Before explaining what a flype is,
we need to define a notion of tangle, one of the fundamental ideas in
knot theory, introduced by J.H. Conway in 1967 (Conway, 1970).
A 1dimensional manifold properly embedded in 3disc is called a tangle (or 2tangle) if it is composed of two arcs and any number of circles. We will use the name of tangle also for a projection of a tangle in 3disc into 2dimensional disc. Intuitively, a tangle in a KL projection is a region in the projection plane a region in the projection plane Â^{2} (or on the sphere S^{3}) surrounded with a circle such that the KL projection intersects with the circle exactly four times. From the intersections four arcs emerge pointing in the compass directions NW, NE, SW, SE. Two tangles are equivalent if one could be transformed into the other by a sequence of Reidmeister moves providing that four endpoints of the strings are fixed and that the strings belonging to the tangle remain inside of the circle. Suppose an alternating KL diagram includes a tangle, as shown afore. Let us fix four ends a, b, c, d and then rotate this tangle by a 2fold rotation (halfturn). The twist on the left in the following figure is moved on the right. Such an operation is called a flype . Tait's Flyping theorem Suppose that L' and L" are two reduced alternating diagrams on the sphere S^{2} of an alternating link L. Then we can change L' into L" by performing a finite number of flypes. The Knot 2000 (K2K) function MutationOfTangle (webMathematica MutationOfTangle) calculates the result of a flype, mutation, vertical mutation, and horizontal mutation applied to some knot tangle. In the case of nonalternating KLs, in addition to the flype it is necessary to introduce a 2pass : KL transformation where a string is simply passed over a tangle. The third Reidemeister move is a special case of a 2pass. However, flypes and 2passes are not sufficient to pass between all minimal diagrams of a nonalternating KL (Hoste, Thistlethwaite and Weeks, 1998). After introducing signs of crossings we can also use signed link graphs : each edge takes a sign +1 or 1, of a vertex it passes through. In this way we have established onetoone correspondence between such edgeweighted KL graphs and signed KL diagrams. Reidemeister moves can be represented as the local moves on signed KL graphs shown on the corresponding figure. Two plane signed KL graphs G_{1} and G_{2} represent the same link L iff G_{1} can be transformed into G_{2} by some finite sequence of the moves I', II", III"' and their inverses. If G is any plane signed KL graph and G' is its plane dual with the signs of the edges multiplied by 1, then the links L(G) and L(G' ) are ambient isotopic.
