Quandle Cohomology and Statesum Invariants of Knotted Curves and Surfaces
Abstract
The 2twist spun trefoil is an example of a sphere that is knotted in 4dimensional space. Here this example is shown to be distinct from the same sphere with the reversed orientation. To demonstrate this fact a statesum 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 3space. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define statesum invariants for knotted circles in 3space and knotted surfaces in 4space. Cohomology groups of various quandles are computed herein and applied to the study of the statesum invariants of classical knots and links and other linked surfaces. Nontriviality of the invariants are proved for variety of knots and links, including the trefoil and figureeight knots, and conversely, knot invariants are used to prove nontriviality of cohomology for a variety of quandles.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 March 1999
 arXiv:
 arXiv:math/9903135
 Bibcode:
 1999math......3135C
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Quantum Algebra;
 57Q45;
 57M25;
 57M05
 EPrint:
 The definition of cohomology has been revised to coincide with that given in the sequels. Other minor and stylistic errors have been corrected