arXiv subjects
Publications and source records attributed to Trevor Oliveira-Smith.
In this paper, we standardize a homotopy $4$-sphere constructed by Dunfield and Gong. As a corollary, we show that the $18$-crossing knot $18_{\text{nh}00000601}$, which is not known to be ribbon, is slice in the standard $4$-ball. Thus, $18_{\text{nh}00000601}$ serves as a potential counterexample to the Slice-Ribbon Conjecture. In addition, we show that the same knot bounds a fibered handle-ribbon disk in $B^{4}$.
The generalized Property R conjecture (GPRC) predicts that if framed surgery on an $n$-component link $L$ in $S^3$ produces $\#^{n} (S^1\times S^2)$, then $L$ is handleslide equivalent to an unlink, the obvious way to construct such a surgery. Many potential counterexamples to the GPRC are known, but obstructing handleslide equivalence is a tricky proposition. In this vein, we disprove a further generalization of the GPRC. It would be reasonable to expect that if an $n$-component link in $S^3$ surgers to the connected sum of $n$ three-manifolds with the homology of $S^1 \times S^2$, then this link should be handleslide equivalent to an $n$-component split link, the obvious way to construct such a surgery. However, we prove that there are 2-component framed links in $S^3$ that surger to a connected sum of homology $S^1\times S^2$'s but that are not handleslide equivalent, or even weakly handleslide equivalent, to a split link.