Mirror of a classical knot

Fix the orientation-reversing involution $r:S^3\to S^3$ given by reflection across a fixed equatorial two-sphere containing a chosen standard unknot $U$. For a smooth knot $K\subset S^3$, its mirror is $\bar{K}:=r(K)$, with the orientation transported by $r$. A diagram of the mirror can be obtained by keeping the projected shadow fixed and interchanging the over- and under-crossing data at every crossing; equivalently, one may reflect the planar shadow while retaining the crossing data. These are alternative mirror conventions, not operations to perform together. Since $r^2=\mathrm{i}\mathrm{d}$ and $r(U)=U$, mirroring twice returns $K$, and $K$ is the unknot exactly when $\bar{K}$ is.

Kronheimer–Mrowka, Khovanov homology is an unknot-detector (2011), §§1.1 and 8.6–8.7

Yang, An Overview of Floer Homologies (2025), §5.7