Quandle Cohomology and State-sum Invariants of Knotted Curves and Surfaces
The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. Here this example is shown to be distinct from the same sphere with the reversed orientation. To demonstrate this fact a state-sum invariant for classical knots and knotted surfaces is developed via a cohomology theory of racks and quandles (also known as distributive groupoids). A quandle is a set with a binary operation --- the axioms of which model the Reidemeister moves in the classical theory of knotted and linked curves in 3-space. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define state-sum invariants for knotted circles in 3-space and knotted surfaces in 4-space. Cohomology groups of various quandles are computed herein and applied to the study of the state-sum invariants of classical knots and links and other linked surfaces. Non-triviality of the invariants are proved for variety of knots and links, including the trefoil and figure-eight knots, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles.