9 problems

An R-link is a link $L$ such that 0-surgery on $L$ yields a connected sum of copies of $S^1\times S^2$. A 0-framed unlink is an unlink whose components all have framing zero, and a…

Let $L=K_1\cup\dots\cup K_n$ be an $n$-component link in $S^3$. A 0-framed surgery on $L$ means surgery with framing zero on every component, and let $\#_n(S^2\times S^1)$ denote t…

Let $W$ and $W'$ be $4$-dimensional $2$-handlebodies. Two such handlebodies are $2$-equivalent when they are related by a finite sequence of isotopies, handle slides, and creations…

Let $L$ be an R-link, and let $U$ be a 0-framed unlink. A Hopf pair is a Hopf link with one component 0-framed and the other dotted, representing a 4-dimensional 1-handle. The weak…

Let $L$ be an R-link, meaning an $n$-component link in $S^3$ whose 0-surgery yields $Y_n=\mathop{\#}^n(S^1\times S^2)$. Let $U$ be a 0-framed unlink. The stable Generalized Propert…

Let $U_n$ be the $n$-component zero-framed unlink in $S^3$. A link $L$ has Stable Property R if, for some integers $r$ and $m$, the split union $L\sqcup U_r$ is handleslide equival…

Let $Q\subset S^3$ be the square knot, and let Property $2R$ mean that whenever $Q$ is a component of a $2$-component link whose integral surgery yields … a sequence of handle slid…

Let $L=L_1\cup L_2$ satisfy the hypotheses of the preceding Kirby-calculus formulation, except that $L_2$ is required only to generate $H_1(S^3-L_1;\mathbb Z)$ rather than…

Let $L=L_1\cup L_2$ be a link in $S^3$, with $L_1$ a dotted $p$-component unlink and $L_2$ a framed link of $p+q$ components. Suppose that $L_2$ normally generates $\pi_1(S^3-L_1)$…