Quandle Cohomology and State-sum Invariants of Knotted Curves and Surfaces
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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.
The definition of cohomology has been revised to coincide with that given in the sequels. Other minor and stylistic errors have been corrected
The definition of cohomology has been revised to coincide with that given in the sequels. Other minor and stylistic errors have been corrected