Mutation (knot theory)

Consider a disc D in the projection plane of the diagram whose boundary circle intersects K exactly four times.

We may suppose that (after planar isotopy) the disc is geometrically round and the four points of intersection on its boundary with K are equally spaced.

The part of the knot inside the disc is a tangle.

There are two reflections that switch pairs of endpoints of the tangle.

They have the same hyperbolic volume (by a result of Ruberman), and have the same HOMFLY polynomials.