11n
128
(K11n
128
)
1
Arc Sequences
7 1 10 8 9 2 4 11 1 7 5
Solving Sequence
8,11 5,9
1 4 7 2 3 6 10
c
8
c
11
c
4
c
7
c
1
c
2
c
6
c
10
c
3
, c
5
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
11
4u
9
+ u
8
+ 6u
7
4u
6
3u
5
+ 4u
4
u
3
+ u
2
+ u 3,
7u
10
4u
9
+ 22u
8
29u
6
+ 17u
5
+ 14u
4
7u
3
+ 3u
2
+ 13b 9u 1,
4u
10
6u
9
+ 7u
8
+ 26u
7
24u
6
20u
5
+ 60u
4
4u
3
41u
2
+ 39a + 32u 34i
I
u
2
= hu
32
u
31
+ ··· 12u + 1, 9.89687 × 10
16
u
31
1.71588 × 10
17
u
30
+ ··· + 5.00890 × 10
17
b 1.23096 × 10
18
,
1.22348 × 10
17
u
31
+ 8.03736 × 10
17
u
30
+ ··· + 5.00890 × 10
17
a + 2.46155 × 10
18
i
There are 2 irreducible components with 43 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
11
4u
9
+· · ·+u3, 7u
10
4u
9
+· · ·+13b1, 4u
10
6u
9
+· · ·+39a34i
(i) Arc colorings
a
8
=
0
u
a
11
=
0.102564u
10
+ 0.153846u
9
+ ··· 0.820513u + 0.871795
0.538462u
10
+ 0.307692u
9
+ ··· + 0.692308u + 0.0769231
a
5
=
1
0
a
9
=
0.205128u
10
+ 0.692308u
9
+ ··· 0.358974u + 0.256410
0.461538u
10
+ 0.307692u
9
+ ··· 0.307692u + 0.0769231
a
1
=
0.435897u
10
0.153846u
9
+ ··· 1.51282u + 0.794872
0.538462u
10
+ 0.307692u
9
+ ··· + 0.692308u + 0.0769231
a
4
=
1
u
2
a
7
=
u
u
3
+ u
a
2
=
0.487179u
10
0.769231u
9
+ ··· + 1.10256u 0.358974
0.384615u
10
+ 0.0769231u
9
+ ··· 0.0769231u + 0.769231
a
3
=
1.12821u
10
1.30769u
9
+ ··· + 4.97436u 3.41026
0.153846u
10
+ 0.769231u
9
+ ··· 1.76923u + 2.69231
a
6
=
0.641026u
10
+ 0.538462u
9
+ ··· 1.87179u + 2.05128
0.461538u
10
+ 0.307692u
9
+ ··· 1.30769u + 1.07692
a
10
=
0.897436u
10
+ 0.153846u
9
+ ··· 2.82051u + 0.871795
0.538462u
10
+ 0.307692u
9
+ ··· + 1.69231u + 0.0769231
a
10
=
0.897436u
10
+ 0.153846u
9
+ ··· 2.82051u + 0.871795
0.538462u
10
+ 0.307692u
9
+ ··· + 1.69231u + 0.0769231
(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 = 1.303899 0.374956I
a = 0.636425 + 0.878595I
b = 1.059853 + 0.766199I
3.56282 6.44913I 4.62110 + 5.90724I
u = 1.303899 + 0.374956I
a = 0.636425 0.878595I
b = 1.059853 0.766199I
3.56282 + 6.44913I 4.62110 5.90724I
u = 1.113084 0.252147I
a = 1.243187 + 0.126313I
b = 0.266277 + 0.765184I
12.65170 1.14869I 5.06516 0.05051I
u = 1.113084 + 0.252147I
a = 1.243187 0.126313I
b = 0.266277 0.765184I
12.65170 + 1.14869I 5.06516 + 0.05051I
u = 0.064226 0.786482I
a = 0.576520 + 0.820028I
b = 0.735500 + 0.606796I
0.44585 + 2.19055I 0.27735 4.50255I
u = 0.064226 + 0.786482I
a = 0.576520 0.820028I
b = 0.735500 0.606796I
0.44585 2.19055I 0.27735 + 4.50255I
u = 0.704223 0.799205I
a = 0.680741 0.256158I
b = 0.482497 + 0.432213I
0.834177 + 0.664427I 5.62999 + 1.84817I
u = 0.704223 + 0.799205I
a = 0.680741 + 0.256158I
b = 0.482497 0.432213I
0.834177 0.664427I 5.62999 1.84817I
u = 1.130351 0.302780I
a = 0.114420 + 1.030366I
b = 0.722191 + 1.091394I
2.43632 + 3.36377I 4.69391 3.63598I
u = 1.130351 + 0.302780I
a = 0.114420 1.030366I
b = 0.722191 1.091394I
2.43632 3.36377I 4.69391 + 3.63598I
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.29327
a = 0.317476
b = 1.36323
4.68995 8.42499
3
II. I
u
2
=
hu
32
u
31
+ · · · 12u + 1, 9.90 × 10
16
u
31
1.72 × 10
17
u
30
+ · · · + 5.01 × 10
17
b
1.23 × 10
18
, 1.22 × 10
17
u
31
+ 8.04 × 10
17
u
30
+ · · · + 5.01 × 10
17
a + 2.46 × 10
18
i
(i) Arc colorings
a
8
=
0
u
a
11
=
0.244262u
31
1.60461u
30
+ ··· + 51.1148u 4.91435
0.197586u
31
+ 0.342566u
30
+ ··· 13.4573u + 2.45755
a
5
=
1
0
a
9
=
3.39545u
31
1.12542u
30
+ ··· + 65.7600u 9.37715
0.771918u
31
0.0178954u
30
+ ··· + 15.0150u 1.00246
a
1
=
0.0466763u
31
1.94718u
30
+ ··· + 64.5722u 7.37190
0.197586u
31
+ 0.342566u
30
+ ··· 13.4573u + 2.45755
a
4
=
1
u
2
a
7
=
u
u
3
+ u
a
2
=
0.590773u
31
1.76202u
30
+ ··· + 59.4915u 6.72272
0.199663u
31
+ 0.213550u
30
+ ··· 14.7748u + 2.74780
a
3
=
4.28025u
31
+ 0.484730u
30
+ ··· + 54.0490u 3.93592
2.66133u
31
0.797429u
30
+ ··· 30.9601u + 3.50979
a
6
=
1.93811u
31
0.942980u
30
+ ··· 25.2538u + 6.08360
1.07477u
31
+ 0.752123u
30
+ ··· + 0.523274u + 0.239711
a
10
=
0.828985u
31
1.68179u
30
+ ··· + 60.1743u 5.88389
0.589274u
31
+ 0.367301u
30
+ ··· 15.2775u + 2.48408
a
10
=
0.828985u
31
1.68179u
30
+ ··· + 60.1743u 5.88389
0.589274u
31
+ 0.367301u
30
+ ··· 15.2775u + 2.48408
(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 = 1.43379 0.69251I
a = 0.437105 + 0.491138I
b = 1.042802 0.178645I
15.2091 0.2793I 8.54524 0.16406I
u = 1.43379 + 0.69251I
a = 0.437105 0.491138I
b = 1.042802 + 0.178645I
15.2091 + 0.2793I 8.54524 + 0.16406I
u = 1.36987 0.42119I
a = 0.157634 0.791867I
b = 1.37290 0.81577I
5.87267 7.08941I 7.09677 + 5.04780I
u = 1.36987 + 0.42119I
a = 0.157634 + 0.791867I
b = 1.37290 + 0.81577I
5.87267 + 7.08941I 7.09677 5.04780I
u = 1.295736 0.161402I
a = 0.125650 0.659439I
b = 1.19979 1.83348I
14.4931 2.6399I 9.17083 + 2.92028I
u = 1.295736 + 0.161402I
a = 0.125650 + 0.659439I
b = 1.19979 + 1.83348I
14.4931 + 2.6399I 9.17083 2.92028I
u = 1.223502 0.038660I
a = 0.702725 + 0.804451I
b = 0.815453 + 0.142941I
5.17336 1.79108I 9.36406 + 4.29945I
u = 1.223502 + 0.038660I
a = 0.702725 0.804451I
b = 0.815453 0.142941I
5.17336 + 1.79108I 9.36406 4.29945I
u = 1.197228 0.303844I
a = 0.499776 + 1.233171I
b = 0.874051 + 0.767936I
1.91400 4.83866I 2.54787 + 6.95016I
u = 1.197228 + 0.303844I
a = 0.499776 1.233171I
b = 0.874051 0.767936I
1.91400 + 4.83866I 2.54787 6.95016I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.170624 1.162780I
a = 0.865413 0.778375I
b = 1.081778 0.698388I
11.45544 6.47625I 4.15728 + 4.26891I
u = 0.170624 + 1.162780I
a = 0.865413 + 0.778375I
b = 1.081778 + 0.698388I
11.45544 + 6.47625I 4.15728 4.26891I
u = 0.162858 0.639240I
a = 1.079956 + 0.870764I
b = 0.529074 + 0.422527I
1.26710 + 1.27122I 4.54373 1.02158I
u = 0.162858 + 0.639240I
a = 1.079956 0.870764I
b = 0.529074 0.422527I
1.26710 1.27122I 4.54373 + 1.02158I
u = 0.082800 0.471723I
a = 1.44494 2.58768I
b = 0.762439 0.930035I
10.18922 + 0.37972I 2.16269 + 0.17573I
u = 0.082800 + 0.471723I
a = 1.44494 + 2.58768I
b = 0.762439 + 0.930035I
10.18922 0.37972I 2.16269 0.17573I
u = 0.138542 1.056330I
a = 0.700432 0.165424I
b = 0.733437 0.365394I
1.06384 + 2.01989I 6.33778 3.45023I
u = 0.138542 + 1.056330I
a = 0.700432 + 0.165424I
b = 0.733437 + 0.365394I
1.06384 2.01989I 6.33778 + 3.45023I
u = 0.164476 0.076390I
a = 2.34687 3.05088I
b = 0.716191 + 0.638089I
1.10961 1.48762I 5.18747 + 2.41383I
u = 0.164476 + 0.076390I
a = 2.34687 + 3.05088I
b = 0.716191 0.638089I
1.10961 + 1.48762I 5.18747 2.41383I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.535771 0.550945I
a = 0.0747958 0.0848161I
b = 0.049461 0.604800I
0.34099 + 1.52893I 0.59741 5.96300I
u = 0.535771 + 0.550945I
a = 0.0747958 + 0.0848161I
b = 0.049461 + 0.604800I
0.34099 1.52893I 0.59741 + 5.96300I
u = 1.195331 0.230217I
a = 0.143715 0.578971I
b = 1.100620 0.617950I
2.64686 + 1.18437I 3.89291 0.66467I
u = 1.195331 + 0.230217I
a = 0.143715 + 0.578971I
b = 1.100620 + 0.617950I
2.64686 1.18437I 3.89291 + 0.66467I
u = 1.219351 0.077393I
a = 0.447699 + 0.856804I
b = 1.37078 + 1.24326I
4.27477 + 2.33689I 7.63645 2.31412I
u = 1.219351 + 0.077393I
a = 0.447699 0.856804I
b = 1.37078 1.24326I
4.27477 2.33689I 7.63645 + 2.31412I
u = 1.224974 0.159550I
a = 1.98616 + 0.81810I
b = 0.670400 0.174363I
13.62097 + 1.88059I 11.51732 4.17618I
u = 1.224974 + 0.159550I
a = 1.98616 0.81810I
b = 0.670400 + 0.174363I
13.62097 1.88059I 11.51732 + 4.17618I
u = 1.34905 0.49788I
a = 0.052323 + 0.917678I
b = 0.839959 + 0.308002I
5.07213 + 3.71163I 7.67058 3.05284I
u = 1.34905 + 0.49788I
a = 0.052323 0.917678I
b = 0.839959 0.308002I
5.07213 3.71163I 7.67058 + 3.05284I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.44332 0.48895I
a = 0.265560 1.125697I
b = 1.38562 0.96493I
16.5743 + 12.2619I 6.15908 5.48895I
u = 1.44332 + 0.48895I
a = 0.265560 + 1.125697I
b = 1.38562 + 0.96493I
16.5743 12.2619I 6.15908 + 5.48895I
7
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
11
+ 6u
9
+ 13u
7
+ 15u
5
+ u
4
+ 9u
3
+ 2u
2
+ 3u + 1)
(u
32
+ u
31
+ ··· 20u + 1)
c
2
(u
11
+ 12u
10
+ ··· + 5u 1)(u
32
+ 47u
31
+ ··· 80u + 1)
c
3
(u
11
u
10
3u
9
3u
8
+ u
7
+ 7u
6
+ 12u
5
+ 15u
4
+ 11u
3
+ 7u
2
+ 3u + 1)
(u
32
28u
30
+ ··· + 154u + 43)
c
4
(u
11
4u
9
+ u
8
+ 6u
7
4u
6
3u
5
+ 4u
4
u
3
+ u
2
+ u 3)
(u
32
+ u
31
+ ··· + 12u + 1)
c
5
(u
11
u
10
+ u
9
u
8
+ 4u
7
+ u
5
4u
4
+ 2u
3
+ u 1)
(u
32
+ 2u
31
+ ··· 2606u + 1291)
c
6
(u
11
+ 6u
9
+ 13u
7
+ 15u
5
u
4
+ 9u
3
2u
2
+ 3u 1)
(u
32
+ u
31
+ ··· 20u + 1)
c
7
(u
11
4u
9
u
8
+ 6u
7
+ 4u
6
3u
5
4u
4
u
3
u
2
+ u + 3)
(u
32
+ u
31
+ ··· + 12u + 1)
c
8
(u
11
+ 2u
10
+ u
9
4u
8
6u
7
2u
6
+ 9u
5
+ 13u
4
+ 9u
3
+ 2u
2
u 1)
(u
32
+ 9u
31
+ ··· + 26u + 1)
c
9
(u
11
9u
10
+ ··· + 17u 3)(u
32
+ 4u
31
+ ··· 2696u + 589)
c
10
(u
11
+ 2u
10
+ 5u
9
+ 4u
8
+ u
7
u
6
6u
5
+ 4u
4
2u
3
+ 3u
2
3u + 1)
(u
32
+ 3u
31
+ ··· + 7138u + 3929)
c
11
(u
11
+ u
10
+ 2u
8
+ 4u
7
+ u
6
+ 4u
4
+ u
3
+ u
2
+ u + 1)
(u
32
+ 2u
31
+ ··· + 34u + 19)
8
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y
11
+ 12y
10
+ ··· + 5y 1)(y
32
+ 47y
31
+ ··· 80y + 1)
c
2
(y
11
20y
10
+ ··· + 121y 1)(y
32
117y
31
+ ··· + 5428y + 1)
c
3
(y
11
7y
10
+ ··· 5y 1)(y
32
56y
31
+ ··· + 323982y + 1849)
c
4
, c
7
(y
11
8y
10
+ ··· + 7y 9)(y
32
25y
31
+ ··· 34y + 1)
c
5
(y
11
+ y
10
+ 7y
9
+ 9y
8
+ 14y
7
+ 6y
6
+ 17y
5
6y
4
+ 6y
3
4y
2
+ y 1)
(y
32
+ 24y
31
+ ··· + 12176136y + 1666681)
c
8
(y
11
2y
10
+ 5y
9
2y
8
+ 4y
7
+ 43y
5
+ 5y
4
+ 7y
3
+ 4y
2
+ 5y 1)
(y
32
+ 9y
31
+ ··· 248y + 1)
c
9
(y
11
9y
10
+ ··· 35y 9)(y
32
62y
31
+ ··· + 1792760y + 346921)
c
10
(y
11
+ 6y
10
+ 11y
9
14y
8
71y
7
83y
6
18y
5
+ 18y
3
5y
2
+ 3y 1)
(y
32
+ 37y
31
+ ··· + 205149034y + 15437041)
c
11
(y
11
y
10
+ 4y
9
6y
8
+ 6y
7
17y
6
6y
5
14y
4
9y
3
7y
2
y 1)
(y
32
10y
31
+ ··· 3322y + 361)
9