Homology Cobordism and Smooth Knot Concordance
Author(s)
Collins, Sarah
Advisor(s)
Hom, Jennifer
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Abstract
To a knot K in the 3-sphere we can associate the 3-manifold that arises from zero-framed Dehn surgery on K. It is a natural question to ask: if two knots have zero-surgeries which are integer homology cobordant via cobordism W (with technical condition that the cobordism preserves the homology class of the positive meridians), does that imply that the knots must be smoothly concordant? In this paper, we give a pair of rationally slice knots which are not smoothly concordant but whose zero-surgeries have such a cobordism between them. One knot in the pair is the figure eight knot, 4_1, which represents a 2-torsion element in the smooth concordance group C; all knots used in previous counterexamples represent infinite order elements. The other knot in the pair is the Mazur pattern of the figure eight knot, which we show must represent a non-torsion element in C and is thus not concordant to 4_1.
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Date
2023-04-27
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Dissertation