8 problems

Let $K\subset S^3$ be a knot, and let $c$ be a crossing of $K$. A crossing is nugatory if its crossing circle bounds a disk in the complement of $K$; a generalized crossing change…

Let $Th(p,q)$ denote a Turk’s head knot, with $p\geq 5$ and $q\geq 4$. A knot is super-Kauffman if it admits a minimal alternating diagram whose every crossing has the Kauffman pro…

Let $K\subset S^3$ be an oriented knot and let $D\subset S^3$ be an oriented disk intersecting $K$ in two points of opposite sign. A crossing change on $K$ is the operation of perf…

Cosmetic crossing conjecture. Changing a nontrivial crossing in a link diagram must change the isotopy type of the link.

Let $K_-$ and $K_+$ be knots such that $K_+$ is obtained from $K_-$ by a negative generalized crossing change on $2n$ strands. Characteristic-two generalized crossing-change inequa…

Cosmetic crossing conjecture. An oriented knot $K$ in $S^{3}$ does not have cosmetic crossings.

Let $L$ be a two-component link, let $b$ be a band, and let $K_b$ be the oriented knot obtained from $L$ by band surgery along $b$. For each $n\in\mathbf{Z}$, let $K_{b+n}$ be the…

Let $K_-$ and $K_+$ be oriented knots in $S^3$ that agree outside a small ball, where $K_-$ has a negative crossing and $K_+$ has a positive crossing inside the ball. Let $L$ be th…