11a
36
(K11a
36
)
1
Arc Sequences
4 1 7 2 10 8 3 5 11 6 9
Solving Sequence
5,10
6
2,11
4 1 3 9 8 7
c
5
c
10
c
4
c
1
c
2
c
9
c
8
c
7
c
3
, c
6
, c
11
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
3
+ a
2
1, u 1, a
2
+ bi
I
u
2
= hu
63
4u
62
+ ··· 3u + 1, 11u
62
+ 34u
61
+ ··· + 2a + 14, 21u
62
+ 65u
61
+ ··· + 4b + 23i
There are 2 irreducible components with 66 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
3
+ a
2
1, u 1, a
2
+ bi
(i) Arc colorings
a
5
=
0
1
a
10
=
a
a
2
a
6
=
a
2
a
2
a
2
=
1
0
a
11
=
a
2
+ a 1
1
a
4
=
1
1
a
1
=
0
1
a
3
=
1
1
a
9
=
a
2
0
a
8
=
a
2
a
2
a
7
=
a
2
a
2
a
7
=
a
2
a
2
(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 = 0.877439 0.744862I
b = 0.215080 1.307141I
4.66906 + 2.82812I 7.71191 2.59975I
u = 1.00000
a = 0.877439 + 0.744862I
b = 0.215080 + 1.307141I
4.66906 2.82812I 7.71191 + 2.59975I
u = 1.00000
a = 0.754878
b = 0.569840
0.531480 4.42382
3
II. I
u
2
= hu
63
4u
62
+ · · · 3u + 1, 11u
62
+ 34u
61
+ · · · + 2a +
14, 21u
62
+ 65u
61
+ · · · + 4b + 23i
(i) Arc colorings
a
5
=
0
u
a
10
=
11
2
u
62
17u
61
+ ··· +
23
2
u 7
21
4
u
62
65
4
u
61
+ ··· + 9u
23
4
a
6
=
41
4
u
62
+
141
4
u
61
+ ··· 25u +
59
4
31
4
u
62
+
99
4
u
61
+ ··· 17u +
37
4
a
2
=
1
0
a
11
=
1
4
u
62
+
3
4
u
61
+ ···
1
2
u +
1
4
u
17
+ 5u
15
+ ··· 4u
4
u
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
u
2
+ 1
u
4
a
9
=
35
4
u
62
+
119
4
u
61
+ ··· 20u +
53
4
1
2
u
62
+
1
2
u
61
+ ··· 2u +
5
2
a
8
=
35
4
u
62
+
119
4
u
61
+ ··· 20u +
53
4
9
4
u
62
+
41
4
u
61
+ ··· 9u +
31
4
a
7
=
57
4
u
62
+
197
4
u
61
+ ··· 35u +
83
4
35
4
u
62
+
127
4
u
61
+ ··· 23u +
57
4
a
7
=
57
4
u
62
+
197
4
u
61
+ ··· 35u +
83
4
35
4
u
62
+
127
4
u
61
+ ··· 23u +
57
4
(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 = 1.260647 0.325376I
a = 0.872350 0.360942I
b = 0.247563 1.265493I
6.64299 + 5.98946I 3.19575 5.55727I
u = 1.260647 + 0.325376I
a = 0.872350 + 0.360942I
b = 0.247563 + 1.265493I
6.64299 5.98946I 3.19575 + 5.55727I
u = 1.246675 0.342536I
a = 0.391390 + 0.754108I
b = 0.49480 + 1.52188I
7.36657 + 0.17993I 4.78991 0.25174I
u = 1.246675 + 0.342536I
a = 0.391390 0.754108I
b = 0.49480 1.52188I
7.36657 0.17993I 4.78991 + 0.25174I
u = 1.210371 0.030629I
a = 0.701798 + 0.568033I
b = 0.12583 + 1.57795I
3.42806 2.58129I 1.79543 + 2.34538I
u = 1.210371 + 0.030629I
a = 0.701798 0.568033I
b = 0.12583 1.57795I
3.42806 + 2.58129I 1.79543 2.34538I
u = 1.182534 0.257010I
a = 0.786900 0.019515I
b = 0.777418 0.263774I
1.01619 + 1.23564I 2.18804 4.79371I
u = 1.182534 + 0.257010I
a = 0.786900 + 0.019515I
b = 0.777418 + 0.263774I
1.01619 1.23564I 2.18804 + 4.79371I
u = 1.158557 0.468666I
a = 0.785776 + 0.383079I
b = 1.56392 + 0.11806I
7.07861 2.97979I 4.67590 + 1.36728I
u = 1.158557 + 0.468666I
a = 0.785776 0.383079I
b = 1.56392 0.11806I
7.07861 + 2.97979I 4.67590 1.36728I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.148084 0.485998I
a = 0.503048 0.854764I
b = 0.79519 3.47615I
6.27660 8.79934I 3.04264 + 6.53675I
u = 1.148084 + 0.485998I
a = 0.503048 + 0.854764I
b = 0.79519 + 3.47615I
6.27660 + 8.79934I 3.04264 6.53675I
u = 1.143908 0.371905I
a = 0.122725 0.463834I
b = 0.536396 0.832404I
3.80293 1.26646I 5.80106 + 0.80985I
u = 1.143908 + 0.371905I
a = 0.122725 + 0.463834I
b = 0.536396 + 0.832404I
3.80293 + 1.26646I 5.80106 0.80985I
u = 1.081169 0.440014I
a = 0.726282 + 0.457965I
b = 1.10450 + 2.68878I
0.60605 3.69700I 2.17014 + 4.90316I
u = 1.081169 + 0.440014I
a = 0.726282 0.457965I
b = 1.10450 2.68878I
0.60605 + 3.69700I 2.17014 4.90316I
u = 0.870881
a = 0.270944
b = 0.619471
1.26098 8.95538
u = 0.817947 0.392240I
a = 1.109270 + 0.241744I
b = 0.03987 1.47069I
2.19626 + 0.72911I 1.158852 + 0.138195I
u = 0.817947 + 0.392240I
a = 1.109270 0.241744I
b = 0.03987 + 1.47069I
2.19626 0.72911I 1.158852 0.138195I
u = 0.706877 0.424114I
a = 0.541742 1.254814I
b = 1.84661 1.59720I
1.85739 4.32227I 0.09659 + 6.54104I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.706877 + 0.424114I
a = 0.541742 + 1.254814I
b = 1.84661 + 1.59720I
1.85739 + 4.32227I 0.09659 6.54104I
u = 0.331068 0.319832I
a = 1.49067 + 1.26115I
b = 1.38167 + 0.91525I
1.55408 + 0.10582I 6.26556 + 0.58371I
u = 0.331068 + 0.319832I
a = 1.49067 1.26115I
b = 1.38167 0.91525I
1.55408 0.10582I 6.26556 0.58371I
u = 0.136404 0.659647I
a = 1.35415 1.18055I
b = 0.93743 1.16572I
3.43377 + 4.42046I 0.17195 2.68359I
u = 0.136404 + 0.659647I
a = 1.35415 + 1.18055I
b = 0.93743 + 1.16572I
3.43377 4.42046I 0.17195 + 2.68359I
u = 0.074618 0.675215I
a = 1.02576 1.24925I
b = 0.339889 + 0.034133I
4.03831 1.31004I 1.39925 + 2.71878I
u = 0.074618 + 0.675215I
a = 1.02576 + 1.24925I
b = 0.339889 0.034133I
4.03831 + 1.31004I 1.39925 2.71878I
u = 0.178721 0.848670I
a = 0.750131 + 0.993339I
b = 0.179316 + 0.127652I
2.92774 4.17173I 0.05467 + 2.38856I
u = 0.178721 + 0.848670I
a = 0.750131 0.993339I
b = 0.179316 0.127652I
2.92774 + 4.17173I 0.05467 2.38856I
u = 0.195316 0.700814I
a = 0.819954 0.047781I
b = 0.0323524 + 0.0016528I
0.08935 2.18002I 0.08533 + 3.20984I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.195316 + 0.700814I
a = 0.819954 + 0.047781I
b = 0.0323524 0.0016528I
0.08935 + 2.18002I 0.08533 3.20984I
u = 0.198021 0.868774I
a = 1.049606 + 0.917230I
b = 0.95994 + 1.19007I
2.02603 9.97382I 1.64006 + 7.22241I
u = 0.198021 + 0.868774I
a = 1.049606 0.917230I
b = 0.95994 1.19007I
2.02603 + 9.97382I 1.64006 7.22241I
u = 0.264864 0.800811I
a = 0.559548 1.099467I
b = 0.81299 1.61264I
3.55985 4.49777I 7.47027 + 4.85592I
u = 0.264864 + 0.800811I
a = 0.559548 + 1.099467I
b = 0.81299 + 1.61264I
3.55985 + 4.49777I 7.47027 4.85592I
u = 0.415444 0.641775I
a = 0.100446 + 1.297986I
b = 0.346139 + 1.236417I
1.88400 + 1.11201I 5.93092 2.67876I
u = 0.415444 + 0.641775I
a = 0.100446 1.297986I
b = 0.346139 1.236417I
1.88400 1.11201I 5.93092 + 2.67876I
u = 0.638145 0.682574I
a = 1.192667 + 0.309590I
b = 0.86573 + 1.12014I
2.75209 3.26890I 5.78541 + 2.79593I
u = 0.638145 + 0.682574I
a = 1.192667 0.309590I
b = 0.86573 1.12014I
2.75209 + 3.26890I 5.78541 2.79593I
u = 0.656494 0.538703I
a = 0.012964 + 0.851835I
b = 0.272117 + 0.549297I
1.61397 + 1.37996I 3.33026 3.84355I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.656494 + 0.538703I
a = 0.012964 0.851835I
b = 0.272117 0.549297I
1.61397 1.37996I 3.33026 + 3.84355I
u = 0.742658 0.034260I
a = 1.13508 1.09861I
b = 0.366755 1.129166I
3.99010 + 2.92514I 5.31377 3.77140I
u = 0.742658 + 0.034260I
a = 1.13508 + 1.09861I
b = 0.366755 + 1.129166I
3.99010 2.92514I 5.31377 + 3.77140I
u = 0.778473 0.655233I
a = 0.764783 0.681937I
b = 1.29948 2.07021I
6.20929 + 2.51179I 9.63450 3.63863I
u = 0.778473 + 0.655233I
a = 0.764783 + 0.681937I
b = 1.29948 + 2.07021I
6.20929 2.51179I 9.63450 + 3.63863I
u = 0.881881 0.575730I
a = 0.840382 0.232270I
b = 0.504565 + 0.299278I
0.93068 + 3.11451I 1.81708 3.79797I
u = 0.881881 + 0.575730I
a = 0.840382 + 0.232270I
b = 0.504565 0.299278I
0.93068 3.11451I 1.81708 + 3.79797I
u = 0.892480 0.637862I
a = 0.390325 + 1.014433I
b = 1.01680 + 2.47664I
2.02729 + 8.31052I 3.92324 8.70014I
u = 0.892480 + 0.637862I
a = 0.390325 1.014433I
b = 1.01680 2.47664I
2.02729 8.31052I 3.92324 + 8.70014I
u = 1.089748 0.505160I
a = 0.852987 + 0.069193I
b = 0.778146 + 0.571791I
0.13818 + 3.33082I 2.52685 2.17772I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.089748 + 0.505160I
a = 0.852987 0.069193I
b = 0.778146 0.571791I
0.13818 3.33082I 2.52685 + 2.17772I
u = 1.141124 0.407113I
a = 0.964140 + 0.331926I
b = 0.199886 + 1.127971I
6.84424 0.78588I 3.59501 1.40789I
u = 1.141124 + 0.407113I
a = 0.964140 0.331926I
b = 0.199886 1.127971I
6.84424 + 0.78588I 3.59501 + 1.40789I
u = 1.149532 0.509940I
a = 0.143550 + 0.520988I
b = 0.390377 + 0.840878I
2.82669 + 6.78405I 3.37376 6.18383I
u = 1.149532 + 0.509940I
a = 0.143550 0.520988I
b = 0.390377 0.840878I
2.82669 6.78405I 3.37376 + 6.18383I
u = 1.153782 0.425567I
a = 0.368537 0.854079I
b = 0.51992 1.47700I
7.38822 + 5.19388I 4.52301 6.32517I
u = 1.153782 + 0.425567I
a = 0.368537 + 0.854079I
b = 0.51992 + 1.47700I
7.38822 5.19388I 4.52301 + 6.32517I
u = 1.157218 0.552536I
a = 0.676408 0.488559I
b = 1.00434 2.45256I
0.91945 + 9.52832I 3.69264 8.35881I
u = 1.157218 + 0.552536I
a = 0.676408 + 0.488559I
b = 1.00434 + 2.45256I
0.91945 9.52832I 3.69264 + 8.35881I
u = 1.198763 0.538883I
a = 0.754252 0.361327I
b = 1.310190 0.342612I
5.96644 + 9.24715I 3.19852 5.45523I
10
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.198763 + 0.538883I
a = 0.754252 + 0.361327I
b = 1.310190 + 0.342612I
5.96644 9.24715I 3.19852 + 5.45523I
u = 1.201634 0.551168I
a = 0.463214 + 0.830081I
b = 0.56304 + 3.29672I
5.0319 + 15.1597I 1.53749 10.20107I
u = 1.201634 + 0.551168I
a = 0.463214 0.830081I
b = 0.56304 3.29672I
5.0319 15.1597I 1.53749 + 10.20107I
11
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
3
(u
63
+ 4u
62
+ ··· 3u 1)
c
2
(u + 1)
3
(u
63
+ 34u
62
+ ··· + 5u + 1)
c
3
, c
7
u
3
(u
63
+ u
62
+ ··· + 12u + 8)
c
4
(u + 1)
3
(u
63
+ 4u
62
+ ··· 3u 1)
c
5
(u
3
u
2
+ 1)(u
63
+ 2u
62
+ ··· + 10u
2
1)
c
6
u
3
(u
63
+ 21u
62
+ ··· 624u 64)
c
8
(u
3
u
2
+ 2u 1)(u
63
+ 2u
62
+ ··· 18u + 9)
c
9
(u
3
+ u
2
+ 2u + 1)(u
63
+ 20u
62
+ ··· + 20u + 1)
c
10
(u
3
+ u
2
1)(u
63
+ 2u
62
+ ··· + 10u
2
1)
c
11
(u
3
u
2
+ 2u 1)(u
63
+ 20u
62
+ ··· + 20u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
3
(y
63
34y
62
+ ··· + 5y 1)
c
2
(y 1)
3
(y
63
6y
62
+ ··· 27y 1)
c
3
, c
7
y
3
(y
63
+ 21y
62
+ ··· 624y 64)
c
5
, c
10
(y
3
y
2
+ 2y 1)(y
63
20y
62
+ ··· + 20y 1)
c
6
y
3
(y
63
+ 37y
62
+ ··· + 167168y 4096)
c
8
(y
3
+ 3y
2
+ 2y 1)(y
63
+ 12y
62
+ ··· 16272y 81)
c
9
, c
11
(y
3
+ 3y
2
+ 2y 1)(y
63
+ 48y
62
+ ··· + 340y 1)
13