Flypes, crossing balls, and standard-position Conway spheres

A 2-tangle diagram is a regular diagram in a disk of two properly embedded tangle arcs and possibly finitely many closed tangle components, with four specified endpoints on the disk boundary; the arcs and circles are disjoint in the represented 3-ball, while their projections may have crossings. A flyping circle is a simple closed curve in the projection sphere whose intersection with the diagram consists of exactly three points, exactly one of which is a crossing; the circle is transverse to both branches at that crossing and to the diagram at the other two points. A small annular neighborhood of this circle has two complementary tangle disks. A flype fixes the tangle in one disk, interchanges which pair of strands crosses in the annular neighborhood, and turns over the tangle in the other disk: its projected tangle is reflected and all its over-under crossing data are reversed. This turnover is a half-turn in three-dimensional space about an axis in the projection sphere. Component orientations are transported by this motion, and the exterior diagram is fixed. The move with the opposite choice of fixed disk is also allowed; a disk containing no crossings is allowed. A flype whose fixed tangle has no crossings is called an entire flype. Diagrams are drawn on the oriented projection sphere; a sphere planar isotopy is an orientation-preserving smooth isotopy of that sphere carrying the decorated diagram. A plane diagram is obtained by choosing a point of the sphere outside the projection and deleting it. At each crossing of a link diagram, a crossing ball is a small 3-ball in which the overpass lies on the upper hemisphere and the underpass lies on the lower hemisphere, with the rest of the link outside that ball. Replacing all crossing disks by upper hemispheres or by lower hemispheres gives the shifted projection spheres $S_{+}$ and $S_{-}$. A saddle disk in a crossing ball is a properly embedded disk whose boundary meets each of the two hemispheres in two arcs and whose interior passes across the equatorial crossing disk. For a closed embedded surface transverse to the link and to $S_{+}$ and $S_{-}$, each component $C$ of its intersection with either shifted sphere has a bubble word: read once around $C$, recording $P$ each time it crosses the link and $S$ each time it passes through a crossing ball along a saddle disk. The surface is in bubble standard position if every such word is nonempty and no word records the same crossing ball more than once. A Conway sphere is an embedded sphere meeting the link transversely in four points; it is in crossing-ball standard position when its intersection with the crossing balls consists of saddle disks and its shifted-sphere intersection curves satisfy the bubble-standard conditions.

The turnover convention is the one in Kindred's Definition 2.9 and Figure 1. In particular, an entire flype permits viewing the whole diagram from the opposite side of the sphere, as in his Definition 2.11. Closed components inside a tangle disk are carried along with its arcs.

Kindred, A geometric proof of the flyping theorem, §2.2 (Definitions 2.9–2.11) and §§3.1–3.2

Menasco and Thistlethwaite, The classification of alternating links (1993)