In topological proofs the phrase "slightly deform" is widely used. To me, although I can accept the idea intuitively, the phrase "slightly deform" does not sound like a strict mathematical concept.
For example, a knot diagram is defined to be a projection of a knot onto some plane — a curve with finitely many self-intersection points called crossings. All crossings are required to be transversal double points and for each crossing there is an additional information of which part of the knot goes above and which one goes below.
But how does one prove that each knot has at least one knot diagram? I suppose that in the proof to such a statement, you probably will see the phrase "slightly deform" or a similar idea. For example, to get rid of triple crossings, one may need to "slightly deform" a part of a knot.
Even if it is intuitively understandable, how does one define "slightly deform" at all? Until it is unambiguously defined, I do not feel comfortable to use it in proofs.
Edit: The following paragraph was quoted from Braid Groups by C. Kassel and V. Turaev.
I find myself very reluctant to accept the claim that "Moreover, if $b$, $b'$ are generic polygonal braids related by a sequence of $\Delta$-moves, then slightly deforming the vertices of the intermediate polygonal braids, we can ensure that these polygonal braids are also generic." How can one prove this statement?