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The 0-surgeries of two knots K1 and K2 are homology cobordant rel meridians if there exists an integer homology cobordism X between them such that the two positive knot meridians are in the same homology class of X. It is a natural question to ask: if two knots have the “same” 0-surgeries in this sense, must they be smoothly concordant? We give a pair of knots as counterexample, with one of concordance order two and the other of infinite order, and along the way expand upon a Floer homology technique for obstructing torsion in the smooth concordance group first introduced by Hom, Kang, Park, and Stoffregen.

FHT Program Seminar: Homology Cobordism And Knot Concordance