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Some geometric aspects of quandle (co)homology and cocycle invariants of knotted curves and surfaces
by
Masahico Saito
University of South Florida
Coauthors: J.S. Carter, D. Jelsovsky, S. Kamada, L. Langford
A cohomology theory of quandles is defined and applications to knot theory will be given. Cocycle conditions are derived from Reidemeister moves, and generalized to a cochain complex. Quandle 2-cocycles are assigned to crossings of classical knot diagrams that are colored by elements of a finite quandle, and 3-cocycles are assigned to triple points of colored diagrams of knotted surfaces. To define the state-sum invariants, the product of all cocycles assigned to crossings is computed, and the sum is taken over all possible colorings of a given knot diagram. The state-sum invariants are used to prove non-invertibility of some twist spun torus knots. Some computational methods of quandle cohomology and the state-sum invariants will be discussed. For example, quotient homomorphisms and colored knot diagrams are used to prove non-triviality of cohomology groups for some quandles.
Date received: January 18, 2000
Copyright © 2000 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # caea-11.