The Cosmetic Crossing Conjecture

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 complement of $K$. Cosmetic Crossing Conjecture. Any crossing change that preserves the isotopy class of a knot must occur at a nugatory crossing. This classical conjecture remains open in general, although it is known for several classes of knots, including two-bridge and fibered knots and various genus-one and alternating knots.