Cross-Handle

A cross-handle is a topological structure which can be thought of as the object produced by puncturing a surface twice, attaching a zip around each puncture travelling in the same direction, pulling the edges of the zips together after one tube first passes through itself it order for the direction of the zips to match up, and then zipping up. In three-space, the cross-handle contains a line of self-intersection.

A cross-handle is homeomorphic to two cross-caps (Francis and Weeks 1999). Dyck's theorem states that handles and cross-handles are equivalent in the presence of a cross-cap.

REFERENCES:

Francis, G. K. and Weeks, J. R. "Conway's ZIP Proof." Amer. Math. Monthly 106, 393-399, 1999.