How do we define the mirror image of a knot in general 3-manifolds How do we define the mirror image of a knot in general oriented 3-manifolds ? For instance for a knot in an irreducible integer homology sphere.

One method would be to express the manifold as an open book decomposition (see http://en.wikipedia.org/wiki/Open_book_decomposition).  Then project the knot onto one of the pages (fibers) of the decomposition recording over an under crossings.  Switch over crossings to undercrossings and vice verse. You might want to see my paper Reidemeister's Theorem in Three Manifolds http://sci-prew.inf.ua/v110/2/S0305004100070353.pdf.
