Quandle cohomology and state-sum invariants of knotted curves and surfaces

• Published in: Transactions of the American Mathematical Society 2003-06, Vol.355 (10), p.3947-3989
• Main Authors: Carter, J. Scott, Jelsovsky, Daniel, Kamada, Seiichi, Langford, Laurel, Saito, Masahico
• Format: Article
• Language: English
• Subjects: Algebra; Algebraic topology; Automorphisms; Braiding; Category theory, homological algebra; Coefficients; Diagrams; Eigenfunctions; Exact sciences and technology; Homomorphisms; Integers; Manifolds and cell complexes; Mathematical surfaces; Mathematics; Research article; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Vertices
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-03-03046-0

The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. A proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed. Our proof is based on a state-sum invariant for knotted surfaces 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 classical knot theory. 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 33-space and knotted surfaces in 44-space. Cohomology groups of various quandles are computed herein and applied to the study of the state-sum invariants. Non-triviality of the invariants is proved for a variety of knots and links, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles.