State-sum invariants for knotted curves and surfaces using quandle cohomology were introduced by Laurel Langford and the authors in 4]. In this paper we present methods to compute the invariants and sample computations. Computer calculations of cohomological dimensions for some quandles are presented. For classical knots, Burau representations together with… (More)
The groups of link bordism can be identified with homotopy groups via the Pontryagin–Thom construction. B.J. Sanderson computed the bordism group of 3 component surface–links using the Hilton–Milnor Theorem, and later gave a geometric interpretation of the groups in terms of intersections of Seifert hypersurfaces and their framings. In this paper, we… (More)
Crane and Frenkel proposed a state sum invariant for triangulated 4-manifolds. They sketched the definition of a Hopf category that was to be used in their construction. Crane and Yetter studied Hopf categories and gave some examples using group cocycles that are associated to the Drinfeld double of a finite group. In this paper we define a state sum… (More)
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… (More)
We propose solving computational problems with DNA molecules by physically constructing 3-dimensional graph structures. Building blocks consisting of intertwined strands of DNA are used to represent graph edges and vertices. Diierent blocks would be combined to form all possible 3-dimensional structures representing a graph. The solution to the Hamiltonian… (More)
φ φ φ φ φ φ y p q r x q p r p*q q*r (p*q)*r (p*r)*(q*r) p*r q*r (x, y) (p, q) (q, r) (p*q, r) (p, r) x y 1 T T 0 T 0 T 0-1 T-1 T T-2 1 T T 0 T 0-1 T-1 T-1 T T-2 φ T φ φ T φ φ φ T r p*q q*r (p*q)*r (p*r)*(q*r) p*r (p, q) (q, r) (p*q, r) (p, r) Abstract Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was… (More)
State-sum invariants for classical knots and knotted surfaces in 4-space are developed via the cohomology theory of quandles. Cohomol-ogy of quandles are computed to evaluate the invariants. Some twist spun torus knots are shown to be non-invertible using the invariants.
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 of the Yang–Baxter… (More)