Geodesic
Classification of Heegaard diagrams of genus three
Fundamentalʹnaâ i prikladnaâ matematika, Tome 11 (2005) no. 5, pp. 91-97.

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We present a classification of the Heegaard diagram of genus 3 and an example of an infinite series of diagrams, which have the same type modeled on the one-dimensional skeleton of the octahedron. 
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     author = {F. Korablev},
     title = {Classification of {Heegaard} diagrams of genus three},
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F. Korablev. Classification of Heegaard diagrams of genus three. Fundamentalʹnaâ i prikladnaâ matematika, Tome 11 (2005) no. 5, pp. 91-97. http://geodesic.mathdoc.fr/item/FPM_2005_11_5_a7/

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[3] Homma T., Ochiai M., Takahashi M., “An algorithm for recognizing $S^3$ in $3$-manifolds with Heegaard splittings of genus two”, Osaka J. Math., 17 (1990), 625–648 | MR