Abstract: 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 state-sums. The invariants are used to derive information on twisted cohomology groups.

"On the Steenrod operations on cyclic cohomology," (with Y. Gouda), Int.J. Math. Math. Sci. 2003, #72, (2003), pp 4539-4545.

Abstract: For a commutative Hopf algebra A over Z/p, where p is a prime integer, we define the Steenrod operations pi in cyclic cohomology of A using a tensor product of a free resolution of the symmetric group Sn and the standard resolution of the algebra A over the cyclic category according to Loday (1992). We also compute some of these operations.

" Homology theory for the set-theoretic Yang-Baxter equation and knot invariants from generalizations of Quandles," (with J.Scott Carter, & Masahico Saito), Fundamenta Mathematicae, 184, (2004), pp 31-54.

Abstract: A homology theory is developed for set-theoretic Yang-Baxter equations, and knot invariants are constructed by generalized colorings by biquandles and Yang-Baxter cocycles.

"Cocycle knot invariants from Quandle modules and Generalized Quandle homology," (with J.Scott Carter, Matias Grana & Masahico Saito), Osaka J. Math., 42 (2005), 499-541.

Abstract: Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was proposed by Andruskiewitsch and Graña. We specialize that theory to the case when there is a group action on the coefficients.

"A lower bound for the number of Reidemeister moves of type III ,"(with J.Scott Carter, Masahico Saito & Shin Satoh), Topology Appl.153 (2006), 2788-2794.

Abstract: We study the number of Reidemeister type III moves using Fox n-coloring of knot diagrams.

"Cohomology of the adjoint of Hopf algebras " (with J. Scott Carter, Alissa Crans & Masahico Saito), Journal of Generalized Lie Theory and Applications, vol 2 (2008), no 1, 19-34.

Abstract: A cohomology theory of the adjoint of Hopf algebras, via deformations, is presented by means of diagrammatic techniques. Explicit calculations are provided in the cases of group algebras, function algebras on groups, and the bosonization of the super line. As applications, solutions to the Yang-Baxter equation are given and quandle cocycles are constructed from groupoid cocycles.

"Cohomology of Categorical self-distributivity" (with J. Scott Carter, Alissa Crans & Masahico Saito), Journal of Homotopy and Related Structures, vol 3 (2008), no 1, 13-63.

Abstract: We define self-distributive structures in the categories of coalgebras and cocommutative coalgebras. We obtain examples from vector spaces whose bases are the elements of finite quandles, the direct sum of a Lie algebra with its ground field, and Hopf algebras. The self-distributive operations of these structures provide solutions to the Yang-Baxter equation, and, conversely, solutions to the Yang-Baxter equation can be used to construct self-distributive operations in certain categories.

Moreover, we present a cohomology theory that encompasses both Lie algebra and quandle cohomologies, in analogous to Hochschild cohomology, and can be used to study deformations of these self-distributive structures. All of the work here is informed via diagrammatic computations.

"Virtual knot invariants from group biquandles and their cocycles " (with J. Scott Carter, Masahico Saito, Dan Silver & Susan Williams), arXiv:math.GT/0703594, To appear in Journal of Knot Theory and its Ramifications.

Abstract: A group-theoretical method, via Wada's representations, is presented to distinguish Kishino's virtual knot from the unknot. Biquandles are constructed for any group using Wada's braid group representations. Cocycle invariants for these biquandles are studied. These invariants are applied to show the non-existence of Alexander numberings and checkerboard colorings.

"Tangle embeddings and quandle cocycle invariants ," (with Kheira Ameur, Tom Rose, Masahico Saito & Chad Smudde), to appear in Experimental Mathematics.

Abstract: To study embeddings of tangles in knots, we use quandle cocycle invariants. Computations are carried out for the table of knots and tangles, to investigate which tangles may or may not embed in knots in the tables.

"Cohomology of Fronenius algebras and the Yang_baxter equation " (with J. Scott Carter, Alissa Crans & Masahico Saito), arXiv:math.GT/0801.2567, To appear in Communications in Contemporary Mathematics, vol. 10 (1), (2008), pp 1-24.

Abstract: A cohomology theory A cohomology theory for multiplications and comultiplications of Frobenius algebras is developed for in low dimensions, in analogy with Hochschild cohomology of bialgebras, based on the deformation theory. Concrete computations are provided for key examples. Skein theoretic constructions give rise to solutions to the Yang-Baxter equation, using multiplications and comultiplications of Frobenius algebras, and 2-cocycles are used to obtain deformations of R-matrices thus obtained.

"Cocycle deformations of Algebraic Identities and R-matrices " (with J. Scott Carter, Alissa Crans & Masahico Saito), arXiv:math.GT/08.

Abstract: For an arbitrary identity L=R between compositions of maps
L and R on tensors of vector spaces V, a general construction of a 2-cocycle condition is given. These 2-cocycles correspond to those obtained in deformation theories of algebras. The construction is applied to a canceling pairings and copairings, with explicit examples with calculations. Relations to the Kauffman bracket and knot invariants are discussed.

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