I
a = 1.54927 + 0.54512I
b = 0.865119 + 0.230334I
3.12136 0.50572I 2.23334 2.05026I
u = 0.548023 0.473980I
a = 0.57075 + 1.36834I
b = 1.19599 0.76251I
3.67375 4.20324I 2.19223 + 6.09439I
u = 0.548023 + 0.473980I
a = 0.57075 1.36834I
b = 1.19599 + 0.76251I
3.67375 + 4.20324I 2.19223 6.09439I
u = 0.424864 0.027630I
a = 0.530561 + 0.409845I
b = 0.884090 0.153437I
2.32372 0.01230I 6.85568 1.19794I
u = 0.424864 + 0.027630I
a = 0.530561 0.409845I
b = 0.884090 + 0.153437I
2.32372 + 0.01230I 6.85568 + 1.19794I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.29805 1.43344I
a = 0.126192 + 0.970531I
b = 1.082079 0.236406I
8.54985 3.07602I 4.56660 + 3.83204I
u = 0.29805 + 1.43344I
a = 0.126192 0.970531I
b = 1.082079 + 0.236406I
8.54985 + 3.07602I 4.56660 3.83204I
u = 0.29587 1.50669I
a = 0.549394 + 1.112712I
b = 1.38896 0.90972I
6.9750 13.8661I 2.04130 + 7.67342I
u = 0.29587 + 1.50669I
a = 0.549394 1.112712I
b = 1.38896 + 0.90972I
6.9750 + 13.8661I 2.04130 7.67342I
u = 0.19425 1.47878I
a = 0.497452 0.822167I
b = 1.68356 + 0.85055I
10.00149 6.93788I 4.75117 + 5.20238I
u = 0.19425 + 1.47878I
a = 0.497452 + 0.822167I
b = 1.68356 0.85055I
10.00149 + 6.93788I 4.75117 5.20238I
u = 0.16112 1.56360I
a = 0.150404 0.488005I
b = 1.062070 0.065113I
8.93627 + 1.59631I 4.88775 3.74200I
u = 0.16112 + 1.56360I
a = 0.150404 + 0.488005I
b = 1.062070 + 0.065113I
8.93627 1.59631I 4.88775 + 3.74200I
u = 0.088735 1.341392I
a = 0.317234 0.556138I
b = 0.734105 + 1.052125I
1.96980 1.69833I 2.59929 + 0.52353I
u = 0.088735 + 1.341392I
a = 0.317234 + 0.556138I
b = 0.734105 1.052125I
1.96980 + 1.69833I 2.59929 0.52353I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.042556 1.284613I
a = 0.893842 0.087105I
b = 1.47619 + 0.50462I
1.29749 1.62987I 2.22422 + 3.54187I
u = 0.042556 + 1.284613I
a = 0.893842 + 0.087105I
b = 1.47619 0.50462I
1.29749 + 1.62987I 2.22422 3.54187I
u = 0.054010 1.410140I
a = 1.08544 + 2.37567I
b = 0.069834 0.547372I
5.39390 + 0.19809I 2.00855 + 3.97238I
u = 0.054010 + 1.410140I
a = 1.08544 2.37567I
b = 0.069834 + 0.547372I
5.39390 0.19809I 2.00855 3.97238I
u = 0.145572 1.361910I
a = 0.320382 + 0.336180I
b = 0.23858 1.78174I
3.71762 + 5.12635I 0.75757 7.21246I
u = 0.145572 + 1.361910I
a = 0.320382 0.336180I
b = 0.23858 + 1.78174I
3.71762 5.12635I 0.75757 + 7.21246I
u = 0.14754 1.48353I
a = 0.479713 + 0.610038I
b = 1.031292 0.652025I
6.24860 + 2.92553I 1.87350 2.25816I
u = 0.14754 + 1.48353I
a = 0.479713 0.610038I
b = 1.031292 + 0.652025I
6.24860 2.92553I 1.87350 + 2.25816I
u = 0.186501 0.360437I
a = 4.27556 2.10586I
b = 0.347010 + 0.520893I
0.113312 0.691322I 2.03417 9.81182I
u = 0.186501 + 0.360437I
a = 4.27556 + 2.10586I
b = 0.347010 0.520893I
0.113312 + 0.691322I 2.03417 + 9.81182I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.31725 1.49380I
a = 0.348472 0.904236I
b = 1.095973 + 0.735684I
3.39122 + 8.09434I 0.70203 5.58722I
u = 0.31725 + 1.49380I
a = 0.348472 + 0.904236I
b = 1.095973 0.735684I
3.39122 8.09434I 0.70203 + 5.58722I
u = 0.334189 0.406515I
a = 0.407485 0.964590I
b = 0.365946 + 0.499179I
0.073174 + 1.047744I 1.21383 6.28305I
u = 0.334189 + 0.406515I
a = 0.407485 + 0.964590I
b = 0.365946 0.499179I
0.073174 1.047744I 1.21383 + 6.28305I
u = 0.491071 0.191924I
a = 0.42507 1.77690I
b = 0.575381 + 1.218444I
1.16688 + 2.84021I 5.40123 7.45362I
u = 0.491071 + 0.191924I
a = 0.42507 + 1.77690I
b = 0.575381 1.218444I
1.16688 2.84021I 5.40123 + 7.45362I
u = 0.668019 0.947602I
a = 0.420524 0.169289I
b = 0.573736 0.158036I
1.20798 + 1.63374I 5.34484 + 3.65075I
u = 0.668019 + 0.947602I
a = 0.420524 + 0.169289I
b = 0.573736 + 0.158036I
1.20798 1.63374I 5.34484 3.65075I
u = 0.875135 0.400189I
a = 0.485093 + 0.952532I
b = 0.771627 0.533290I
2.73210 + 3.80447I 3.64129 6.83498I
u = 0.875135 + 0.400189I
a = 0.485093 0.952532I
b = 0.771627 + 0.533290I
2.73210 3.80447I 3.64129 + 6.83498I
8
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)(u
40
+ 2u
39
+ ··· + 2u + 1)
c
2
(3u
2
+ 1)(3u
40
+ 19u
39
+ ··· + 64u + 32)
c
3
(u
2
+ u + 1)(u
40
+ 2u
39
+ ··· + 2u + 1)
c
4
(3u
2
3u + 1)(3u
40
+ 10u
39
+ ··· + 192u + 103)
c
5
(u
2
u + 1)(u
40
+ 2u
39
+ ··· + 2u + 1)
c
6
(u 1)
2
(u
40
+ 3u
39
+ ··· + 31u + 9)
c
7
u
2
(u
40
+ 3u
39
+ ··· + 156u + 36)
c
8
(u + 1)
2
(u
40
+ 3u
39
+ ··· + 31u + 9)
c
9
, c
10
(u
2
u + 1)(u
40
+ 2u
39
+ ··· + 2u + 1)
9
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
2
+ y + 1)(y
40
22y
39
+ ··· 4y + 1)
c
2
(3y + 1)
2
(9y
40
181y
39
+ ··· 13312y + 1024)
c
3
, c
9
, c
10
(y
2
+ y + 1)(y
40
+ 38y
39
+ ··· 4y + 1)
c
4
(9y
2
3y + 1)(9y
40
+ 38y
39
+ ··· + 78084y + 10609)
c
6
, c
8
(y 1)
2
(y
40
21y
39
+ ··· + 389y + 81)
c
7
y
2
(y
40
15y
39
+ ··· 7416y + 1296)
10