Is $K_1\# K_2$ isotopic to $K_1\#\overline{K_2}$?

Question

Let $K_1$ and $K_2$ be two oriented knots (with fixed embeddings in $S^3$). Then we have a well-defined oriented connect sum $K_1\#K_2$. We can also take $K_2$ with the opposite orientation to form $K_1\#\overline{K_2}$. Ignoring their orientations, are these two knots isotopic? Is this some kind of "generalized Reidemeister 1 move"?

Answer 1

No, these are not isotopic. The simplest example is actually the simplest connected sums: The granny knot $3_1\# 3_1 $ and the square knot $3_1\# \overline{3_1}$ are different knots. But these are difficult to detect as different with most of the quick methods, but they do have different Jones Polynomials. In Rolfsen, these were distinguished by checking that the signature of the square knot is zero (and is slice) and the granny knot is not.