Abstract
In this paper we study embeddings of the Fano plane as a bipartite graph. We classify all possible embeddings especially focusing on those with non-trivial automorphism group. We study them in terms of rotation systems, isomorphism classes and chirality. We construct quotients and show how to obtain information about face structure and genus of the covering embedding. As a by-product of the classification we determine the genus polynomial of the Fano plane.
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The authors would like to warmly thank Jürgen Wolfart for many useful hints and discussions and for carefully reading the final version of this paper.
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Appendix: Embeddings of the Fano plane with non-trivial automorphism group
Appendix: Embeddings of the Fano plane with non-trivial automorphism group
In the following table sixteen non-isomorphic embeddings of the Fano plane \((\varGamma _D,R)\) are listed. For the construction we choose the Frobenius difference set \(D=\{1,2,4\}\). The automorphism group of the embeddings is of order at least three and it contains the Frobenius automorphism. The remaining non-isomorphic embeddings with non-trivial automorphism group can be obtained mirroring the listed ones.
The first column of the table gives rotation systems such that the Frobenius automorphism is contained in \(Aut(\varGamma _D,R)\) (see Sect. 4). The sign \(+\) and − denote respectively permutations (124) and (142) of the difference set D. The next two columns contain the resulting embeddings and their quotients with the corresponding face valencies. Embeddings are subdivided into four groups having the same quotient. The last column lists the numbers of the figures in which the quotients are sketched.
\(\rho _0\) | \(\rho _1,\rho _2,\rho _4\) | \(\rho _3,\rho _6,\rho _5\) | (\(\varGamma _D,R\)) | (\(\overline{\varGamma },\overline{R}\)) | Figs. |
|---|---|---|---|---|---|
\(\varrho _0\) | \(\varrho _1,\varrho _2,\varrho _4\) | \(\varrho _3,\varrho _6,\varrho _5\) | |||
+ | + | + | (7, 7, 7) | (7) | 6a |
− | − | − | Regular | ||
− | + | + | (7, 7, 7) | (7) | |
+ | − | − | |||
+ | + | + | (21) | (7) | |
+ | − | − | |||
− | + | + | (21) | (7) | |
− | − | − | |||
+ | + | + | (3, 3, 3, 3, 9) | (1, 3, 3) | 7b |
− | + | + | |||
− | + | + | (3, 3, 3, 3, 9) | (1, 3, 3) | |
+ | + | + | |||
+ | + | + | (3, 3, 3, 3, 3, 3, 3) | (1, 3, 3) | |
+ | + | + | Regular | ||
− | + | + | (3, 9, 9) | (1, 3, 3) | |
− | + | + | |||
+ | + | + | (7, 7, 7) | (7) | 6b |
− | − | + | |||
− | + | + | (7, 7, 7) | (7) | |
+ | − | + | |||
+ | + | + | (21) | (7) | |
+ | − | + | |||
− | + | + | (21) | (7) | |
− | − | + | |||
+ | + | + | (3, 6, 12) | (1, 2, 4) | 7a |
− | + | − | |||
− | + | + | (3, 6, 12) | (1, 2, 4) | |
+ | + | − | |||
+ | + | + | (3, 6, 12) | (1, 2, 4) | |
+ | + | − | |||
− | + | + | (3, 4, 4, 4, 6) | (1, 4, 2) | |
− | + | − |
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Catalano, D.A., Sarti, C. Fano plane’s embeddings on compact orientable surfaces. Beitr Algebra Geom 58, 635–653 (2017). https://doi.org/10.1007/s13366-017-0330-1
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DOI: https://doi.org/10.1007/s13366-017-0330-1