6 problems

Cosmetic crossing conjecture. If changing the crossing of $K$ at $c$ preserves $K$ as an oriented knot, then $c$ is nugatory.

Let $T$ be a Conway tangle with the specified connectivity and no closed components, and let $\{K_n\}_{n\in\mathbb{Z}}$ be the associated family of knots. Say that $T$ is horizonta…

Let $T$ be a Conway tangle, and let $K_0$ and $K_1$ be the knots obtained from the associated family of tangle fillings. Say that $T$ is horizontally split when it admits the corre…

Let $c$ be a crossing circle for a knot $K$, and let $K_n$ denote the knot obtained by performing $-1/n$-Dehn surgery on $c$, equivalently by adding $n$ full right-handed twists at…

Let $K$ be a knot, and let a crossing circle be the circle encircling the two strands at a crossing. A crossing is nugatory if its crossing circle bounds an embedded disk in the co…

Generalized cosmetic crossing conjecture. No knot admits a generalized cosmetic crossing.