A braid-like isotopy for links in 3–space is an isotopy which uses only those Reidemeister moves which occur in isotopies of braids. We define a refined Jones polynomial and its corresponding Khovanov homology which are, in general, only invariant under braid-like isotopies.
"A Jones polynomial for braid-like isotopies of oriented links and its categorification." Algebr. Geom. Topol. 5 (4) 1535 - 1553, 2005. https://doi.org/10.2140/agt.2005.5.1535