11a
257
(K11a
257
)
1
Arc Sequences
4 8 1 2 11 10 3 5 6 7 9
Solving Sequence
3,7
8
2,11
10 6 5 4 9 1
c
7
c
2
c
10
c
6
c
5
c
4
c
9
c
11
c
1
, c
3
, c
8
Representation Ideals
I = I
u
1
\
I
v
1
I
u
1
= hu
53
u
52
+ ··· + 32u + 32,
5.54210 × 10
74
u
52
+ 1.00634 × 10
75
u
51
+ ··· + 4.34993 × 10
75
a + 1.36685 × 10
76
,
8.12564 × 10
74
u
52
1.64015 × 10
75
u
51
+ ··· + 4.34993 × 10
75
b 2.82274 × 10
76
i
I
v
1
= h−b
4
b
3
b + v, b
5
+ 2b
4
+ 2b
3
+ 3b
2
+ 2b + 1, ai
There are 2 irreducible components with 58 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
53
u
52
+· · ·+32u+32, 5.54×10
74
u
52
+1.01×10
75
u
51
+· · ·+4.35×10
75
a+
1.37 × 10
76
, 8.13 × 10
74
u
52
1.64 × 10
75
u
51
+ · · · + 4.35 × 10
75
b 2.82 × 10
76
i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
8
=
u
u
a
2
=
u
2
+ 1
u
2
a
11
=
0.127407u
52
0.231346u
51
+ ··· + 2.51377u 3.14224
0.186799u
52
+ 0.377052u
51
+ ··· + 0.365170u + 6.48917
a
10
=
0.127407u
52
0.231346u
51
+ ··· + 2.51377u 3.14224
0.315075u
52
+ 0.626133u
51
+ ··· 0.385768u + 9.81524
a
6
=
0.176149u
52
0.169235u
51
+ ··· + 6.50283u + 1.36817
0.129352u
52
+ 0.328232u
51
+ ··· 0.630136u + 5.95277
a
5
=
0.174728u
52
+ 0.273011u
51
+ ··· 2.46734u + 2.48051
0.164893u
52
+ 0.230733u
51
+ ··· 2.39046u + 2.37255
a
4
=
0.126769u
52
+ 0.172366u
51
+ ··· 2.09467u + 0.996624
0.0997625u
52
0.239566u
51
+ ··· 0.350812u 2.52208
a
9
=
0.0520344u
52
+ 0.146799u
51
+ ··· 1.71844u + 0.936085
0.0850408u
52
+ 0.0104995u
51
+ ··· 1.42076u 1.36118
a
1
=
0.00983469u
52
0.0422783u
51
+ ··· + 0.0768807u 0.107964
0.216696u
52
+ 0.369654u
51
+ ··· 1.66698u + 3.41074
a
1
=
0.00983469u
52
0.0422783u
51
+ ··· + 0.0768807u 0.107964
0.216696u
52
+ 0.369654u
51
+ ··· 1.66698u + 3.41074
(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 = 1.188691 0.195565I
a = 0.094810 + 0.604257I
b = 0.345791 0.896343I
5.17639 3.49609I 4.63095 + 4.25730I
u = 1.188691 + 0 .195565I
a = 0.094810 0.604257I
b = 0.345791 + 0 .896343I
5.17639 + 3 .49609I 4.63095 4.25730I
u = 0.855231 0.401653I
a = 1.36023 0 .49535I
b = 2.26240 0.51589I
2.25469 + 0 .67746I 3.01493 1.14045I
u = 0.855231 + 0 .401653I
a = 1.36023 + 0 .49535I
b = 2.26240 + 0.51589I
2.25469 0.67746I 3.01493 + 1.14045I
u = 0.664472 0.233706I
a = 1.77243 0 .98321I
b = 0.441092 + 0.997387I
5.56278 5.20660I 8.08747 + 4.37223I
u = 0.664472 + 0 .233706I
a = 1.77243 + 0 .98321I
b = 0.441092 0.997387I
5.56278 + 5 .20660I 8.08747 4.37223I
u = 0.62157 1.36984I
a = 0.526383 + 0.032505I
b = 2.16537 + 0.14998I
8.94669 + 9 .97082I 4.89802 7.16229I
u = 0.62157 + 1 .36984I
a = 0.526383 0.032505I
b = 2.16537 0.14998I
8.94669 9.97082I 4.89802 + 7.16229I
u = 0.542183 1.242343I
a = 0.371271 + 1.005181I
b = 0.989057 0.612601I
0.57753 + 4 .59375I 1.46970 3.14047I
u = 0.542183 + 1 .242343I
a = 0.371271 1.005181I
b = 0.989057 + 0 .612601I
0.57753 4.59375I 1.46970 + 3.14047I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.492745
a = 2.37990
b = 0.0734104
2.04678 5.78295
u = 0.473285 0.433941I
a = 0.852569 0.110133I
b = 0.434150 0.031178I
0.824240 + 0.967983I 5.68123 4.85193I
u = 0.473285 + 0 .433941I
a = 0.852569 + 0.110133I
b = 0.434150 + 0 .031178I
0.824240 0.967983I 5.68123 + 4.85193I
u = 0.435882 1.186922I
a = 0.589814 + 0.966814I
b = 2.85835 + 1 .34356I
2.63587 + 9 .48987I 2.19993 7.65628I
u = 0.435882 + 1 .186922I
a = 0.589814 0.966814I
b = 2.85835 1.34356I
2.63587 9.48987I 2.19993 + 7.65628I
u = 0.391683 0.808198I
a = 0.49058 + 1.44909I
b = 0.331394 + 1 .155264I
2.96531 2.94902I 0.90325 + 1.99862I
u = 0.391683 + 0 .808198I
a = 0.49058 1.44909I
b = 0.331394 1.155264I
2.96531 + 2 .94902I 0.90325 1.99862I
u = 0.37407 1.44840I
a = 0.455980 0.212206I
b = 1.92907 + 0 .40766I
10.80814 + 1.96242I 7.50155 0.08639I
u = 0.37407 + 1 .44840I
a = 0.455980 + 0.212206I
b = 1.92907 0.40766I
10.80814 1.96242I 7.50155 + 0.08639I
u = 0.296375 1.071012I
a = 0.052619 0.628437I
b = 0.238161 0.850981I
1.05950 + 2 .27821I 0.19319 2.78604I
4
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.296375 + 1 .071012I
a = 0.052619 + 0.628437I
b = 0.238161 + 0 .850981I
1.05950 2.27821I 0.19319 + 2.78604I
u = 0.183707 1.120874I
a = 0.261584 0.844218I
b = 0.68375 1.70437I
1.20946 + 2 .41405I 1.58592 4.08041I
u = 0.183707 + 1 .120874I
a = 0.261584 + 0.844218I
b = 0.68375 + 1.70437I
1.20946 2.41405I 1.58592 + 4.08041I
u = 0.112836 0.967417I
a = 0.38926 1 .45859I
b = 1.02729 + 1.05497I
2.89086 + 4 .92095I 0.55924 4.00619I
u = 0.112836 + 0 .967417I
a = 0.38926 + 1 .45859I
b = 1.02729 1.05497I
2.89086 4.92095I 0.55924 + 4.00619I
u = 0.009419 1.108929I
a = 0.610995 0.184702I
b = 2.01125 + 0.11837I
3.50201 + 0 .91026I 5.44392 1.30086I
u = 0.009419 + 1.108929I
a = 0.610995 + 0.184702I
b = 2.01125 0.11837I
3.50201 0.91026I 5.44392 + 1.30086I
u = 0.059792 1.053412I
a = 0.589497 + 0.386440I
b = 0.549337 0.381676I
3.20353 1.98495I 3.73935 + 3.73440I
u = 0.059792 + 1.053412I
a = 0.589497 0.386440I
b = 0.549337 + 0.381676I
3.20353 + 1 .98495I 3.73935 3.73440I
u = 0.090176 1.135126I
a = 0.328587 + 1.188059I
b = 2.11046 + 1.46912I
1.51168 4.33800I 0.23892 + 2.49464I
5
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.090176 + 1.135126I
a = 0.328587 1.188059I
b = 2.11046 1.46912I
1.51168 + 4 .33800I 0.23892 2.49464I
u = 0.29619 1.46580I
a = 0.018015 + 0.911867I
b = 1.72500 + 0 .48191I
6.20214 + 1 .77875I 2.88575 1.86526I
u = 0.29619 + 1.46580I
a = 0.018015 0.911867I
b = 1.72500 0.48191I
6.20214 1.77875I 2.88575 + 1.86526I
u = 0.343484 0.504413I
a = 0.432801 0.568812I
b = 0.872344 0.560597I
1.84631 + 0 .73042I 6.32026 + 1.11956I
u = 0.343484 + 0.504413I
a = 0.432801 + 0.568812I
b = 0.872344 + 0.560597I
1.84631 0.73042I 6.32026 1.11956I
u = 0.382495 1.166080I
a = 0.611148 0.022214I
b = 2.14436 + 0 .16052I
2.67007 5.91475I 2.65705 + 7.38929I
u = 0.382495 + 1.166080I
a = 0.611148 + 0.022214I
b = 2.14436 0.16052I
2.67007 + 5 .91475I 2.65705 7.38929I
u = 0.46482 1.42497I
a = 0.392089 0.602633I
b = 1.22102 1.36037I
8.05943 5.77245I 3.63249 + 4.48533I
u = 0.46482 + 1.42497I
a = 0.392089 + 0.602633I
b = 1.22102 + 1 .36037I
8.05943 + 5 .77245I 3.63249 4.48533I
u = 0.475600 0.956286I
a = 0.492810 + 1.223395I
b = 0.900856 + 0.121424I
4.99265 1.07686I 5.54603 + 2.94632I
6
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.475600 + 0.956286I
a = 0.492810 1.223395I
b = 0.900856 0.121424I
4.99265 + 1 .07686I 5.54603 2.94632I
u = 0.55462 1.35931I
a = 0.057201 0.557594I
b = 0.401082 0.573266I
7.38726 5.90070I 2.56652 + 2.20051I
u = 0.55462 + 1.35931I
a = 0.057201 + 0.557594I
b = 0.401082 + 0.573266I
7.38726 + 5 .90070I 2.56652 2.20051I
u = 0.574960 0.227723I
a = 0.059130 + 1.078041I
b = 0.132159 0.455097I
0.17010 + 2 .09238I 3.71444 4.57317I
u = 0.574960 + 0.227723I
a = 0.059130 1.078041I
b = 0.132159 + 0.455097I
0.17010 2.09238I 3.71444 + 4.57317I
u = 0.637313 0.488996I
a = 1.52789 0.89025I
b = 1.44620 0.05526I
6.40428 3.28310I 8.64938 + 3.46229I
u = 0.637313 + 0.488996I
a = 1.52789 + 0.89025I
b = 1.44620 + 0 .05526I
6.40428 + 3 .28310I 8.64938 3.46229I
u = 0.65952 1.35739I
a = 0.585757 + 0.760011I
b = 3.12350 + 1.01978I
3.5199 13.7301I 0.47152 + 7.72319I
u = 0.65952 + 1.35739I
a = 0.585757 0.760011I
b = 3.12350 1.01978I
3.5199 + 13.7301I 0.47152 7.72319I
u = 1.126197 0.022940I
a = 0.725938 0.060946I
b = 0.203956 + 0.084885I
3.24184 + 0 .00358I 1.53150 + 0.33065I
7
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.126197 + 0.022940I
a = 0.725938 + 0.060946I
b = 0.203956 0.084885I
3.24184 0.00358I 1.53150 0.33065I
u = 1.211771 0.269117I
a = 1.045860 0.460989I
b = 0.75903 + 1 .61278I
0.01422 + 7 .04483I 0.31193 4.88753I
u = 1.211771 + 0.269117I
a = 1.045860 + 0.460989I
b = 0.75903 1.61278I
0.01422 7.04483I 0.31193 + 4.88753I
8
II. I
v
1
= h−b
4
b
3
b + v, b
5
+ 2b
4
+ 2b
3
+ 3b
2
+ 2b + 1, ai
(i) Arc colorings
a
3
=
1
0
a
7
=
b
4
+ b
3
+ b
0
a
8
=
b
4
+ b
3
+ b
0
a
2
=
1
0
a
11
=
0
b
a
10
=
b
2
+ b
b
a
6
=
b
4
+ 2b
3
+ 2b
2
+ 3b + 1
b
2
+ b + 1
a
5
=
b
4
+ 2b
3
+ 2b
2
+ 3b + 1
1
a
4
=
b
4
+ 2b
3
+ 2b
2
+ 3b + 2
1
a
9
=
b
3
b
2
1
b
4
+ b
3
+ b
a
1
=
b
4
2b
3
2b
2
3b 1
1
a
1
=
b
4
2b
3
2b
2
3b 1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
9
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 0.766826
a = 0
b = 1.58802
0.756147 2.80750
v = 0.339110 0.822375I
a = 0
b = 0.438694 0.557752I
1.31583 1.53058I 0.02714 + 4.76366I
v = 0.339110 + 0.822375I
a = 0
b = 0.438694 + 0 .557752I
1.31583 + 1.53058I 0.02714 4.76366I
v = 0.455697 + 1.200152I
a = 0
b = 0.232705 1.093812I
4.22763 4.40083I 4.43089 + 2.80751I
v = 0.455697 1.200152I
a = 0
b = 0.232705 + 1.093812I
4.22763 + 4.40083I 4.43089 2.80751I
10
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
5
(u
53
+ 6u
52
+ ··· + 5u + 1)
c
2
, c
7
u
5
(u
53
+ u
52
+ ··· + 32u 32)
c
3
, c
4
(u + 1)
5
(u
53
+ 6u
52
+ ··· + 5u + 1)
c
5
(u
5
+ 3u
4
+ ··· u 1)(u
53
+ 6u
52
+ ··· + 5u + 1)
c
6
(u
5
u
4
+ ··· + u + 1)(u
53
+ 2u
52
+ ··· u + 1)
c
8
(u
5
u
4
+ ··· + u 1)(u
53
+ 2u
52
+ ··· 353u 505)
c
9
, c
10
(u
5
+ u
4
+ ··· + u 1)(u
53
+ 2u
52
+ ··· u + 1)
c
11
(u
5
u
4
+ ··· + u 1)(u
53
+ 12u
52
+ ··· + 577u + 73)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
4
(y 1)
5
(y
53
52y
52
+ ··· 3y 1)
c
2
, c
7
y
5
(y
53
+ 33y
52
+ ··· 3584y 1024)
c
5
(y
5
y
4
+ ··· + 3y 1)(y
53
+ 54y
51
+ ··· + 3y 1)
c
6
, c
9
, c
10
(y
5
5y
4
+ ··· y 1)(y
53
48y
52
+ ··· 5y 1)
c
8
(y
5
+ 3y
4
+ ··· y 1)(y
53
24y
52
+ ··· 2730661y 255025)
c
11
(y
5
+ 3y
4
+ ··· y 1)(y
53
+ 12y
52
+ ··· 67257y 5329)
12