Invertible knot

There are only five knot symmetry types, indicated by chirality and invertibility: fully chiral, reversible, positively amphichiral noninvertible, negatively amphichiral noninvertible, and fully amphichiral invertible.

In 1962 Ralph Fox conjectured that some knots were non-invertible, but it was not proved that non-invertible knots exist until Hale Trotter discovered an infinite family of pretzel knots that were non-invertible in 1963.

No general method is known that can distinguish if a given knot is invertible.

By imposing the stronger condition that the homeomorphism also be an involution, i.e. have period 2 in the homeomorphism group of the 3-sphere, we arrive at the definition of a strongly invertible knot.

The pretzel knot 7, 5, 3 is non-invertible, as are all pretzel knots of the form (2p + 1), (2q + 1), (2r + 1), where p, q, and r are distinct integers, which is the infinite family proven to be non-invertible by Trotter.

The simplest non-trivial invertible knot, the trefoil knot . Rotating the knot 180 degrees in 3-space about an axis in the plane of the diagram produces the same knot diagram, but with the arrow's direction reversed.
The non-invertible knot 8 17 , the simplest of the non-invertible knots.