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In page Heegaard splitting:
"There is an analogous notion of thin position, defined for knots, for Heegaard splittings. The complexity of a connected surface S, c(S), is defined to be ; the complexity of a disconnected surface is the sum of complexities of its components. The complexity of a generalized Heegaard splitting is the multi-set {c(S_i)}, where the index runs over the Heegaard surfaces in the generalized splitting. These multi-sets can be well-ordered by lexicographical ordering (monotonically decreasing). A generalized Heegaard splitting is thin if its complexity is minimal.
