Virtual Doodles



Updated 6th October, 2018

Virtual Doodles

Starting from labelled peer codes representing distinct link immersions we may label immersion crossings as flat or virtual in all possible combinations to obtain labelled peer codes for a set of initial virtual doodle diagrams. Each initial virtual doodle diagram may then be reduced to a minimal virtual doodle diagram by removing all Reidemeister I and Reidemeister II configurations. Note that the assignment of immersion crossing labels may result not only in "simple" Reidemeister II configurations, those designated FR2 and VR2 in [1], but also Reidemeister I or Reidemeister II detours. These are simple Reidemeister configurations where the edge(s) involved in the corresponding Reidemeister move are replaced by a (virtual) detour. In such cases, we either move, or completely remove, the detours in order to reduce the diagram by carrying out the resulting, simple, Reidemeister move. Note also that by removing Reidemeister configurations and detours, we may reduce both the number of flat and virtual crossings and it is possible that two initial doodle diagrams, possibly having a different number of (immersion) crossings, reduce to the same minimal virtual doodle diagram.

Minimal virtual doodles are defined up to detour moves, so to remove all duplicates we first remove all renumbering duplicates and then remove doodles with the same Gauss data. The removal of renumbering duplicates includes the removal of orientation reverses, so we end up with labelled peer codes for representatives of distinct equivalence classes of unoriented, minimal virtual doodle diagrams (Diagramminstrict+rev in [2]).

The lists presented below contain the the representatives of distinct unoriented, minimal virtual doodle diagrams obtained from initial virtual doodle diagrams derived from a list of all immersions (including non-prime immersions) of up to ten crossings having a single component.

The file vlist-release-29-12-17.tar contains the C++ source code for the programme vlist used to create the above lists, together with a Makefile.

The lists of distinct unoriented minimal virtual doodles were calculated by concatenating the lists of realizable link immersions with up to ten crossings into a file called realizable-non-prime-up-to-10, then executing the command line vlist -m 10 1 realizable-non-prime-up-to-10.

This produces a set of files containing the labelled peer code of each equivalence class but the lists are not indexed sequentially and the files do not show the canonical left preferred Gauss data for each class. To produce the files shown above, these initial lists were concatenated into a file doodle-input-codes and the command line vlist -ds 10 1 doodle-input-codes executed.

Virtual doodle diagrams

We present below diagrams of the distinct non-trivial virtual doodles that have been identified with three, four and five flat crossings. These diagrams were rendered by metapost using source code produced by the draw programme.

d3.1

d4.1 d4.2 d4.3

d4.4 d4.5 d4.6

d4.7 d4.8 d4.9

d4.10 d4.11 d4.12

d4.13 d4.14 d4.15

d4.16 d4.17 d4.18

d4.19

d5.1 d5.2 d5.3

d5.4 d5.5 d5.6

d5.7 d5.8 d5.9

d5.10 d5.11 d5.12

d5.13 d5.14 d5.15

d5.16 d5.17 d5.18

d5.19 d5.20 d5.21

d5.22 d5.23 d5.24

d5.25 d5.26 d5.27

d5.28 d5.29 d5.30

d5.31 d5.32 d5.33

d5.34 d5.35 d5.36

d5.37 d5.38 d5.39

d5.40 d5.41 d5.42

d5.43 d5.44 d5.45

d5.46 d5.47 d5.48

d5.49 d5.50 d5.51

d5.52 d5.53 d5.54

d5.55 d5.56 d5.57

d5.58 d5.59 d5.60

d5.61 d5.62 d5.63

d5.64 d5.65 d5.66

d5.67 d5.68 d5.69

d5.70 d5.71 d5.72

d5.73 d5.74 d5.75

d5.76 d5.77 d5.78

d5.79 d5.80 d5.81

d5.82 d5.83 d5.84

d5.85 d5.86 d5.87

d5.88 d5.89 d5.90

d5.91 d5.92 d5.93

d5.94 d5.95 d5.96

d5.97 d5.98 d5.99

d5.100 d5.101 d5.102

d5.103 d5.104 d5.105

d5.106 d5.107 d5.108

d5.109 d5.110 d5.111

d5.112 d5.113 d5.114

d5.115 d5.116 d5.117

d5.118 d5.119 d5.120

d5.121 d5.122 d5.123

d5.124 d5.125 d5.126

d5.127 d5.128 d5.129

d5.130 d5.131 d5.132

d5.133 d5.134 d5.135

d5.136 d5.137 d5.138

d5.139 d5.140 d5.141

d5.142 d5.143 d5.144

d5.145 d5.146 d5.147

d5.148 d5.149 d5.150

d5.151 d5.152 d5.153

d5.154 d5.155 d5.156

d5.157 d5.158 d5.159

d5.160 d5.161 d5.162

d5.163 d5.164 d5.165

d5.166 d5.167 d5.168

d5.169 d5.170 d5.171

d5.172 d5.173 d5.174

d5.175 d5.176 d5.177

d5.178 d5.179 d5.180

d5.181 d5.182 d5.183

d5.184 d5.185 d5.186

d5.187 d5.188 d5.189

d5.190 d5.191 d5.192

d5.193 d5.194 d5.195

d5.196 d5.197 d5.198

d5.199 d5.200 d5.201

d5.202 d5.203 d5.204

d5.205 d5.206 d5.207

d5.208 d5.209 d5.210

d5.211 d5.212 d5.213

d5.214 d5.215 d5.216

d5.217 d5.218 d5.219

d5.220 d5.221 d5.222

d5.223 d5.224 d5.225

d5.226 d5.227 d5.228

d5.229 d5.230 d5.231

d5.232 d5.233 d5.234

d5.235 d5.236 d5.237

d5.238 d5.239 d5.240

d5.241 d5.242 d5.243

d5.244 d5.245 d5.246

d5.247 d5.248 d5.249

d5.250

Planar Doodles

The first doodle that appears in the list of distinct non-trivial virtual doodles presented above is, in fact a planar doodle. It is one of a family of n-poppy doodles described on [1]. By adapting the search methodology used for virtual doodles to consider only flat crossings it has been possible to take the link immersions of up to three components described on the link immersions page and search for planar doodles.

The search results in just six planar doodles, the first four are examples of the n-poppy family for n ≥ 4, together with the associated family involving one additional crossing. These planar doodles all have one component.

The 4-poppy The 4-poppy+1

The 5-poppy The 5-poppy+1

There is only one planar doodle with two components having ten crossings or less:

    

Finally, the search identified the Borromean doodle, having three components:

    

References:

[1] A. Bartholomew, R. Fenn, N. Kamada, S. Kamada. Doodles on Surfaces, arXiv:1612.08473v2, Journal of Knot Theory and its Ramifications (to appear).

[2] A. Bartholomew, R. Fenn, N. Kamada, S. Kamada. On Gauss codes of virtual doodles, arXiv:1806.05885v1, special edition: Self-distributive system and quandle (co)homology theory in algebra and low-dimensional topology, Journal of Knot Theory and its Ramifications (to appear).

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