Research Interests

• Low dimensional Topology
• Knot Theory
• Quantum algebra
• K-theory

Articles

"A note on lambda-operations in orthogonal K-theory" Proc. Amer. Soc. 128, #1, (2000), pp 1-4

Abstract: In Comment. Math. Helv. 55 (1980), 233-254, Kratzer defined Lambda operations on classical algebraic K-theory by using exterior powers of representations and a splitting principle (R. G. Swan, Proc. Sympos. in Pure Math. 21 (1971), 155-159). Because hyperbolic forms are not stable under exterior powers, we instead use a larger class of symmetric bilinear forms to define the operation of exterior powers on the classifying space of the orthogonal K-theory.

"On S^3-Equivariant Homology," Int. J. Math. Math. Sci. 26, # 4, (2001), pp193-197.

Abstract: We prove that the group \$S^3\$ (norm 1 quaternions) cannot be a geometric realization of a crossed simplicial group and construct an exact sequence connecting \$S^3\$-equivariant homology of an \$S^3\$-space with its \$\textrm{Pin}(2)\$-equivariant homology.

" Twisted Quandle homology Theory and cocycle knot invariants," (with J.Scott Carter, & Masahico Saito), Algebr. Geom. Topol. 2 (2002), pp 95-135.

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.

"Extensions of Quandles and cocycle knot invariants," (with Marina A. Nikiforou, J.Scott Carter, & Masahico Saito), J. Knot Theory and its Ramifications, vol 12, n 6 (2003), pp 725-738.

Abstract: 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.

"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.

First, quandle modules are used to generalize Burau representations and Alexander modules for classical knots. Second, 2-cocycles valued in non-abelian groups are used in a way similar to Hopf algebra invariants of classical knots. These invariants are shown to be of quantum type. Third, cocycles with group actions on coefficient groups are used to define quandle cocycle invariants for both classical knots and knotted surfaces. Concrete computational methods are provided and used to prove non-invertibility for a large family of knotted surfaces. In the classical case, the invariant can detect the chirality of 3-colorable knots in a number of cases.

"A lower bound for the number of Reidemeister moves of type III ,"(with J.Scott Carter, Masahico Saito & Shin Satoh), Topology Appl.153 (2006), pp 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, pp 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 3 (2008), no 1, pp 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.

"Tangle embeddings and quandle cocycle invariants ," (with Kheira Ameur, Tom Rose, Masahico Saito & Chad Smudde), Experimental Mathematics 17 (2008), no 4, pp 487-497.

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, Karadayi Enver & Masahico Saito), Communications in Contemporary Mathematics 10 (2008) Supp. 1, pp 791-814.

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.

"Virtual knot invariants from group biquandles and their cocycles " (with J. Scott Carter, Masahico Saito, Dan Silver & Susan Williams), J. Knot Theory and its Ramifications, vol 18, n 7 (2009), pp 957-972.

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.

"Cocycle deformations of Algebraic Identities and R-matrices " (with J. Scott Carter, Alissa Crans & Masahico Saito), arXiv:math.GT/0802.2294.

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.

"Cohomology and Formal Deformations of Alternative Algebras " (with N. Makhlouf), arXiv:math.RA/0907.1584.

Abstract: The purpose of this paper is to introduce an algebraic cohomology and formal deformation theory of left alternative algebras. Connections to some other algebraic structures are given also.

"Deformations of Hom-Alternative and Hom-Malcev algebras " (with N. Makhlouf), arXiv:1006.2499v1 [math.RA]

Abstract: The aim of this paper is to extend Gerstenhaber formal deformations of algebras to the case of Hom-Alternative and Hom-Malcev algebras. We construct deformation cohomology groups in low dimensions. Using a composition construction, we give a procedure to provide deformations of alternative algebras (resp. Malcev algebras) into Hom-alternative algebras (resp. Hom-Malcev algebras). Then it is used to supply examples for which we compute some cohomology invariants.

"N-Degeneracy in rack homology and link invariants" (With S. Nelson), arXiv:1007.3550v1 [math.GT]

Abstract: We consider rack homology for racks with rack rank N>1. For such racks, we prove that N-degenerate chains form a sub-complex of the classical complex defining rack homology. This allows a homology theory analogous to quandle homology for non-quandle racks with finite ranks. If our rack has rack rank N=1 then it is a quandle and the homology theory coincides with the CKJLS homology theory. We use nontrivial cocycles to define invariants of knots and provide examples of calculations for knots up to 8 crossings and links up to 7 crossings.