We show that the lower bounds for Betti numbers given in [CJKS1] are equalities for a class of racks that includes dihedral and Alexander racks. We confirm a conjecture from the same paper by defining a splitting for the short exact sequence of quandle chain complexes. We define isomorphisms between Alexander racks of certain forms, and we also list the second and third homology groups of some dihedral and Alexander quandles.

We introduce a new class of quantum enhancements we call biquandle brackets, which are customized skein invariants for biquandle colored links. Quantum enhancements of biquandle counting invariants form a class of knot and link invariants that includes biquandle cocycle invariants and skein invariants such as the HOMFLY-PT polynomial as special cases, providing an explicit unification of these apparently unrelated types of invariants. We provide examples demonstrating that the new invariants are not determined by the biquandle counting invariant, the knot quandle, the knot group or the traditional skein invariants.

The forbidden moves can be combined with Gauss diagram Reidemeister moves to obtain move sequences with which we may change any Gauss diagram (and hence any virtual knot) into any other, including in particular the unknotted diagram.In 1996 Kauffman [1] introduced the theory of virtual knots, extending the topological concept of "knots" to include general Gauss codes. In 1999 Goussarov, Polyak and Viro [2] described virtual knots in terms of Gauss diagrams, which provide a visual way to represent Gauss codes.Consider a classical knot diagram K ⊂ R 2 as an immersion K : S 1 → R 2 of the circle in the plane with crossing information specified at each double point. A Gauss diagram for a classical knot diagram is an oriented circle considered as the preimage of the immersed circle with chords connecting the preimages of each double point. We specify crossing information on each chord by directing the chord toward the undercrossing point and decorating each with with signs specifying the local writhe number. A virtual knot is an equivalence class of Gauss diagrams under the relations in Figure 2, which are the classical Reidemeister moves written in terms of Gauss diagrams. Note that there are several variations of move III depending on the orientations of the strands; we only depict two.Not every Gauss diagram corresponds to a classical knot. Indeed, a Gauss diagram determines a 4-valent graph with crossing information specified at the

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We define ambient isotopy invariants of oriented knots and links using the counting invariants of framed links defined by finite racks. These invariants reduce to the usual quandle counting invariant when the rack in question is a quandle. We are able to further enhance these counting invariants with 2-cocycles from the coloring rack's second rack cohomology satisfying a new degeneracy condition which reduces to the usual case for quandles.

Abstract. We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure. These invariants coincide with the classical Yang-Baxter cocycle invariants for classical knots but provide extra information about virtual knots and links. In particular, they provide a method for detecting non-classicality of virtual knots and links.

We introduce several algebraic structures related to handlebody-knots, including G-families of biquandles, partially multiplicative biquandles and group decomposable biquandles. These structures can be used to color the semiarcs in Y -oriented spatial trivalent graph diagrams representing S 1 -oriented handlebody-knots to obtain computable invariants for handlebody-knots and handlebody-links. In the case of G-families of biquandles, we enhance the counting invariant using the group G to obtain a polynomial invariant of handlebody-knots.

Finite quandles with n elements can be represented as n × n matrices. We show how to use these matrices to distinguish all isomorphism classes of finite quandles for a given cardinality n, as well as how to compute the automorphism group of each finite quandle. As an application, we classify finite quandles with up to 5 elements and compute the automorphism group for each quandle.