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K-stable equivalence for knots in Heegaard surfaces


Speaker: Alice Stevens, UC Davis
Location: 2112 MSB
Start time: Tue, Nov 18 2008, 4:00PM

Let K_i be a knot embedded in a Heegaard surface S_i of a closed, orientable, 3-manifold M. We say that the pairs (S_1, K_1) and (S_2, K_2) are equivalent in M if there is an ambient isotopy of M that maps one pair to the other. We define K-stabilization, which is similar to stabilization in the standard theory of Heegaard splittings, and we prove the following version of the Reidemeister-Singer theorem: if (S_1, K) and (S_2, K) are embeddings of the knot K in Heegaard surfaces S_i of M such that the surface slope of K in S_1 is equal to the surface slope of K in S_2, then there is a third pair (S, K) which is a K-stabilization of both. The main tool used in the proof is amalgamation of Heegaard splittings.