11a
259
(K11a
259
)
1
Arc Sequences
4 7 1 2 10 9 3 11 5 6 8
Solving Sequence
5,9
10 6 7
2,11
4 1 3 8
c
9
c
5
c
6
c
10
c
4
c
1
c
3
c
8
c
2
, c
7
, c
11
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
5
a
4
2a
3
+ a
2
+ a + 1, u 1, a
4
2a
2
+ bi
I
u
2
= hu
44
+ 6u
43
+ ··· 3u 1, 93u
43
397u
42
+ ··· + 16a + 9, 101u
43
+ 473u
42
+ ··· + 4b 76i
There are 2 irreducible components with 49 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
= ha
5
a
4
2a
3
+ a
2
+ a + 1, u 1, a
4
2a
2
+ bi
(i) Arc colorings
a
5
=
0
1
a
9
=
a
a
4
+ 2a
2
a
10
=
a
a
4
+ 2a
2
+ a
a
6
=
a
2
a
4
2a
2
a
a
7
=
a
4
2a
2
0
a
2
=
1
0
a
11
=
a
3
+ a
1
a
4
=
1
1
a
1
=
0
1
a
3
=
1
0
a
8
=
a
4
2a
2
0
a
8
=
a
4
2a
2
0
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 1.21774
b = 0.766826
4.04602 9.19250
u = 1.00000
a = 0.309916 0.549911I
b = 0.339110 + 0.822375I
1.97403 1.53058I 11.97286 + 4.76366I
u = 1.00000
a = 0.309916 + 0.549911I
b = 0.339110 0.822375I
1.97403 + 1.53058I 11.97286 4.76366I
u = 1.00000
a = 1.41878 0.21917I
b = 0.455697 + 1.200152I
7.51750 + 4.40083I 16.4309 2.8075I
u = 1.00000
a = 1.41878 + 0.21917I
b = 0.455697 1.200152I
7.51750 4.40083I 16.4309 + 2.8075I
3
II. I
u
2
= hu
44
+ 6u
43
+ · · · 3u 1, 93u
43
397u
42
+ · · · + 16a +
9, 101u
43
+ 473u
42
+ · · · + 4b 76i
(i) Arc colorings
a
5
=
0
u
a
9
=
5.81250u
43
+ 24.8125u
42
+ ··· 7.87500u 0.562500
101
4
u
43
473
4
u
42
+ ··· + 43u + 19
a
10
=
5.81250u
43
+ 24.8125u
42
+ ··· 7.87500u 0.562500
10.4375u
43
49.9375u
42
+ ··· + 18.6250u + 8.93750
a
6
=
4.25000u
43
20.6250u
42
+ ··· + 11.8750u + 2.87500
4.06250u
43
19.5625u
42
+ ··· + 9.37500u + 3.56250
a
7
=
2u
43
29
4
u
42
+ ··· +
31
4
u 5
49
4
u
43
105
2
u
42
+ ··· +
75
4
u + 6
a
2
=
1
0
a
11
=
0.0625000u
43
0.312500u
42
+ ··· + 4.12500u + 0.0625000
u
7
3u
5
+ 2u
4
+ 2u
3
4u
2
+ u
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
3
+ 2u
u
3
+ u
a
8
=
11u
43
52u
42
+ ··· +
25
2
u +
59
4
87
4
u
43
197
2
u
42
+ ··· +
129
4
u + 16
a
8
=
11u
43
52u
42
+ ··· +
25
2
u +
59
4
87
4
u
43
197
2
u
42
+ ··· +
129
4
u + 16
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.60536 0.25067I
a = 0.894144 + 0.261319I
b = 1.321439 0.259202I
18.5896 1.6346I 17.3078 + 0.4945I
u = 1.60536 + 0.25067I
a = 0.894144 0.261319I
b = 1.321439 + 0.259202I
18.5896 + 1.6346I 17.3078 0.4945I
u = 1.57148 0.25916I
a = 0.280263 0.436514I
b = 0.224586 + 0.384308I
11.90459 4.53656I 13.93261 + 0.49755I
u = 1.57148 + 0.25916I
a = 0.280263 + 0.436514I
b = 0.224586 0.384308I
11.90459 + 4.53656I 13.93261 0.49755I
u = 1.56869 0.31082I
a = 0.937609 + 0.018974I
b = 1.80427 + 2.62560I
17.5761 12.5201I 16.2387 + 6.2588I
u = 1.56869 + 0.31082I
a = 0.937609 0.018974I
b = 1.80427 2.62560I
17.5761 + 12.5201I 16.2387 6.2588I
u = 1.56570
a = 0.934799
b = 1.34957
13.6319 18.1159
u = 1.56049 0.28855I
a = 0.344825 + 0.391638I
b = 1.35142 1.29990I
11.4356 8.9717I 12.8397 + 6.3159I
u = 1.56049 + 0.28855I
a = 0.344825 0.391638I
b = 1.35142 + 1.29990I
11.4356 + 8.9717I 12.8397 6.3159I
u = 1.43714 0.02392I
a = 0.623420 + 0.133365I
b = 0.775931 0.403203I
6.64112 0.04136I 14.06391 0.57797I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.43714 + 0.02392I
a = 0.623420 0.133365I
b = 0.775931 + 0.403203I
6.64112 + 0.04136I 14.06391 + 0.57797I
u = 1.43422 0.16711I
a = 0.931610 + 0.076783I
b = 0.02744 2.45105I
8.90196 7.03042I 14.4696 + 6.0464I
u = 1.43422 + 0.16711I
a = 0.931610 0.076783I
b = 0.02744 + 2.45105I
8.90196 + 7.03042I 14.4696 6.0464I
u = 1.42769 0.10914I
a = 0.146426 0.477138I
b = 0.61601 + 1.78174I
4.26770 3.54538I 9.88379 + 4.33460I
u = 1.42769 + 0.10914I
a = 0.146426 + 0.477138I
b = 0.61601 1.78174I
4.26770 + 3.54538I 9.88379 4.33460I
u = 0.432601 0.117164I
a = 3.21805 0.36389I
b = 1.036250 0.494033I
6.51912 4.63399I 7.99025 + 3.83362I
u = 0.432601 + 0.117164I
a = 3.21805 + 0.36389I
b = 1.036250 + 0.494033I
6.51912 + 4.63399I 7.99025 3.83362I
u = 0.328694 0.089361I
a = 1.45518 1.33027I
b = 0.425840 0.055664I
0.76310 1.71420I 4.36493 + 4.23791I
u = 0.328694 + 0.089361I
a = 1.45518 + 1.33027I
b = 0.425840 + 0.055664I
0.76310 + 1.71420I 4.36493 4.23791I
u = 0.058722 0.367772I
a = 0.72279 + 3.23795I
b = 0.999076 + 0.832027I
2.14292 0.55015I 5.79202 + 0.04414I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.058722 + 0.367772I
a = 0.72279 3.23795I
b = 0.999076 0.832027I
2.14292 + 0.55015I 5.79202 0.04414I
u = 0.192197 0.469108I
a = 1.084462 0.214093I
b = 0.915109 + 0.557744I
0.99729 + 1.64755I 2.31020 6.18875I
u = 0.192197 + 0.469108I
a = 1.084462 + 0.214093I
b = 0.915109 0.557744I
0.99729 1.64755I 2.31020 + 6.18875I
u = 0.262321 0.630465I
a = 0.98151 1.67813I
b = 0.36904 1.64466I
3.43627 + 4.29473I 9.95227 6.87080I
u = 0.262321 + 0.630465I
a = 0.98151 + 1.67813I
b = 0.36904 + 1.64466I
3.43627 4.29473I 9.95227 + 6.87080I
u = 0.406553
a = 0.750061
b = 0.367645
0.758185 13.9250
u = 0.545081 0.886664I
a = 0.93671 + 1.08365I
b = 1.55211 + 1.24103I
10.66986 + 8.11106I 14.5153 5.8460I
u = 0.545081 + 0.886664I
a = 0.93671 1.08365I
b = 1.55211 1.24103I
10.66986 8.11106I 14.5153 + 5.8460I
u = 0.550396 0.837192I
a = 0.785743 0.000797I
b = 1.35075 0.49759I
4.52260 + 4.83043I 11.16325 6.23604I
u = 0.550396 + 0.837192I
a = 0.785743 + 0.000797I
b = 1.35075 + 0.49759I
4.52260 4.83043I 11.16325 + 6.23604I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.609155 0.800806I
a = 0.194482 0.746623I
b = 0.861504 + 0.157766I
4.71947 + 0.64919I 11.99618 + 0.22434I
u = 0.609155 + 0.800806I
a = 0.194482 + 0.746623I
b = 0.861504 0.157766I
4.71947 0.64919I 11.99618 0.22434I
u = 0.663066 0.832758I
a = 0.705911 + 1.211855I
b = 0.419372 + 0.018820I
11.03427 2.36395I 15.4118 + 0.2932I
u = 0.663066 + 0.832758I
a = 0.705911 1.211855I
b = 0.419372 0.018820I
11.03427 + 2.36395I 15.4118 0.2932I
u = 0.878617 0.388577I
a = 1.239097 0.631785I
b = 0.494749 0.896750I
5.28913 0.51604I 15.8220 + 1.1476I
u = 0.878617 + 0.388577I
a = 1.239097 + 0.631785I
b = 0.494749 + 0.896750I
5.28913 + 0.51604I 15.8220 1.1476I
u = 1.082710 0.136687I
a = 0.166207 + 0.347319I
b = 0.918567 0.721472I
1.44721 + 0.71558I 5.57948 + 1.23300I
u = 1.082710 + 0.136687I
a = 0.166207 0.347319I
b = 0.918567 + 0.721472I
1.44721 0.71558I 5.57948 1.23300I
u = 1.237686 0.186516I
a = 1.078592 + 0.054463I
b = 1.52093 + 2.21918I
6.11126 + 2.71120I 13.25852 2.28581I
u = 1.237686 + 0.186516I
a = 1.078592 0.054463I
b = 1.52093 2.21918I
6.11126 2.71120I 13.25852 + 2.28581I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.47584 0.02352I
a = 0.318559 0.428804I
b = 0.915789 + 0.405933I
6.79823 + 2.11103I 12.35870 3.20933I
u = 1.47584 + 0.02352I
a = 0.318559 + 0.428804I
b = 0.915789 0.405933I
6.79823 2.11103I 12.35870 + 3.20933I
u = 1.50758 0.04579I
a = 0.983775 + 0.122092I
b = 2.43837 1.44143I
13.05821 + 5.28645I 15.7286 3.0190I
u = 1.50758 + 0.04579I
a = 0.983775 0.122092I
b = 2.43837 + 1.44143I
13.05821 5.28645I 15.7286 + 3.0190I
9
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
5
(u
44
+ 6u
43
+ ··· 3u 1)
c
2
, c
7
u
5
(u
44
+ u
43
+ ··· 32u + 32)
c
3
, c
4
(u + 1)
5
(u
44
+ 6u
43
+ ··· 3u 1)
c
5
(u
5
+ u
4
+ ··· + u 1)(u
44
+ 2u
43
+ ··· 3u 1)
c
6
(u
5
3u
4
+ ··· u + 1)(u
44
+ 6u
43
+ ··· 175u + 53)
c
8
(u
5
u
4
+ ··· + u 1)(u
44
+ 6u
43
+ ··· 57u 9)
c
9
, c
10
(u
5
u
4
+ ··· + u + 1)(u
44
+ 2u
43
+ ··· 3u 1)
c
11
(u
5
+ u
4
+ ··· + u + 1)(u
44
+ 6u
43
+ ··· 57u 9)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
4
(y 1)
5
(y
44
46y
43
+ ··· 17y + 1)
c
2
, c
7
y
5
(y
44
33y
43
+ ··· 512y + 1024)
c
5
, c
9
, c
10
(y
5
5y
4
+ ··· y 1)(y
44
42y
43
+ ··· + y + 1)
c
6
(y
5
y
4
+ ··· + 3y 1)(y
44
18y
43
+ ··· 41755y + 2809)
c
8
, c
11
(y
5
+ 3y
4
+ ··· y 1)(y
44
+ 42y
43
+ ··· 3555y + 81)
11