Abstract. 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 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. 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.
Geometric representations of cycles in quandle homology theory are given in terms of colored knot diagrams. Abstract knot diagrams are generalized to diagrams with exceptional points which, when colored, correspond to degenerate cycles. Bounding chains are realized, and used to obtain equivalence moves for homologous cycles. The methods are applied to prove that boundary homomorphisms in a homology exact sequence vanish.
Abstract.A homology theory is developed for set-theoretic Yang-Baxter equations, and knot invariants are constructed by generalized colorings by biquandles and YangBaxter cocycles. Kauffman [24,25] re-focused attention on algebraic structures that are defined via diagrams. Advantages of using virtual knots were observed for the bracket polynomial [25] and Vassiliev invariants [19]. The fundamental (Wirtinger) groups of virtual knots were studied [27,34] and their geometric interpretations were given [22]. Generalizations of Alexander polynomials were studied in relations to virtual knots [33,35]. The theory of racks and quandles (in particular the homology theory thereof) as exposed in [14,16,17] was used to define state-sum invariants for knotted surfaces, as well as for classical and virtual knots [5]. A generalization of quandles, called biquandles, is proposed in [26]. Examples include a generalized Burau matrix used in [33] and [35]. The set-theoretic solutions to the Yang-Baxter equations are studied in detail in the papers [11,12,29,36]. Their affine solutions appear among our birack matrices. Some of these solutions also appeared in [35] and [10].
Introduction. The introduction of virtual knots byIn this paper a homology theory for the YBE is constructed, and cocycles are used to define knot invariants via colorings of (virtual) knot diagrams by biquandles and a state-sum formulation. This paper is organized into two sections; Section 2 develops algebraic theories, and Section 3 gives applications to knot theory. In Section 2, the Yang-Baxter sets are reviewed and colorings of cubical complex by Yang-
Quandle cocycles are constructed from extensions of quandles. The theory is parallel to that of group cohomology and group extensions. An interpretation of quandle cocycle invariants as obstructions to extending knot colorings is given, and is extended to links component-wise.
A movie description of a surface embedded in 4-space is a sequence of knot and link diagrams obtained from a projection of the surface to 3-space by taking 2-dimensional cross sections perpendicular to a fixed direction. In the cross sections, an immersed collection of curves appears, and these are lifted to knot diagrams by using the projection direction from 4-space. We give a set of 15 moves to movies (called movie moves) such that two movies represent isotopic surfaces if and only if there is a sequence of moves from this set that takes one to the other. This result generalizes the Roseman moves which are moves on projections where a height function has not been specified. The first 7 of the movie moves are height function parametrized versions of those given by Roseman. The remaining 8 are moves in which the topology of the projection remains unchanged.
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