Narrow Results

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.

We define invariants of surface-links, called generalized fundamental classes, and determine the values which the generalized fundamental classes of surface-links represented by diagrams with two or three triple points can take.

In this paper we give a lower bound of triple point numbers of special family of 2-knots colored by the dihedral quandle of order 5.

We prove that if G is an abelian group of odd order then there is an isomorphism from the second quandle homology to G ∧ G, where ∧ is the exterior product. In particular, for , k odd, we have . Nontrivial allows us to use 2-cocycles to construct new quandles from T(G), and to construct link invariants. Computation of is also the first, fundamental step in the direction of computing homology of Takasaki quandles in general.

We extend the notion of geometric intersection numbers 0 and 1, for circles in the Dehn quandle, to general quandles, assuming existence of elements having certain algebraic properties in the latter. These properties enable the construction of many useful 2 and 3 cycles in Dehn quandle homology. Particularly, we construct a special 2-cycle homology representative and an operation which promotes it to higher dimensional cycles of the same type, in the Dehn quandle, and show how to do so in any quandles admitting such elements. For such quandles, this gives some non-triviality results in the quandle homology. We look at a secondary rack formed by tuples of elements of the Dehn quandle and using the ideas above, obtain some basic non-triviality results in its homology as well. We also give some initial possibilities for its application in representing monodromy of singularities of elliptic fibrations.

This paper presents the first complete calculation of the cohomology of any nontrivial quandle, establishing that this cohomology exhibits a very rich and interesting algebraic structure. Rack and quandle cohomology have been applied in recent years to attack a number of problems in the theory of knots and their generalizations like virtual knots and higher-dimensional knots. An example of this is estimating the minimal number of triple points of surface knots [E. Hatakenaka, An estimate of the triple point numbers of surface knots by quandle cocycle invariants, Topology Appl139(1–3) (2004) 129–144.]. The theoretical importance of rack cohomology is exemplified by a theorem [R. Fenn, C. Rourke and B. Sanderson, James bundles and applications, Proc. London Math. Soc. (3)89(1) (2004) 217–240] identifying the homotopy groups of a rack space with a group of bordism classes of high-dimensional knots. There are also relations with other fields, like the study of solutions of the Yang–Baxter equations.

We examine some questions regarding which torus links admit colorings by elements of a Dehn quandle.

Foams in all dimensions are defined by being modeled on a dual structure found in an $n$-dimensional simplex. Their crossings are studied from geometric and homological points of view. A process of defining invariants thereof is outlined. Interesting research level problems are proposed. From considerations of foams, a partial order on certain families of forests is given via Hasse diagrams.

We give elementary proofs of certain relations in the mapping class group of a closed surface of genus 2, MCG(F_{2, 0}). We generalize portions of these to relations in quandles with certain types of elements, associated to a relatively broad class of groups (including mapping class groups), and derive further similar quandle relations. We show these quandle relations correspond to 2-cycles in the homology of racks of tuples of quandle elements, and thence to families of commutation relations back in the groups. The recurrence of some of these phenomena within higher level structures is also explored, as are multiple types of modifications yielding different relations. The constructions are quite malleable in this respect.

It is a classical result in reduced homology of finite groups that the order of a group annihilates its homology. Similarly, we have proved that the torsion subgroup of rack and quandle homology of a finite quasigroup quandle is annihilated by its order. However, it does not hold for connected quandles in general. In this paper, we define an $m$-almost quasigroup ($m$-AQ) quandle which is a generalization of a quasigroup quandle and study annihilation of torsion in its rack and quandle homology groups.

In 1993, Fenn, Rourke and Sanderson introduced rack spaces and rack homotopy invariants, and modifications to quandle spaces and quandle homotopy invariants were introduced by Nosaka in 2011. In this paper, we define the Cayley-type graph and the extended quandle space of a quandle in analogy to rack and quandle spaces. Moreover, we construct the shadow homotopy invariant of a classical link and prove that the shadow homotopy invariant is equal to the quandle homotopy invariant multiplied by the order of a quandle.