Sam Nelson

On symplectic quandles

We study the structure of symplectic quandles, quandles which are also R-modules equipped with an antisymmetric bilinear form. We show that every finite dimensional symplectic quandle over a finite field F or arbitrary field F of characteristic other than 2 is a disjoint union of a trivial quandle and a connected quandle. We use the module structure of a symplectic quandle over a finite ring to refine and strengthen the quandle counting invariant.

ON THE ORBIT DECOMPOSITION OF FINITE QUANDLES

We study the structure of finite quandles in terms of subquandles. Every finite quandle

Augmented biracks and their homology

Q

VIRTUAL YANG-BAXTER COCYCLE INVARIANTS

decomposes in a natural way as a union of disjoint

RACK SHADOWS AND THEIR INVARIANTS

Q

Rack module enhancements of counting invariants

-complemented subquandles; this decomposition coincides with the usual orbit decomposition of

ALEXANDER QUANDLES OF ORDER 16

Q

THE 2-GENERALIZED KNOT GROUP DETERMINES THE KNOT

. Conversely, the structure of a finite quandle with a given orbit decomposition is determined by its structure maps. We describe a procedure for finding all non-connected quandle structures on a disjoint union of subquandles.

Quantum enhancements and biquandle brackets

We introduce augmented biracks and define a (co)homology theory associated to augmented biracks. The new homology theory extends the previously studied Yang–Baxter homology with a combinatorial formulation for the boundary map and specializes to N-reduced rack homology when the birack is a rack. We introduce augmented birack 2-cocycle invariants of classical and virtual knots and links and provide examples.

Polynomial knot and link invariants from the virtual biquandle

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.