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 1-dimensional manifold properly embedded in 3-disc is called a tangle (or 2-tangle) 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 3-disc into 2-dimensional disc.

Intuitively, a tangle in a KL projection is a region in the projection plane a region in the projection plane 2 (or on the sphere S3) 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 end-points 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 2-fold rotation (half-turn). 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 S2 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 non-alternating KLs, in addition to the flype it is necessary to introduce a 2-pass : KL transformation where a string is simply passed over a tangle. The third Reidemeister move is a special case of a 2-pass. However, flypes and 2-passes are not sufficient to pass between all minimal diagrams of a non-alternating 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 one-to-one correspondence between such edge-weighted 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 G1 and G2 represent the same link L iff G1 can be transformed into G2 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. 

 

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