Mutation

Consider a Knot as being formed from two Tangles. The following three operations are called mutations.

1. Cut the knot open along four points on each of the four strings coming out of $T_2$, flipping $T_2$ over, and gluing the strings back together.

2. Cut the knot open along four points on each of the four strings coming out of $T_2$, flipping $T_2$ to the right, and gluing the strings back together.

3. Cut the knot, rotate it by 180°, and reglue. This is equivalent to performing (1), then (2).

Mutations applied to an alternating Knot projection always yield an Alternating Knot. The mutation of a Knot is always another Knot (a opposed to a Link).

References

Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, p. 49, 1994.