Masahico Saito

Quandle cohomology and state-sum invariants of knotted curves and surfaces

The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. A proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed. Our proof is based on a state-sum invariant for knotted surfaces developed via a cohomology theory of racks and quandles (also known as distributive groupoids). A quandle is a set with a binary operation - the axioms of which model the Reidemeister moves in classical knot theory. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define state-sum invariants for knotted circles in 3-space and knotted surfaces in 4-space. Cohomology groups of various quandles are computed herein and applied to the study of the state-sum invariants. Non-triviality of the invariants is proved for a variety of knots and links, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles.

STABLE EQUIVALENCE OF KNOTS ON SURFACES AND VIRTUAL KNOT COBORDISMS

We introduce an equivalence relation, called stable equivalence, on knot diagrams and closed generically immersed curves on surfaces. We give bijections between the set of abstract knots, the set of virtual knots, and the set of the stable equivalence classes of knot diagrams on surfaces. Using these bijections, we define concordance and link homology for virtual links. As an application, it is shown that Kauffmans example of a virtual knot diagram is not equivalent to a classical knot diagram.

Surfaces in 4-Space

1 Diagrams of Knotted Surfaces.- 2 Constructions of Knotted Surfaces.- 3 Topological Invariants.- 4 Quandle Cocycle Invariants.- Epilogue.- Append.- References.

GEOMETRIC INTERPRETATIONS OF QUANDLE HOMOLOGY

Geometric representations of cycles in quandle homology theory are given in terms of colored knot diagrams. Abstract knot diagrams are generalized to diagrams with exceptional points which, when colored, correspond to degenerate cycles. Bounding chains are realized, and used to obtain equivalence moves for homologous cycles. The methods are applied to prove that boundary homomorphisms in a homology exact sequence vanish.

Three dimensional DNA structures in computing.

We show that 3-dimensional graph structures can be used for solving computational problems with DNA molecules. Vertex building blocks consisting of k-armed (k = 3 or 4) branched junction molecules are used to form graphs. We present procedures for the 3-SAT and 3-vertex-colorability problems. Construction of one graph structure (in many copies) is sufficient to determine the solution to the problem. In our proposed procedure for 3-SAT, the number of steps required is equal to the number of variables in the formula. For the 3-vertex-colorability problem, the procedure requires a constant number of steps regardless of the size of the graph.

Quandle Homology Groups, Their Betti Numbers, and Virtual Knots

Abstract Lower bounds for the Betti numbers for homology groups of racks and quandles will be given using the quotient homomorphism to the orbit quandles. Exact sequences relating various types of homology groups are analyzed. Geometric methods of proving non-triviality of cohomology groups are also given, using virtual knots. The results can be applied to knot theory as the first step towards evaluating the state-sum invariants defined from quandle cohomology.

Cocycle knot invariants from quandle modules and generalized quandle homology

Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was proposed by Andruskiewitsch and Grana. 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.

Twisted quandle homology theory and cocycle knot invariants

The quandle homology theory is generalized to the case when the coecient groups admit the structure of Alexander quandles, by includ- ing an action of the innite 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. Ex- plicit formulas for cocycles corresponding to extensions are given, and used to prove non-triviality of cohomology groups for some quandles. The cor- responding generalization of the quandle cocycle knot invariants is given, by using the Alexander numbering of regions in the denition of state- sums. The invariants are used to derive information on twisted cohomology groups. AMS Classication 57N27, 57N99; 57M25, 57Q45, 57T99

REIDEMEISTER MOVES FOR SURFACE ISOTOPIES AND THEIR INTERPRETATION AS MOVES TO MOVIES

A movie description of a surface embedded in 4-space is a sequence of knot and link diagrams obtained from a projection of the surface to 3-space by taking 2-dimensional cross sections perpendicular to a fixed direction. In the cross sections, an immersed collection of curves appears, and these are lifted to knot diagrams by using the projection direction from 4-space. We give a set of 15 moves to movies (called movie moves) such that two movies represent isotopic surfaces if and only if there is a sequence of moves from this set that takes one to the other. This result generalizes the Roseman moves which are moves on projections where a height function has not been specified. The first 7 of the movie moves are height function parametrized versions of those given by Roseman. The remaining 8 are moves in which the topology of the projection remains unchanged.

State-sum invariants of knotted curves and surfaces from quandle cohomology

State-sum invariants for classical knots and knotted surfaces in 4-space are developed via the cohomology theory of quandles. Cohomology groups of quandles are computed to evaluate the invariants. Some twist spun torus knots are shown to be noninvertible using the invariants.