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.

We give a generating set of the generalized Reidemeister moves for oriented singular links. We then introduce an algebraic structure arising from the axiomatization of Reidemeister moves on oriented singular knots. We give some examples, including some non-isomorphic families of such structures over non-abelian groups. We show that the set of colorings of a singular knot by this new structure is an invariant of oriented singular knots and use it to distinguish some singular links.2000 Mathematics Subject Classification. Primary 57M25.

We present a set of 26 finite quandles that distinguish (up to reversal and mirror image) by number of colorings, all of the 2977 prime oriented knots with up to 12 crossings. We also show that 1058 of these knots can be distinguished from their mirror images by the number of colorings by quandles from a certain set of 23 finite quandles. We study the colorings of these 2977 knots by all of the 431 connected quandles of order at most 35 found by Vendramin. Among other things, we collect information about quandles that have the same number of colorings for all of the 2977 knots. For example, we prove that if Q is a simple quandle of prime power order then Q and the dual quandle Q* of Q have the same number of colorings for all knots and conjecture that this holds for all Alexander quandles Q. We study a knot invariant based on a quandle homomorphism f : Q1 → Q0. We also apply the quandle colorings we have computed to obtain some new results for the bridge index, the Nakanishi index, the tunnel number, and the unknotting number. In an appendix we discuss various properties of the quandles in Vendramin’s list. Links to the data computed and various programs in C, GAP and Maple are provided.

The quandle homology theory is generalized to the case when the coefficient groups admit the structure of Alexander quandles, by including an action of the infinite cyclic group in the boundary operator. Theories of Alexander extensions of quandles in relation to low dimensional cocycles are developed in parallel to group extension theories for group cocycles. Explicit formulas for cocycles corresponding to extensions are given, and used to prove non-triviality of cohomology groups for some quandles. The corresponding generalization of the quandle cocycle knot invariants is given, by using the Alexander numbering of regions in the definition of statesums. The invariants are used to derive information on twisted cohomology groups.