How do we define the mirror image of a knot in general 3-manifolds

Question

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.

Answer 1

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.

Answer 2

I don't think there can be a general definition.

The reason there even exists a definition in $S^3$ is that there exists an orientation reversing homeomorphism $h:S^3 \to S^3$ and $h$ is unique up to isotopy; the mirror image of a knot $K \subset S^3$ is then $h(K)$. But there are many oriented 3-manifolds which have no orientation reversing homeomorphism, and many which have a non-unique orientation reversing homeomorphism, so this definition will not apply in general.

Perhaps there are special definitions. For instance, suppose $M$ is an oriented manifold and $K$ is a knot contained in an open ball $B \subset M$, then there is a kind of "local mirror" of $K$: the ball $B$ has a unique orientation reversing homeomorphism $h$ of $B$ up to isotopy and using it you can produce a mirror $h(K)$. This should be well-defined independent of the choice of $B$.