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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)
We propose molecular models for homologous DNA recombination events that are guided by either double-stranded RNA (dsRNA) or single-stranded RNA (ssRNA) templates. The models are applied to explain DNA rearrangements in some groups of ciliates, such as Stylonychia or Oxytricha, where extensive gene rearrangement occurs during differentiation of a somatic(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)
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)
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)
The state-sum invariants for knots and knotted surfaces defined from quandle cocycles are described using the Kronecker product between cycles represented by colored knot diagrams and a cocycle of a finite quandle used to color the diagram. Such an interpretation is applied to evaluating the invariants. Algebraic interpretations of quandle cocycles as(More)