Homology cobordism, smooth concordance, and the figure eight knot

Sally Collins

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arXiv · 2210.10308

Homology cobordism, smooth concordance, and the figure eight knot

Sally Collins

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Abstract

The $0$-surgeries of two knots $K_1$ and $K_2$ are homology cobordant rel meridians if there exists a $\mathbb{Z}$-homology cobordism $X$ between them such that the two knot meridians are in the same homology class in $H_{1}(X,\mathbb{Z})$. In this paper, we give a pair of rationally slice knots which are not smoothly concordant but whose $0$-surgeries are homology cobordant rel meridians. One knot in the pair is the figure eight knot, which has concordance order two; all previous examples of such pairs of knots are infinite order.

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Sally Collins. 2022-10-19. Homology cobordism, smooth concordance, and the figure eight knot. https://arxiv.org/abs/2210.10308

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