11n
168
(K11n
168
)
1
Arc Sequences
7 5 1 9 2 9 3 11 7 4 8
Solving Sequence
8,11
9
1,3
4 5 2 6 7 10
c
8
c
11
c
3
c
4
c
2
c
5
c
7
c
10
c
1
, c
6
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
12
+ u
10
2u
9
+ 4u
8
3u
7
+ 5u
6
5u
5
+ 6u
4
3u
3
+ 3u
2
3u + 1,
4u
11
13u
10
15u
9
+ 3u
8
13u
6
31u
5
+ 13u
4
13u
3
+ u
2
+ 25b 15u + 7,
19u
11
7u
10
+ 15u
9
58u
8
+ 75u
7
82u
6
+ 116u
5
143u
4
+ 118u
3
86u
2
+ 25a + 65u 77i
I
u
2
= hu
24
u
23
+ ··· 4u + 1, 7.13253 × 10
25
u
23
2.22737 × 10
25
u
22
+ ··· + 2.25715 × 10
25
b 1.14477 × 10
26
,
1.78850 × 10
27
u
23
9.59380 × 10
26
u
22
+ ··· + 5.64286 × 10
26
a 4.28995 × 10
27
i
I
u
3
= hu
25
7u
23
+ ··· 8u + 1,
20025629876734u
24
+ 32815384984813u
23
+ ··· + 605049147419665b + 74356614651533,
239751606214929u
24
74356614651533u
23
+ ··· + 605049147419665a 4360379155201633i
There are 3 irreducible components with 61 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
12
+ u
10
+ · · · 3u + 1, 4u
11
13u
10
+ · · · + 25b + 7, 19u
11
7u
10
+ · · · + 25a 77i
(i) Arc colorings
a
8
=
0
u
a
11
=
0.760000u
11
+ 0.280000u
10
+ ··· 2.60000u + 3.08000
4
25
u
11
+
13
25
u
10
+ ··· +
3
5
u
7
25
a
9
=
2
5
u
11
+
1
5
u
10
+ ··· +
13
5
u
2
+
11
5
6
25
u
11
+
7
25
u
10
+ ··· +
2
5
u +
2
25
a
1
=
0.760000u
11
+ 0.280000u
10
+ ··· 2.60000u + 3.08000
u
a
3
=
1
0
a
4
=
7
25
u
11
4
25
u
10
+ ···
4
5
u +
6
25
u
2
a
5
=
2.68000u
11
1.96000u
10
+ ··· 6.80000u + 3.44000
0.160000u
11
0.520000u
10
+ ··· 1.60000u + 1.28000
a
2
=
3
5
u
11
+
4
5
u
10
+ ··· u +
14
5
4
25
u
11
+
13
25
u
10
+ ··· +
3
5
u
7
25
a
6
=
6
5
u
11
7
5
u
10
+ ··· 2u
2
5
0.680000u
11
0.960000u
10
+ ··· + 0.200000u + 0.440000
a
7
=
u
u
a
10
=
0.640000u
11
0.0800000u
10
+ ··· 0.400000u + 2.12000
0
a
10
=
0.640000u
11
0.0800000u
10
+ ··· 0.400000u + 2.12000
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 = 1.02625 1.07318I
a = 0.379524 + 0.699564I
b = 0.55207 + 1.84022I
1.85485 5.48541I 0.93921 + 8.00832I
u = 1.02625 + 1.07318I
a = 0.379524 0.699564I
b = 0.55207 1.84022I
1.85485 + 5.48541I 0.93921 8.00832I
u = 0.603205 0.775464I
a = 0.818055 + 0.258374I
b = 0.555767 0.051210I
4.01613 1.54035I 3.24789 1.08803I
u = 0.603205 + 0.775464I
a = 0.818055 0.258374I
b = 0.555767 + 0.051210I
4.01613 + 1.54035I 3.24789 + 1.08803I
u = 0.351412 0.975594I
a = 0.973934 + 0.121616I
b = 0.53868 + 1.54266I
5.37548 + 4.45869I 5.23379 2.73299I
u = 0.351412 + 0.975594I
a = 0.973934 0.121616I
b = 0.53868 1.54266I
5.37548 4.45869I 5.23379 + 2.73299I
u = 0.504389 0.932104I
a = 1.035620 + 0.479382I
b = 0.58266 + 1.61135I
6.48342 + 2.87353I 6.21901 3.05514I
u = 0.504389 + 0.932104I
a = 1.035620 0.479382I
b = 0.58266 1.61135I
6.48342 2.87353I 6.21901 + 3.05514I
u = 0.522926 0.172175I
a = 2.24735 + 0.14288I
b = 0.050723 0.197669I
1.62052 0.67051I 10.59498 6.84644I
u = 0.522926 + 0.172175I
a = 2.24735 0.14288I
b = 0.050723 + 0.197669I
1.62052 + 0.67051I 10.59498 + 6.84644I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.953548 0.611838I
a = 0.247131 + 1.274676I
b = 0.40159 + 1.58203I
2.21233 + 5.51031I 2.13932 5.32316I
u = 0.953548 + 0.611838I
a = 0.247131 1.274676I
b = 0.40159 1.58203I
2.21233 5.51031I 2.13932 + 5.32316I
4
II.
I
u
2
= hu
24
u
23
+· · ·4u+1, 7.13×10
25
u
23
2.23×10
25
u
22
+· · ·+2.26×10
25
b
1.14 × 10
26
, 1.79 × 10
27
u
23
9.59 × 10
26
u
22
+ · · · + 5.64 × 10
26
a 4.29 × 10
27
i
(i) Arc colorings
a
8
=
0
u
a
11
=
3.16949u
23
+ 1.70016u
22
+ ··· 12.4493u + 7.60243
3.15998u
23
+ 0.986808u
22
+ ··· 11.1571u + 5.07176
a
9
=
0.861960u
23
+ 0.299637u
22
+ ··· + 1.65849u + 2.10829
4.80135u
23
1.75071u
22
+ ··· + 18.8922u 6.65574
a
1
=
3.16949u
23
+ 1.70016u
22
+ ··· 12.4493u + 7.60243
2.16949u
23
+ 0.700165u
22
+ ··· 8.44930u + 3.60243
a
3
=
1
0
a
4
=
5.49414u
23
1.46269u
22
+ ··· + 20.0162u 6.86876
9.09657u
23
2.89563u
22
+ ··· + 34.9044u 13.8292
a
5
=
3.63374u
23
+ 2.73697u
22
+ ··· 21.4409u + 11.6546
0.641384u
23
+ 0.526275u
22
+ ··· 2.70213u + 1.98666
a
2
=
3.16949u
23
+ 1.70016u
22
+ ··· 11.4493u + 7.60243
2.16949u
23
+ 0.700165u
22
+ ··· 7.44930u + 3.60243
a
6
=
2.86671u
23
0.0380303u
22
+ ··· + 5.51479u 0.352725
3.87240u
23
0.903452u
22
+ ··· + 14.5861u 4.77613
a
7
=
u
u
a
10
=
2.07983u
23
0.101523u
22
+ ··· + 5.27527u + 0.219252
6.01922u
23
2.15187u
22
+ ··· + 22.5090u 8.54478
a
10
=
2.07983u
23
0.101523u
22
+ ··· + 5.27527u + 0.219252
6.01922u
23
2.15187u
22
+ ··· + 22.5090u 8.54478
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
5
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.63727 1.77021I
a = 0.377830 + 0.348770I
b = 1.53343 + 2.07647I
7.81112 2.53747I 14.4387 + 1.7127I
u = 1.63727 + 1.77021I
a = 0.377830 0.348770I
b = 1.53343 2.07647I
7.81112 + 2.53747I 14.4387 1.7127I
u = 1.007826 0.931272I
a = 0.433427 + 0.982322I
b = 0.81094 + 1.44716I
2.89796 6.22910I 8.04009 + 11.28166I
u = 1.007826 + 0.931272I
a = 0.433427 0.982322I
b = 0.81094 1.44716I
2.89796 + 6.22910I 8.04009 11.28166I
u = 0.708011 0.906125I
a = 0.293372 0.135901I
b = 0.909352 0.401261I
3.58234 3.33657I 2.17703 + 1.92424I
u = 0.708011 + 0.906125I
a = 0.293372 + 0.135901I
b = 0.909352 + 0.401261I
3.58234 + 3.33657I 2.17703 1.92424I
u = 0.625850 0.653229I
a = 0.516479 + 1.293486I
b = 0.205505 + 0.872305I
1.08693 4.46082I 0.35199 + 4.72827I
u = 0.625850 + 0.653229I
a = 0.516479 1.293486I
b = 0.205505 0.872305I
1.08693 + 4.46082I 0.35199 4.72827I
u = 0.624755 0.225337I
a = 0.34040 1.70022I
b = 1.43910 2.88422I
6.29691 5.40399I 10.52298 + 8.56336I
u = 0.624755 + 0.225337I
a = 0.34040 + 1.70022I
b = 1.43910 + 2.88422I
6.29691 + 5.40399I 10.52298 8.56336I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.097566 0.372381I
a = 0.27117 + 2.62197I
b = 0.231733 0.249277I
1.93604 0.91968I 3.53074 + 7.18196I
u = 0.097566 + 0.372381I
a = 0.27117 2.62197I
b = 0.231733 + 0.249277I
1.93604 + 0.91968I 3.53074 7.18196I
u = 0.387233 0.109050I
a = 2.29511 + 1.04619I
b = 0.307647 + 0.322980I
1.93604 + 0.91968I 3.53074 7.18196I
u = 0.387233 + 0.109050I
a = 2.29511 1.04619I
b = 0.307647 0.322980I
1.93604 0.91968I 3.53074 + 7.18196I
u = 0.659423 0.044313I
a = 0.12761 + 1.87166I
b = 1.58313 + 2.58787I
7.81112 + 2.53747I 14.4387 1.7127I
u = 0.659423 + 0.044313I
a = 0.12761 1.87166I
b = 1.58313 2.58787I
7.81112 2.53747I 14.4387 + 1.7127I
u = 0.828793 0.821143I
a = 0.099131 0.302865I
b = 1.121497 0.775020I
3.58234 3.33657I 2.17703 + 1.92424I
u = 0.828793 + 0.821143I
a = 0.099131 + 0.302865I
b = 1.121497 + 0.775020I
3.58234 + 3.33657I 2.17703 1.92424I
u = 0.968657 0.487749I
a = 0.184274 + 1.345956I
b = 0.13493 + 2.04810I
2.89796 + 6.22910I 8.04009 11.28166I
u = 0.968657 + 0.487749I
a = 0.184274 1.345956I
b = 0.13493 2.04810I
2.89796 6.22910I 8.04009 + 11.28166I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.07597 2.21108I
a = 0.446808 + 0.140338I
b = 1.71010 + 1.37700I
6.29691 5.40399I 10.52298 + 8.56336I
u = 1.07597 + 2.21108I
a = 0.446808 0.140338I
b = 1.71010 1.37700I
6.29691 + 5.40399I 10.52298 8.56336I
u = 1.281207 0.495305I
a = 0.053181 + 0.915737I
b = 0.61063 + 1.86425I
1.08693 + 4.46082I 0.35199 4.72827I
u = 1.281207 + 0.495305I
a = 0.053181 0.915737I
b = 0.61063 1.86425I
1.08693 4.46082I 0.35199 + 4.72827I
8
III. I
u
3
=
hu
25
7u
23
+· · ·8u+1, 2.00×10
13
u
24
+3.28×10
13
u
23
+· · ·+6.05×10
14
b+
7.44 × 10
13
, 2.40 × 10
14
u
24
7.44 × 10
13
u
23
+ · · · + 6.05 × 10
14
a 4.36 × 10
15
i
(i) Arc colorings
a
8
=
0
u
a
11
=
0.396251u
24
+ 0.122894u
23
+ ··· 11.2978u + 7.20665
0.0330975u
24
0.0542359u
23
+ ··· + 2.37940u 0.122894
a
9
=
0.417628u
24
+ 0.795035u
23
+ ··· + 7.05883u + 4.93555
0.231391u
24
+ 0.287294u
23
+ ··· + 8.32205u 0.917929
a
1
=
0.396251u
24
+ 0.122894u
23
+ ··· 11.2978u + 7.20665
u
a
3
=
1
0
a
4
=
0.122894u
24
+ 0.0330975u
23
+ ··· + 4.03664u + 1.39625
u
2
a
5
=
2.06508u
24
+ 0.405501u
23
+ ··· + 46.6705u 2.21745
0.142970u
24
0.157822u
23
+ ··· + 2.29680u 0.772813
a
2
=
0.363154u
24
+ 0.0686576u
23
+ ··· 9.91837u + 7.08376
0.0330975u
24
0.0542359u
23
+ ··· + 2.37940u 0.122894
a
6
=
0.190330u
24
0.0224550u
23
+ ··· + 17.1015u + 2.98856
0.186231u
24
0.288318u
23
+ ··· + 1.09172u 0.332384
a
7
=
u
u
a
10
=
0.227555u
24
+ 0.490589u
23
+ ··· + 3.18314u + 5.44330
0.0413182u
24
0.0171523u
23
+ ··· + 4.44636u 0.410187
a
10
=
0.227555u
24
+ 0.490589u
23
+ ··· + 3.18314u + 5.44330
0.0413182u
24
0.0171523u
23
+ ··· + 4.44636u 0.410187
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
9
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.36183 0.94506I
a = 0.263317 0.832190I
b = 0.52709 2.42294I
9.79096 7.40216I 7.26136 + 3.90619I
u = 1.36183 + 0.94506I
a = 0.263317 + 0.832190I
b = 0.52709 + 2.42294I
9.79096 + 7.40216I 7.26136 3.90619I
u = 1.113724 0.401444I
a = 0.234408 0.754921I
b = 0.69166 1.42578I
3.12369 0.06362I 5.51965 0.12740I
u = 1.113724 + 0.401444I
a = 0.234408 + 0.754921I
b = 0.69166 + 1.42578I
3.12369 + 0.06362I 5.51965 + 0.12740I
u = 1.055804 0.669349I
a = 0.469212 0.757255I
b = 0.29832 1.83739I
3.62398 0.55768I 7.12692 + 1.89809I
u = 1.055804 + 0.669349I
a = 0.469212 + 0.757255I
b = 0.29832 + 1.83739I
3.62398 + 0.55768I 7.12692 1.89809I
u = 1.019350 0.663007I
a = 0.593594 0.596790I
b = 0.143171 0.218436I
4.81720 2.41116I 4.79174 + 2.48268I
u = 1.019350 + 0.663007I
a = 0.593594 + 0.596790I
b = 0.143171 + 0.218436I
4.81720 + 2.41116I 4.79174 2.48268I
u = 1.015419 0.130266I
a = 0.83193 + 1.20740I
b = 0.491171 + 1.314251I
7.96688 + 3.80546I 9.00823 2.72232I
u = 1.015419 + 0.130266I
a = 0.83193 1.20740I
b = 0.491171 1.314251I
7.96688 3.80546I 9.00823 + 2.72232I
10
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.578442
a = 0.243739
b = 0.496888
0.898624 11.0258
u = 0.149236 0.123502I
a = 4.88788 + 2.29476I
b = 0.268131 0.287572I
1.91022 0.92265I 6.06720 + 6.51070I
u = 0.149236 + 0.123502I
a = 4.88788 2.29476I
b = 0.268131 + 0.287572I
1.91022 + 0.92265I 6.06720 6.51070I
u = 0.647135 0.820882I
a = 0.536581 0.115750I
b = 0.661020 0.221272I
3.61375 + 2.05944I 4.90254 7.19693I
u = 0.647135 + 0.820882I
a = 0.536581 + 0.115750I
b = 0.661020 + 0.221272I
3.61375 2.05944I 4.90254 + 7.19693I
u = 0.657268 0.350420I
a = 0.94713 1.32247I
b = 0.340946 1.195624I
1.98209 2.65903I 3.09005 + 4.34714I
u = 0.657268 + 0.350420I
a = 0.94713 + 1.32247I
b = 0.340946 + 1.195624I
1.98209 + 2.65903I 3.09005 4.34714I
u = 0.907003 0.057306I
a = 0.69801 + 1.60596I
b = 0.50201 + 1.33113I
8.86649 + 2.64359I 10.06958 2.50086I
u = 0.907003 + 0.057306I
a = 0.69801 1.60596I
b = 0.50201 1.33113I
8.86649 2.64359I 10.06958 + 2.50086I
u = 0.991209 0.696352I
a = 0.766008 0.588258I
b = 0.202015 + 0.068381I
4.29953 + 9.06645I 3.72304 7.02542I
11
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.991209 + 0.696352I
a = 0.766008 + 0.588258I
b = 0.202015 0.068381I
4.29953 9.06645I 3.72304 + 7.02542I
u = 1.12346 1.07222I
a = 0.377394 0.626248I
b = 0.34283 2.05189I
1.38725 + 4.87941I 6.34329 + 0.12140I
u = 1.12346 + 1.07222I
a = 0.377394 + 0.626248I
b = 0.34283 + 2.05189I
1.38725 4.87941I 6.34329 0.12140I
u = 1.38003 0.93988I
a = 0.207747 0.871241I
b = 0.66793 2.36844I
8.8717 + 15.0132I 5.71790 7.82964I
u = 1.38003 + 0.93988I
a = 0.207747 + 0.871241I
b = 0.66793 + 2.36844I
8.8717 15.0132I 5.71790 + 7.82964I
12
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u + 1)
24
(u
12
3u
11
+ ··· + 2u + 5)
(u
25
22u
24
+ ··· + 40960u 4096)
c
2
(u
12
5u
11
+ ··· 3u + 1)
(1 + 3u
2
+ 3u
3
+ 6u
4
+ 8u
5
+ 12u
6
+ 16u
7
+ 21u
8
+ 20u
9
+ 13u
10
+ 5u
11
+ u
12
)
2
(u
25
8u
24
+ ··· 3u + 2)
c
3
, c
7
(u
12
+ u
10
+ 2u
9
+ 4u
8
+ 3u
7
+ 5u
6
+ 5u
5
+ 6u
4
+ 3u
3
+ 3u
2
+ 3u + 1)
(u
24
+ u
23
+ ··· + 4u + 1)(u
25
7u
23
+ ··· 8u 1)
c
4
(u
12
u
11
+ ··· + 11u + 5)(u
24
u
23
+ ··· 162u + 27)
(u
25
+ u
24
+ ··· 80u 85)
c
5
1
(1 + 3u
2
+ 3u
3
+ 6u
4
+ 8u
5
+ 12u
6
+ 16u
7
+ 21u
8
+ 20u
9
+ 13u
10
+ 5u
11
+ u
12
)
2
(u
12
+ 5u
11
+ ··· + 3u + 1)(u
25
8u
24
+ ··· 3u + 2)
c
6
(u
12
+ u
10
3u
8
u
7
2u
6
3u
5
+ 9u
4
+ 2u
3
4u
2
+ 1)
(u
24
+ 3u
23
+ ··· + 400u + 109)(u
25
+ 15u
23
+ ··· u + 1)
c
8
1
(1 + 2u u
2
3u
3
+ 10u
4
22u
5
+ 28u
6
30u
7
+ 25u
8
16u
9
+ 9u
10
3u
11
+ u
12
)
2
(u
12
+ 5u
11
+ ··· + 20u + 5)(u
25
+ 8u
24
+ ··· + 55u + 4)
c
9
(u
12
+ u
10
3u
8
+ u
7
2u
6
+ 3u
5
+ 9u
4
2u
3
4u
2
+ 1)
(u
24
+ 3u
23
+ ··· + 400u + 109)(u
25
+ 15u
23
+ ··· u + 1)
c
10
(u
12
4u
10
+ u
9
+ 9u
8
u
7
10u
6
+ u
5
+ 7u
4
4u
3
+ 2u
2
2u + 1)
(u
24
+ u
23
+ ··· 774u + 135)(u
25
8u
23
+ ··· 18u + 28)
c
11
(u
12
5u
11
+ ··· 20u + 5)
(1 + 2u u
2
3u
3
+ 10u
4
22u
5
+ 28u
6
30u
7
+ 25u
8
16u
9
+ 9u
10
3u
11
+ u
12
)
2
(u
25
+ 8u
24
+ ··· + 55u + 4)
13
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
(y 1)
24
(y
12
3y
11
+ ··· 104y + 25)
(y
25
4y
24
+ ··· + 92274688y 16777216)
c
2
, c
5
1
(1 + 6y + 21y
2
+ 51y
3
+ 102y
4
+ 136y
5
+ 100y
6
+ 60y
7
+ 45y
8
+ 10y
9
+ 11y
10
+ y
11
+ y
12
)
2
(y
12
+ 7y
11
+ ··· + 11y + 1)(y
25
+ 8y
24
+ ··· 51y 4)
c
3
, c
7
(y
12
+ 2y
11
+ ··· 3y + 1)(y
24
+ 3y
23
+ ··· 8y + 1)
(y
25
14y
24
+ ··· + 22y 1)
c
4
(y
12
7y
11
+ ··· 11y + 25)(y
24
21y
23
+ ··· + 93636y + 729)
(y
25
23y
24
+ ··· + 73550y 7225)
c
6
, c
9
(y
12
+ 2y
11
+ ··· 8y + 1)(y
24
+ 15y
23
+ ··· 39228y + 11881)
(y
25
+ 30y
24
+ ··· + 3y 1)
c
8
, c
11
(y
12
+ 7y
11
+ ··· + 70y + 25)
(1 6y + 33y
2
+ 115y
3
+ 82y
4
72y
5
136y
6
44y
7
+ 57y
8
+ 70y
9
+ 35y
10
+ 9y
11
+ y
12
)
2
(y
25
+ 16y
24
+ ··· + 753y 16)
c
10
(y
12
8y
11
+ ··· + 2y
2
+ 1)(y
24
17y
23
+ ··· 49896y + 18225)
(y
25
16y
24
+ ··· + 4188y 784)
14