mathematically, knots are a set of closed, non-intersecting curves which exist in three-dimensional space. the earrings above are math knots.
this is also a math knot:
this is not a math knot:
topology ("rubber sheet" geometry) focuses on geometrical properties that are unchanged by an infinite amount of twisting, bending, pulling, shrinking, or stretching. angles, length, width, etc. don't matter - everything is elastic. a sphere is topologically equivalent to a cube (scube?); a donut is topologically equivalent to a coffee mug. because the heart of "rubber sheet" geometry is the relationship between an object's points and these properties, topology is reliant upon set theory and the theories of limits and continuum.
knot theory, a branch within topology, centers on the shape of our tiffany's earrings: links and knots. an ambient isotopy (stretching, shrinking, pulling, bending, twisting, or magic tricking the knot) cannot reduce the curve to an unknot (something that isn't a knot).
oh, and cutting the knot is forbidden. but why would you want to? the earrings are so pretty as is.
