4 problems

Let $K$ be a knot with unknotting number one. For an unknotting crossing $x$ of $K$, let $K_x$ and $K'_x$ be the associated Piccirillo friends. Piccirillo friend conjecture. The fo…

Let $K$ be a knot with an unknotting crossing $x$, and let $K_x$ denote its Piccirillo friend, constructed from $K$ and $x$. A twisted Whitehead double is a knot obtained by applyi…

A fibered and strongly quasi-positive knot (or a fibered and strongly quasi-negative knot) is a knot having the stated properties; a friend is a knot associated to it by the trace-…

Let $K$ and $K'$ be knots. For a knot $K$, its 0-trace $X_0(K)$ is the 4-manifold obtained by attaching a 0-framed 2-handle to $B^4$ along $K$. Abe's conjecture. If $K$ and $K'$ ha…