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Thinning genus two Heegaard spines in $$S^ 3$$. (English) Zbl 1048.57002
A Heegard spine of compact 3-manifold is a graph whose regular neighbourhood has as its closed complement a handlebody. One can construct such a spine by adding a number of arcs to a tunnel number one link. It is a well-known theorem of Waldhausen that any Heegard splitting of $$S^3$$ can be isotoped to a standard one of the same genus. Suppose the trivalent graph $$\Gamma$$ is an arbitrary Heegard spine of $$S^3$$ (the notion of thin position is defined). Suppose a height function is given on $$S^3$$ and we isotope $$\Gamma$$ in $$S^3$$ to make it as thin as possible. What can we say about the positioning of $$\Gamma$$? The authors answer the question for genus two Heegard spines: once such a spine is put in thin position, some simple edge (not a loop) will be level. The similar question for higher genus is still open.

##### MSC:
 57M10 Covering spaces and low-dimensional topology 57N10 Topology of general $$3$$-manifolds (MSC2010)
##### Keywords:
Heegard splitting; spine; thin position
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##### References:
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