Updated 18th September, 2023.
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]).
A consequence of the search algorithm is that, having distinguished a set of virtual doodles with k crossings from immersions with n crossings, one cannot discount the possibility of finding another distinct doodle with k crossings from an immersion having m > n crossings. A virtual doodle with k crossings will only be encountered by the search algorithm when considering immersions with k+v crossings, where v is the virtual crossing number of the doodle.
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.
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 
It is possible to adapt the search methodology used for virtual doodles to consider only flat crossings and search for planar doodles but such an approach is limited in its success and is also very time consuming. Consideration of the link immersions of up to three components described on the link immersions page using this approach results in just six planar doodles.
Given that every doodle is the shadow of a knot, one can also search for minimal planar doodles by considering the shadow of the prime knots listed in the Thistlethwaite-Hoste tables included in knotscape. Lists of the Gauss codes and planar diagram data for these knots may be found on the Gauss codes page. Whilst this is a fast and effective method of identifying planar doodles, it is limited to doodles with one component and it is currently unkown whether there is a non-trivial planar doodle that is not the shadow of a non-trivial knot.
An enumeration of all reduced prime minimal planar doodles is possible by searching for the dual graph of a doodle as described in [3]. In this context prime means that a planar doodle is 3-connected when regarded as a 4-valent graph, we also distinguish as super-prime those planar doodles that are 4-connected.
The software used for the search is available on the search software page, together with a Makefile. Note that this software requires the use of the Open Graph Drawing Framework (OGDF) [4] to test for planarity of dual graphs and to provide a planar embedding of planar dual graphs. I am deeply grateful to the authors of the OGDF for making their work publicly available and my work much easier! For this project the version of OGDF used was Dogwood, February 2nd, 2022. It also requires access, locally or via a symbolic link, to the braid programme in order to de-duplicate the lists it produces.
As described in [3], dual graphs for prime doodles are constructed from a disc with an even number of vertices on the boundary and the disc sub-divided by adding additional interior edges and vertices to create four-sided regions. This approach is valid since, for a prime doodle, the boundary of the dual graph is an embedded circle for any choice of infinite region (see [3]). Focussing on prime doodles also means that we do not have to consider bigons in the dual graph, since removing an endpoint from each of the doodle edges corresponding to a bigon's edges in a dual graph would disconnect the doodle. This is significant simplification since it means we need consider at most one edge between a pair of vertices in the dual graph. A additional consequence of this is that the search avoids the infinite but uninteresting family of "satellite" doodles, as described below.
Given any minimal planar doodle one can add a small circular component that surrounds an individual crossing and create another minimal doodle. This construction yields an infinite family of doodles that are not of great interest, so doodles produced by the search that contain such removable vertex circles are discarded. Similarly, given any two minimal doodles one could extend an edge from one to encircle a boundary crossing of the other and construct another minimal doodle, again not of great interest when attempting to identify distinct doodles. This latter family is, however not 3-connected and therefore implicitly avoided by the search. Finally, it is possible to construct a neighbourhood, N, of a point in the interior of an edge of a doodle that meets the doodle in an arc. Then, it's possible to add another doodle, called a satellite doodle, within N so that the resultant doodle is minimal. Satellite doodles are so named because they may be constructed on any edge of the host doodle. Clearly any number of satellites could be added, and recursively. As noted above, this family is avoided by not having to consider bigons in the dual graph.
The search has been completed for doodles having up to fourteen crossings, a search lasting approximately 14 days using a single linux processing core. The following table summarises the results of this search, together with the results of the search based on the Thistlethwaite-Hoste knot lists for fifteen and sixteen crossings. In each cell of the table the first number gives the number of distinct doodles that are prime but not super-prime (3-connected but not 4-connected), the second number is the number of super-prime doodles. The results obtained via the dual graph for single component doodles have been verified as being consistent with the distinct minimal doodles determined by the shadow of Thistlethwaite's knots up to fourteen crossings.
| number of crossings | reduced prime minimal planar doodles with m components | |||
|---|---|---|---|---|
| m=1 | m=2 | m=3 | m=4 | |
| 6 | 0,1 | |||
| 7 | ||||
| 8 | 0,1 | |||
| 9 | 1,0 | |||
| 10 | 0,1 | 0,1 | ||
| 11 | 1,0 | 1,1 | ||
| 12 | 1,2 | 1,2 | 1,1 | 0,2 |
| 13 | 2,3 | 4,1 | 2,2 | 2,0 |
| 14 | 5,12 | 8,14 | 8,4 | 0,1 |
| 15 | 20,18 | |||
| 16 | 55,81 | |||
The following lists contain the the unoriented left-preferred gauss code for each doodle, as defined in [2].
S6^3_1 
S8^1_1 
P9^1_1 
S10^1_1 
S10^2_1 
P11^1_1 
P11^3_1 
S11^3_1 
P12^1_1 
P12^2_1 
P12^3_1 
S12^1_1 
S12^1_2 
S12^2_3 
S12^2_2 
S12^3_1 
S12^4_1 
S12^4_2 
P13^1_1 
P13^1_2 
P13^2_1 
P13^2_2 
P13^2_3 
P13^2_4 
P13^3_1 
P13^3_2 
P13^4_1 
P13^4_2 
S13^1_1 
S13^1_2 
S13^1_3 
S13^2_1 
S13^3_1 
S13^3_2 
P14^1_1 
P14^1_2 
P14^1_3 
P14^1_4 
P14^1_5 
P14^2_1 
P14^2_2 
P14^2_3 
P14^2_4 
P14^2_5 
P14^2_6 
P14^2_7 
P14^2_8 
P14^3_1 
P14^3_2 
P14^3_3 
P14^3_4 
P14^3_5 
P14^3_6 
P14^3_7 
P14^3_8 
S14^1_1 
S14^1_2 
S14^1_3 
S14^1_4 
S14^1_5 
S14^1_6 
S14^1_7 
S14^1_8 
S14^1_9 
S14^1_{10} 
S14^1_{11} 
S14^1_{12} 
S14^2_1 
S14^2_2 
S14^2_3 
S14^2_4 
S14^2_5 
S14^2_6 
S14^2_7 
S14^2_8 
S14^2_9 
S14^2_{10} 
S14^2_{11} 
S14^2_{12} 
S14^2_{13} 
S14^2_{14} 
S14^3_1 
S14^3_2 
S14^3_3 
S14^3_4 
S14^4_1 
[1] A. Bartholomew, R. Fenn, N. Kamada, S. Kamada. Doodles on Surfaces, arXiv:1612.08473v2, Journal of Knot Theory and its Ramifications Vol. 27 No 12 (2018), 1850071..
[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 special edition: Self-distributive system and quandle (co)homology theory in algebra and low-dimensional topology, Journal of Knot Theory and its Ramifications Vol. 27, No. 11 (2018) 1843013.
[3] A. Bartholomew, R. Fenn. Planar Doodles: Their Properties, Codes and Classification, arXiv:2308.08834
[4] M. Chimani, C. Gutwenger, M. Jünger, G. W. Klau, K. Klein, P. Mutzel. The Open Graph Drawing Framework (OGDF). Chapter 17 in: R. Tamassia (ed.), Handbook of Graph Drawing and Visualization, CRC Press, 2014.