Homology cobordism, smooth concordance, and the figure eight knot

TL;DR

This paper constructs a pair of rationally slice knots, including the figure eight knot, that are not smoothly concordant but have homology cobordant 0-surgeries rel meridians, challenging previous examples limited to infinite order knots.

Contribution

It provides the first example of such pairs where one knot has finite concordance order, specifically the figure eight knot with order two.

Findings

The pair of knots are rationally slice but not smoothly concordant.

Their 0-surgeries are homology cobordant rel meridians.

This is the first example with a finite order knot, the figure eight knot.

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.